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VOLUME 78, NUMBER 1 PHYSICAL REVIEW LETTERS 6JANUARY 1997
Anisotropy of the Electron and Hole Drift Mobility in KNbO
3
and BaTiO
3
Pietro Bernasconi, Ivan Biaggio, Marko Zgonik, and Peter Günter
Nonlinear Optics Laboratory, Institute of Quantum Electronics, Swiss Federal Institute of Technology,
ETH Hönggerberg, CH-8093 Zürich, Switzerland
(Received 6 September 1996)
We determine the anisotropy of the mobility of photoexcited charge carriers in single crystals of
KNbO
3
and BaTiO
3
by means of a purely optical method. In orthorhombic KNbO
3
the mobility
anisotropy is measured for both electrons and holes giving m
a
兾m
c
苷 1.05 6 0.06, m
b
兾m
c
苷 2.9 6 0.3
for holes and m
a
兾m
c
苷 1.15 6 0.09, m
b
兾m
c
苷 1.9 6 0.2 for electrons. In BaTiO
3
we find for holes
m
a
兾m
c
苷 19.6 6 0.6. Our experiment also demonstrates that the acoustic phonon contribution to the
dielectric permittivity is substantially different for spatially modulated electric fields as compared to the
homogeneous case. [S0031-9007(96)02051-0]
PACS numbers: 71.38.+i, 72.20.Fr, 72.20.Jv, 77.84.Dy
The drift mobility of photoexcited holes in tetragonal
barium titanate (BaTiO
3
) shows an anisotropy of about
a factor of 20 between the two different crystallographic
directions. A recent work [1] has pointed out that this
anisotropy has the same temperature dependence as the
anisotropy of the dielectric permittivity, a fact which lead
to speculation upon a possible linear relation between the
drift mobilities and the dielectric permittivities. Several
hypotheses have been examined trying to explain these
phenomena but none was completely satisfactory. The de-
scription of the charge transport by small-polaron hopping
was not completely successful [1], either, although it was
demonstrated that in BaTiO
3
this process is dominant [2].
We now present new measurements of the anisotropy of
the photoexcited charge carrier mobility in potassium nio-
bate (KNbO
3
), another anisotropic polar material of the
same family and with characteristics similar to those of
BaTiO
3
.
These two materials belong to the oxygen-octahedra
ferroelectrics with perovskite structure and are particularly
interesting for electro-optical, nonlinear optical as well as
for photorefractive applications [3–5]. Although at dif-
ferent temperatures, both crystals have the same sequence
of structural phase transitions, from the cubic to the tetra-
gonal, orthorhombic, and finally to the low temperature
rhombohedric phase. In the lower symmetry phases, both
crystals are ferroelectric and show a spontaneous polar-
ization P
S
associated with a weak lattice distortion. At
room temperature KNbO
3
is orthorhombic (point group
mm2) with the crystallographic b axis in the pseudocu-
bic [010] direction, while the a and c axes lie along the
pseudocubic [101] and [
101] directions. At room tem-
perature, BaTiO
3
has a tetragonal structure (point group
4 mm) with the axes parallel to the cubic ones. In both
materials the c axis is oriented parallel to P
S
.
We determine the anisotropy of the mobility in the bulk
of the material from the photoconductivity measured by
a purely optical method. Using light energies smaller
than the material band gap, the photoconductivity is due
to charge photoexcitation from energy levels introduced
by impurities or intentional doping. In contrast to direct
photoconductivity measurements, our results are not influ-
enced by the quality of contacts between crystal and elec-
trodes or by parasitic buildup of screening charges close
to the electrodes themselves or by surface conductivity,
all of which are particularly inconvenient effects in highly
insulating materials like KNbO
3
and BaTiO
3
.
We measure the bulk photoconductivity by detecting
the decay of the space-charge electric fields generated
by the photoconductivity itself. Two interfering laser
beams first create a spatially sinusoidal charge carrier
excitation rate which leads to a charge displacement from
the brighter to the darker regions of the interference
pattern. At steady state and small modulation depth of the
interference pattern, this process converges to a sinusoidal
charge distribution inside the impurity centers [6,7] and
the resulting dc electric space-charge field modulates the
refractive index of the crystal via the linear electro-
optic (Pockels) effect. The amplitude of the generated
phase grating is monitored by the diffraction of a weak
probe beam incident at the Bragg angle. The diffracted
intensity is proportional to the square of the space-charge
field amplitude [6,8]. We study the decay of the phase
grating when it is erased by a strong homogeneous
illumination. By measuring the grating decay times
for various orientations of the grating wave vector, we
determine the anisotropy of the photoconductivity and
thus of the charge mobility.
