2548
J. Opt. Soc. Am. B / Vol. 20, No. 12 / December 2003
Wilson et al.
Second-harmonic amplitude and phase
spectroscopy by use of
broad-bandwidth femtosecond pulses
P. T. Wilson, Y. Jiang, R. Carriles, and M. C. Downer
Department of Physics, University of Texas at Austin, Austin, Texas 78712-1081
Received March 14, 2003; revised manuscript received July 2, 2003; accepted August 1, 2003
We present an in-depth experimental study of frequency-domain (FD) methods for measuring second-harmonic
(SH) amplitude and phase spectra of surfaces by use of a 60-nm bandwidth femtosecond source and spectral
dispersion of generated SH light. We directly compare FD with conventional scanning approaches, in which
a narrowband laser is tuned over resonant features, by applying them to common Si1⫺x Gex , Si1⫺x⫺y Gex Cy , and
Si(001) – SiO2 – Cr metal–oxide–semiconductor (MOS) samples. FD methods yield chirp-independent ␹ (2) amplitude spectra in good agreement with more time-consuming conventionally measured spectra. FD interferometric SH (FDISH) phase spectroscopy avoids the need for an interferometer scan at each frequency and yields
detailed, reproducible phase spectra of the MOS capacitor. To validate the measured phase spectra, we reproduce their bias-dependent features in detail with a model of a resonant electric-field-induced SH polarization superposed coherently upon a field-independent background. © 2003 Optical Society of America
OCIS codes: 240.4350, 190.4350, 240.6490, 320.7110.
1. INTRODUCTION
When surfaces or thin films are characterized by spectroscopic ellipsometry (SE), typically both the amplitude and
the phase of the reflected light are measured over spectral
bandwidths as wide as ប ␻ ⬃ 5 eV. 1 This permits complete characterization of the complex linear dielectric
function ⑀ (z, ␻ ) within the skin depth region. For realtime monitoring and control applications, data at different wavelengths are acquired in parallel by use of a
broadband light source, a spectrometer, and a high-speed
computer to analyze reflected amplitude and phase.2
Such wide-bandwidth, high-speed capabilities are routine
features of commercial SE instrumentation.
For centrosymmetric materials such as elemental semiconductors,
second-harmonic
generation
(SHG)
spectroscopy3 is much more specific than SE to surfaces
and interfaces4,5 and to bulk regions pervaded by electric
fields,5 strain gradients,6 and other centrosymmetrybreaking features. This is so because, in the electric dipole approximation, second-order nonlinear susceptibility
␹ ( 2 ) is nonzero only where centrosymmetry is broken.4
However, in contrast to SE, typically only SHG amplitude
at a single wavelength (or within a narrow bandwidth) is
measured, because the conventional approaches to acquiring SHG spectral amplitude and phase are prohibitively time-consuming for many applications. One acquires conventional SHG surface spectra by measuring
the intensity of the reflected SHG signal while tuning a
narrowband pulsed source laser. Conventional phase
measurements are made by a time-domain interferometric method in which, at each wavelength, the SHG signal
interferes with a reference SH signal generated in a spectrally flat nonlinear material by a split-off part of the incident beam.7,8 Because of the sensitivity of pulsed laser
sources to alignment, the tuning process alone can take
0740-3224/2003/122548-14$15.00
several minutes to several hours,9 depending on signal
level and spectral range. This time is increased greatly
when the spectral phase of surface SHG is measured, because an interferometer must be scanned at each
wavelength.10
Recently, broadband infrared (IR) sources were used together with an intense narrowband visible pump to acquire surface vibrational spectra of adsorbed
molecules11–13 and buried interfaces14,15 by sumfrequency generation (SFG), a close ␹ ( 2 ) relative of SHG.
The IR pulse spanned one or more single-photon vibrational resonances [Fig. 1(a)] and was temporally synchronized with the narrowband pump. When the latter was
nonresonant with higher levels—as required to avoid convolving one- and two-photon-resonant structures—the reflected SFG spectrum was recorded with parallel multichannel detection with no moving parts, thus
dramatically reducing data acquisition time compared
with that for spectra acquired by serial tuning of a narrowband IR source. The signal-to-noise ratio also improved because drifts associated with laser tuning were
eliminated. Amplitude, but not phase, spectra were acquired.
In a recent Letter we briefly introduced an analogous
surface SHG method that measures both amplitude and
phase spectra in nearly real time, using a single broadband coherent source (a 15-fs Ti:sapphire laser pulse).16
Unlike broadband SFG, broadband SHG required no temporal synchronization and resembled a linear SE system.
SHG amplitude spectra were recorded by dispersion of
the reflected SHG signal in a spectrometer equipped with
an array detector. SHG phase spectra were recorded by
a frequency-domain interferometric second-harmonic
(FDISH) method: A spectrally flat nonlinear reference
film was inserted into the incident beam before the
© 2003 Optical Society of America
Wilson et al.
Vol. 20, No. 12 / December 2003 / J. Opt. Soc. Am. B
Fig. 1. (a) Broadband SFG by an IR continuum pulse that is
resonant with one-photon transition(s) and a narrowband visible
pulse.10,11 Two-photon (IR ⫹ VIS) processes should be nonresonant. (b) Broadband SHG by use of a single incident continuum
pulse. Left, direct SHG processes that dominate in the absence
of one-photon resonances near the fundamental frequencies.
Right, SFG that becomes important in the presence of such resonances.
sample, generating a reference SH pulse that interfered
with the sample SH pulse in a spectrometer to yield
frequency-domain (FD) interference fringes. Fourier
analysis of these interferograms yielded the SH phase
spectrum of the sample surface relative to the reference
film.
In general, a mixture of direct SHG [Fig. 1(b), left] and
SFG [Fig. 1(b), right] processes contribute to the measured broadband SHG spectrum. Specifically, incident
field E( ␻ ) creates a SH polarization
P
共2兲
共2␻兲 ⫽
1
共 2␲ 兲2
冕
⬁
⫺⬁
␹
共2兲
共 2 ␻ , ⍀, 2␻ ⫺ ⍀ 兲
⫻ E 共 ⍀ 兲共 2 ␻ ⫺ ⍀ 兲 d⍀
(1)
that convolves SHG (⍀ ⫽ ␻ ) and SFG (⍀ ⫽ ␻ ) and depends on the chirp of E(⍀). The resultant signal is thus
of little use in extracting material spectra. If, however,
we restrict broadband SHG to situations in which material resonances are absent near the fundamental
frequencies—analogous to the requirement that the narrowband visible pulse in broadband SFG be
nonresonant—then ␹ ( 2 ) is constant across the bandwidth
of E(⍀) and factors out of the integral, leaving
P 共 2 兲共 2 ␻ 兲 ⫽
1
共2␲兲
2
␹ 共 2 兲共 2 ␻ 兲
冕
⬁
⫺⬁
E 共 ⍀ 兲 E 共 2 ␻ ⫺ ⍀ 兲 d⍀.
(2)
Now P (2 ␻ ) is directly proportional to the material
spectrum ␹ ( 2 ) (2 ␻ ). Chirp dependence, contained in the
integral term, is no longer convolved with the material
spectrum and can thus be normalized by a single independent measurement of the broadband SHG spectrum of a
spectrally flat reference sample such as crystalline
quartz. Recently16 it was indeed demonstrated that normalized broadband SHG amplitude and phase spectra
mirror expected qualitative ␹ ( 2 ) trends: e.g., alloy shift of
(2)
2549
the E 1 peak of Si1⫺x Gex , and ␲ phase shift of SHG from a
metal–oxide–semiconductor (MOS) structure on crossing
flatband voltage.
In this paper we present a quantitative foundation for
broadband SHG spectroscopy. Quantitative evaluation is
essential because, in reality, ␹ ( 2 ) (2 ␻ , ⍀, 2␻ ⫺ ⍀) is
never completely ⍀ independent across the incident pulse
spectrum. For example, ⍀-dependent absorption across
the indirect bandgap of silicon is present in all the
samples studied here. A conceptual framework for evaluating the influence of weak ⍀ dependence on broadband
SHG spectroscopy is therefore presented below. Explicit
quantitative measurements of the dependence of the SHG
spectrum on the chirp of the incident broadband pulse are
then presented, for the first time to our knowledge. A
normalization procedure that yields robust, chirpindependent SHG spectra that agree quantitatively in
their resonant structure at SH frequencies with conventional scanning spectra is demonstrated. Finally, we report for what we believe is the first time a detailed, quantitative comparison of FDISH and conventional SH phase
spectroscopy, using a Cr–SiO2 – Si MOS capacitor that
permits convenient control of SHG phase by varying applied bias. This comparison establishes the consistency
of FDISH and conventional SH phase measurements and
also highlights strengths and weaknesses of each method.
