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Phys. Rev. Lett. 78(5), 891 (February 3, 1997).
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Band Mobility of Photoexcited Electrons
Robert W. Hellwarth and
Jouni P. Partanen**
University of Southern California
Departments of Physics and Electrical Engineering
(Received 10 May 1996)
We determine the band mobility of photoexcited electrons in cubic n-type
We measure a room temperature mobility of 3.4 ± 0.5 cm2/(Vs) that decreases
monotonically to 1.7 ± 0.3 cm2/(Vs) as the temperature is increased to 200°C.
We show that electrons in Bi12SiO20 form large polarons. Our results are predicted
by strong coupling polaron theory if the band mass of the electrons is chosen to be
2.0 ± 0.1 electron masses. We determine the electron-phonon coupling constant
and effective longitudinal optical phonon frequency required for this prediction
from the available infrared reflectivity spectrum of Bi12SiO20.
63.20.Kr, 72.20.Fr, 72.10.Di, 42.70.Nq
The mobility of a photoexcited electron drifting in the conduction band
of transparent n-type cubic Bi12SiO20 crystals (n-BSO) at
300 K has been reported to lie between 3 and 5 cm2/(Vs)
[1,2,3]. If the usual
independent-collision model were used to describe this drift, the
thermal mean free path (~ 0.2 nm) would be less then the De Broglie
wavelength of the electron, and the collision rate times the Planck
constant (the uncertainty in the electron energy) would be an order of
magnitude greater than the thermal energy. In this "strong coupling"
case the Boltzmann equation cannot be expected to apply. We report
measurements of mobility vs. temperature in n-BSO which can be
explained well by a "polaron" theory that avoids the limitations of
the Boltzmann equation [4,5,6,7].
An electron in a polar crystal polarizes the lattice in its
neighbourhood. The electron moving with its accompanying lattice
distortion is called a polaron. We argue that a photoelectron in
n-BSO constitutes the clearest case of a strongly coupled "large"
polaron, i.e. a polaron whose wave function extends over many atoms,
so that the electron can be thought of as interacting with phonons
rather than with independent atoms. Previous examples of large
polarons were found in a variety of alkali halides, where mobility
values lie above ~12 cm2/(Vs), the values for KCl and KBr at
330°C . These materials are closer to the case of
independent collisions. Band mobilities lower than in n-BSO, like
the value of 0.5±0.1 cm2/(Vs) found in orthorhombic KNbO3
, correspond to "small" polarons. Small polarons have a
wavefunction extending less then an interatomic distance and move by
hopping or tunneling . Many even smaller mobility values
are reported for various insulators, but these generally reflect the
effects of shallow traps [1,11].
n-BSO has a sufficiently large linear electro-optic effect, so that
charge separations can be easily seen. Therefore, we determine the
electron mobility using the holographic time of flight (HTOF) method
[9,11,12,13,14], in which two
interfering laser beams excite a spatially sinusoidal pattern of
charge carriers in the bulk of the sample. The evolution of this
pattern is measured optically by observing the development of the
space-charge-induced index change caused by the separation of the free
carriers from their excitation point. HTOF techniques offer two
distinct advantages over regular time-of-flight methods: (1) The
length scale is set by the spatial period of the sinusoid and is
easily varied. (2) The movement of the photoelectrons is measured
optically, allowing a higher time resolution (given by the laser pulse
length, which is 30 ps in our case). HTOF measurements can be
performed in the presence of an applied electric field
(drift mode) , or without any applied field
(diffusion mode) [9,14], where
purely thermal diffusion of the photoexcited charge
distribution is measured. In this work we use the
HTOF technique in diffusion mode, which has a number of additional
advantages: (3) no electrodes are needed, (4) no particular sample
shapes are required, and (5) no uncertainties are introduced because
of possible internal electric field variations caused by trapped
The experimental four-wave-mixing configuration is shown in
Fig. 1. We use a frequency-doubled Nd:YAG laser that
produces 30 ps, 532 nm pulses at a repetition rate of 5 Hz. A beam
splitter sends part of a pulse into a delay line, to act as a probe
beam, and the other part is split again into two write pulses that
arrive simultaneously in the sample. They excite a sinusoidal electron
distribution, ~sin (Kg z), from donor sites with energy
levels near the middle of the 3.2 eV band gap into the conduction band.
Their charge is compensated initially by that of the newly created
ionized donor sites. Diffusion tends to make the conduction band
electron distribution uniform, uncovering the space charge field
Esc of photoionized donor sites with the following time dependence
where t -1 = t0-1 + tD-1.
t0 is the average time for photoexcited electrons to remain in the
conduction band before going to uniformly distributed traps of unknown
origin. tD is the diffusion time:
Here e is the unit charge, Kg is the magnitude of the sinusoidal
space-charge modulation wavevector, µ is the band mobility, k is
Boltzmann's constant, and T is the absolute temperature. This
analysis assumes that a small fraction of the donors are photoexcited,
and that the evolution of the photoelectron pattern is dominated by
diffusion, as justified experimentally below.