The amplitude of the space-charge field grating decays
exponentially. The exponential time constant t is given
by [5]
t共I兲 苷
e
eff
e
0
emn共I兲
µ
1 1 k
2
g
兾k
2
D
1 1 k
2
g
兾k
2
0
∂
, (1)
where e
eff
and e
0
are the effective [9] and the vacuum
dielectric constants, e is the unit charge, n is the density
of free charge carriers with mobility m, and k
g
is the
modulus of the grating vector. The Debye screening
106 0031-9007兾96兾78(1)兾106(4)$10.00 © 1996 The American Physical Society
VOLUME 78, NUMBER 1 PHYSICAL REVIEW LETTERS 6JANUARY 1997
wave vector k
0
and the inverse diffusion length k
D
are
two material constants which depend on the impurity
concentration [5]. The density of photocarriers n共I兲 is
responsible for the light intensity dependence of Eq. (1).
This formula is valid for sinusoidal modulations of the
space-charge distribution obtained by a low contrast
illuminating interference pattern.
We choose the magnitude of k
g
to be much smaller
than both k
0
and k
D
. In this limit the term between
parentheses on the right hand side of Eq. (1) can be safely
neglected. The decay time constant is then directly given
by the dielectric relaxation time 共e
eff
e
0
兲兾共emn兲. Provided
that the number density of photoexcited charges n共I兲
stays constant, we can write the following expression for
angular dependence of the decay time constant:
t共q 兲 苷 t
q 苷0
e
eff
共q 兲
e
eff
c
m
c
m共q 兲
, (2)
where q is the angle between the grating wave vector
and the c axis of the crystal and m
c
兾m共q 兲 describes
the angular dependence of the mobility. In Eq. (2) an
effective permittivity e
eff
must be used because of the
sinusoidal spatial modulation of the internal electric field.
In such cases e
eff
is neither the one corresponding to a
strain-free (clamped) crystal, nor the one corresponding
to a stress-free (unclamped) crystal. The calculation of
the effective dielectric constant for a particular direction
of the space-charge field has to take into account the
piezoelectric contributions generated by the mechanical
response of the crystal to the internal sinusoidal electric
space-charge field [9].
In general, the mobility can be described by a diagonal
second rank tensor [10]. Although the static electric
field E is always parallel to the grating wave vector,
the drift current is generally not. Only the component
of the current density parallel to the grating wave vector
is responsible for the decay of the space-charge grating.
By calculating this component for every direction of the
grating wave vector, we can compute the effective scalar
mobility as a function of q . The mobility ratio appearing
in Eq. (2) is then given by
m共q 兲
m
c
苷 cos
2
共q 兲 1
m
a,b
m
c
sin
2
共q 兲 , (3)
where the subscripts a , b, and c denote the axes along
which the mobility is considered.
In contrast to previous studies [1,11] where only
two extremal points were measured, we determine the
full dependence of the decay times t of the space-
charge grating on the angle q between k
g
and the c
axis. We choose a very large grating period of 45 mm
(k
g
苷 0.14 mm
21
) in order to neglect the terms in the
parentheses on the right hand side of Eq. (1). Taking into
account that k
0
, k
D
. 1.25 mm
21
, the error introduced is
less than 1%. The measurements have been performed
in many different KNbO
3
crystals including pure, iron
doped, p-type and n-type samples, and in a nominally
pure BaTiO
3
probe. The crystals, cut with the surfaces
normal to the crystallographic axes, are mounted so
that they can be rotated around the surface normal
bisecting the directions of the two beams which produce
the grating (I
w1
苷 0.03 W兾cm
2
and I
w2
苷 0.21 W兾cm
2
).
