Download Picosecond dynamics of surface electron transfer processes: Surface

Picosecond dynamics of surface electron transfer processes: Surface
restricted transient grating studies of the n-Ti0 2 /H 2 0 interface
J. J. Kasinski, L. A. Gomez-Jahn, K. J. Faran,a) S. M. Gracewski,a) and R. J.
Dwayne Miller
Department of Chemistry and the Laboratory for Laser Energetics. University ofRochester. Rochester. New
York 14627
(Received 2 March 1988; accepted 28 September 1988)
The surface restricted transient grating is demonstrated as a sensitive probe of ultrafast surface
reaction dynamics. Studies of doped single crystal n- Ti0 2 (00 1) surfaces in air demonstrate
linear trapping processes, assigned to crystal defects within the surface deformation layer, that
limit carrier lifetimes to 5 ns. Direct in situ grating studies at photochemically active n-Ti0 2/
H 0 interfaces demonstrate that the dominant mechanism of interfacial electron transfer in
this system involves thermalized hole carriers at the atomic surface. The d~namics are.
consistent with adsorbed OH- as the initial hole acceptor. In addition, optical generatIOn of
coherent surface acoustic modes is demonstrated. A detailed theory is presented for the grating
excitation ofthe surface acoustics. Acoustic propagation in the H 20 half-space of the Ti0 2/
H 2 0 liquid interface gives evidence for a phase change of the water layer at the polar Ti0 2
(001) surface to a solid phase.
I. INTRODUCTION
The abrupt phase discontinuity defined by a surface alters significantly the physics of a chemical reaction pathway.
The static potential of the solid state surface has an ordering
effect on the molecular system at the interface which, in
tum, lowers activation barriers and enables the surface to act
as a catalyst. Other surface reaction processes also involve
direct participation of the surface through electronic interactions between the solid state lattice and the molecular adsorbates. In this regard, one of the most fundamental steps in
a surface reaction sequence is that of electron transfer. In the
presence of a phase boundary, the problem of understanding
surface mediated electron transfer becomes more complicated than that in a homogeneous phase. In the context of solid
interfaces, particular attention must be paid to the solid state
aspects of the electron transfer step. The main problem is to
accurately connect the electronic wave function of the periodic potential defined by the solid state lattice with that of
the localized potential of the molecular acceptor which is
strongly coupled to a continuum of nuclear bath modes in
the adjacent gas or liquid phase. In consideration of solidliquid interfaces, the problem is further complicated by solvent dynamics. The orientating effects of the static potential
of the surface should significantly alter the structure of polar
fluids at the interfacial boundary. This restructuring is important as the overall dynamics of surface electron transfer
processes are greatly influenced by the solvent repolarization which follows the buildup of charge density on an acceptor site. In the bulk, solvent reorganization is fairly well described by continuum models for dielectric relaxation. 1
However, at a surface, the high frequency dielectric properties are unknown and presumably different than the bulk as a
result of this interaction between the polar solvent molecules
and the charged surface. Similar problems occur in properly
defining the atomic surface layer of the solid state lattice.
a)
Department of Mechanical Engineering, University of Rochester, Rochester, New York 14627.
J. Chern. Phys. 90 (2), 15 January 1989
The break in lattice structure at the surface generally leads to
bond rehybridization of the surface atoms and the formation
of surface states which are not strongly coupled with the
band states of the bulk solid state lattice. 2 Surface states are
also formed by lattice vacancies, defects and chemical impurities. These surface states may act as either localized intermediates in the interfacial electron transfer process or
compete with molecular acceptors as a charge transfer acceptor. The latter process leads to surface degradation. 3 A
detailed understanding of surface mediated electron transfer
processes needs to encompass both the coupling of two distinctly different phases through interfacial electron tunneling and the effects associated with the restructuring that occur on both sides of the atomic interface.
The most successful treatments of surface mediated
electron transfer assumed that electron transfer processes
occur directly from the first few atomic layers of the solid
state surface to acceptors directly adjacent to the surface.
The free electron or hole vacancy in the solid state is assumed thermally equilibrated with the lattice and highly localized at the surface, i.e., thermalized surface band edge
electron transfer processes. 4 However, in the context of electron transfer processes at semiconductor surfaces, it has
been recently pointed out that efficient electron tunneling
can occur from sites well below the surface at higher energies
than the surface valence or conduction band edges, i.e., hot
carrier injection. 5 Assuming isoenergetic states for the electron on both sides of the interface, the predicted dynamics
for the two different electron transfer mechanisms differ by
over two orders of magnitude. Ifhot carrier transfer mechanisms are dominant, formation of oxidized or reduced product states at the interface should occur on a 100 fs time scale.
However, this mechanism requires that solvent repolarization at the interface occur on the same time scale to prevent
back electron transfer. In aqueous systems, this would require near bulk dielectric properties at the interface. In contrast, thermalized surface electron transfer rates will be rate
limited by either the transit time of the carrier to the surface
0021-9606/89/021253-17$02.10
@ 1989 American Institute of Physics
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Kasinski et al.: Surface electron transfer processes
or the energy relaxation of the carrier through optical
phonon emission. The space charge region (depletion layer)
ofthe semiconductor is expected to be highly quantized due
to the confining potential formed by the space charge field
and the Helmholtz barrier. 6 These quantization effects
should change the thermalization process from single
phonon events to multiphonon emission which is much
slower. 5.7 From this line of reasoning, thermalized band edge
electron transfer should occur on a 10-100 ps time scale or
slower. Processes faster than this time scale would have a
significant electron transfer component from thermally unequilibrated levels.
Based on the large difference in the dynamics for the two
mechanisms of interfacial electron transfer, it is possible to
determine the dominant operating mechanism at the interface using time domain techniques. This determination
would uniquely define the spatial and energetic coordinates
of the electron transfer step which is essential to a detailed
understanding of this event. To date, there have been few
experimental studies of this kind. The best attained resolution has been in the nanosecond to subnanosecond time scale
which is insufficient to resolve the physics of the actual interfacial electron transfer step. 8 Herein, we describe a surface
restricted transient grating method as a new approach to
studying surface reaction dynamics which is capable of optical pulse width limited resolution (1O- 13 _1O- 12s) along
with submonolayer sensitivity. This method is used in conjunction with efficient semiconductor liquid junctions to
provide both an optical trigger for the electron transfer process and to act as an electric field focusing element for carriers photogenerated within the space charge region. The field
focusing of the junction makes the technique sensitive to the
atomic surface layer. The information content of the surface
grating image is high, as will be discussed in detail below.
The surface grating technique can be used to selectively measure interfacial electron transfer dynamics or the competing
processes (e.g., surface state trapping) by control of the
junction. In addition, we have demonstrated for the first
time that coherent surface acoustic waves can be optically
generated by the grating image. The surface wave deformation offers a unique, highly surface specific, probe of the
interface structure and interactions whereas the frequency
response of the surface acoustics provide a sensitive measurement of carrier thermal relaxation processes at the semiconductor surface. The large amount of information that can
be obtained from surface restricted grating spectroscopy
makes this technique a multifaceted probe of both surface
reaction dynamics and structure.
The specific system used in these studies was the n- Ti0 2
aqueous liquidjunction (Ti0 2IH2 OIOH-) which has been
predicted to show significant unthermalized electron tunneling or hot carrier effects. 5 These predictions were based on
an extremely small effective mass of the hole carrier (the
minority carrier) which is involved in the interfacial charge
transfer step. The hole carrier effective mass of Ti0 2 has
reported hole mass values of 0.01 me. 9 Such small minority
carrier masses would lead to strong quantization of the space
charge region. Space charge quantization is essential in preventing thermal relaxation of the hole carrier from decou-
pling supraband edge electronic energy levels from the elec.
tron transfer mechanism. From this consideration, the
n- Ti0 2!H2 junction is an important test case for interfacial charge transfer models based on long range tunneling of
thermally unequilibrated carriers. This system is also important in its own right as this was the first system to demonstrate photodissociation of water to oxygen and hydrogen
under solar fluences. Its important role in the development
of semiconductor liquid junctions make it an important
model system to understand interfacial charge transfer and
surface reaction processes in general.
°
II. SURFACE RESTRICTED TRANSIENT GRATING
SPECTROSCOPY OF SURFACES
A. Studies of carrier dynamics
Surface reactions are considerably more complex than
reactions in a homogeneous phase. The overall reaction often involves numerous intermediates and reaction product
channels. For this reason, the major emphasis in understanding surface chemistry has been on the atomic structure
of the interface at single crystal surfaces. Models are based
on correlating reaction mechanisms with structure. An alternative approach to surface reactivity is to study the dynamics of the reaction process directly by using picosecond
spectroscopic techniques. The optical pulse sequences used
in picosecond spectroscopy provide very narrow filters for
the study of reaction processes. Provided the steps in the
reaction process yield optically distinct intermediates, each
individual step can be studied selectively and the overall reaction mechanism can be determined. 10 The main problem
to overcome in extending picosecond spectroscopy to the
study of surface reaction dynamics is a technical one. Optical
techniques, with a few exceptions such as surface enhanced
spontaneous Ramanll(a) or coherent anti-Stokes Raman
scattering in waveguidesll(b) and second harmonic generation,12 are generally not surface specific. Bulk contributions
to the optical probe normally dominate the spectroscopy.
An additional problem, exclusive to time domain techniques, is that the surface reaction sequence must be optically triggered in phase to give a well defined time origin for the
reaction sequence. Transient grating spectroscopy, used in
conjunction with semiconductor liquid junctions, represents
a solution to the above technical problems.