The space charge field of (1) modulates the refractive index
of the material via the linear electro-optic (Pockels) effect. We
detect the resulting phase grating by diffracting the time-delayed
probe pulse from it (see Fig. 1). We note that the
build-up time of the space-charge field (1) does not depend on
the write fluence , and that it is given by the
diffusion time when it is much shorter than the free carrier lifetime.
We know that photoexcitation of holes is not influencing our
measurement, for, if hole transport were significant, we would not
observe the time dependence in (1).
The sample is homogeneously illuminated all the time by a 514 nm Argon
Ion laser beam with an intensity of approximately 0.1 W/cm2. In
the 0.2 s interval between two measurements, this illumination erases
the space-charge induced grating created by the write pulses. During
the measurement time (a few ns) the erase beam deposits six orders of
magnitude less energy than the write pulses. We verified that it does
not affect our results.
We use three nominally undoped n-BSO samples labelled SU1, CT1, and
CT3. These crystals are well characterized by many experiments as
described in Refs. 15,16. The SU1 sample has
been used in previous HTOF experiments in the drift mode
The result of a typical measurement is shown in Fig. 1.
When the probe pulse precedes the write pulse there is only a small
signal. A relatively strong signal caused by third order nonlinear
effects is observed when the three pulses are present in the crystal
at the same time. When the probe pulse is delayed (positive times in
the figure), one sees the diffracted signal increase as the
photoexcited electrons diffuse away from the positive ions at which
they were bound, thus creating a space-charge field.
The signal observed before zero-delay has amplitude E0 and is due
to scattered light from the probe pulse and to the remains of a
grating that is not completely erased by the Argon Laser beam
(scattered light from the write pulses was taken into account
separately). Although it is very small, this background grating needs
to be taken into account when measuring the build-up time. We fit the
detected diffraction efficiency h to h(t) ~
|Esc(t)+E0|2 by adjusting E0 and t [see (1)].
The relative phases of Esc and E0 are uncertain, especially at
the low intensities where we performed the build-up time measurements.
For each data set we perform two least squares fits: once imposing a
phase-shift of 90° between E0 and Esc and once imposing
a 0 phase shift. To obtain a final value for the rise time, we
average the results of the two fits, which differ by 20% at most.
In order to determine the mobility accurately, we must establish that
we are in the low-fluence limit discussed above. To do this we
measure the fluence-dependence of both the build-up time and the
magnitude of the signal. Fig. 2a shows the grating
amplitude and the build-up time t as a function of write fluence
in sample CT3. For small fluences, the grating amplitude grows
linearly with the fluence of the write pulses and the build-up time is
a constant independent on fluence. As the write fluence F
approaches 10 mJ/cm2 saturation effects appear .
From the data in Fig. 2a one derives a photoexcitation
cross section s between ~10-18 and 10-17
cm2, consistent with an earlier estimate . We
performed all the mobility measurements described below with a write
fluence of the order of 1 mJ/cm2.
The dependence of the build-up times on grating spacing 2 p/Kg is
shown in Fig. 2b. The data points were obtained by
averaging several measurements. The build-up time is proportional to
the square of the grating spacing. This confirms that the electron
avoids shallow and deep traps for so long that the build-up time
t corresponds to the electron diffusion time
tD and the
grating build-up is dominated by diffusion (see (1),
t =tD). The electron lifetime t0 in the SU1-sample has
been independently determined to be 26 ± 2 ns .
The solid line in Fig. 2b is the result of a
least-square fit of (2) to the data of both samples, with
only the mobility as a free parameter. It gives a mobility value
of 3.4 ± 0.5 cm2/(Vs). The build-up times in the SU1 and
CT3 samples are the same within the experimental error. This
suggests that the mobility is an intrinsic property of the material.
This idea is also supported by our measurements in the CT1 crystal
at the two largest grating spacings, where we found the build-up
time to agree with the results in Fig. 2b.
Our mobility value is consistent with other, less accurate results
reported previously [1,2,3].
A mobility of 50 cm2/(Vs) derived from grating build-up time
in a similar experiment  is erroneous because
trapping, not diffusion, dominated the dynamics .
Fig. 3 shows the temperature dependence of the band mobility
in CT3 and SU1. The sample was enclosed in a temperature-controlled
oven with small apertures for the laser beams. We set up the
experiment at room temperature and selected an appropriate write
fluence around 1 mJ/cm2 as described above. We raised the
temperature, stopping every 20-30 degrees to stabilize it and perform
a measurement. Care was taken not to change the adjustment of the
experimental setup when changing the temperature of the crystal. The
data in Fig. 3 was obtained at a grating spacing of 0.38
µm. Other measurements performed at longer grating spacings from
0.43 to 0.57 µm give the same temperature dependence, within
The observed decrease of the mobility with rising temperature and the
polar nature of the n-BSO lattice suggest that the band mobility is
controlled by interaction with longitudinal optical (LO) phonons
[4,5,6,7]. The interaction of an electron
with a polarizable lattice has been studied in Refs.