These two beams and the erasing beam (I 苷 3.5 W兾cm
2
)
are provided by an Ar
1
laser operating at 488 nm. All
polarizations are always adjusted perpendicular to the c
axis to reduce possible beam coupling or beam fanning
effects [12]. The monitor beam (He-Ne laser at 633 nm,
I 苷 0.01 W兾cm
2
) is polarized along the c axis. While
rotating the samples, we readjust all the polarizations by
means of l兾2 plates in order to ensure the same writing
and, in particular, identical erasing conditions. In this way
n共I兲 remains unchanged and independent of q .
After the grating has reached the steady state, the
grating decay is induced by blocking the two writing
beams and by simultaneously switching on the strong
erasing beam. All the decay curves measured for different
q follow a single exponential function over at least ten
decay time constants. We perform a least-squares fit of
the angular dependence of t using Eq. (2), in which the
angular dependence is governed by the single parameter
m
a,b
兾m
c
. The scaling parameter t共0兲 does not influence
the form of the curves; e
eff
共q 兲兾e
eff
q 苷0
is known [9,13].
Figure 1 shows the grating decay time vs grating
wave vector orientation in the BaTiO
3
crystal, while
Figs. 2 and 3 present the results for KNbO
3
. The solid
lines correspond to the least squares fit using Eq. (2).
The agreement is remarkable. The mobility anisotropies
obtained from the fits are summarized in Table I. The
measurement performed in the hole conducting BaTiO
3
crystal agrees with the value obtained in Refs. [1,11,14].
Experiments in electron conducting samples of BaTiO
3
could not be performed because no suitable crystal was
FIG. 1. Decay time of the photoinduced space-charge grating
in a nominally pure BaTiO
3
crystal as a function of the angle
between k
g
and the c axis of the crystal. The solid line
represents the fit with Eq. (2) using e
eff
共q 兲. The dashed curve
is the fit obtained when using the uncorrected clamped or
unclamped dielectric constants.
107
VOLUME 78, NUMBER 1 PHYSICAL REVIEW LETTERS 6JANUARY 1997
FIG. 2. Decay time of the photoinduced space-charge modu-
lation in an iron doped hole conducting KNbO
3
crystal as a
function of the angle between k
g
and the c axis of the crystal.
Angular dependence in the a-c plane (above) and in the b-c
plane (below). The solid line represents the fitted curve calcu-
lated from Eq. (2).
available. The only previous measurements in KNbO
3
have been published in Ref. [15], but they are affected by
a poor accuracy [16]. Since the absolute value of the drift
band mobility along the c axis for the electrons in KNbO
3
is m
c
苷 0.5 6 0.1 cm
2
共Vs兲
21
[17] we have m
a
苷 0.6 6
0.2 cm
2
共Vs兲
21
and m
b
苷 1.0 6 0.3 cm
2
共Vs兲
21
. These
values are similar to m
c
苷 0.13 6 0.03 cm
2
共Vs兲
21
and
m
a
苷 1.2 6 0.3 cm
2
共Vs兲
21
found in BaTiO
3
by Hall
experiments [18]. Unfortunately, no data are available on
the absolute hole band mobility for these two materials.
Our measurements lead to several observations. The
mobility ratios do not depend appreciably on the doping
concentration since a concentration of the trapping centers
of the order of 10
25
per unit cell can influence the charge
density n共I兲 but not, or only to a negligible extent, the
charge carrier mobility. In KNbO
3
, the mobility ratio
m
a
兾m
c
for both electrons and holes is very similar and
close to unity, while the ratio m
b
兾m
c
shows a larger
anisotropy. In BaTiO
3
, m
a
兾m
c
is even larger.
FIG. 3. Decay time of the photoinduced space-charge modu-
lation in an iron doped electron conducting KNbO
3
crystal as a
function of the angle between k
g
and the c axis of the crystal.
Angular dependence in the a-c plane (above) and in the b-c
plane (below). The solid line represents the fitted curve calcu-
lated from Eq. (2).
The low mobilities observed in both materials can be
understood on the basis of a small-polaron model [19].