The transient grating technique has been extensively developed over the years. 13 The general features of the transient grating technique as applied to semiconductor surfaces
are shown in Fig. 1. Two time coincident above band gap
excitation pulses are used to image an optical interference
pattern on the semiconductor surface. The above band gap
excitation promotes an electron from the valence band to the
conduction band forming an electron-hole pair which degenerates into free carriers (see Fig. 7). The spatial modulation of the carrier popUlation creates a spatially sinusoidal
variation in the material index of refraction which exactly
mimics the optical interference pattern. A holographic diffraction grating is formed which can be probed by monitoring the diffraction efficiency as a function of time with a
variably delayed probe pulse. The grating image decays ac-
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Kasinski et al.: Surface electron transfer processes
\ 1+1
FIG. 1. Surface restricted transient grating spectroscopy. An optical interference pattern generated by two crossed excitation pulses is holographically encoded in 'he semiconductor surface as a diffraction grating in the form
of electron-h. ,Ie pairs. The carrier population dynamics are monitored by
their diffraction efficiency at below band gap frequencies into the ± 1 order
of diffraction. The crystal surface is part of a liquid junction (n-type is
shown). The space charge field of the junction focuses the minority carrier
to the surface, conserving the grating image, to give virtually atomic surface
selectivity. Nonradiative carrier relaxation processes in the grating image
also excite single frequency surface acoustic modes-shown as a sinusoidal
surface displacement of height WOo
cording to the carrier population and thus measures directly
the dynamics of carrier trapping and interfacial electron
transfer processes.
The initial surface selectivity to these processes is given
by the very short optical penetration depth of semiconductors to above band gap excitation. Depending on the excitation wavelength, these values range from 100--2000 A for
both direct and indirect gap semiconductors. In the case of
indirect gap semiconductors, the excitation must involve a
direct transition, i.e., blue of the band edge. The surface selectivity is highly enhanced by the use of semiconductor liquidjunctions. 3,14 An intense space charge field is associated
with liquid junctions which is exactly analogous to abrupt
metal-semiconductor Schottky junctions. The space charge
field arises from electron exchange processes that equilibrate
the chemical potential across the solid-liquid interface. The
overall effect is that majority carriers are depleted at the
surface leaving ionized impurities in the surface layer forming a space charge electric field. The width of the space
charge field and hence the electric field depends on the background carrier concentration. The width can be varied from
a 100 A to 1000 A range by varying the semiconductor doping level, i.e., fields from lO4 to - 106 VI cm. The width of the
space charge region is comparable to the optical 11e penetration depth. Electron-hole pairs optically generated within
the space charge field are separated by the electric field.
These pick up a drift velocity component in the space charge
field which causes vectorial transport of the minority carrier
1255
to the surface and the majority carrier to the bulk. The enormous electric fields present within the space charge region
drives the minority carrier to the surface on a picosecond to
subpicosecond timescale. In the hot carrier model, this
transport process is assumed to be ballistic. In either case,
the carrier is rapidly transported to within tunneling distances from the surface. By this field focusing effect, the
grating image becomes essentially atomic surface selective.
This surface selectivity is achieved in the absence of any nonlinear interaction of the optical field with the atomic surface,
in contrast to second harmonic generation or surface enhanced Raman scattering. In addition to enhancing the surface selectivity, the semiconductor liquid junction provides
an optical trigger for the interfacial electron transfer process.
The separation of the electron hole pairs prevents surface
induced or bulk electron-hole pair recombination from
complicating the dynamics. Minority carriers optically
created on resonance with molecular acceptors will tunnel
across the interface either through a hot carrier or thermally
equilibrated charge transfer mechanism. The quantum yield
for interfacial charge transfer is well characterized at these
interfaces and is near unity.
The transient grating signal is determined by the diffraction efficiency of the grating image. Light diffraction of the
surface grating is described within the Raman-Nath limit to
light diffraction, i.e.,13(a)
/
~ = rJ~ (21Ttlnd lAp),
(1)
/0
where / m is the diffracted light intensity of the mth order, J m
is an mth order Bessel function, d is the grating thickness, Ap
is the probe wavelength, and tlli is the spatially periodic
change in the complex index of refraction. In principle the
grating diffraction efficiency can be probed by either above
band gap or below band gap probes. The high absorptivity of
semiconductors to above band gap light would necessitate
monitoring the grating in reflection rather than in transmission. However, the larger background scattered light and
strong absorption of the probe limit the use of above gap
probes. For below band gap probes, the amplitude component to the thin grating is negligible and the diffraction efficiency (1]) in the first order is described by a thin phase
grating. 13(a),15
(2)
where only changes in the index of refraction (tln) contribute to the signal. The spatial geometry of the excitation and
probe beams of the grating is defined by
.
sm () ± 1
.
Ap
= sm (}j ± A
'
(3)
where () ± 1 and (}j are the angles of the first orders of diffraction of the probe beam relative to the surface normal (see
Fig. 1). A is the grating fringe spacing which is given by
A=
Aex
,
(4)
2 sin (}ex
whereA ex is the wavelength of the above band gap excitation
J. Chern. Phys., Vol. 90, No.2, 15 January 1989
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Kasinski et al.: Surface electron transfer processes
1256
and (}ex is the half-angle between the two grating excitation
pulses.
In consideration of just the photogenerated carrier dynamics, the index of refraction change upon the photoexcitation of electron-hole pairs can be approximated by the
Drude model for the optical properties of the quasifree carriers. The periodic index of refraction change is then given
byl6
2
2nm eh(t)p€O
'
(5)
where N is the number density of photogenerated carriers, e
is the fundamental electron charge, meh is the effective reduced mass of the electron-hole pair
[meh
= (11m: + limn -\], and (t)p is the radial probe frequency. This periodic change in the real part of the index of refraction from photogenerated free carriers is found to be the
dominant term in the light diffraction process as will be discussed below. 15, 16 There are other possible optical effects following electron hole pair generation, such as band gap renormalization or bound exciton resonances, that can also
contribute to the grating. \3 In the case of Ti02 which is a
high dielectric, indirect gap, semiconductor these optical
components are negligible in comparison to the free carrier
absorption and dispersion components described by Eq. (5)
for probes well red of the band gap origin.
The important feature of the grating signal is that both
the electron and hole carrier densities contribute to the signal. The exact contribution from each carrier depends inversely on the carrier reduced mass [Eq. (5)]. Depending
on the relative magnitudes of the effective masses, the grating measures the population dynamics of either the electron
or hole carriers. For Ti0 2 , the grating image would be expected to monitor fairly exclusively the hole carrier dynamics based on reported hole and electron effective masses (m~
~0.01 me' m: ~30 me ).5,9 In other semiconductors the carrier effective masses are nearly the same such that both carrier population densities contribute equally to the signal.
The grating decay gives a direct measurement of electron
and hole carrier population dynamics. The optically generated carriers may decay by either solid state recombination
processes or interfacial charge transfer. The decrease in the
grating image is irrespective of the image conservation in the
interfacial electron transfer event or surface state trapping.
The subsequent change in optical properties of the molecular
acceptors or surface trap sites, at the probe wavelengths
used, is at least an order of magnitude smaller than the depletion of the free carrier grating image within the semiconductor surface. Basically, the surface state traps and molecular
charge acceptors represent localized states which are offresonance from the probe and are less polarizable than the
free carriers. Thus, in the high quantum limit for interfacial
electron transfer, the transient grating signal selectively
measures the surface electron transfer dynamics through the
charge transfer depletion of the minority carrier population.
B. Theoretical treatment of the optical generation of
surface acoustics
Experimentally we have determined that the grating image also excites a single frequency surface acoustic mode.
These results will be discussed below. The exact coupling
and spatial relationship of the surface displacements need to
be determined to assign the surface acoustics. This section
presents an analytical solution for elastic waves generated at
a surface by transient grating excitation. Assuming any other boundaries of the solid are far enough away from the surface of interest such that reflections do not return during the
experimental time scale, the solid can be modeled as a halfspace with a coordinate system as shown in Fig 1. The solution will be derived for a homogeneous, isotropic, linearly
elastic halfspace with a traction free surface. The excitation
grating will generate a heating pattern within the interference region which will be sinusoidal along the solid surface
and will decay exponentially with depth. A previous treatment of the problem neglected the finite optical penetration
on the surface displacement which is critical to surface vs
bulk acoustic excitation. 17 We make the following assumptions: ( 1) the excitation pulse and carrier thermal relaxation
times are much shorter than the acoustic time constant, such
that the heat deposition is considered to be instantaneous;
(2) thermal diffusion will be neglected as it is at least two
orders of magnitude slower in affecting material displacement than sound propagation. Therefore, the induced temperature rise a T will have a step function time dependence
aT(x,z,t)
= Tmax e- az (1 +sinbx)H(t),
(6)
where Tmax is the maximum temperature rise, a is the depth
decay constant or absorptivity, b = 211'/A, and H(t) is the
unit step function.
c. Problem formulation
The resulting displacements u and w in the x and z directions, respectively, satisfy the governing equations,
(A
2
ax axaz
au
=p at
2 a2w)
2
+ J-l) (-a u + ----;::2
azax az- + p,V w aw
a2w)
u++ J-l) (-a2
- + J-lV 2U -
a'(3,.1,
a(an
+ 2p,)--
ax
2
(7a)
2 '
(A
a'(3,.1,
a(an
+ 2J-l) -
az
2
=p
al 2
'
(7b)
throughout the region z> 0, where
V2=~+~
2
ax
ar
is the two-dimensional Laplacian operator, p is the density,
a' is the thermal expansion coefficient, and A and J-l are the
Lame constants. Traction-free boundary conditions are applied atz = 0
= 0,
where 1'ij (iJ
1'ZZ
1'zx
= 0,
x,y,z)
are the components ofthe stress tensor l' and are related to the displacements for a linear-elastic,
isotropic media byl8
1'ij
=,.1,
aUk
(au.' +-'
au.)
k2: -aX- {kj i j +J-l aX ax; ,
3
I
(8)
j
where {jij is the Kronecker delta. The radiation condition
must also be satisfied as z -> 00.