Taken together, these measurements establish the quantitative validity of acquiring surface SHG intensity and
phase spectra with a broadband source. Subtleties of the
FDISH measurement technique are also described in
much greater detail than previously.
2. EXPERIMENTAL PROCEDURE
A. Light Sources
A mode-locked Ti:sapphire (Ti:S) laser [Kapteyn–
Murnane Laboratories Model TS (Ref. 17)] provided 15-fs,
5–10-nJ fundamental incident pulses at a 76-MHz repetition rate with a typical bandwidth of 60 nm (FWHM) centered at 775 nm (␭ SH ⫽ 3.2 eV) for most broadband SHG
measurements. For some measurements, different laser
optics sets were substituted to shift the center wavelength as much as 30 nm to produce better SHG data at
the blue or red end of the spectrum. We controlled the
chirp of the output pulses by passing the pulses through a
pair of extracavity prisms and characterized it by
frequency-resolved
optical
gating.18
Preliminary
measurements19 of SHG spectra used 0.1-nJ, ⬃4-ps incident pulses with bandwidths as wide as 200 nm that were
produced by focusing of the Ti:s pulses into a tapered20 or
photonic crystal21 fiber to generate a white-light continuum.
To acquire conventional scanning spectra for comparison we converted the Ti:S laser to a narrow-bandwidth
(⬍10 nm) tunable source by inserting a slit into the cavity
at a position where intracavity prisms spatially dispersed
its spectrum. In addition, intracavity fused-silica prisms
were replaced by LaKL21 glass to stabilize mode locking
with reduced bandwidth. Pulse duration increased to
⬃100 fs, with 5–10 nJ pulse energy. Translation of the
slit permitted us to tune from 720 to 870 nm, using two
sets of cavity mirrors.
2550
J. Opt. Soc. Am. B / Vol. 20, No. 12 / December 2003
Wilson et al.
For qualitative comparison with SHG spectra, linear
pseudodielectric functions 具 ⑀ 1,2( ␻ ) 典 of some samples were
measured with either a phase-modulated spectral
ellipsometer22 (Verity Instruments, Inc., Model VeriFILM) or a rotating-analyzer ellipsometer equipped with
a computer-controlled MgF2 wave plate as a
compensator.23 Details of the SE measurement and
analysis procedures are described elsewhere.2,23
B. Acquisition of SHG spectra
1. SH Amplitude Spectra: Broadband and Scanning
Methods
For measurements of broadband SHG amplitude spectra
without phase information, the incident 60-nm bandwidth pulses impinged at f/20, 45° incidence, and p polarization on the sample. The reflected SH light was polarization analyzed and spectrally filtered before entering a
spectrometer (Acton Model SP300i) through an f/numbermatching lens. Its spectrum was recorded by a liquidnitrogen-cooled photon-counting CCD array (Princeton
Instruments Model 1100PB). Glass plates were inserted
into the incident beam to vary its chirp. For normalization against spectral structure of the incident pulse, we
acquired reference spectra by substituting a standard
sample (hereafter termed simply ‘‘standard’’) for the
sample under study.
For conventional scanning spectra, the narrowband
Ti:S laser was used, and a photon-counting photomultiplier tube (PMT) replaced the spectrometer and the CCD
for SHG detection. To normalize against pulse variations
during tuning, a portion of the incident beam was split off
to generate a SH signal in a Z-cut quartz standard at each
wavelength.
For the current studies, fluences of 10⫺4 ⬍ F
⬍ 10⫺3 J/cm2 impinged upon the sample’s surface for
both methods. Because of their different pulse characteristics and detectors, absolute SHG intensities measured by broadband and scanning methods could not be
compared accurately. Thus to compare broadband and
scanning SHG spectra for a series of samples we scaled
one set of spectra by a single multiplicative factor to
achieve the best fit.
2. FDISH Phase Spectra
To acquire phase information we altered the broadband
procedure described above only by inserting a thin nonlinear reference film (SnO2 or In:SnO2 ; thickness, approximately one SHG coherence length l coh ⬃ 15 ␮m) and
its glass substrate (thickness, d ⬃ 2 mm) into the incident beam before the sample [Fig. 2(a)]. A SH reference
pulse was generated in the SnO2 and then delayed from
the fundamental by ␶ ⬃ 1 ps by the group-velocity dispersion of the glass plate. Alternatively, the incident pulse
could be reflected from a nonlinear reference (e.g., crystalline quartz or GaAs) and then passed through an ⬃2-mmthick glass plate en route to the sample [Fig. 2(a), inset].
These two pulses were then reflected from the sample,
where the fundamental pulse generated a sample SH
pulse that led the reflected reference SH pulse into the
spectrometer. The CCD then recorded a FD interference
pattern24
Fig. 2. (a) Fabry–Perot configuration for FDISH spectroscopy.
The reference SH pulse is generated by a 15-fs pulse in a SnO2
film and then delayed from the fundamental in a glass substrate.
Inset (dashed box), alternative double-reflection scheme for generating a reference SH pulse and temporally separating it from
the fundamental input. (b) Interferogram measured at the spectrometer detector array.
I共 2␻, ␶ 兲 ⫽
冏 冕
1
⬁
2␲
⫺⬁
关 E samp共 t 兲
⫹ E ref 共 t ⫺ ␶ 兲兴 exp共 2i ␻ t 兲 dt
冏
2
⫽ 兩 E samp共 2 ␻ 兲 ⫹ E ref共 2 ␻ 兲 exp共 ⫺2i ␻ ␶ 兲 兩 2
⫽ 兩 E samp共 2 ␻ 兲 兩 2 ⫹ 兩 E ref共 2 ␻ 兲 兩 2
⫹ 2 兩 E samp共 2 ␻ 兲 兩兩 E ref共 2 ␻ 兲 兩 cos共 ⌬ ␾ 共 2 ␻ 兲
⫹ 2␻␶ 兲,
(3)
as illustrated in Fig. 2(a). Fourier analysis of such FD
interferograms24 yielded the spectral phase difference
⌬ ␾ (2 ␻ ) ⫽ ␾ samp(2 ␻ ) ⫺ ␾ ref(2 ␻ ) between the two pulses.
As in a Fabry–Perot interferometer, the two interfering
pulses propagate collinearly and interact with identical
optical elements, with no moving parts. As a result, very
high-contrast frequency-domain interferograms see [Fig.
2(b)] were generated consistently. The Fabry–Perot configuration with SnO2 reference was used for results in
this paper.
We tested other configurations (e.g.,
Mach–Zehnder),25 but the Fabry–Perot configuration consistently yielded the highest contrast and the mostreliable interferograms.
A delay ␶ ⬃ 1 ps compromised between maximizing
fringe density (and thus the accuracy of phase retrieval)
and defining individual fringes, limited by CCD pixel density (we used ⬃10 pixels/fringe). In the Fabry–Perot con-
Wilson et al.
Vol. 20, No. 12 / December 2003 / J. Opt. Soc. Am. B
figuration ␶ can be determined very accurately (⫾1 fs), because it results entirely from group delay in a glass plate
of well-characterized thickness d and index n( ␻ ), with a
small (usually negligible) correction for the dispersion of
air between reference and sample. This accuracy is important because, in Fourier analyzing the interferograms,
one must subtract the uninteresting linear function 2␻␶
in Eq. (3) from the final result to obtain the correct linear
contribution to the expansion, ⌬ ␾ (2 ␻ ) ⫽ ⌬ ␾ (2 ␻ 0 )
⫹ 2( ␻ ⫺ ␻ 0 )⌬ ␾ ⬘ (2 ␻ 0 ) ⫹ 2( ␻ ⫺ ␻ 0 ) 2 ⌬ ␾ ⬙ (2 ␻ 0 ) ⫹ ¯ of
the material-dependent spectral phase difference. Generally a FDISH cannot provide an accurate value of the dc
term ⌬ ␾ (2 ␻ 0 ); however, because ␶ must be known with
subcycle accuracy, a supplemental conventional SH phase
measurement at one ␻ 0 is needed, as discussed further in
Subsection 3.C below. The advantage of the FDISH is
that it provides rapid, accurate measurement of spectral
and sample-dependent variations in ⌬␾.
For silicon samples, the incident beam had to be focused to produce an adequate SH signal. Focusing elements within the interferometer invariably introduced
spectral phase artifacts and reduced fringe contrast.
