Japanese Journal of Applied Physics
Vol. 45, No. 3B, 2006, pp. 1926–1930#2006 The Japan Society of Applied Physics

Nonlinear Dependences in Pulse-Pair-Excited Scanning Tunneling Microscopy
Osamu T AKEUCHI, Masahiro AOYAMA, Hiroyuki K ONDO, Atsushi T ANINAKA,
Yasuhiko T ERADA and Hidemi SHIGEKAWA
Institute of Applied Physics, 21st Century COE, CREST-JST, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan(Received August 1, 2005; accepted November 24, 2005; published online March 27, 2006)

The mechanism underlying an ultrafast decay of a sample parameter appears that in a time-resolved tunnel current signal
obtained by pulse-pair-excited scanning tunneling microscopy (PPX-STM) was theoretically and experimentally investigated.
It was shown that the nonlinear dependence of the tunnel current on the sample parameter allows us to observe the decay
process in a PPX-STM experiment even when the sample parameter itself does not have a nonlinear response to the lightintensity. This result promises the applicability of PPX-STM measurement to a wide variety of samples. On the other hand,such a nonlinear dependence makes the analysis of the obtained data very diﬃcult. Since the apparent decay time in the signal
can be diﬀerent from the actual value, the analysis must be carried out with detailed knowledge of dependence of the tunnel
current on the sample parameter. [DOI: 10.1143/JJAP.45.1926]
KEYWORDS: scanning tunneling microscopy, ultrafast mesurement, femtosecond pulsed laser

1.

Introduction

In order to satisfy the limitless requirement for miniaturizing and speeding up electronic devices, development offemtosecond–angstrom technology has been proposed to bea helpful solution and is gathering attention. The technology
combines ultrashort-pulsed lasers and scanning tunnelingmicroscopy (STM) to realize a measurement techniquewhich has both extremely high temporal and spatial
resolutions. Among several setups proposed for realizing
such measurements,1–10,12) pulse-pair-excited STM (PPXSTM) has recently made a signiﬁcant stride by introducing
the delay-time-modulation method,8–10,12) in which thetunnel gap of an STM apparatus is illuminated by asequence of ultrashort laser pulse pairs and the variation intunnel current is measured as a function of the delay time
between the two pulses in each pulse pair. As a result,picosecond-order ultrafast response in tunnel current was
successfully detected for a low-temperature-grown GaN As1 sample.
As already pointed out,9,10) however, the analysis of thetime-resolved tunnel current signal obtained by PPX-STM isvery complex. This is because the obtained time-resolved
tunnel current signal does not exactly correspond to theultrafast variation in tunnel current caused by the laser
illumination. Instead, it represents the variation in thetemporarily averaged tunnel current as a function of delay
time. The dependence of tunnel current on delay time
generally involves complex physical processes. The detailed
physical processes that occur in a sample must be understood to extract the ultrafast physical property of the sample
from the obtained signal. This paper aims to clarify a basic
concept for the analysis of time-resolved tunnel current
obtained by PPX-STM, where the nonlinear response of thetunnel current to the light illumination in the sample plays amajor role. Finally, the experimentally obtained data isinterpreted along with the discussion.
2.

Pulse-Pair-Excited STM

100 MHz, a delay time between two paired pulses of 0 –500 ps and an averaged intensity of 1–100 mW, is targeted
onto a spot with a diameter 50 mm at the tunnel junction in
STM. A weak feedback control is applied for the separation
between the STM tip apex and the sample surface tomaintain the averaged tunnel current constant at a reference
value of 0.1–10 nA with a bias voltage of 10 –1000 mV.
Since the bandwidth of the STM preampliﬁer is limited, thetunnel current is temporarily averaged so that the possible
tunnel current spikes generated by the laser pulse pair cannot
be resolved. The averaged tunnel current is, however, stillaﬀected by the laser illumination and is measured as afunction of the delay time between the two pulses. Since thevariation in tunnel current is generally very small, the signal
needs to be processed by a lock-in ampliﬁer to reduce thenoise level. In the recently proposed shaken-pulse-pairexcited STM (SPPX-STM) method, the lock-in detection is
synchronized with the small periodic modulation of thedelay time. This new method enables high-sensitivity
measurement without introducing the elongation and shrinkage problem of the STM tip due to the heat generated by the
exciting laser pulses even under a high-intensity excitation
condition.8) Although some other techniques have also been
proposed, the obtained signal in any PPX-STM method is
the temporally averaged tunnel currnt that varies as afunction of delay time, It ðtd Þ.
Next, we consider how the ultrafast response of theultrasmall region of the sample is reﬂected in the It ðtd Þ
signal. When a laser pulse impinges the sample surface, thetransient tunnel current It ðt; td Þ is generated in response to it;i.e., the tunnel current increases or decreases due to thesample excitation brought about by the laser pulse anddecays in a time scale from femtoseconds to hundreds ofpicoseconds resulting in an ultrashort tunnel current pulse.
The averaged tunnel current is related to the transient tunnel
current as
Z TrepIt ðtd Þ ¼ frep
It ðt; td Þ dt;
ð1Þ
0First, we describe the typical experimental setup for PPXSTM. A sequence of laser pulse pairs, with a pulse width of
10 –100 fs, a wavelength of 800 nm, a repetition rate ofwhere the integral is taken over the repetition period of thesequence of laser pulses, Trep ¼ 1= frep . As is discussed indetail below, when the averaged tunnel current shows a1926