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1257
Kasinski sf a/.: Surface electron transfer processes
In the applied temperature field, only the term which
varies sinusoidally with x will generate surface acoustic
waves. Results presented below correspond to this term. The
term which is independent of x will cause a uniform, onedimensional expansion which can easily be determined and
added to the solution. However, since this uniform expansion is independent of x, it will not diffract the detection
pulse and therefore cannot be observed experimentally. In
the derivation, the complete details of which are presented
elsewhere, 19 the closed-form solution for time harmonic excitation is first obtained by integral transform methods. The
final solution for the applied temperature field with a unit
step time dependence is then obtained by integrating the
time harmonic solutions appropriately over all frequencies.
The displacements are represented in the form,
U
= Kb cos bx
2rr
I'1m
Joo
m_O
-
[A'( w ) e - r,z + A" ()
w Yte - r,z
Exponential temperature decay
...
'~~
*
I
1
q
II)
+"1
C
IDo
~1
J:l:;j
5}-1
0
0
'I'
o
I'
o
d+-______
-,______-.________.-______,
0.0
1.0
2.0
3.0
4.0
Time
FIG. 2. Theoretical results for surface displacement. Calculated displacement is in units of time normalized to the surface acoustic period. The surface displacement is normalized relative to the acoustic wave vector
(21TIA).
00
(9)
+ A" (w)b 2e - r,z + ae - az]
( 10)
where
(lla)
[4b2Yta- (2b 2 -w 2Ic;)]IR(w),
2
( llb)
A "(w) = 2(2b 2 - w /c;)( YI - a)R(w),
2
2
R(w) = (2b -w2Ic;) -4b YIYt (the Rayleigh wave
A'(w)
=
(llc)
equation),
YI = (b 2 _W 2/cy)1/2,
K = a'(3..i + 21t) T
..i + 21t
max'
Yt = (b 2 _W 2/c;)1/2,
(lId)
(lIe)
and C1 and C t are the bulk longitudinal and transverse wave
speeds, respectively.
The complicated nature of the above solution reflects
the fact that the surface displacement involves both transverse and longitudinal motion which gives rise to two coupled wave equations [Eqs. (7 a) and (7b)]. The theoretical
calculations of the surface displacement are shown in Fig. 2.
There are two main features to this solution. The first is that
the maximum surface displacement occurs at one half an
acoustic cycle from t = 0, the thermal impulse, in analogy to
previous treatments of bulk acoustics. 20 The position of the
surface expansion (negative z displacement) is spatially
coincident with the constructive regions of optical interference. This point becomes important in assigning the sign of
the free carrier optical properties. In addition, there is a very
interesting feature in the surface displacement which appears as a discontinuity in the temporal behavior near t = O.
This rise in the surface displacement is caused by the thermally driven lattice expansion of the region defined by the
very short optical penetration depth. The time scale of this
expansion is determined by the length scale (a - I) over
which the surface layer is optically heated, and the speed of
sound along the surface normal. In the case of Ti0 2 , the
ratio of the optical penetration depth to the acoustic wavelength (a - IIA) is I :20. This difference explains the much
faster rise in the surface displacement from the nonpropagating thermal expansion relative to the surface displacement
from the excited coherent surface acoustic mode.
Further, the theoretical results demonstrate that for
highly confined heating of the surface layer, the material
displacement will lead to selective excitation of a pure surface acoustic wave (SAW). Also, the results show the excited SAW has two counterpropagating components due to the
symmetry of the thermal expansion process along the grating wave vector. A standing SAW is excited which leads to
the beat patterns observed in Fig. 2 in the surface displacement. The resulting wave field depends on the ratio alb
which determines the relative weighting ofthe bulk longitudinal contribution to the surface displacement [see Eqs. (9)
and (10)]. As the optical penetration depth and thermal
heated surface layer increase in thickness, there is a gradual
transition to the excitation of bulk longitudinal acoustic
modes as expected. For Ti02 at 355 nm excitation
(a::::: 8 X 104 cm - I, b = 2.4 X 104 em - I), thermal relaxation
is expected to lead to selective excitation of a surface acoustic
wave.
The effects of finite nonradiative relaxation times can
also be incorporated into the problem. As shown previously,21 the thermalization dynamics can be measured in the
limit the time constants are comparable to a quarter acoustic
cycle. Thermalization times equal to or greater than an
acoustic cycle would lead to cancellation effects and elimination of coherence in the SAW excitation. This effect can be
put in numerically or analytically. Quantitative measurements of carrier thermalization can be made by determining
the frequency response of the coherent SAW amplitude. The
temporal resolution is limited to -30 GHz (15 ps) using
prism coupling to achieve the highest possible SAW frequencies.
As with the detection of the carrier dynamics, the optically generated SAWs are observed by light diffraction. The
propagation of coherent SAW modes on the surface leads to
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1258
Kasinski et al.: Surface electron transfer processes
OPTICAL FIBER
L2
O.S., M.L., Nd:YAG
Gl
FIG. 3. Experimental set up.
The n-Ti02 crystals are
housed in a three electrode
liquid junction photocell
shown at the three beam
crossing point. PC = Pockels
POL = polarizer,
cell,
L = lens, A. /2 = half-wave
plate, V.D. = variable delay
P.O. = photodiode,
line,
G = grating, B.S. = beam
splitter, 2 X = KTP doubling crystal, 3 X = RDP
summing crystal.
L7
the formation of a diffraction grating in the form of a corrugated surface. The spatially sinusoidal surface displacement
diffracts light in superposition to the carrier phase grating,
which turns out to be an extremely sensitive method of detecting surface acoustics. This point will be discussed at
length below.
III. EXPERIMENTAL
The experimental set up is shown in Fig 3. The laser is a
cw pumped, Q-switched and mode locked Y AG. The novel
feature is the use of a fiber optic based pulse compressor to
develop a very stable source of 3 ps, /-LJ pulses at repetition
rates up to 1 kHz without amplification. Details of the fiber
optic grating pulse compressor have been previously described. 22 A microwave triode driven Pockels cell was used
to select a single pulse from the Q-switched pulse train envelope. The selected and compressed 1.064/-Lm pulse was doubled in a KTP crystal and the 0.532 /-Lm harmonic was
further summed in an RDP crystal with a pulse from the
rejected IR pulse train off the Pockels cell polarizer. This
summing procedure gave approximately 1 /-LJ pulses both at
355 and 532 nm. The surface grating was written on n-Ti0 2
single crystal (00 1) surfaces (Commercial Crystal Labs)
using above band gap excitation at 355 nm with a 5· angle
between the excitation beams to avoid carrier diffusion effects. With this small angle, carrier diffusion along the grating wave vector limits the grating image to a 1/e lifetime of 1
/-Ls which is a negligible effect on the nanosecond time scale of
the experiment. The 532 nm pulse was used as the below
band gap probe which was brought in at normal incidence to
monitor the grating dynamics. The first order diffracted
probe signal was isolated with an iris and detected with an
UV filtered PIN photodiode. The timing between the grating
excitation pulse sequence and the probe pulse was adjusted
with a motorized corner cube drawn along a precision lathe
bed with a voltage readout proportional to the position displacement. The diffracted probe signal and probe pulse delay
were processed with a lock-in and X-Yrecorder combination. Data was collected at 500 Hz with the excitation beam
chopped at halfthe laser repetition rate and single shot excitation conditions were varied from 3 X 1013 photons/cm2 to
3 X 10 15 photons/cm2. Under these conditions the diffraction efficiencies varied from <10 - 8 to 10 - 6, respectively.
The semiconductor liquid junction was constructed using the n- Ti0 2 crystal as an optical window of a three electrode aqueous cell with a saturated calomel electrode (SCE)
and platinum mesh counter electrode. The Ti0 2 crystal was
placed over a hole drilled through a fused quartz window
and was mounted using a high pH resistant epoxy. The
aqueous electrolyte used in all cases was 0.01-1 M NaOH
and 1M Na2S04 in distilled water. The n-Ti0 2 crystals were
mechanically polished to A. /10 optical quality and then
doped in a hydrogen furnace. The hydrogen pressure was
kept at 1 atm and the doping controlled by varying the exposure time to temperatures ranging from 400-600 ·C. The
samples became deep blue at high dopings and virtually opaque at carrier concentrations above 10 19 carriers/cm 3 • This
concentration was the highest usable concentration for a 532
nm probe. The surfaces were chemically etched in concentrated H2 S04 after the reduction step, washed in distilled
water and methanol, and air dried. Carrier concentrations,
along with flat band potentials, were measured by MottSchottky plots prior to the experiment. The carrier concentrations and flat band potentials were determined from the
linear portion of these plots. 23 Ohmic contacts were made
with an indium/gallium eutetic rubbed onto the back surface such that a 5 mm strip was left open for probe transmis-
J. Chern. Phys., Vol. 90, No.2, 15 January 1989
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Kasinski et al.: Surface electron transfer processes
1259
facial electron transfer. However, the higher order recombination processes may become dominant at high doping levels. This point is especially relevant given that in order to
minimize carrier trapping losses and maximize the quantum
yield for interfacial electron transfer, semiconductor crystals
are usually highly doped in the 10 18/cm 3 to 10 19/cm 3 range.