Best results were obtained with a single focusing mirror
before the entire interferometer. A focus of ⬃f/20 compromised maximizing signal and minimizing beam divergence between the reference and the sample. We found
that reference–sample separation l up to several Rayleigh
lengths (⬃1 cm for f/20) did not affect the extracted phase
spectrum. Within this limit the positions of reference
and sample relative to beam waist could be freely adjusted to equalize reference and sample intensities and
thus optimize fringe contrast. Although our SnO2 reference was not spectrally flat over the measured range, any
series of FDISH measurements was straightforwardly
normalized by a single, independent FDISH measurement of the relative spectral phase of the reference and of
a spectrally flat standard (usually crystalline quartz).
The accuracy and reproducibility of this normalization
did not require precisely matched sample positions. Of
course, in principle the quartz standard could be used directly. In practice the SnO2 film proved more convenient,
despite the extra normalization step required, because it
yielded a stronger reference signal, closer to our sample
signals, thus improving fringe contrast during a measurement series. We could then conveniently make fine adjustments to equalize reference and sample intensities at
any time by translating the SnO2 film along the propagation axis without otherwise adjusting the optical configuration.
The various contributions to ⌬␾ can be written as
2551
I
from electric-field-induced
tric fields are present, ␾ samp
second-harmonic (EFISH) generation. The contributions
␹
F
␾ ref
⫹ ␾ ref
to ␾ ref are zero for a spectrally flat reference or
P
occurs
for normalized data. A dc propagation delay ␾ ref
when, as in our case, the reference is generated in transP
mission through a non-phase-matched film. ␾ ref
⬃ ␲ for
film thickness l coh . On reflection from the sample, the
R
( ␻ ), which is straightforwardly calreference acquires ␾ ref
culated from the linear Fresnel equations. The last two
terms in Eq. (4) are propagation delays between fundamental and SH reference pulses in air and glass, respectively.
3. Conventional SH Phase Spectra
The Fabry–Perot FDISH configuration in Fig. 2(a) was
converted to the conventional scanning SH interferometer
configuration shown in Fig. 3 by four modifications.
First the 10–15-fs incident pulse was replaced with a narrow, bandwidth tunable 100-fs pulse. Second, the spectrometer and the CCD were replaced with a PMT. Third,
the reference film was inverted such that incident pulses
first passed through the glass substrate and then generated a SH signal in the SnO2 film at the back surface,
thus eliminating delay between fundamental and reference SH pulses in the glass. Finally, the reference was
mounted upon a scanning stage that varied by a distance
l over a range 0.5 ⬍ l ⬍ 10 cm, slightly more than a coherence
length
l air
coh ⫽ (␭ SH/2)/ 关 n air(2 ␻ ) ⫺ n air( ␻ ) 兴
⬇ 6 cm, over which the dispersion of air delays the reference SH pulse from the fundamental by ␲. Thus reference and sample SH pulses remain temporally overlapped
but vary in phase by ⬃␲ during the scan. The detector
signal is proportional to cos(⌬␾ ); ⌬␾ is given by Eq. (4),
with d ⫽ 0.
C. Samples
Coherently strained, native-oxidized Si1⫺x Gex and
Si0.87⫺y Ge0.13Cy samples were grown by ultrahigh-vacuum
chemical-vapor deposition upon Si(001) substrates. The
former were obtained from Lawrence Laboratory, Inc.; the
latter were grown at the University of Texas Microelectronics Research Center.26 Film thickness and composition were measured by x-ray diffraction and secondary
ion mass spectrometry. Germanium film composition
varied over the range 0 ⬍ x ⬍ 0.15, causing the promi-
␹
F
I
⌬ ␾ 共 2 ␻ 兲 ⫽ 关 ␾ samp
⫹ ␾ samp
⫹ ␾ samp
兴
␹
F
P
R
⫺ 关 ␾ ref
⫹ ␾ ref
⫹ ␾ ref
⫹ ␾ ref
兴
⫹
⫹
␻l
c
关 n air共 2 ␻ 兲 ⫺ n air共 ␻ 兲兴
␻d
c
关 n glass共 2 ␻ 兲 ⫺ n glass共 ␻ 兲兴 .
␹
␾ samp
(4)
␾ samp is separated into phase
of the nonlinear susF
ceptibility, ␾ samp
of the Fresnel factors, and, when dc elec-
Fig. 3. Configuration for conventional SH phase measurement.
The reference SH pulse is generated by narrowband 100-fs
pulses in SnO2 film and then delayed from the fundamental in
air between the reference and the sample.
2552
J. Opt. Soc. Am. B / Vol. 20, No. 12 / December 2003
Wilson et al.
nent E 1 resonance to vary from ⬃3.4 to ⬃3.0 eV within
the tuning range of our SH frequencies. We compared
broadband and scanning methods by measuring the
composition-dependent E 1 position and line shape. Film
thickness (⬃300 nm) was below the critical thickness for
strain relaxation by means of misfit dislocations27 but
much greater than the escape depth of SH radiation (⬍30
nm). Thus neither the buried SiGe–Si interface nor the
Si substrate contributed to the SH signals. In the
Si0.87⫺y Ge0.13Cy sample, carbon composition varied over
the range 0 ⬍ y ⬍ 0.015 and served to compensate partially for the strain that is present in the C-free SiGe
films.28
We fabricated MOS capacitors from n-type (borondoped, 0.01 ⍀-cm) Si(001) with a 19-nm thick thermal oxide layers. A 3-nm semitransparent chromium gate electrode and an ohmic aluminum back side electrode were
evaporated onto the samples. Bias voltages of ⫺10 ⬍ V
⬍ ⫹10 V were applied between the Cr and the grounded
Al electrodes. The SHG response from the Cr and oxide
layers was verified to be negligible compared with the
bias-independent SHG response E 2INT
␻ from the Si–SiO2
interface and with the bias-dependent EFISH response
from the space-charge region. The total SHG inE 2EFISH
␻
tensity can be written as
BQ
EFISH 2
I 2 ␻ ⫽ 共 E 2INT
兲 ,
␻ ⫹ E 2␻ ⫹ E 2␻
(5)
E 2BQ
␻
is a bias-independent bulk quadrupole rewhere
sponse that also contributes weakly to the signal. Variation of the applied bias, and therefore of space-charge
field E 0 (z), causes variations in the azimuthal anisotropy,
amplitude, and spectrum of the SHG signal that have
been studied in detail elsewhere.29 For phase measurements, the most important effect of varying applied bias
I
is that EFISH phase ␾ samp
shifts by ␲ when space-charge
field E 0 (z) changes sign at the flat-band voltage V fb .
Capacitance–voltage analysis of the MOS structures used
in the present study showed that V fb ⬃ ⫺1.5 V.
3. EXPERIMENTAL RESULTS AND
DISCUSSION
Fig. 4. (a) Chirp-dependent SHG spectrum of a Si MOS capacitor. Inset, incident laser spectrum. (b) Chirp-dependent SHG
spectrum of quartz. (c) Chirp-independent SHG spectrum of a
Si MOS capacitor normalized to quartz.
A. Chirp Dependence and Normalization of Broadband
SHG Spectra
Figure 4(a) shows several broadband SHG amplitude
spectra measured in reflection ( p in/p out) from a Si MOS
capacitor biased at V fb . The Ti:S laser and extracavity
prisms were adjusted to produce negatively chirped ( ␾ ⬙
⫽ ⫺230 fs2 ) pulses with the spectrum shown in the inset.
Increasingly thick (6.35, 8.35, 14.7, and 21.05 mm) fusedsilica plates were then inserted into the beam path to increase chirp in four increments up to ⫹530 fs2 without altering the spectrum, focus, or power of the incident beam.
As expected, unchirped pulses yielded the highest SHG
spectral amplitude. Finite chirp of either sign sharply
reduced the amplitude by a factor (1 ⫹ a 2 ) ⫺1/2, where a
⫽ 1/2␴ 2 ␾ ⬙ is a dimensionless chirp parameter for a pulse
of e ⫺1 bandwidth ␴, and slightly distorted the shape of the
SHG spectrum. Figure 4(b) shows a sequence of SHG
spectra acquired from a quartz standard in the same way.
Figure 4(c) shows the result of dividing the five spectra in
Fig. 4(a) by the corresponding reference spectra in Fig.
4(b). Within experimental noise, the five normalized
spectra are indistinguishable from one another. The position and shape of the 3.3-eV peak, a surface-modified E 1
feature that results from a direct L-valley transition that
is resonant with SH photon energy h ␯ SH , agrees well
with conventional scanning SHG spectroscopy presented
elsewhere 6,30 and in Subsection 2.B below. The spikes
that are evident in the normalized spectra for the most
strongly chirped pulses (300 and 530 fs2) result from low
SHG signal from both sample and reference at h ␯ SH
⬎ 3.3 eV. These results illustrate robust normalizability against variations in the incident pulse chirp that was
observed for all samples studied here.