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O. T AKEUCHI et al.

delay time dependence, the nonlinear response of thetransient current It ðt; td Þ to the laser intensity plays animportant role.
Let us assume that the two pulses in a pulse pair have thesame intensity, as a simple case. Then, when td is suﬃciently
long such that the two pulses excite the sample independently, there should be two identical current pulses in It ðt; td Þ,each of which corresponds to its respective laser pulse in thepulse pair. In contrast, when td is very short, the pulse pairaﬀects the sample as if they are only one pulse with twofold
intensity, causing only one current pulse in It ðt; td Þ. Thus, if
It ðt; td Þ has any nonlinearity on the laser pulse intensity, It ðtd Þ
takes on diﬀerent values in these two cases. Moreover, when
the delay time is increased from zero to a large value, It ðtd Þvaries from the former value to the latter value reﬂecting thetime scale of the ultrafast transition of the excited sample
state.
3.

Nonlinear Dependence of Tunnel Current
on Photointensity

What is the nonlinear dependence of transient tunnel
current, It ðt; td Þ, on light intensity? Indeed, the relationshipbetween light intensity and tunnel current is generally very
complex. Thus, it seems rather improbable that they arelinearly related. We classify such a nonlinear dependence
into two types: (1) nonlinear dependences of physicalparameters of the samples on light intensity, and (2)nonlinear dependences of tunnel current on the physicalparameter of the samples. Here, the physical parameter ofthe sample can be anything that inﬂuences the tunnel
current, such as photocarrier intensity, plasmon intensity,
electron temperature, electric polarization, and lattice vibration amplitude.
The former case, the nonlinear dependence of a sample
parameter on light intensity, is easier to understand. For
example, the photocarrier generation in a semiconductor
sample often shows a saturable nonlinearity with respect toan intense light illumination. This saturable response iscaused by the depletion of originally occupied electronic
states and by the occupation of originally empty electronic
states due to photoexcitation. Previously reported pumpprobe reﬂectivity measurements show that the absorption of
the second pulse can be slightly suppressed (R=R ¼
106 {103 ) by this eﬀect. In principle, PPX-STM can besensitive to such a phenomenon. If the absorption of thesecond pulse is suppressed, the total amount of photocarrier
generation due to illumination of two pulses is also suppressed, which will aﬀect the temporally averaged tunnel
current. In reality, however, a deviation in tunnel current due
to this eﬀect is impossible to detect. Note that when theintensity of the current pulse due to the second pulse
deviates by 106 {103 , the ratio of the deviation to the total
current, including the static tunnel current and the current
pulse due to the ﬁrst laser pulse, is more than one order ofmagnitude smaller. The inevitable noise level of the tunnel
current detection is usually 104 even with the help of thelock-in detection technique. Consequently, in order to makeuse of the type-1 nonlinearity, the nonlinearity must be
suﬃciently large and the change in the physical parametermust aﬀect the tunnel current very eﬃciently. Multistep
excitation of the sample state might be a good candidate thathas such properties, where the probability of excitation islargely emphasized when the second pulse arrives within the
lifetime of the midstate.
The second type of nonlinearity seems to appear morecommonly, which is the focus of this paper. The tunnel
current is known to be aﬀected by the physical parameters of
the sample discussed above and its dependence generally has
strong nonlinearities. For example, an increase in carrier
density in a semiconductor sample due to light illumination
varies the amount of surface bandbending. This is known as
the photovoltage eﬀect, which can be measured by STM at ahigh spatial resolution.13–15) While photocarrier generation
rate is almost linearly dependent on light intensity, photovoltage is nonlinearly dependent on carrier density. In
particular, photovoltage saturates when the original surface
bandbending is fully compensated by an intense illumination. Then, the photovoltage modiﬁes the eﬀective bias
voltage at the tunnel junction. Since the current-voltage
dependence for a semiconductor sample is also strongly
nonlinear, tunnel current depends on photovoltage nonlinearly. Consequently, the dependence of tunnel current onlight intensity shows a strong nonlinearity on a semiconductor sample.
4.