Increased doping decreases the width of the space charge
region and, thereby, increases the magnitude of the electric
field that drives the minority carriers to the surface. This
effect minimizes the residence time of the carriers in the depletion layer where trapping rates are much higher than in
the bulk. 25 ,26 There is normally a trade off in quantum efficiency between decreased residence time within the depletion layer and the fraction of light absorbed within that region. In addition to this consideration, the hot carrier model
of interfacial electron transfer requires high doping levels to
provide narrow space charge regions which are highly quantized. 5 At high doping levels, Auger recombination may become the dominant recombination mechanism that limits
electron-hole carrier lifetimes. Therefore, the exact nature
of the solid state processes that compete with surface mediated electron transfer need to be experimentally addressed.
Transient grating results for an n- Ti0 2 crystal with a
donor concentration of 8.2X 10 18 carriers/cm 3 is shown in
Fig. 4. The grating decays are nonexponential as expected
for either surface state trapping or higher order recombination processes. (The lie effective decay time is 5.0 ns). An
excitation power dependence was conducted in the range
from 3 X 1020 photogenerated carriers/cm 3 down to 10 18/
cm 3• 19 Above 3 X 10 19 photogenerated carriers/cm 3, the decays are dominated by a fast higher order recombination
component whose relative intensity and decay rate decreases
with decreasing excitation power. At excitation conditions
below 3 X 10 19/cm 3, the decays were no longer power dependent and, for a number of crystals with donor concentrations
sion. The platinum counter electrode was connected to the
ohmic contact with silver epoxy. The cell was operated under either short circuit or open circuit conditions with the
Ti0 2 potential monitored relative to the SeE reference electrode. The potential was maintained with a well regulated
variable power supply. To avoid space charge accumulation
effects, the liquid junction cell was translated at 5 cm/s in
front of the excitation beams to ensure that each laser shot
sampled a surface fully equilibrated with the aqueous redox
couple. The signal intensity was found not to vary for different points along the crystal surface which indicates a uniform surface preparation. Surface inhomogenieties are on a
much shorter length scale than the laser spot sizes.
IV • RESULTS AND DISCUSSION
A. Carrier dynamics
1. Surface trapping studies in air
The first experiments were conducted on n-Ti0 2 in air
as a control. Under ideal conditions, there is no junction
formed and the valence and conduction bands are flat. This
study determines the effective rates of electron-hole pair recombination processes that compete for interfacial electron
transfer on the solid side of the interface. Under these conditions, the lifetime of the photogenerated electron-hole pairs
within the semiconductor can be expressed as an expansion, 15 i.e., 1lreh = AN + BN 2 + eN J + "', where the linear term refers to trapping of carriers at either bulk or surface traps, the bimolecular term refers to radiative electronhole recombination, and the last term refers to Auger recombination. Generally, radiative lifetimes and bulk trapping
processes by impurities occur on a 100 ns time scale. With
the very short optical penetration at above band gap excitation (-1000 A, at 355 nm), surface state trapping processes
are expected to be the dominant competing pathway to inter-
FIG. 4. Transient grating signal from
(doping
concentration
= 8.2X 1OIs/cc) in air (nonexponential with a 5 ns "1/e" time). This decay
is essentially independent of doping
concentration, sample, and excitation
power (-1 X 10 14 photons/em'). The
small periodic oscillations visible are
due to surface acoustic waves.
n-TiO,
o
2
3
4
:I
6
7
TIME (nsec)
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Kasinski et al.: Surface electron transfer processes
1260
ranging from 1016_1019/cm3, were identical (within 5% for
the 1/e decay time) to that of Fig. 4. These results indicate
that higher order electron-hole recombination processes are
not dominant for carrier densities less than 10 19/cm 3. The
grating decay, then, must be due to linear trapping processes. Previous studies ofidentically prepared crystals have
found bulk recombination dynamics to be on the near J.Ls
I
II (t)
= Ji {
*
1Te2
2
2A.p nmeh {i}p€o
Sa""
time scale. S(d) Therefore, the nonexponential decays must be
due to trapping processes associated with a high trapping
region near the surface.
These dynamics can be described by diffusion subject to
recombination at the surface with a constant surface recombination velocity. The predicted grating signal of this model
can be calculated from Eqs. (12)-( 14 ).15
f1N(z,t)dz } ,
(12)
0
f1N(Z,t)=No ex p(- 4rDt __t__ ~)(w[a(Dt)1/2_
Z
]+w[a(Dt)1/2+
Z
]
A2
7R
4Dt
2(Dt) 1/2
2(Dt) 1/2
_
W(x)
2S/D
(S /D) - a
= exp(x2 )erfc(x),
{w[a(Dt)1/2+
Z
2(Dt) 1/2
]
-
(14)
where II (t) is the diffracted light intensity, J I is the first
order Bessel function, f1N(z,t) is the peak-null electronhole pair density difference, No is the initial excess carrier
density at the surface, D is the ambipo1ar diffusion constant,
Sis the surface recombination velocity, and all other parameters are as defined previously. In our case, S is the only
unknown constant. A typical numerical result is compared
to grating data for a 4X 1O ls /cm 3 doped n-Ti0 2 sample in
Fig. 5. In this figure, the surface acoustic modulation of the
signal, which is discussed below, was subtracted from the
data. This calculation is a convolution ofEq. (12) with the
grating pulse shape response. Ti0 2 is a somewhat radical
case in that its ambipolar diffusion constant (0.0134 cm 2/s)
is extremely small compared to that of other semiconductors. S(d).27 For this value of D, calculations with Eq. ( 12) are
fairly insensitive to S. In fact, curves with S ranging from 104
to 107 cm/s are identical to within 1% of that of Fig. 5. Only
for S < 104 cm/s does the numerical result significantly deviate from the data. In the n- Ti0 2 in-air case, then, this is a
mathematical fit with essentially no adjustable parameters.
From the good agreement of the fit, the data is well described
by this model of nonradiative trapping and recombination at
the surface with a surface recombination velocity ;;;. 104
cm/s.
The theoretical fits are much more sensitive to changes
in the ambipolar diffusion constant ( ± 10%) than the surface recombination velocity. The good agreement between
the theoretical fit and the observed grating dynamics illustrates that the carrier dynamics for the in-air studies are
under flat band conditions or essentially zero space charge
field. In general, the presence of impurities on semiconductor surfaces in air leads to the formation of charged surface
states and a significant space charge field even in the absence
of a liquid junction. The surface charge corresponds closely
to the typical surface state densities of 1012/cm2 • For Ti0 2 ,
the dielectric constant is very large such that this level of
surface charge would represent less than 0.02 eV of band
bending. The width of this space charge field (-60 A)
would affect the transport of less than 5% of the optically
generated carriers. It is for this reason that flat band dynam-
w[S (Dt)1/2
Z
]})
D
+ 2(Dt) 1/2 '
(13)
I
ics are observed for the in air studies of Ti0 2 • In contrast, in
the presence of aleV liquid junction, the surface charge is
on the order of 10 14/cm 2 with a space charge field of several
hundred Angstroms. In this case, a significant fraction of the
carriers are generated within the space charge field. The inair studies serve as a control for the affects of the space
charge field on the hole carrier dynamics.
The above surface recombination dynamics also suggest
something about the exact nature of the surface state traps in
Ti0 2 • To determine whether or not the traps are chemisorbed (OH- )s' control experiments were conducted under
flat band conditions where the surface coverage of chemisorbed (OH-)s was varied. This was controlled by varying
the pH of the aqueous solution used to treat the surface.
Similar experiments were conducted with the crystals in
contact with an aqueous electrolyte under open circuit conditions where an applied external voltage bias was used to
keep the crystal under flat band conditions. The Ti0 2 potential relative to the seE reference electrode changes in a
(A)
(8)
o
246
TIME (NSEc)
8
FIG. 5. Comparison of transient grating data and surface state trapping
calculations. (A) Calculated signal response from Eqs. (\ 2), (\3), and
(14) using S = 105 cm/s. This is essentially independent of S for S> 10'
cm/s. (B) Actual transient grating data from n-TiO, (doping concentration = 4X IO ls/cc) in air. Excitation power = I X 10 14 photons/cm' (photogenerated carriers = I X 10 19 /cc). The surface acoustic wave response
has been subtracted out.
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Kasinski et a/.: Surface electron transfer processes
known manner due to hydroxide exchange with TiO z at the
surface to produce a chemisorbed hydroxide layer
(OHs- ).28 Surface bound OHs- should form mid gap
states 29 which would significantly increase the density of
surface state traps and should lead to enhanced surface state
trapping. However, to within signal-to-noise limitations,
identical results were found in all crystals studied irregardless ofOHs- coverage. The results were highly reproducible
from crystal to crystal with only small variations in the grating decays observed (as mentioned before, variations in the
1/e decay time were less than 5% ). Direct comparison of the
carrier dynamics of the same crystal before and after surface
treatment showed identical dynamics. The lack of a significant effect of surface adsorbed hydroxide demonstrates that
the nonexponential population decays observed cannot be
attributed to trapping processes due to chemisorbed OHs- .