The normalizability shown in Fig. 4 validates the approximation of ⍀-independent ␹ ( 2 ) that led to Eq. (2).
This is plausible for the present samples and spectral
range and for many semiconductors and insulators that
use IR–visible pulses, because only weak indirect transitions occur near ⍀. The effect of a weakly ⍀-dependent
Wilson et al.
Vol. 20, No. 12 / December 2003 / J. Opt. Soc. Am. B
␹ ( 2 ) may be estimated analytically from Eq. (1). For a
wide range of crystals, ␹ ( 2 ) (2 ␻ , ⍀, 2␻ ⫺ ⍀) can be written in the separable form ⌬ m ⌶(2 ␻ )Z(⍀), where ⌶(2 ␻ )
⫽ ␹ ( 1 ) (2 ␻ ) contains the material response to the SHG
frequency, Z(⍀) ⫽ ␹ ( 1 ) (⍀) ␹ ( 1 ) (2 ␻ ⫺ ⍀) contains the response to the fundamental frequencies, and ⌬ m is Miller’s
frequency-independent coefficient.31 We assume input
pulses with Gaussian spectrum E(⍀) ⫽ E 0 exp关(⍀
⫺ ␻l)2/␴ 2兴exp关⫺i␾(⍀)兴, spectral phase ␾ (⍀) ⫽ ␾ 0 ⫹ (⍀
⫺ ␻ l ) ␾ ⬘ ( ␻ l ) ⫹ 1/2(⍀ ⫺ ␻ l ) 2 ␾ ⬙ ( ␻ l ) restricted to second
order (linear chirp), carrier frequency ␻ l chosen such that
␾ ⬘ ( ␻ l ) ⫽ 0, and ␾ 0 set arbitrarily to zero. Equation (1)
then yields SH polarization:
P 共 2 兲 共 2 ␻ 兲 ⫽ exp关 2i ␾ 共 ␻ 兲兴
⫻
冕
⬁
⌬ mE 0
2
共 2␲ 兲2
⌶共 2␻ 兲
关 Z 共 ␻ ⫹ ␦ 兲 ⫹ Z 共 ␻ ⫺ ␦ 兲兴
0
⫻ exp关 ⫺共 ␻ ⫹ ␦ ⫺ ␻ t 兲 2 / ␴ 2 兴
⫻ exp关 ⫺共 ␻ ⫺ ␦ ⫺ ␻ l 兲 2 / ␴ 2 兴
⫻ exp共 ⫺2ia ␦ 2 / ␴ 2 兲 d␦ ,
(6)
where we set ⍀ ⫽ ␻ ⫹ ␦ . Assuming the laser frequency
variation ␦ Ⰶ ␻ , we can expand Z( ␻ ⫾ ␦ ) ⫽ Z( ␻ )
⫾ ␦ Z ⬘ ( ␻ ) ⫹ 1/2␦ 2 Z ⬙ ( ␻ ) to second order and integrate to
obtain
P 共 2 兲 共 2 ␻ 兲 ⫽ exp关 2i ␾ 共 ␻ 兲兴
␴ ⌬ m兩 E 共 ␻ 兲兩 2
冋
共 2␲ 兲2
⫻ ⌶共 2␻ 兲Z共 ␻ 兲 1 ⫹
冋
␲
2 共 1 ⫹ ia 兲
册
1/2
Z ⬙共 ␻ 兲
␴2
8 共 1 ⫹ ia 兲 Z 共 ␻ 兲
册
.
(7)
2 ⫺1/2
scaling of 兩 P (2 ␻ ) 兩 emerges
The overall (1 ⫹ a )
naturally. Z( ␻ ) influences P ( 2 ) (2 ␻ ) in two ways. First,
it provides an overall multiplying factor. In this role it
influences scanning and broadband SHG measurements
in the same way. Second, however, it enters the a- and
␴-dependent term 关 ␴ 2 /8(1 ⫹ ia) 兴 (Z ⬙ /Z) which, when the
term is comparable to unity, creates a nonnormalizable
chirp-dependent distortion that is unique to broadband
SHG spectra. To estimate this distortion, we can approximate ␹ ( 1 ) ( ␻ ) of the interfacial SH-generating region
with the bulk susceptibility for which tabulated values32
are readily available. Inasmuch as index of refraction
n(␭) rather than ␹ ( 1 ) ( ␻ ) is usually tabulated, it is convenient to write
␴2
Z ⬙共 ␻ 兲
8 共 1 ⫹ ia 兲 Z 共 ␻ 兲
⫽
(2)
共 ⌬␭ 兲 2
n 2 /2
冋
2
n⬙
1 ⫹ ia n ⫺ 1 n
2
⫹
冉 冊册
3n 2 ⫺ 1 n ⬘
n2 ⫺ 1
n
⫹
2 n⬘
␭ n
2
.
(8)
For an unchirped 0.77-␮m laser pulse of half-bandwidth
⌬␭ ⫽ 30 nm incident upon Si, ratio (8) is less than 10⫺3
(and smaller with increasing chirp a) and is thus negligible compared to the value 1 in Eq. (7). This is consistent with the nearly perfect normalizability evident in
2553
Fig. 4. This analysis holds for both the amplitude and
the phase of the SHG. Ratio (8) remains less than 10⫺2
for ⌬␭ ⫽ 100 nm, suggesting that excellent normalizability is retained with pulses as short as 5 fs. Similar numbers are obtained for near-IR pulses incident upon many
semiconductors and insulators, suggesting that broadband SHG spectroscopy should be a robust technique applicable to a wide range of materials. Before any such
application is made, however, the normalizability test illustrated in Fig. 4 should first be performed.
B. Comparison of Broadband and Conventional SHG
Amplitude Spectra
1. Si 1⫺x Ge x
Figures 5(a)–5(e) show normalized SHG amplitude spectra acquired in reflection from the oxidized Si1⫺x Gex
samples by use of broadband (continuous curves) and conventional scanning (filled squares) methods. p-in/p-out
polarization configuration was used, with the incident
plane oriented at ␾ ⫽ 22.5° from the [110] direction, corresponding to a minimum in the fourfold anisotropic SHG
intensity [Fig. 5(d), left inset].
The broadband spectra were acquired with a single set
of Ti:S laser optics that produced output centered at 775
nm. Each spectrum required only ⬃5 s of data acquisition time and was highly reproducible (typically better
than ⫾1%), except in the wings of the spectrum, where incident intensity was low. Each scanning spectrum required 15–30-min acquisition time. As a result the data
were subject to errors from two sources: (1) changes in
beam pointing during tuning that affected sample and
quartz reference signals differently and (2) timedependent electrostatic charging of the oxide during tuning. The latter effect is caused by multiphoton excitation
of electrons from the SiGe valence band to the oxide conduction band, initiating their slow diffusion to oxide surface traps.33 The resultant electric field buildup gradually enhances the p-polarized signal by EFISH generation
over several minutes, as illustrated in the upper-right inset of Fig. 5(d). Inasmuch as charge builds up on the
same time scale as a spectral scan, it can introduce artifacts into spectral line shapes unless steps are taken to
minimize its effect. Both broadband and scanning SHG
spectra in Fig. 5 were acquired from samples charged to a
common saturation level to eliminate time dependence in
the latter. Consequently the Si–SiO2 interface and bulk
EFISH generation both contribute to the SHG signal.
With more difficulty, we can minimize time dependence
under uncharged conditions by slowly translating the
sample during laser tuning, using low-intensity pulses
(consistent with adequate SHG signal); placing the
sample in UHV (Ref. 33) also help to minimize buildup of
time-dependent EFISH. With such precautions, scanning spectra typically are reproduced within ⫾20% when
they are repeated with decreasing and increasing laser
wavelengths.
The two sets of spectra agree quantitatively in most details. In each set, maximum SHG intensity occurs at the
same two photon energies to within ⫾0.02 eV for given x
and shifts monotonically from ⬃3.32 to ⬃3.20 eV as x increases from 0 to 0.125. The relative amplitudes of the
2554
J. Opt. Soc. Am. B / Vol. 20, No. 12 / December 2003
Wilson et al.
high-quality SHG data over the range 3.2 ⬍ h ␯ SH
⬍ 3.45 eV, agreement between broadband and scanning
data improved markedly in this range, as will become evident from the MOS structure data below.