How Nonlinearity Aﬀects PPX-STM Measurement

Let us consider how such a nonlinear dependence type-2
appears in a PPX-STM measurement. We assume that when
one laser pulse with unit intensity and a negligible pulse
width impinges onto a sample at time zero, some physicalparameter of the sample nðtÞ shows an exponential decay
process as0t<0nðtÞ ¼ n0 ðtÞ ¼:
ð2Þ
N expðt=tÞ 0  t
By assuming that the dependence of the physical property onlight intensity is linear, the decay process caused by a pulse
pair with delay time td can be expressed as
nðt; td Þ ¼ An0 ðtÞ þ Bn0 ðt  td Þ;

ð3Þ

where the intensity of the ﬁrst pulse is A and that of thesecond pulse is B under the condition td > 0. Additionally,
we assume that tunnel current is a function of this sample
parameter, i.e.,
It ðt; td Þ ¼ It ðnðt; td ÞÞ:ð4Þ

In reality, It ðt; td Þ and nðtÞ should contain a constant that
corresponds to their value in the dark condition. Here, we
neglect this steady component and concentrate on thephotoinduced component for simplicity. With these assumptions, the temporally averaged tunnel current can beobtained by eq. (1).
4.1 Polynomial dependence
At ﬁrst, we examine the cases in which the tunnel current
shows polynomial dependences on the sample parameter,
i.e., It ðnÞ ¼ nk . The condition k ¼ 1 corresponds to the linear
dependence of tunnel current on the sample parameter andk > 1 corresponds to a diverging nonlinear dependence. In
the case of k ¼ 1, we obtain

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O. T AKEUCHI et al.

Z
It ðtd Þ ¼ ðA þ BÞ frepn0 ðtÞ dt;

ð5Þ

which suggests that the temporally averaged tunnel current
is always constant without any dependence on delay time. In
contrast, k ¼ 2 leads toZ2
2
It ðtd Þ ¼ ðA þ B Þ frep n0 ðtÞ2 dt
Zð6Þþ 2AB frep n0 ðtÞn0 ðt  td Þ dt:
While the ﬁrst term again does not depend on delay time, thesecond term does. However, the dependence is always as aneven function; i.e., even when the magnitudes of A and B are
diﬀerent, the plot of the time-resolved tunnel current shows
perfect symmetry. In the case of k ¼ 3, we obtainZ
It ðtd Þ ¼ ðA3 þ B3 Þ frep n0 ðtÞ3 dt
Zð7Þþ 3AB frep n0 ðtÞn0 ðt  td Þ
 fAn0 ðtÞ þ Bn0 ðt  td Þg dt;
where the second term depends on delay time. This time, thedependence is not an even function. Thus, when A and B are
diﬀerent, the plot becomes asymmetric. Likewise, thecondition k > 3 results in an asymmetric function except
for some special cases.
Figure 1 shows the calculated curves for It ðtd Þ with k ¼
2 { 5, A ¼ 1=4, B ¼ 1 and t ¼ 1. Even with the assumptionof an exponential decay process for the parameter nðtÞ, thetime-resolved tunnel current signal does not decay as an
exponential function. Moreover, the apparent decay constantvaries with the nonlinear factor k. In general, the larger the k,
the smaller the apparent decay constant. As predicted above,diﬀerent intensities for the two pulses in the pulse pairresults in symmetric and asymmetric curves in the cases ofk ¼ 2 and k > 2, respectively. The result for k ¼ 1 is notshown because it is a constant function. In the case of k > 2,
the apparent time constant is smaller when the stronger pulse
reaches the sample ﬁrst. For comparison, the decay process
assumed for n0 ðtÞ is also plotted with a broken line on thesame time scale with an arbitrary vertical scale. In thisFig. 1. Time-resolved tunnel current signal expected for polynomial
dependences of tunnel current on exponentially decaying sample
parameter n: It ðtÞ ¼ nðtÞk for k ¼ 2 { 5 (solid lines). The decay processof nðtÞ is shown with an arbitrary vertical scale as well (broken line).