At high pH, SOH should be ;> 104 cm/s while at low pH, SOH
should be <103cm/ s. In order for (OHs-) trapping sites to
explain the decay at the lowest pH studied, the capture cross
section for chemisorbed OHs would have to be a very unrealistically high -2X 1O-!2 cm 2. Therefore, the observed
trapping sites must be intrinsic to the semiconductor. The
lack of a chemisorption effect also indicates that the observed trapping sites are not at the atomic surface.
The above results are in accord with theoretical and experimental studies. Ultrahigh vacuum photoemission studies of Ti0 2 (001) have shown that, on defect free surfaces,
there are no intrinsic mid gap states. 30 This result has been
rationalized by theoretical calculations of the electronic
states of finite two dimensional surfaces constructed in the
(001) structure. 3 ! The oxygen and titanium atoms that constitute the atomic surface are found to have energies lying
within the valence and conduction bands, respectively. In
the context of carrier trapping processes, these localized sites
would be isoenergetic with band states and would not be able
to trap electron or hole carriers out of their extended band
states. The energetics of the surface atoms can be understood
by the highly ionic nature of the Ti02 lattice. There is
smaller covalent character to the lattice structure than in
silicon or III-V semiconductors such as GaAs. Intrinsic
midgap surface states are formed in these covalent crystals
due to bond rehybridization and associated interactions occurring at the break in lattice structure defined by the surface. 2 In the TiO z crystal structure, the valence band is
formed principally by the oxygen 2p orbitals and the conduction band by titanium 3d orbitals with very little mixing. 3 !
There are not significant changes at the surface in electronic
structure that would lead to high density, large cross section
intrinsic surface states. In contrast to defect free surfaces, it
has been determined that the principal midgap states of
Ti0 2 are surface defects involving oxygen vacancies at the
surface. 30 Such states leave the titanium in a + 3 oxidation
state which would represent a hole carrier trapping center.
These same defect states are created in the bulk of the crystal
during carrier doping. If oxygen vacancies were effective
trapping centers, a carrier concentration dependence should
have been observed. However, this was not the case.
The observed initial nonexponential decay of the carrier
population can be explained by a spatially distinct region of
1261
high trapping. The lack of a chemisorbed hydroxide effect
and the lack of an effect on oxygen vacancies (doping induced) indicates that this trapping site is not the atomic
surface layer but most likely the deformation layer which
extends approximately 100 Afrom the surface. 32 This region
is caused both by mechanical polishing and by the inhomogeneous nature of the doping process in TiO z that create
strain. The trapping centers are believed to be structural defects within the deformation layer which would have been
common to all the crystals studied. The surface is more
prone to structural defects than the bulk. Fluctuations in
local structure would create strain in the lattice and create
energy levels outside the band states that would act as trapping centers. More studies will be needed to unambiguously
identify the exact nature of trapping centers near the surface
of Ti0 2 • This is especially true since the observed dynamics
are dominated by the ambipolar diffusion of carriers to the
surface region. With faster carrier transport to the surface, a
more pronounced surface treatment effect may be observed.
The intrinsic surface state trapping rate constants and surface effects can be sorted out using different excitation wavelengths with shorter· optical penetration depths such that the
majority of the carriers are generated within the 100 Adeformation profile. However, the most important conclusion to
be drawn from the above studies is that the linear trapping
processes near the surface limit carrier lie lifetimes to 5 ns.
Interfacial electron transfer processes involving hole minority carriers are in competition with these trapping processes.
Quantum yield measurements, based on the photoanodic
current, have determined that the interfacial charge transfer
is greater than 80% efficient for minority carriers optically
generated within the depletion region. 25 Thus, an upper limit
can be placed on the interfacial electron transfer dynamics.
The results shown in Figs. 4 and 5 indicate that the first step
in the surface electron transfer process must be subnanosecondo
2. Interfacial electron transfer studies
After measuring the carrier dynamics in air under flat
band conditions, a liquid junction was constructed with the
crystal in place. Results are shown in Fig. 6 for the same
doping level (_10!9/cm 3 ) used for the in-air results depicted in Fig. 4. The data is noisier than the in-air studies due to
the increased scattered light from the aqueous electrolyte.
The results are shown for closed circuit conditions ( -0 V vs
SeE) at pH = 13.5. The same results were found for open
circuit. The grating decay consists of two distinct decay
components. There is an initial fast decay component of 460
ps and a slower component of 4.8 ns. The slow component is
comparable to the semiconductor in-air studies. It is due to
bulk electron-hole pairs generated outside the space charge
region under field free conditions. The slow decay is due to
diffusion of these bulk electron-hole pairs into the space
charge region where recombination rates with trapping
centers or interfacial electron transfer are high.
The fast decay component is the most important feature
of the signal. This decay is assigned to the depletion of minority carriers generated within the depletion region due to
interfacial electron transfer. The magnitude of this fast de-
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Kasinski et al.: Surface electron transfer processes
1262
FIG. 6. Transient grating signal from
(doping
concentration = 8.2X 10 18 eel in 0.32 M NaOH
in Hz 0. Semiconductor at 0 V vs seE.
Excitation power 6x 10 13 photons/cm z. The initial fast component
of 460 ps is due to thermalized band
edge electron transfer. The long 4.8 ns
component is due to diffusion of carriers generated outside the space charge
region into the depletion layer.
n-TiO z
I+j
o
.Z
.4
.6
.8
LO
I.Z
1.4
1.6
1.8
TIME (nsec)
cay component is approximately 50%. This component is
expected based on Eq. (5) and the fraction of minority carriers generated within the space charge field, i.e., the absorptivity coefficient a::::: 8 X 104 cm - I at 355 nm 32.33 and the
space charge region as determined from Mott-Schottky
measurements was 500 A. Previous measurements of photoanodic current have found a quantum efficiency of greater
than 80% at 299 nm at which virtually all the carriers are
generated within the space charge region. 25 Based on the
high quantum yield for interfacial hole transfer, the fast
component must be almost exclusively due to minority carrier depletion at the surface. Although this rate is ten times
faster than the solid state surface recombination processes
observed in the in-air studies, it is still much slower than that
expected for unthermalized hole tunneling processes or hot
carrier affects. A substantial fraction of the hole carriers are
optically generated within the space charge region
( - 30% ). Within the hot carrier model, the very small carrier mass leads to resonance tunneling across the interface to
a molecular acceptor from within the space charge region.
This requires ballistic transport of the carrier across the
space charge region. Some inelastic phonon scattering is inevitable; nevertheless, for a very small hole mass, the intrinsic
hole mobility prior to thermalization would have to be very
large. All the minority carriers generated within the space
charge region would undergo rapid transport to the surface
on a picosecond to subpicosecond time scale. Interfacial
charge transfer on a similar time scale is expected within the
hot carrier model. A significant decay component should
have been observed that was either pulse width limited with
a decrease in signal amplitude or of a few picoseconds. This
was not the case. The observed 460 ps decay component
demonstrates that unthermalized electron transfer processes
are not occurring at this surface, in contrast to previous predictions.
The above conclusion assumes that minority carrier
thermalization within the expected quantized space charge
region is occurring on a time scale faster than 500 ps. To
verify this, we have measured the thermalization based on
the frequency dependence of coherent surface acoustic generation. The depth of SA W modulation was identical both at
flat band and with a junction at 2 GHz frequencies at the
lowest excitation conditions possible (3X 1Ols/cm3 photocarriers). This result demonstrates that thermalization in
the space charge region is occurring on a time scale faster
than 250 ps (1/4 acoustic period). Slower thermalization
times than this would have caused a phase shift in the acoustic oscillations. Further, from measurements of the absolute
diffraction efficiency, Eq. (5) places a lower limit on m~ of3.
This measurement is in good agreement with another estimate for the hole mass based on a theoretical analysis of the
band structure correlated to reflectivity data. 32 This measured effective hole mass is much too large to effectively
quantize the space charge region. This determination of m~
is on the conservative side as the Drude model typically predicts smaller carrier optical constants than that found experimentally.34 The discrepancy is primarily from the neglect of the semiconductor band structure in calculating the
carrier optical properties with the Drude model. The large
hole mass measured negates any strong space charge quantization effects. The major conclusion to be drawn is that hot
carrier effects are not significant in this system. Electron
transfer processes must be occurring from thermalized carriers directly at the surface.
This conclusion must take into consideration the excitation conditions of the experiment. From studies of lower
doped samples, excitation conditions equivalent to the background carrier level doping show near flat band dynamics,
i.e., the grating decays are very similar to the in-air control
studies. The excitation conditions necessary for adequate
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Kasinski et a/.: Surface electron transfer processes
signal to noise for the highly doped sample shown in Fig. 6
was on the order of 6X 1018/cm3 (this was higher than the
lower doped samples due to the opaque nature of the crystals
at high doping). Under these excitation conditions, the
bands would significantly unbend as hole minority carriers
are driven to the surface. The resulting hole accumulation at
the surface would screen the field of the Helmholtz layer. As
a result, the electric field responsible for level quantization
would be depleted and the level spacing and carrier thermalization perturbed. However, a significant fraction of the carriers would have experienced the full field strength prior to
screening. More importantly, the large measured hole mass
eliminates any prospect for significant space charge region
quantization such that this is not an important effect.