A qualitatively similar redshift of the E 1 critical point
with increasing x has been extensively documented in linear reflectance spectroscopy of Si1⫺x Gex alloys,34 although
the SHG spectrum is shifted by ⬃0.1 eV because of bond
distortions that are unique to the interface.6 Figure 5(e)
shows the real part ⑀ 1 ( ␻ ) of the pseudodielectric function
of the same Si1⫺x Gex samples, derived from SE measurements with the Verity ellipsometer. The E 1 shifts are
very similar in magnitude to those observed in SHG, confirming their common origin in composition-dependent
L-valley transitions. A more-quantitative SE–SHG comparison is not possible at present because the firstprinciples theory of the surface modifications of E 1 is relatively undeveloped.
Fig. 5. (a)–(e) Normalized SHG amplitude spectra of oxidized
Si1⫺x Gex samples acquired by broadband (continuous curves)
and conventional scanning (filled squares) methods. The insets
in (d) show the dependence of SHG intensity at h ␯ SH ⫽ 3.0 eV on
azimuthal sample rotation (left) and time (right). (f ) Real part
of linear pseudodielectric functions of Si1⫺x Gex samples used for
the SHG spectra in (a)–(e) acquired by spectroscopic ellipsometry.
normalized SH signals for different values of x also agree
well for the two sets of spectra. Finally, E 1 line shapes
agree well, except at the edges (e.g., 2ប ␻ ⬎ 3.3 eV for x
⫽ 0, 0.027), where the broadband signal is weak.
The comparison in Fig. 5 shows the strengths and
weaknesses of each technique. The scanning technique,
though it is much slower, yields somewhat more reliable
SHG data in the extrema of the Ti:S laser tuning range
(2ប ␻ ⬍ 2.9 eV or ⬎ 3.3 eV), as the argon-ion pump laser power can be increased there to compensate for
weaker gain. The broadband technique, however, yields
much denser, more-reliable data in most of the tuning
range (3.0 ⬍ 2ប ␻ ⬍ 3.3 eV). When the broadband Ti:S
laser was operated with a bluer spectrum, permitting
2. Si 0.87⫺y Ge 0.13C y
The superior reproducibility of broadband SHG spectroscopy makes it the preferred technique for monitoring very
small shifts of spectral features that fall within the bandwidth of the generated SH light. To illustrate this, in
Fig. 6 we compare normalized broadband (continuous
curves) and scanning (filled squares) SHG spectra of the
oxidized Si0.87⫺y Ge0.13Cy samples. The broadband data
reveal an unambiguous monotonic blueshift (h⌬ ␯ SH
⬃ 0.05 eV) of the SHG E 1 peak with increasing C content, accompanied by slight, but reproducible, changes in
line shape. Though it is also discernible in the scanning
SHG data, the shift is considerably less clear—and less
useful for quantitative analysis—because of the poorer
signal-to-noise ratio of these data. SE measurements of
the same samples by the rotating-analyzer ellipsometer23
reveal a qualitatively similar C-dependent E 1 blueshift,
as shown by the extracted linear dielectric spectrum ⑀(␻)
of our samples in Fig. 6(d). A similar E 1 shift was observed in previous SE of Si1⫺x⫺y Gex Cy samples with different Ge content.23 These small SHG and SE E 1 shifts
can be attributed qualitatively to compositional dependence of the L-valley gap and in part to relief of strain.
Broadband SHG spectroscopy of the compositiondependent E 1 peak may provide a useful technique for
in situ monitoring of the growth of Si1⫺x Gex or
Si1⫺x⫺y Gex Cy films with graded compositional profiles,
which are a fundamental component of some high-speed
devices.35 SE analysis often yields profiles that are not
unique, that disagree with Auger depth profiling, or
both.35 Broadband SHG would be sensitive to nearsurface composition and is fast enough to permit quasireal-time data acquisition.
3. Si – SiO 2 – Cr MOS Structure
Figure 7(a) compares broadband (continuous curves) and
scanning (discrete data points) SHG spectra of the MOS
structure acquired in p-in–p-out configuration with incident plane along [110]. For the broadband data the Ti:S
laser was operated at a bluer central wavelength (⬃750
nm, or ␭ SH ⬃ 3.3 eV) than for the SiGe data, so that the
bluer Si E 1 feature could be better characterized. At biases near V fb , broadband and scanning SHG spectra
Wilson et al.
Vol. 20, No. 12 / December 2003 / J. Opt. Soc. Am. B
2555
ases with 兩 V 兩 ⬎ 3 V. No time-dependent effects were
present with this sample, and the discrepancy was not reduced despite careful minimization of alignment drifts
during tuning. It therefore appeared to be a systematic
feature of the data obtained at large bias.
This discrepancy can be attributed to the preferential
effect of screening of the space-charge region by photoinjected minority carriers36–38 on the scanning SHG data.
For our range of fluences, linear and two-photon absorption of the incident pulses injects electron–hole (e-h) pairs
of density in the range 3 ⫻ 1017 ⬍ n e-h ⬍ 5 ⫻ 1018 cm⫺3
Fig. 6. (a)–(c) Normalized SHG amplitude spectra of oxidized
Si0.87⫺y Ge0.13Cy samples acquired by broadband (continuous
curves) and conventional scanning (filled squares) methods. (d)
Real part of linear dielectric functions ⑀ 1 ( ␻ ) of Si0.87⫺y Ge0.13Cy
samples used for the SHG spectra in (a)–(c), acquired by spectroscopic ellipsometry.
agreed quantitatively in line shape [see data for V
⫽ ⫺2.45 V; Fig. 7(a)]. At large negative biases, however, the E 1 peak measured by scanning was consistently
redshifted from the same broadband-measured feature
[see the data for V ⫽ ⫺7 V, Fig. 7(a)]. Moreover, the in⫺7 V ⫺2.45 V
tensity ratio I SHG
/I SHG , evaluated at the spectral peak,
was ⬃30% smaller in the scanning data than in the
broad-band data. In Fig. 7(a) the relative amplitude was
scaled to produce a minimum ␹ 2 fit for the near-flat-band
case. With this assumption, the largest discrepancy between the two V ⫽ ⫺7 V spectra occurred in the region
3.35 ⬍ h ␯ SH ⬍ 3.45 eV. Figure 7(b) compares the bias
dependence of the SHG signal within this region (h ␯ SH
⫽ 3.37 eV), again scaling the relative amplitudes to be
equal at the flat band. The discrepancy between broadband and scanning signals was significant for negative bi-
Fig. 7. (a) Broadband (continuous curves) and scanning (filled
squares and filled circles) SHG amplitude spectra of a
Si–SiO2 – Cr MOS structure at large negative bias (⫺7 V) and
near V fb (⫺2.45 V). (b) Bias dependence of the SHG signal at
h ␯ SH ⫽ 3.37 eV acquired by broadband (squares) and scanning
(circles) methods. The vertical scales in (a) and (b) are mutually
consistent.
2556
J. Opt. Soc. Am. B / Vol. 20, No. 12 / December 2003
within the absorption depth. These electrons and holes
separate under the influence of the space-charge field on a
time scale 20 fs ⬍ ␻ p ⫺1 ⬍ 80 fs, where ␻ p 2 ⫽ 4 ␲ n e-he 2 /
⑀␮ * is the plasma frequency for e-h pairs of reduced effective mass ␮ * photoinjected into Si with a dielectric
constant ⑀ ⫽ 11.9. As the carriers are hot on this time
scale, ␮ * is much closer to free-electron mass m e than to
the band-edge effective masses. We used ␮ * ⬃ 0.5 m e
for the above estimates. Separation of electrons and
holes partially screens the space-charge field, thus reducing the EFISH contribution to the SHG signal. Because
the broadband data were acquired with pulses of duration
␶ p ⬃ 15 fs, the space-charge field remains essentially unscreened during the pulse. The scanning data, however,
were
acquired
with
pulses
of
duration
␶p
⬃ 100 fs, during which time separating e-h pairs substantially screen the EFISH contribution.
This interpretation is confirmed by photomodulated
SHG measurements with the 100-fs source on the same
MOS structure reported elsewhere.38 Briefly, the long
Ti:S pulses were split into weak SH-generating pulses
(F ⬃ 0.5 ⫻ 10⫺4 J/cm2 ), and into energetic pump pulses,
similar in fluence (F ⬃ 10⫺3 J/cm2 ) to the pulses used for
the data in Fig. 7, to inject carriers independently. The
relative change ⌬I SH(V)/I SH(V) in bias-dependent SHG
intensity with pump off and pump on is of the same sign
as, and similar in magnitude to, the broadband and scanning data, respectively, in Fig. 7(b). Moreover, the pumpoff and pump-on curves cross at the flat-band voltage,38
validating our assumption of equating the broadband and
scanning SHG amplitudes at V fb . Calculations of the
photomodulated EFISH,38 treating the injected excess minority carriers in a quasi-Fermi-level approximation,39
are also consistent with the sign and magnitude of the
photoeffect that is present in both sets of data.38
This interpretation is also consistent with the spectral
characteristics of the two sets of ⫺7-V data in Fig. 7(a).