In the one dimensional case, the small-polaron hopping
mobility is of the type [20]
m 苷
ea
2
J
2
¯hk
B
T
p
4E
A
k
B
T兾p
exp共E
A
兾k
B
T兲 , (4)
where a is the interatomic distance, E
A
is the activation
energy, and J is the overlap integral of the electronic
wave functions of the adjacent sites. The anisotropy
of the overlap integral can be considered as the origin
of the mobility anisotropy [21]. For highly localized
wave functions, the overlap integral can be assumed to
vary exponentially as J ~ exp共2a兾r
0
兲 where r
0
is the
localization length. Thus using Eq. (4) we can write
m
a,b
m
c
苷
µ
a
a,b
a
c
∂
2
exp关22共a
a,b
2 a
c
兲兾r
0
兴 . (5)
In KNbO
3
the lattice constants along the a, b, and
108
VOLUME 78, NUMBER 1 PHYSICAL REVIEW LETTERS 6JANUARY 1997
TABLE I. Photoconductivity type (P.C.), dopant concentra-
tion (Nom. pure: Nominally pure), and drift mobility ratios
in single crystals of KNbO
3
and BaTiO
3
.
Sample P.C. Doping
m
a
兾m
c
m
b
兾m
c
KNbO
3
p Fe: 150 ppm
a
1.05 6 0.06 2.9 6 0.3
p Nom. pure · · · 3.2 6 0.4
p Nom. pure · · · 2.9 6 0.3
n Fe: 20 ppm
a
1.15 6 0.09 1.9 6 0.2
BaTiO
3
p Nom. pure 19.6 6 0.6
n Nom. pure
b
9.2 6 5.1
c
a
Concentration in Fe兾Nb atomic ratio in the crystal.
b
Reduced in flowing hydrogen [19].
c
As determined from Hall measurements in Ref. [19].
c axes are a
a
苷 5.6896 Å, a
b
苷 3.9692 Å, and a
c
苷
5.7256 Å which, combined with the mobility ratios, give
a polaron localization r
0
between 0.5 and 3.5 Å. These
values would be compatible to the small-polaron model.
However, this is not the case for BaTiO
3
where the
polaron localization was calculated to be 0.03 Å [1],
which appears to be unphysically small.
In BaTiO
3
another puzzling relation was noticed be-
tween mobility and dielectric permittivity andMahgerefteh
et al. advanced the hypothesis that the two quantities are
not independent. In their work they found m
c
兾m
a
苷
Re
c
兾e
a
where R is a constant close to 2 and temperature
independent over the whole tetragonal phase. Our mea-
surements have been carried out at room temperature but
the results demonstrate that R has no universality pretence
since its values in KNbO
3
are close to 4 and 9 for m
c
兾m
a
and m
c
兾m
b
, respectively.
As already pointed out above, the solid curves in all the
figures have been calculated using the effective dielectric
permittivity which takes into account the piezoelectric as
well as the elastic contributions in the case of a sinusoidal
space-charge field. To show the importance of using the
correct expression for e
eff
, in Fig. 1 we also plot the curve
of Eq. (2) when e
eff
is calculated from either the clamped
or the unclamped values. An evident disagreement with
the measured data is obtained. Our measurement of the
dielectric relaxation time as a function of the direction of
the space-charge field is thus the first direct experimental
confirmation of the validity of the expressions for the
effective dielectric constant given in Ref. [9].
In conclusion, we have shown that optical methods
can provide a reliable tool for the characterization of
the charge drift mobility in photoconductive materials.
The measurement of the decay time of a photoinduced
space-charge fields as a function of the crystal orientation
allows the precise determination of the charge carrier drift
mobility ratios.
The localization length of the quasifree charge carri-
ers is calculated in KNbO
3
according to the small-polaron
model. Its value is consistent with the theory, while that
obtained for BaTiO
3
is too small. A further comparison of
the results in BaTiO
3
and in electron and hole conducting
KNbO
3
shows that the value which relates the mobility
ratio with the corresponding dielectric permittivity ratio is
a material parameter. The explanation of the temperature
independence of this parameter noticed in BaTiO
3
, and
the complete description of the charge transport mecha-
nisms in anisotropic polar materials such as KNbO
3
and
especially BaTiO
3
, are still open questions.
We are grateful to Hermann Wüest and Michael Ewart
for crystal growth and post-growth treatment and to
Jaroslav Hajfler for expert preparation of KNbO
3
samples.
The authors also thank Germano Montemezzani for help-
ful comments and enlightening suggestions.
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