The large hole effective mass also eliminates the need to
take into consideration space charge quantization on carrier
transport to the surface. Classical continuum treatments of
the carrier transport to the surface can be used. At present,
the intrinsic hole mobility is uncertain. However, from recent optical measurements of the ambipolar diffusion constant, the hole mobility must be at least 1 cm2IV S.8(d) Thus
the minority carriers generated within the depletion layer
are driven to the surface in less than 40 ps under the influence of the space charge electric field (E field = 2X 105
V/ cm, 500 A space charge width). 26 The hole carrier transport to the surface from within the depletion layer is faster
than the observed 460 decay component. Thus, the decay of
the minority carrier population within the depletion layer is
determined by the slow interfacial electron transfer rate at
the surface. The minority carriers become thermalized with
the lattice and the carrier acceptor is most likely either chemisorbed hydroxide ions (OHs- ) at Ti + 3 surface lattice
sites or the adsorbed layer of hydroxide (OH- (aq» which
comprises the Helmholtz layer. 25 Surface state trapping in
principle could also be an intermediate process for the interfacial transfer step. However, the defect centers are predominantly within the space charge region and not exclusively at
the atomic surface as determined from the in-air control
studies. These states would act as efficient trapping and recombination centers inhibiting interfacial electron transfer
and are unlikely, on the basis of the large quantum yield for
interfacial charge transfer. In addition, the effective surface
density of surface defect trapping centers is on the order of
10 12/ cm2 which is two orders of magnitUde smaller than the
hydroxide layer.
In contrast, hole filling from OH- adlayer can account
for the observed dynamics. The highly delocalized nature of
the hole vacancy in combination with the high OH- surface
coverage (> 10 141cm2) insures that electron tunneling
across the interface will be determined by the activation barrier to attain isoenergetic states and not by spatial diffusion
of carriers to a reactive acceptor site. The energetic and spatial position of the minority carrier is well defined by the
determination that the transfer involves thermalized minority carriers. The transfer dynamics in this case will be dominated by processes occurring at the atomic surface valence
band edge. Thus, the spatial and energetic coordinates of the
electron transfer process are well defined. The electron
transfer rate, involving OH-, can then be estimated from
1263
single site, single energy expressions for electron transfer.
These rates are defined b y35.36
k
(E - E
= vK(r)exp _
v
-A. ')2
(15)
redox
4kTA.'
'
where k is the electron transfer rate constant, v is the frequency of the nuclear reaction coordinate, K(r) is the electron transfer tunneling parameter, E v is the energy of the
surface valence band edge, E redox is the redox energy level
and A. ' is the solvent reorganization energy. The energetics of
the transfer process involving OH- (aq) are depicted in Fig.
7. The exact energy position of the valence band edge was
determined by Mott-Schottky measurements of the flat
band potential. The measured flat band potential of - 1.1 V
vs SCE agrees with previous studies. 23 •28 The redox potential
ofthe OH- (aq)/OHo(aq) was taken from the most recent
electrochemical data on this reaction 37 and serves as a reference for the energetics of 0 H - ( aq). The solvent reorganization was calculated according to expressions from Marcus
and the constants of water to be 1 ± 0.1 eV. 35 The changes in
nuclear coordinates in lattice relaxation of the solid state are
negligible in comparison to solvent relaxation. The frequency of this nuclear reaction coordinate in the bulk aqueous
layer would be determined to a good approximation by the
longitudinal dielectric relaxation time. 38 Although the structure of the aqueous interface and its dielectric properties are
unknown, the predicted electron transfer rates using the dielectric relaxation frequency of bulk water for v in Eq. (15),
vary from ten picoseconds to several hundred picoseconds
-1.1 U"". SC[
E
'f;~'~~';'
.................
T-
(ON-ION")
5.U
....---- .ed..
+ 1.0 U
"". $C[
A..l.U
+1.9 UUI. SC[
1_
+2.0U
UI. $[[
E
FIG. 7. Energy level diagram for the n-Ti02 /OH- ,H2 0 system. The semiconductor is on the left, electrolyte on the right. The arrows indicate unthermaIized long range electron tunneling processes ( T T ) and surface band edge
thermalized electron transfer ( Tsc ) from the hydroxide ion to the hole site.
The minority carrier acceptor distribution is weighted in favor of thermalized charge transfer processes.
J. Chem. Phys., Vol. 90, No.2, 15 January 1989
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1264
Kasinski et al.: Surface electron transfer processes
within a range of acceptable reorganization energies (0.91.1 eV) and OH- (aq) distances (2-5 .A.) from the surface.
These transfer times are in qualitative agreement with the
observed dynamics and supports OH- (aq) as the initial
hole acceptor. Earlier estimates ofOHs- place the energy of
that hole carrier acceptor at 0.6 eV above OH-(aq)25.29,39
which would make electron transfer to that site 1-2 orders of
magnitude slower than OH- (aq). However, the exact energetics of the interface are not well enough characterized to
enable a distinction between OHs- and OH- (aq).
The energy position of OH- (aq) also explains the absence of observable hot carrier effects. The OH- (aq) energetics overlap almost exactly with the surface valence band
edge which is ideal for thermalized band edge processes. The
density ofOH- (aq) ions isoenergetic with hot carrier states
is small. The more ideal case for the OH- (aq) energetics
with regard to unthermalized electron tunneling processes
would be with the OH- (aq) energy level centered supraband edge, i.e., at energies below the surface valence band
edge by -0.5 eV. Both the energetics of the OH- (aq) energy level distribution and the large hole mass are factors operating against unthermalized, electron tunneling processes at
this surface.
B. Optically generated surface acoustic wave studies of
the TI0 2 surface
The small oscillations seen in the free carrier grating
studies discussed above are due to the optical excitation of
surface acoustics. The evidence for this assertion will be given below. To increase the SAW amplitude modulation above
the carrier grating dynamics, the excitation power was increased to optically generate 1020 carriers/cm3. Under these
excitation conditions, higher order electron-hole recombination processes begin to dominate the carrier dynamics.
These results are shown in Fig. 8. In this particular study,
the pulse compressor was not used and the excitation pulse
durations were 140 ps. The initial decay in the grating is
nearly pulse width limited. The onset of the faster recombination is quite dramatic, occurring over a factor of 5 in carrier concentration, from 2 X 10 19/cm 3 to 1 X 1020 electronhole pairs/cm 3. The observed power dependence indicates
that these faster recombination processes are due to higher
order recombination. A comparison of the results in Fig. 4
with these results illustrates that the SA W amplitude has
increased significantly as indicated by the increased depth of
signal modulation. In going from carrier excitation levels of
3 X 10 1S/cm 3 to 1 X 102°/cm3, the SAW amplitude increases
by an order of magnitude. The increase in the SAW amplitude demonstrates that more energy is being deposited into
the lattice per optically generated carrier with the onset of
the faster recombination processes. This increase in the nonradiative rate is proof that Auger recombination processes
begin dominating the carrier recombination at these higher
carrier concentration levels. 24
From the spatial geometry of the acoustic excitation
mechanism, the acoustic wavelength must match the optical
interference pattern. In addition, the theoretical work in Sec.
II has shown that the diffracted probe signal should show an
(AI - A2 cos CtJt) oscillatory dependence. This predicted
dependence reflects the standing wave character of the optically generated SAWs. This effect is observed in Fig. 8 and
the beat pattern agrees with the theoretical predictions. This
result would have been expected from previous studies of
optically generated bulk acoustics. 20 From the acoustic beat
frequency (vacoustic)' the velocity can be readily measured
( cs = A Vacoustic ). In this manner, the speed of sound is found
to be 4.84 ± 0.01 X 105 cm/s. This speed of sound measurement is in excellent agreement with the known surface
acoustic velocity of 4.807X 105cm/s based on SAW excitation using conventional means. 40 Since all other acoustic
modes of Ti0 2 have substantially higher velocities,41 the optically excited acoustic mode can only be attributed to a surface acoustic wave. Thus, this result demonstrates that surface restricted transient gratings can be used to selectively
excite coherent surface acoustic waves. Previous studies using optical excitation of surface acoustics did not establish
coherence in the SAW excitation. 17.42 Coherence in this case
is defined as the collective excitation of the material displacements in phase. The results in Fig. 8 show clean sinusoidal
oscillations which are indicative of the coherent excitation of
a single frequency SAW.
The SAW diffraction efficiency at the optical excitation
level used was 1 X 10 - 7. From this measurement, the amplitude of the surface displacement can be measured and compared to the theoretical work in Sec. II. However, the SAW
light diffraction is superimposed on top of a free carrier
phase grating component, as observed by the SA W oscillations occurring on top of a large baseline offset which is slowly decaying nonexponentially (see Fig. 8). A detailed analysis of the light diffraction process needs to incorporate
diffraction from both the carrier population grating l5 .16 and
the surface acoustics. 43 The total diffraction, with those effects included, is given by
'T/
"
= ( -21T)2[dll.N(X,z,t)n eh + ~
Ap
+
(1
2
00
ll.n(X,z,t)jjdZ)
o
2
r
4
;
TIME (NSEc)
(n l
0
-
n 2 )w;
(16)
6
8
FIG. 8. Transient grating excitation of surface acoustics at n-Ti02 surfaces
(doping concentration = 4X IOI8/ee ) in air. Excitation power = 2X 10"
photons/cm2 (photogenerated carriers = 2 X l(j2o/cc).
J. Chem. Phys .• Vol. 90. No.2. 15 January 1989
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Kasinski sf al.: Surface electron transfer processes
where n 1 - n2 is the index of refraction difference across the
interface, n2 referring to the solid phase index of refraction, i
refers to the different surface acoustic modes that propagate
along the interface,
is the surface displacement of each
mode which leads to surface corrugation at the acoustic
wavelength, and anjf refers to density modulations in the
index of refraction as related through the material photoelastic parameters. In the case ofTi02 in air, there is only one
surface acoustic mode to consider.