EFISH, because of its bulk origin, contributes an E 1 peak
at the bulk E 1 critical point energy 3.37 eV,30,40 whereas
the Si–SiO2 interface contributes the redshifted E 1 peak
that is evident in the ⫺2.45-V data in Fig. 7(a). Partial
screening of the EFISH contribution would thus be expected to redshift and attenuate the composite E 1 feature,
as observed. Thus the broadband SHG measurement
performed with 15-fs pulses yields a more accurate measurement of the EFISH response by avoiding femtosecond
carrier screening.
In addition to femtosecond transient screening, background steady-state screening of the space-charge field,
caused by pulse-to-pulse accumulation of photoinduced
carriers,41 likely also influences both the scanning and
the broadband SHG data. Carrier accumulation is expected because the interval between pulses is 13 ns,
whereas carriers recombine in ⬃3 ␮s. Eliminating and
studying this contribution to screening require will lowerrepetition-rate sources. The use of a strong independent
carrier-generation beam in combination with broadband
SHG should provide a powerful method for acquiring photomodulated SHG spectra in either the transient or the
steady-state screening regime. Such spectra should exhibit sharp, derivativelike structures analogous to those
in linear modulation spectroscopy.42
Wilson et al.
C. Comparison of Broadband and Conventional SHG
Phase Spectra
The savings in data acquisition time that result from use
of broadband methods is squared for measurements of
SHG phase spectra, because an interferometer scan as
well as a laser wavelength scan is avoided.
1. Conventional SH Phase Measurements of the MOS
Capacitor
Scanning SH interferometry data for the MOS capacitor
were acquired with the same two biases (⫺7 and ⫺2.45
V), and the same polarization configuration and sample
orientation, used for the data in Fig. 7(a). Figure 8
shows the ⫺7-V data. At each bias, reference–sample
separation l was scanned from 0 to 6 cm for each of 16 laser wavelengths in the range 720 ⬍ ␭ ⬍ 870 nm. Several hours were needed for acquiring data at each bias.
Maxima (minima) in SH intensity correspond to constructive (destructive) interference between reference and
sample SH pulses. For 870 ⬎ ␭ ⬎ 770 nm (i.e., 2.85
⬍ h ␯ SH ⬍ 3.2 eV), the position of the maximum, and
thus the SH phase, is nearly independent of both wavelength and bias. For shorter wavelengths, the SH phase
depends on both wavelength and bias. To document the
bias dependence, we also acquired conventional SH phase
data (not shown) at fixed ␭ ⫽ 736 nm (h ␯ SH ⫽ 3.37 eV)
for 12 biases in the range ⫺9 ⬍ V ⬍ ⫹ 4V. Qualitita-
Fig. 8. Conventional time-domain interferometric SH scans of a
Si–SiO2 – Cr MOS capacitor at bias ⫺7 V for 16 laser wavelengths.
Wilson et al.
Vol. 20, No. 12 / December 2003 / J. Opt. Soc. Am. B
tively, this contrast reflects the dominance of EFISH over
interface SHG near the bulk E 1 resonance (3.4 eV).
To extract the material-dependent component ⌽ SH
[first two bracketed terms in Eq. (4)] of the total phase difference ⌬␾(2␻) we fitted the curves in Fig. 8 to the equation
I 共 t 兲 ⫽ I samp ⫹
I ref
1⫹z
2
2
/z R
⫹
⫻ cos共 ⌬kl ⫺ ⌽ SH兲 ,
␣ 冑I sampI ref
冑1 ⫹ z 2 /z 02
(9)
where I samp and I ref are signal intensities from sample
and reference, respectively, ␣ is a coherence parameter
constrained to 0 ⬍ ␣ ⬍ 1, and ⌬kl is the third term in
Eq. (4). Rayleigh length z R and spatial frequency ⌬k
were calculated from the focusing conditions and from
n air , respectively. The resultant SH phase spectra
⌽ SH( ␻ ) are shown by filled squares and filled circles in
Fig. 9(a).
Bias dependence ⌽ 3.37 eV(V) at h ␯ SH
Fig. 9. (a) SHG phase spectra of the MOS capacitor as measured by FDISH (continuous curves) at several biases and by
conventional SH scanning interferometry at ⫺2.45 and ⫺7 V
(discrete points). Because FDISH does not accurately determine the dc component of phase, the seven FDISH phase spectra
have been placed vertically such that their average phase is zero
at h ␯ SH ⫽ 2.87 eV. (b) Bias-dependent phase shift of SHG at
h ␯ SH ⫽ 3.37 eV measured by FDISH (squares) and by the conventional method (circles).
2557
⫽ 3.37 eV [filled circles in Fig. 9(b)] reveals a gradual
monotonic shift totaling ⌬⌽ SH ⫽ 2.5 ⫾ 0.5 rad between
⫺10 and ⫹4 V, roughly consistent with the ⬃␲ phase shift
I
expected for EFISH phase ␾ samp
on reversal of the dc
field. ⌽ SH was determined from fits of single oscillation
periods (see Fig. 8). This restriction comes about because
air is weakly dispersive, whereas reference movement is
constrained between the sample and the focusing mirror.
The focusing required for adequate signal thus sets the
upper limit on interferometer scan length. The error
bars on ⌽ SH points in Fig. 9 reflect the variances of these
fits but not errors from laser and alignment drifts or air
currents through the interferometer. We were unable to
obtain reliable SH interferometry scans for the quartz
standard because it produced weaker SH signals and reflected the reference pulse inefficiently. Thus the ⌽ SH
data in Fig. 9 retain reference phase contributions contained in the second term in Eq. (4).
2. FDISH Measurements of the MOS Capacitor
Figures 10(b)–10(h) show FDISH data for the MOS capacitor at seven biases obtained by use of the same polarization configuration and sample orientation as for previous data. For the data in Fig. 10(a) the quartz standard
was substituted for the MOS capacitor and provided overall normalization. The reference pulse was generated in
the SnO2 film on glass in all cases. No adjustments of
any kind were made to the Ti:S laser or optical configuration during the data acquisition. Remarkably, each interferogram in Fig. 10 encodes the same phase information as the entire set of conventional SH phase data in
Fig. 8. Yet each required only 5–10 s of data acquisition
time. The complete bias-dependent series shown in Fig.
10 could be comfortably acquired within 2 min. Consequently the FDISH data are virtually immune from drifts
in laser operation or optical alignment and can be extensively repeated to ensure reproducibility.
At biases closer to V fb , fringe contrast was lower in the
short-wavelength portion of the interferograms. This is
so because the bias-dependent EFISH contribution, which
dominates this portion of the SH spectrum, approaches
its minimum amplitude at V fb , making this portion of the
signal pulse weaker than the reference pulse. Fringe
contrast was also relatively low for the quartz sample because it reflects less than 4% of the SHG from the SnO2 .
Nevertheless, the contrast was adequate throughout to
permit reliable SH phase spectra normalized to the spectrally flat quartz standard to be extracted.
To highlight the most important phase shifts that occurr in the data of Fig. 10, in Fig. 11 we directly superpose
two interferograms obtained with the MOS capacitor biased well above (⫹4 V) and well below (⫺10 V) V fb . The
change in direction of the dc field shifts fringes in the
EFISH-dominated region (␭ SH ⬍ 370) by ⬃␲, as shown
by the interchange of peaks and valleys in the FD fringes.
The fringes in the Si–SiO2 interface-dominated regime
(␭ SH ⬎ 370), however, shift only slightly.
The continuous curves in Fig. 9(a) are SH phase spectra ⌽ FDISH(2h ␯ SH) extracted from FD interferograms at
seven fixed biases. The artificial linear contribution 2␻␶
[Eq. (3)] has been subtracted out, and air and glass contribute negligibly to the dispersion of ⌬ ␾ (2 ␯ SH). Thus
2558
J. Opt. Soc. Am. B / Vol. 20, No. 12 / December 2003
Fig. 10. FDISH interferograms for (a) the quartz standard and
for (b)–(h) a MOS capacitor at several biases. The small extra
peak at 375 nm for ⫺3.79 V results from bad CCD pixels. There
is no phase discontinuity. Such infrequent defects are filtered
from the data before phase spectra are extracted.