5,6,7 starting from the "Fröhlich
Hamiltonian" which has three material parameters: the electron-phonon
coupling constant a, a single LO phonon frequency nLO,
and the effective mass of the electron in the conduction band. In the
alkali halide crystals where much of the polaron theory was applied
, a single LO phonon frequency is appropriate. However,
n-BSO has many polar optical vibrational branches with frequencies
between 50 cm-1 and 900 cm-1 . In order to
apply the extensive predictions of the existing polaron models
[4,5,6,7] to our results, we have devised two
mathematical schemes to imitate the phonon structure in n-BSO by a
single "average" or "effective" LO phonon branch having frequency
nLO Hz and oscillator strength W Hz. These are related by
(W2/nLO2)=einf-1 - edc-1. Here
einf=5.7 is the long wavelength limit of the electronic
contribution to the refractive index squared, and edc=50
is the clamped dc dielectric constant. Our schemes determine
nLO and W from the experimental infrared reflectivity
spectrum of n-BSO at room temperature . In terms of
nLO and W, the dimensionless coupling constant
a defined by Fröhlich is
where Ry=13.6 eV is one Rydberg of energy, h is Planck's
constant, m* is the effective mass of an electron in the conduction
band (with no phonons), and me is the electron mass. Using the
frequency dependence of the dielectric constant of n-BSO we obtain,
from one scheme, W/c = 195 cm-1 and nLO/c = 504
cm-1, and thus a coupling constant
a= 2.25 (m*/me)1/2.
A second scheme gives W/c= 195 cm-1 and
nLO/c = 495 cm-1 instead, which makes negligible
difference in the predictions.
Since our experiment was performed above room temperature, we use the
general result (Eqs. (46) and (47)) of Ref. 7,
obtained before the low temperature limit was taken:
where µ is the mobility,
b = hnLO/(k T),
a2=(b/2)2 +Rb coth(b v/2),
b= Rb /sinh(bb v/2),
R=(v2 - w2)/(w2 v),
and v,w are temperature dependent variational parameters
[5,6]. We find that putting b=0 in
(4) makes less than 0.1% error throughout our
temperature range. This is useful because K(a,0)=K1(a)/a, where
K1 is a modified Bessel function . We note that
(4), where K1(a)/a can be substituted for K(a,b),
becomes equal to Eq. (24) in Ref. 21 in the limit
of small a.
We determined the parameters v and w at every temperature by
following the free energy minimization procedure described in
Ref. 6. Table 1 gives the values of the v,w
parameters in the temperature range we investigated. Using these
values we estimate a polaron radius of approximately 0.6 nm
. The n-BSO unit-cell is 1.0 nm large and contains 66
atoms. The sphere defined by the polaron radius contains ~ 60
atoms. Therefore, the continuum approximation of Refs.
5,6,7 can be applied.
The only unknown parameter in (3) and (4) is
the electron effective band mass m*. The prediction of
(4) at T=300 K corresponds to our room-temperature
mobility value of 3.4 cm2/(Vs) when setting m*=2.01 me. From
m* and (4) we can predict the temperature dependence
of the mobility. The result is shown in Fig. 3 together with
two other curves obtained using m*=1.7 me and m*=2.3 me.
The agreement with the m*=2.0 me curve is very good. No
parameter was adjusted to fit the experimental temperature dependence,
which is given mostly by the effective phonon frequency nLO.
In conclusion, we have presented comparative and thermal evidence that
photoelectrons in nominally undoped, n-type Bi12SiO20 are
large, strongly coupled polarons; their observed band mobility is
intrinsic. Their small mobility makes photoelectrons in n-type
Bi12SiO20 the clearest example of such polarons.
We would like to acknowledge the support of the Air Force Office of
Scientific Research under Grant No. F49620-94-1-0139.
Table 1:Variational parameters v and w for various coupling
constants a and temperature parameters b.
Figure 1: Diffracted signal vs. probe-pulse delay time in sample CT3,
at a grating spacing of 0.38 µm. The inset shows the experimental
arrangement. Two pulses w1 and w2 (532 nm, 30 ps) arrive in the
sample at the same time. The probe pulse, which is counterpropagating
to w1, is diffracted by the space-charge grating and produces a signal
pulse counterpropagating to w2. The beam splitter BS sends the signal
pulse into the detection system. The beam diameters are 1.7 mm for
the write beams and 0.5 mm for the probe beam. All beams are
vertically polarized. The grey arrow represents a cw erasing beam at
514 nm. It illuminates the crystal from the top with an intensity of
approximately 0.1 W/cm2.
Figure 2: (a) Grating build-up time and amplitude vs. write pulse
fluence in sample CT3 for a grating spacing of 0.56 µm. The
dashed line gives a linear dependence of the grating amplitude on
fluence. (b) Log-log plot of the grating build-up time t vs.
grating spacing in samples SU1 and CT3 at 1 mJ/cm2 fluence. The
solid line is obtained from Eq. (2) using a mobility value of
Figure 3: Electron band mobility vs. temperature in sample SU1
and CT3, and predictions of Eq. (4).
* Permanent address:
Nonlinear Optics Laboratory,
Institute of Quantum Electronics, Swiss Federal Institute of Technology,
ETH-Hönggerberg, CH-8093 Zürich.
** Also with:
3D Systems, 26081 Avenue Hall,
Valencia, CA 91355.
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