From the symmetry of the (001) surface and the photoelastic tensor for Ti0 2 , only the Pl122 photoelastic effect
contributes to the signal. 44,45 The sign of this constant is positive,45 which means the index of refraction decreases with a
decrease in density or positive dilation strain. The surface
corrugation effect and the photoelastic effect are opposing
effects in this case. The photoelastic effect partially cancels
the phase modulation of the light off the spatially harmonic
surface displacement. However, the surface corrugation effect is larger than the bulk density contributions at £r incidence. 46 In addition, the surface contribution to the diffraction efficiency is further enhanced by the optically induced
thermal strain component which is nonpropagating. This
thermal strain is induced by the nonradiative relaxation of
the optically generated carriers which is highly localized at
the surface by the short optical penetration depth ( - 1000 A
or A/to) at the optical excitation wavelength. In contrast,
the acoustic strain is driven by the initial thermal expansion
and radiates away from the surface with the acoustic strain
amplitUde decaying exponentially away from the surface
with a 1/e decay profile of approximately one acoustic wavelength. Thus, the acoustic strain does not spatially coincide
with the thermally induced strain. This additional contribution by the surface thermal strain layer further determines
that the diffraction efficiency is dominated by surface corrugation.
The initial surface displacement, which is spatially coincident with the peaks in the optical interference pattern, is
calculated to lead to negative z displacements, i.e., a bump on
the surface. The effective optical pathlength is, therefore,
increased in transmission in spatial regions corresponding to
peaks in the optical interference pattern. In contrast, the
Drude model for the optical properties of electron-hole carriers used to derive Eq. (5) predicts that the index of refraction should decrease, thereby decreasing the effective optical
pathlength in this same region. Therefore, the diffraction of
this SAW should appear as a modulation which decreases
the free carrier phase grating diffraction. This is observed
experimentally (see Fig. 8). The first peak in the SAW diffraction appears at one half an acoustic cycle from t = 0 and
clearly shows up as a decrease in the signal. This comparison
of the SAW light diffraction, which has a known phase,
uniquely determines the sign of the free carrier phase grating. This result demonstrates that the Drude model for the
free carriers predicts the correct sign for the free carrier optical properties, even though the magnitUde of the change in
absorption and dispersion is probably not correct due to the
neglect of the semiconductor band structure in this model.
From the depth of the SAW modulation of the carrier
phase grating, the surface displacement w' is found to be
w;
1265
0.5 ± 0.2 A. This determination takes into account the carrier phase grating cross term in Eq. (16). The large error bar
in this measurement results from the uncertainty in the photoelastic constants of Ti0 2 • The measured surface displacement is in excellent agreement with the theoretically calculated value of 0.4 ± 0.2 A for the experimental conditions
employed. In this case, the error bar is a consequence of
using an isotropic approximation. Since the SAW propagation is planar, the principle uncertainty is in the thermal
expansion coefficient. This agreement is as close as could be
expected given the large uncertainty in the photoelastic contribution to diffraction. This good agreement demonstrates
that the theory developed in Sec. II is quantitatively correct
and can be used to calculate accurately photoinduced surface strain and SAW propagation.
These same optical SAW studies were conducted in the
presence of an aqueous interface. The results are shown in
Fig. 9. The results are significantly different than the above
studies conducted in air. The solid-air surface represents a
free half space to SAW propagation; whereas the SAW studies in the presence of the liquid interface involve nontraction
free propagation. At the interface between two elastic media,
there are only certain material combinations for which interfacial modes, called Stonely waves, exist. These conditions
involve large impedance mismatches in the shear velocity
components at the interface. 47 This condition is met at liquid
interfaces as there is virtually no shear component to acoustic propagation in liquids. In the presence of solid-liquid
interfaces, there is expected to be two interfacial modes
which co-propagate. The high frequency mode localized in
the solid half space is referred to as the generalized Rayleigh
mode and propagates with a velocity approximately equal to
the SAW mode of the traction free solid but decays slowly as
energy leaks off into the liquid. The second mode, referred to
as the Stonely mode, is an interfacial wave with its energy
highly localized in the liquid phase which propagates 1%-
N
T
E
N
5
I
T
Y
o
2
4
6
TIME (nsGlc)
8
10
FIG. 9. Transient grating excitation of interfacial acoustics at the
n-Ti02 /H20 interface. n-Ti02 (doping concentration = 4X 1018/cm 3) in
1 M Na2 SO. and 0.054 M NaOH in H 2 0. Semiconductor is at - 0.19 V vs
SeE and the excitation power = 2X 1015 photons/cm2 (photogenerated
carriers = 2 X 1020/ cm 3 ). There are three acoustic components in the data.
The most clearly visible are the high frequency Ti02 Rayleigh wave and the
low frequency modulation of the Rayleigh beat signal due to the aqueous
interfacial Stonely wave.
J. Chern. Phys., Vol. 90, No.2, 15 January 1969
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1266
Kasinski sf al.: Surface electron transfer processes
2% slower than the bulk longitudinal modes in the liquid. 47
Thus, two frequency components are expected. All these results are observed experimentally. There are negative signal
oscillations corresponding to the SA W mode observed in the
air studies with the same velocity observed. The amplitUde of
this wave is attenuated by the liquid layer (decay time of
-4-5 ns) whereas in the absence of the aqueous interface it
was relatively unattenuated. Similarly, there is a slower velocity component of 1.471 X 105 cm/s which is 1.3% slower
than the bulk speed of sound in water which corresponds to
the Stonely aqueous interfacial mode.
However, rather than just two acoustic modes, Fourier
transform analysis of the data as well as time domain inspection has revealed a third velocity component. This feature of
the signal is better illustrated in Fig. 10 where the interfacial
aqueous wave component of the signal has been subtracted
from the data. Here the third component is clearly visible as
a positive shoulder on the Ti0 2 SAW wave (indicated in the
figure by arrows). This result shows that the third component induces a phase grating of the opposite sign relative to
the Ti0 2 SAW component. The sign of the phase grating
developed from this anomolous interfacial mode is the same
as that observed for the aqueous interfacial acoustic mode.
This 180· change in the phase grating on the aqueous side of
the interface is expected due to the 180· phase difference in
the way the liquid acoustic modes modulate the surface corrugation. The fact that light diffraction from this anomolous
acoustic mode is 180· phase shifted from the Ti0 2 solid
SAW mode demonstrates that this mode is localized within
the water half space of the interface. In addition, this mode
attenuates much faster than any other surface mode which
further demonstrates that it is an independent surface mode.
It is a true surface acoustic mode uncoupled from the solid
Rayleigh or liquid interfacial modes. The interesting feature
of this acoustic mode is that its speed of sound is
4.4 ± 0.2 X 105 cm/s. This velocity is much too fast to be
explained by the bulk phase of water. This fast water velocity
component corresponds more closely to a solid wave speed
which indicates that the water layer has undergone a phase
structure change at the highly ionic Ti0 2 interface. This
point will be discussed further below.
There are two possible excitation mechanisms for the
aqueous interfacial modes. Either these modes are excited by
the propagation of the solid state SAW itself or by thermal
energy transfer across the interface. In the solid SAW excitation mechanism of the aqueous modes, the first maxima in
the material strain would be phase shifted from its normal
position at one half an acoustic cycle from t = 0 (which is
the signature of a thermal excitation mechanism). This is
not observed, although the exact position of the first maxima
of the high velocity water mode is very imprecise and this
statement cannot be applied to this mode with certainty.
Moreover, each oscillation in the surface due to the solid
SAW propagation would excite the interfacial aqueous
modes out of phase due to the much higher acoustic frequency of this mode relative to the aqueous modes. This effect
would not lead to coherent excitation and would cause cancellation of the standing wave acoustic beats. This is also not
observed. Therefore, the excitation mechanism must involve
thermal energy transfer across the interface. These modes
are amplified in their diffractive power by the presence of the
carrier phase grating. The quadratic cross terms of the liquid
N
T
E
I
N
N
s
T
E
N
T
y
S
I
T
0
2
4
6
8
TIME(nsec)
10
CA)
Y
FIG. 10. Surface acoustic studies
of n-Ti02 (doping concentration
= 4X 10 '"Icc).
Excitation
power = 2x 10'5 photons/cm2
(photogenerated
carriers
= 2X IO'°/cc). (A) In air. (B)
In I M Na2SO. and 0.054 M
NaOH in H 20. Semiconductor
at - 0.19V vs SeE. Longest period acoustic component has
been subtracted. The conditions
are the same as Figs. 8 and 9 for
(A) in air and (B) in I M Na,
SO. and 0.054 M NaOH in H 20,
respectively.
The
Stonely
aqueous acoustic component is
subtracted from the data in (B).
The high frequency H2 wave in
curve (B) appears as positive
shoulders on the Ti02 Rayleigh
mode, i.e., 180· out of phase (indicated by arrows). This is better
observed in the inset where the
two curves are directly overlapped. The bottom curve in the
inset is Ti02 in air.
°
(8)
2
4
6
TIMECnsec)
8
10
J. Chern. Phys., Vol. 90, No.2, 15 January 1989
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Kasinski et al.: Surface electron transfer processes
SAWs with the carrier phase grating in Eq. (16) act to amplify the surface acoustic diffraction in a manner very similar
to optical heterodyne detection with a reference optical signal. 48 It is for this reason that these liquid interfacial modes
are even observable.