Fig. 11. Direct comparison of FDISH interferograms for a MOS
capacitor at biases above and below V fb . The relative amplitudes have been scaled to high-light the ⬃␲ phase shift in the
spectral region (␭ SH ⬍ 370 nm) where EFISH dominates.
the plotted ⌽ FDISH(2 ␯ SH) reflects only the first two bracketed terms in Eq. (4). Because ⌽ FDISH(2 ␯ SH) is normal␹
F
ized to quartz, ␾ ref
and ␾ ref
do not contribute. Moreover,
the variations in ⌽ FDISH with ␯ SH and V proved highly reproducible in all details. Thus the FDISH phase spectra
are the better subject for the theoretical model described
Wilson et al.
in Subsection 3.C.3, below. However, FDISH does not determine the absolute dc component of ⌽ FDISH as accurately as conventional SH interferometry, as discussed in
Subsection 2.B.2. The ⌽ FDISH curves in Fig. 9(a) have
been placed vertically such that their average phase at
h ␯ SH ⫽ 2.88 eV is zero. In addition to separating them
from the conventional SH phase spectra for viewing, this
placement is consistent with theoretical model of the
sample contribution to ⌽ FDISH . The physically real dc
offset of ⌽ SH almost certainly results from a phase mismatch between the fundamental and the reference SH
pulses in the SnO2 reference film, not from the sample.
P
As this film was ⬃l coh thick, a phase offset ␾ ref
⬃ ␲ is indeed expected.
Several features of the ⌽ FDISH(h ␯ SH , V) spectra in Fig.
9(a) present the principal challenges for theoretical modeling. First, the frequency dependence of ⌽ FDISH(h ␯ SH)
and that of ⌽ SH(h ␯ SH) are qualitatively similar: approximately flat phase below the E 1 resonance (h ␯ SH
⬍ 3.2 eV) and increasing phase on passing through the
E 1 resonance (h ␯ SH ⬎ 3.2 eV). Naı̈vely, a phase shift of
␲ between the SH polarization response and driving field
is indeed expected when the SH frequency passes through
a resonance. However, the FDISH measurements clearly
show that phase change ⌽ SH(3.45 eV) – ⌽ SH(2.85 eV) significantly exceeds ␲ for V ⬍ V fb and is ⬍␲ for V ⬎ V fb .
Second, the sensitivity to bias varies strongly with ␯ SH .
The ⌽ FDISH(h ␯ SH , V) spectra separate cleanly into two
groups, one for V ⬍ V fb (four spectra at ⫺7 ⭐ V
⭐⫺2 V) and a second for V ⬎ V fb (three spectra at 0
⭐ V ⭐ ⫹4 V). Bias dependence is weak within each
group. However, ⌽ SH(V ⬍ V fb) ⫺ ⌽ SH(V ⬎ V fb) varies
from ⬃␲ near 3.4 eV, where EFISH dominates, to much
smaller values at lower energies. Third, the weak bias
dependence within the V ⬍ V fb group reveals a subtle but
reproducible pattern. As bias varies from ⫺7 V (strong
EFISH) to ⫺2 V (weak EFISH), the ⌽ FDISH spectra cross
close to the E 1 resonant energy of 3.37 eV; i.e., ⌽ FDISH decreases (increases) with decreasing V, below (above) the
crossing point. A similar crossing may also be present in
the V ⬎ V fb group but is less clear because the three spectra nearly coincide. This important detail was highly reproducible in the FDISH data but hardly evident in the
corresponding conventional SH phase spectra at ⫺7 and
⫺2.45 V because of a poorer signal-to-noise ratio.
The filled squares in Fig. 9(b) show the bias dependence
of ⌽ FDISH within the EFISH-dominated region (h ␯ SH
⫽ 3.37 eV). The total shift ⌬⌽ FDISH ⫽ 2.5 ⫾ 0.1 is similar to the scanning SH interferometry result. However,
the change is more abrupt and is at a bias consistent with
V fb ⬃ ⫺1.5 V determined from capacitance–voltage measurements. We believe that the more gradual phase shift
measured by the conventional method results from the
preferential effect of space-charge screening by photoinjected carriers on this measurement, as discussed in connection with Fig. 7. By minimizing the screening artifact, FDISH evidently provides the more-accurate
measurement of dc field effects on SH phase.
3. Simple Model of Second-Harmonic Phase Spectra
To validate the measured FDISH phase spectra in Fig.
9(a) we now show that their major features are explained
Wilson et al.
by modeling the observed SH field Ẽ 2TOT
␻ as a coherent superposition of a dc-field-independent background SH field
BQ
E 2FI␻ , which combines E 2INT
␻ ⫹ E 2 ␻ in Eq. (5), and a resoEFISH
nant EFISH field Ẽ 2 ␻
. The phasor diagrams in Fig.
12(a) illustrate the main idea. In each diagram we have
represented E 2FI␻ , approximated for simplicity as a real
constant, by a phasor that terminates in a filled circle.
Complex field Ẽ 2EFISH
⫽ E EFISH exp(i␾EFISH), represented
␻
by thinner arrows, varies with frequency and bias.
Counterclockwise rotation corresponds to increasing
phase and thus, for resonant contributions, to increasing
frequency. At large negative bias (see the phasor diagram labeled V Ⰶ V fb), ␾ EFISH varies from ⬃␲/2 below
Vol. 20, No. 12 / December 2003 / J. Opt. Soc. Am. B
2559
resonance (2.88 eV)—nonzero because of the effective
depth of the EFISH polarization beneath the Si–SiO2
interface—to nearly 3␲/2 above resonance (3.64 eV).
Thus ␾ EFISH undergoes the usual increase of ␲ through
resonance. However, when E 2FI␻ adds coherently to
Ẽ 2EFISH
, it pulls the resultant phasor Ẽ 2TOT
␻
␻ ⫽ E TOT
⫻ exp(␾TOT), represented by thicker arrows, toward the
real axis at the two extreme frequencies. Consequently,
␾ TOT increases by more than ␲ through resonance
[heavier solid curve in Fig. 12(a)], as observed. At
smaller negative bias (see the phasor diagram labeled V
⬍ V fb), amplitudes E EFISH of the EFISH phasors are reduced, whereas ␾ EFISH and E FI remain unchanged. Thus
E FI pulls the resultant phasor even farther toward the
real axis at the extreme frequencies, and ␾ TOT increases
by an even larger amount [Fig. 12(a), thinner solid
curves]. Consequently phase spectra ␾ TOT for the
smaller and larger negative bias cross near resonance, explaining another key observation. Finally, at positive
bias (see the phasor diagram labeled V ⬎ V fb), the Ẽ 2EFISH
␻
phasors reverse direction with respect to those at V
⬎ V fb . Now ␾ EFISH varies from ⬃⫺␲/2 at 2.88 eV to
⬃⫹␲/2 at 3.64 eV. Thus, as E FI again pulls the resultant
phasor toward the real axis at these frequencies, ␾ TOT
changes only from a small negative to a small positive
value [dashed curves in Fig. 12(a)]. The overall change is
much less than ␲, as observed. The observed weak bias
dependence within the negative-and positive-bias groups,
the ⬃␲ gap between the groups near 3.4 eV, and the
smaller gap between the groups at lower energies also
emerge naturally from this model.
Relaxing the approximation of real, constant E 2FI␻
changes quantitative details, but not major qualitative
features, of the model. This is so because (1) at frequencies (h ␯ ⬍ 3.2 eV) well below the E 1 resonance, E 2FI␻ essentially is a real constant and (2) within the resonance,
兩 E 2FI␻ 兩 Ⰶ 兩 E 2EFISH
兩 for large biases, so any spectral depen␻
dence of amplitude and phase of E 2FI␻ has little effect on
the sum E 2FI␻ ⫹ E 2EFISH
. Thus, to minimize the number
␻
of adjustable parameters, we keep E 2FI␻ a real constant for
the remaining discussion.
We now summarize quantitative details that underlie
the calculated ␾ TOT curves in Fig. 12(a). The total SH
field generated by the MOS capacitor is
E TOT共 2 ␻ 兲 ⫽ E 2FI␻ ⫹ E 2EFISH
␻
⫽ 关 F 2FI␻ ␹ 共FI2 兲 ⫹ F̃ 2 ␻˜␹ 共 3 兲 Ĩ EFISH兴 F ␻2 E ␻2 ,
Fig. 12. Calculated phase of the SH field from a MOS capacitor
for comparison with FDISH measurements in Fig. 9(a). (a) Total phase ␾ TOT of the SH field for negative biases V Ⰶ V fb
(heavier solid curve) and V ⬍ V fb (lighter solid curves) and positive biases V ⬎ V fb (heavier dashed curve) and a larger bias
(lighter dashed curve). Insets, phasor diagrams depicting the
addition of E FI (phasors with filled-circle ends) and E EFISH
(lighter arrows) in a complex plane at three frequencies and
three biases. (b) Contributions to phase of E 2EFISH
from suscep␻
tibility ␹ (3) modeled as a Lorentzian oscillator (thin solid curve,
␾ ␹ ), EFISH integral I (short-dashed curve, ␾ I ), and Fresnel factor F 2 ␻ (long-dashed curve, ␾ F ), and their sum ␾ EFISH ⫽ ␾ ␹
⫹ ␾ I ⫹ ␾ F (thicker solid curve). The thinner solid curve that
R
of the SH
nearly coincides with ␾ F shows the phase change ␾ ref
reference pulse on reflection from the MOS sample. Inset, amplitude of the EFISH field.