Fig. 2. Time-resolved tunnel current signal expected for exponential
dependence of tunnel current on exponentially decaying sample parameter n: It ðtÞ ¼ exp½nðtÞ (solid line). The decay process of exp½nðtÞ isshown with an arbitrary vertical scale as well (broken line).

example, the signal intensity decreases when k is increased.
This is, however, not a generic law. Substituting diﬀerent
values for A and B, the relative intensity changes markedly.
4.2 Exponential dependence
What happens when the nonlinearity is even higher thanthe polynomial dependence? An exponential dependence,It ðtÞ ¼ exp½nðtÞ, also gives a diverging nonlinearity similar
to the polynomial dependences but the nonlinearity becomesmuch higher when n  1. Figure 2 shows the result for
A ¼ 4, B ¼ 16, and t ¼ 1 with a solid line. The broken line
corresponds to It ðAn0 ðtÞÞ with an arbitrary vertical scale. The
coincidence of these two curves is not accidental. When the
nonlinearity is very high, i.e., ðdIt =dnÞ=ðIt =nÞ  1 at n ¼
Bn0 ð0Þ, the integrated intensity of the function It ðBn0 ðt  td ÞÞ
is dominated by the peak intensity at t ¼ td under thecondition of B  A. Introducing another less intense
function of An0 ðtÞ, the peak intensity of the function It ðnÞ
becomes It ðAn0 ðtd Þ þ Bn0 ð0ÞÞ. When the dependence isexponential, this value is proportional to It ðAn0 ðtd ÞÞ.
As a summary of the diverging nonlinearities, when the
nonlinearity is very strong, the time-resolved tunnel current
is related to the decay process of the sample parameter in a
simple form given by It ðAn0 ðtd Þ þ Bn0 ð0ÞÞ. Knowing thefunction It ðnÞ, we can directly obtain the decay process ofthe sample parameter nðtÞ from the experimental result. On
the other hand, if the dependence is second-order polynomial, the result is an autocorrelation function of nðtÞ. In
moderate cases, the time-resolved tunnel current signal has ashape between these two cases.
4.3 Saturable nonlinear dependence
Finally, we examine the saturable nonlinearities. Figure 3
shows the case of It ðtÞ ¼ nðtÞ1=k with k ¼ 2 { 4, A ¼ 2, B ¼ 8
and t ¼ 1 (solid lines). In this case, the time-resolved
tunnel current has a dip at zero delay time, with apparent
decay constant larger than the actual decay constant of thephysical parameter (broken line). The plot is asymmetric
and, surprisingly, the plot can be convex upward when thestrong pulse reaches the sample ﬁrst. As the intensity ratio
between two pulses increases, the plot becomes almost

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O. T AKEUCHI et al.

Fig. 3. Time-resolved tunnel current signal expected for saturable
dependences of tunnel current on exponentially decaying sample
parameter n: It ðtÞ ¼ nðtÞ1=k for k ¼ 2 { 4 (solid lines). The decay processof nðtÞ is shown with arbitrary vertical scale as well (broken line).

constant in the range of small negative values of td . This isbecause, in the case of a strong saturable nonlinearity, asmall increase in n does not contribute to It when theabsolute value of n is large. Thus, when the intensity of thetwo pulses are very diﬀerent, the contribution of the weakpulse becomes negligible if it temporarily overlaps with thestronger pulse. Thus, the constant value of
R 1It for small
negative values of td is equal to It ð1Þ  0 It ðAn0 ðtÞÞ dt
when A  B. In addition,
the plot for positive td values
R1
approaches It ð1Þ  td It ðAn0 ðtÞÞ dt. In comparison with adiverging nonlinearity, the saturable nonlinearity correlates
the tunnel current nonlocally to the decay of the sample
parameter. Consequently, the interpretation of the experimental result tends to be more diﬃcult.
5.