Further studies of the TiOz surface in the presence of
other liquids were also conducted. Figure 11 shows a comparison between SAW generation at an aqueous interface
and ethanol liquid interface. The ethanol interface shows the
two expected interfacial modes involving preferential propagation in the solid and liquid bound half spaces. The slow
velocity of the liquid interfacial mode agrees with the expected interfacial Stonely mode. Two important points result
from the study of the ethanol interface. First, the high speed
liquid interfacial mode is not observed at the ethanol interface. Second, the depth of modulation due to the Stonely
interfacial ethanol mode is nearly identical with that of the
aqueous interface. The first point illustrates that the high
velocity liquid mode is unique to the structure of the aqueous
interface and not an acoustic sideband between the two normal interfacial acoustic modes. Nonlinear harmonics can
also be ruled out from this study as nonlinear harmonics
would diffract light into the second order or higher and not
into the first order of diffraction that was monitored. 43 The
fact that the depth of signal modulation due to the liquid
ethanol interfacial mode did not significantly increase demonstrates the diffraction process is dominated by surface corrugation effects. If photoelastic contributions were significant, a pronounced increase in signal should have been
observed since the photothermal effect in ethanol is five
times larger than in water. 49 The photoelastic effect in Eq.
(16), integrated over the acoustic 1/e energy profile of the
stationary state wave, is comparable to surface corrugation
effects on diffraction. The absence of a significant photoelastic component to diffraction means that the energy localization of the interfacial liquid acoustic mode is much shorter
range under these transient observation conditions than that
calculated based on analytical solutions to steady state prop-
N
T
E
N
S
I
T
Y
CA)
(8)
o
2
4
TIME
6
8
10
(nsec)
FIG. 11. Surface acoustic studies at n-Ti02 interfaces. Conditions same as
Fig. 9 except (A) represents a neat ethanol interface. Curve (B) is for nTiO,in 1 M NazSO. and 0.054 M NaOH in H 2 0 (Fig. 9). The high frequency component observed in the water layer is not seen at the ethanol
interface.
1267
agation. Interference effects that would normally localize
the energy within one acoustic wavelength under stationary
conditions, would not have been established in just I! to 2
acoustic cycles which is the observation time of this interfacialliquid mode. In addition, impulsive material propagation radially away from the surface, following the surface
harmonic temperature jump, would lead to a multitude of
regions of constructive and destructive interference that
would cancel photoelastic effects on this short time scale.
The necessarily very short range localization of the acoustic
energy at the surface with regard to effects on light diffraction further highlight the importance of the carrier phase
grating in amplifying the diffraction.
The most important point in all the studies of SAW
propagation at interfaces is the unprecedented observation
of a high velocity component traveling in the liquid half
space. This result implies that there has been significant restructuring of water at the interface. In fact, the observed
speed of sound is very close to the speed of sound in ice at
o·C (within 10%).50 The correlation of the wave speed with
a solid or glass phase indicates that the highly ionic Ti02
surface has strongly perturbed the water layer. The static
potential of the surface undoubtedly preferentially orients
the water molecules and removes slip-plane translational degrees of freedom which are essential to liquid phase structure. This perturbation would extend several atomic layers
from the surface. This restructured water layer would represent a bounded thin elastic layer. A full theoretical treatment
of acoustic propagation in such a structure needs to be
worked out. It has already been shown analytically that
acoustic propagation in thin elastic layers on a solid half
space are dominated by mode propagation in the solid
state. 51 If the layer is much thinner than an acoustic wavelength, the thin elastic layer has no effect on the surface
acoustics. However, these analytical solutions require a stationary state solution to interfacial wave propagation. The
excitation of the acoustic mode in a restructured water layer
in the above experiment is by impulsive thermal energy
transfer from the Ti0 2 surface into the tightly bound
aqueous layer. The fact that this mode is not a stationary
solution to the interfacial wave problem means that any
transient mode propagation would be very heavily damped,
which is exactly what is observed. The high velocity water
mode only lives for q acoustic cycles.
The length scale of the restructured aqueous layer is an
important factor. There has been some theoretical work on
similar problems which have treated the density modulations at homogeneous solid-liquid interfaces at phase transition critical points. 52 More relevant is the work by Rossky
and co-workers on the structure of water at heterogeneous
solid-aqueous interfaces where the density perturbation is
found to extend a few atomic layers. 53 The depth profile of
the restructured water layer can be estimated from the experiment by comparing the amplitUde of the more classical
aqueous interfacial mode to the anomolous high velocity
component. The static thermal strain of the former mode is
localized by slow thermal diffusion to less than 300 A from
the surface within one-half acoustic cycle. Based on the observed relative diffraction efficiency of the two modes, the
J. Chem. Phys., Vol. 90, No.2, 15 January 1989
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Kasinski et at: Surface electron transfer processes
1268
restructured water layer would have to be - 30 A thick (7lO monolayers) to account for the data. This length scale is
in approximate agreement with theoretical estimates of the
surface effect on liquid density. The layer may be more extensively restructured in the presence of the n- Ti02/H2
interface than expected, not only due to the highly ionic nature of Ti0 2, but also to the formation of the liquid junction
at the interface. Very intense electric fields are associated
with the formation of a double layer of charge at this interface. Within the solid half space, the space charge fields
which develop upon junction formation are on the order of
lO5_lO6 V/cm over a space charge width of a few hundred
Angstroms. An opposite field develops in the accumulation
of hydroxide ions within the aqueous interface. 36 The width
of the counter-ion accumulation in the water layer is an order of magnitude smaller. This very high electric field and
the accumulation of hydroxide ions ( > lOl4/cm2 at the surface boundary) would have a significant affect on the liquid
structure of the interface which would extend the perturbation of the surface. In addition, the formation of a gel layer at
the Ti0 2/H 2 interface at high hydroxide coverage has
been postulated from other experiments. 54
Further theoretical work on transient acoustic mode
propagation along thin bounded layers will be needed to fully characterize the high velocity interfacial water mode.
However, the wave speed demonstrates that the water layer
is strongly interacting with the surface. This strong interaction and restructuring of the water layer would have a pronounced effect on the dielectric relaxation at the interface
which in tum would affect the dynamics for interfacial electron transfer. The effect would become most pronounced in
cases where the electron transfer step involves surface band
edge levels centered at the maximum in the electronic energy
bandwidth of the molecular acceptor, as in the present case.
In this case, the electron transfer step could be rate limited
by dielectric relaxation processes. 55 Liquid restructuring effects and slower dielectric relaxation at interfaces would also
limit the possibility oftaking advantage of hot carrier interfacial transfer processes in surface chemistry. Water restructuring effects would also dramatically affect molecular diffusion parallel to the surface. Both dielectric relaxation and
constraints on molecular diffusion are important considerations in understanding chemical processes at solid-liquid
interfaces.
°
°
°
ACKNOWLEDGMENTS
v. CONCLUDING REMARKS
The surface restricted grating method has been demonstrated as a multifaceted probe of both reaction dynamics
and phase structure at single crystal surfaces. The high temporal resolution enables detailed studies of surface processes
in situ at working single crystal surfaces. For highly doped
Ti0 2 crystals, the main competing pathway for charge carriers at the surface are linear recombination processes involving crystal defect centers within the surface strain layer. Auger recombination processes are not effective in carrier
recombination until photocarrier levels in excess of5 X lO19/
cm 3. The in situ investigation of minority carrier depletion
within the n-Ti0 2/H2 liquid junction space charge region
°
demonstrates that interfacial charge transfer involves predominantly thermalized hole vacancies within kT of the
atomic surface valence band edge. This is the main conclusion to be drawn from this study. This conclusion is further
supported by SAW measurements of carrier thermalization
and the grating measurement of the effective hole mass,
mt;,3, which rule out unthermalized charge transfer processes at Ti0 2 interfaces. The OH-;- (aq) adlayer is believed
to be the initial acceptor based on the observed dynamics of
460 ps for the minority carrier depletion and estimates of the
surface energetics. The exact energetics of the interface will
be needed to completely distinguish between chemisorbed
OHs- and OH-(aq) in the outer Helmholtz plane as the
predominant hole acceptor. A detailed theoretical treatment
of the three dimensional aspects of the problem incorporating both minority carrier transport and interfacial transfer
also needs to be addressed to make a definitive assignment of
the initial hole carrier acceptor based on dynamics.
In addition to the study of carrier dynamics, the grating
can be used to selectively excite surface acoustic modes. Our
theoretical treatment of the photothermal coupling to the
surface modes predicts very closely the observed surface displacements. Use of the SAW grating as a reference has enabled a determination of the free carrier optical properties
which agrees with the Drude model. In the presence of the
aqueous interface, optically generated SAWs provided new
information about the collective atomic structure of the interface. A transient high velocity component in the water
half-space gives evidence for a liquid/solid phase transition
at the highly polar Ti0 2 surface i.e., a rigid layer of H2 at
the Ti0 2 interface. Due to the inhomogeneous nature of the
surface, this solid water layer would also have to be highly
inhomogeneous.
The high information content of optically written surface gratings provide a new approach to the study of surface
reaction processes. This work can be extended to other surfaces to provide a detailed experimental analysis of both surface mediated electron transfer dynamics, the role of surface
states in the process, and interfacial phase structure. All
these factors are important to understanding surface chemistry and the electron transfer mechanism in general but also
have particular importance in controlling the surface chemistry that limits the applications of semiconductor interfaces.
This work was supported by the Department of Energy,
Office of Basic Sciences (81-049) and the National Science
Foundation (S.M.G.). R.J.D.M. is a: recipient of a NSF
Presidential Young Investigator Award and an A. P. Sloan
Research Fellowship.
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