(2)
␹ FI
(10)
is an effective second-order susceptibility, ˜␹ ( 3 )
where
is the third-order susceptibility of Si, Ĩ EFISH
⫽ 兰 0⬁ E 0 (z)exp i关(k2␻ ⫹ k␻)z兴dz is the EFISH integral29
over space-charge field E 0 (z) and SH and the fundamental k-vectors in Si (with the Si–SiO2 interface at z ⫽ 0),
and F̃ 2 ␻ and F ␻ are Fresnel factors. F̃ 2 ␻ , ˜␹ ( 3 ) , and
Ĩ EFISH are complex quantities whose product can be expressed as F 2 ␻ ␹ ( 3 ) I EFISH exp关i(␾F ⫹ ␾␹ ⫹ ␾I)兴. Inasmuch
as F ␻ is related to refractive indices n ␻Si and n ␻ox of Si and
SiO2 at ␻ and is thus essentially real, ␾ F (2 ␻ ) ⫹ ␾ ␹ (2 ␻ )
with respect to
⫹ ␾ I (2 ␻ , V) is the total phase of E 2EFISH
␻
driving field E ␻ .
We evaluated the three contributions to ␾ EFISH as follows: For a p-polarized SH field generated at the
2560
J. Opt. Soc. Am. B / Vol. 20, No. 12 / December 2003
Si–SiO2 interface we calculated the Fresnel factor5
F̃ 2 ␻ ⫽ 4 ␲ (n ␻Si cos ␪2T␻ ⫺ ñ2Si␻ cos ␪␻T) 兵 关 (n␻Si) 2 ⫺ (ñ 2Si␻ ) 2 兴 (n 2ox␻
⫻ cos ␪2T␻ ⫹ ñ2Si␻ cos ␪2R␻)其⫺1, using tabulated32 complex reT,R
fractive indices ñ ␻Si,ox
,2␻ for Si and SiO2 and angles ␪ ␻ ,2␻ of
transmitted (T) and reflected (R) SH and fundamental
waves evaluated from Snell’s law with incidence angle
␪ ␻I ⫽ 29° inside the oxide, following refraction from a
laboratory incidence angle of 45°. The resultant phase
␾ F is plotted as a long-dashed curve at the bottom of Fig.
12(b). The thinner solid curve that nearly coincides with
R
␾ F is the phase change ␾ ref
of the SH reference pulse on
R
⬇ 0. ␹ ( 3 )
reflection from the sample. Thus ␾ F ⫺ ␾ ref
2
2
was modeled as a Lorentzian 关 ␻ 0 ⫺ (2 ␻ ) ⫺ 2i ␻ ␥ 兴 ⫺1
with E 1 resonant frequency ប ␻ 0 ⫽ 3.37 eV and broadening parameter 2ប ␻ ␥ ⫽ 0.1 eV chosen to fit the width of
the measured SH amplitude spectrum. The resultant
phase ␾ ␹ , plotted as the lighter solid curve in Fig. 12(b),
changes by ␲ as 2␻ tunes through the E 1 resonance. We
evaluated Ĩ EFISH by using space-charge field E 0 (z)
⫽ E int0 ⫺ 2 ␰ z that varies linearly with depth z, as in the
Schottky model,43 which is reasonably valid for the dopant density and biases used here.29 Writing interface
field E int0 ⫽ 2 ␰ W in terms of width W of the depletion
layer integrates Ĩ EFISH analytically to
2␰
i⌬ 1 ⫺ ⌬ 2
再
⫺W ⫹
exp关共 i⌬ 1 ⫺ ⌬ 2 兲 W 兴 ⫺ 1
i⌬ 1 ⫺ ⌬ 2
冎
,
where
⌬ 1 ⫹ i⌬ 2 ⬅ k 2 ␻ ⫹ 2k ␻ ⫽ (2 ␲ /␭ SH) 关 ñ Si(2 ␻ )
⫹ n Si( ␻ ) 兴 is determined from tabulated32 refractive indices of Si. Phase ␾ I ⫽ tan⫺1(II EFISH /RĨ EFISH) now depends only on W. The ␾ I spectrum plotted in Fig. 12(b)
was evaluated by use of the maximum width43 W m
⬇ 0.04 ␮m for a room-temperature, negatively biased Si
MOS structure with donor density N D ⫽ 1018 cm⫺3 , although ␾ I was insensitive to variations in W of a factor of
2. For positive bias, ␰ and Ĩ EFISH change sign, so
␾ I (h ␯ SH) shifts uniformly by ␲. ␾ I is offset from zero by
⬃0.4␲ rad, reflecting the effective depth of the EFISH polarization beneath the Si–SiO2 interface. This depth is
determined by the convolution of depletion width W, coherence length ⌬ 1 ⫺1 , and absorption depth ⌬ 2 ⫺1 embodied in Ĩ EFISH . Total phase ␾ EFISH for V ⬍ V fb is plotted
as a heavy solid curve in Fig. 12(b). The corresponding
amplitude F 2 ␻ ␹ ( 3 ) I EFISH (to within a multiplicative constant) is plotted in the inset.
To complete the model, only a single adjustable parameter is required:
amplitude ratio ␣ ⬅ E 2EFISH
(h ␯
␻
⬇ 3.0 eV)/E 2FI␻ below resonance. In Fig. 12(a) we show
the calculated phase spectra ␾ TOT for ␣ ⫽ 1 (large 兩 V 兩 ) or
0.7 (smaller 兩 V 兩 ) for negative V ⬍ V fb (heavy and light
solid curves, respectively) and positive V ⬎ V fb (heavy
and light dashed curves, respectively). All trends of the
calculated ⌽ TOT(h ␯ SH , V) agree well with, and thus validate the accuracy of, the corresponding ⌽ FDISH data in
Fig. 9(a). The model provides no physical basis within
the sample for the offset ⌽ SH ⬃ ⫺2 rad from zero measured below the E 1 resonance, thus validating the interpretation that it arises in the reference film.
Wilson et al.
4. CONCLUSIONS
Through experiments with SiGe, SiGeC, and Si–SiO2
MOS structure samples we have demonstrated techniques for rapidly acquiring surface SHG amplitude and
phase spectra with no moving parts by using broadbandwidth (⌬␭ ⬃ 60 nm) femtosecond laser pulses. The
techniques are useful for probing spectral structure at the
SH frequencies when spectral structure at the fundamental frequencies is relatively featureless (e.g., semiconductors, insulators). As long as mild restrictions on incident
pulse bandwidth and the frequency dependence of ␹ ( 2 )
within this bandwidth are satisfied, the measured spectra
are normalized without knowledge of incident pulse chirp
by a single, independent measurement of a spectrally flat
reference sample and are found to agree with conventional scanning spectra within experimental error. We
have presented experimental and theoretical algorithms
for testing whether these restrictions are satisfied for a
given laser source and sample. These algorithms suggest that broadband SHG spectroscopy should retain
chirp-independent normalizability for fundamental bandwidths exceeding 100 nm on a wide class of semiconductor
and insulating samples. For a MOS sample, we have
validated the measured spectra in detail through an optical physics model of coherently superposed resonant
EFISH and SH background polarizations. Because
broadband techniques acquire data much faster than conventional scanning methods, they avoid artifacts caused
by laser and alignment drifts and slow changes in sample
conditions (e.g., laser-induced electrostatic charging) during scanning. Because they use extremely short pulses,
they also avoid artifacts caused by laser-induced surface
dynamics on a ⬎100-fs time scale (e.g., carrier screening).
Specific applications should include real-time monitoring
of surface SHG spectra as surface conditions (e.g., temperature, adsorption, epitaxial growth, etching) change
and probing of surface SHG spectra in femtosecond-timeresolved experiments.
ACKNOWLEDGMENTS
This study was supported by the Robert Welch Foundation (grant F-1038), the National Science Foundation
(NSF; grant DMR-0207295), and the NSF Physics Frontier Center Program (grant PHY-0114336). We are grateful to S. Banerjee for providing the SiGeC samples and to
S. Zollner for providing spectroscopic ellipsometry measurements of these samples.
M. C. Downer’s
@physics.utexas.edu.
e-mail
address
is
downer
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