Experimental Results and Discussion

Here we experimentally examine the light intensity
dependence of the time-resolved tunnel current signal
obtained by SPPX-STM and interpret them along with theabove discussion. The experimental setup was as previously
described in detail.10) The sample was low-temperaturegrown GaN As1 ( ¼ 0:36%) and the measurement was
performed in air. The reference value of the tunnel current
was 0.1 nA with a sample bias voltage of 2:0 V. The
sequence of laser pulse pairs had a repetition rate of 80 MHz,an average power of 10 mW, a center wavelength of
800 nm, and a pulse width of 25 fs. Delay time modulation
had an amplitude of 0.7 – 5 ps at 20 Hz.
Figure 4 shows the result. The relative intensities of thetwo pulses were varied from 1 : 1 to 4 : 0:125. In allmeasurements, the stronger pulse hit the sample later when
the delay time was positive. The vertical axis indicates thetunnel current deviation from the steady value. Since SPPXSTM does not provide the absolute value of deviation, theorigin of the vertical axis was selected arbitrarily. A positive
deviation represents an increase in tunnel current. It can beseen that the all curves show asymmetries for positive andnegative delay time regions, even for the intensity ratio of
1 : 1. This was probably due to the diﬃculty in precise
control of light intensity at the tunnel junction. Although thelight spots of the two pulses in the pulse pair were carefully

Fig. 4. (a), (b) Experimentally obtained time-resolved tunnel current
signals for GaN As1 with varying relative intensities of two pulses
from 1 : 1 to 4 : 0:125. (c) Schematics illustrating existence of long-living
negative component that explains apparant diﬀerence in signal amplitudes on negative and positive sides of delay time.

adjusted at the tunnel junction, the slight diﬀerence in the
spot diameter or the spot position caused the diﬀerence inthe eﬀective intensity of the two laser pulses. We also pointout that such a precise registration of the laser spots relative
to the tunnel junction can temporally ﬂuctuate slightly
during the long acquisition time of the time-resolved tunnel
current signal. Even so, the plots show a systematic variation
against the change in the intensity ratio.11)
During the attempt to interpret the experimental result inaccordance with the above discussion, we noticed theexperimental result has several discrepancies from that
expected in the above discussion. First, the data for theintensity ratio 1 : 2 was almost perfectly ﬁtted by the
combination of a step function and two double exponentialfunctions that decays towards positive and negative inﬁnitiesof delay time as reported previously.12) The time constants of1929

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O. T AKEUCHI et al.

the two double exponential functions are almost the samebut the amplitudes are diﬀerent. The amplitude of the step
function was equal to this diﬀerence such that the two
double exponential curves are connected continuously atzero delay time. Although the data for other intensity ratios
contained too much noise for curve ﬁtting analysis becauseof the shorter averaging times, the amplitude instead of theapparent time constant diﬀers between the negative andpositive sides as well. Indeed, the curve in one side seems tooverlap that of the other side with multiplied by a suitablescaling factor. Secondly, the increase of the intensity of thestronger pulse aﬀected the tunnel current in the opposite
direction for the positive and negative regions of delay timein Fig. 4(a), although the numerical models always increased
the signal amplitudes with an increase in the intensity ofwhichever excitation pulse.
Regarding the ﬁrst issue, the step function in the
numerical model was suggested to represent the existenceof a decay process longer than the measured delay time
range in the previous study.12) However, such a long-living
component should not appear as a step function in a PPXSTM experiment. Since the repetition periodicity of thepulse pair was 12.5 ns, the condition td ¼ 6:25 ns is
physically equivalent to td ¼ þ6:25 ns. Consequently, atime-resolved tunnel current must have the same value atthese points. Thus, instead of the step function, theexperimental curve might consist of, at least, one longliving component that has a dip at zero delay time as
illustrated in Fig. 4(c). Within the limited range of the delaytime, the signal appears to have diﬀerent amplitudes inpositive and negative sides. Separation of such multiple
components with unknown shape is fundamentally very
diﬃcult. Future experiments with diﬀerent wavelengths for
excitation might overcome this diﬃculty by eliminatingsome of these components. A wider scan of delay time andhigher precision for data acquisition must be achived as well.
The second issue is due to the diﬀerent probe-sample
separation for PPX-STM measurement with a diﬀerent
excitation intensity. When the excitation intensity ischanged, the STM probe thermally elongates or shrinks.
Thus, the signal intensity for a diﬀerent excitation intensity
cannot be directly compared. At this moment, no appropriate
solution for this problem is known.
6.

Conclusions

We examined how an exponentially decaying sample
parameter appears in time-resolved tunnel current via anonlinear dependence between the parameter and tunnel
current. It was turned out that PPX-STM has a potential todetect the ultrafast decay processes of any sample parameterthat aﬀects the tunnel current, even when the dependence ofthe parameter on the light intensity is perfectly linear. The
light illumination on an STM specimen often suﬀers fromthe insuﬃcient light intensity at the tunnel junction to induce
the nonlinear response of samples on light intensity. PPXSTM was suggested to be widely applicable even for suchcases.
On the other hand, the nonlinear dependence between
tunnel current and the sample parameter makes the analysisof the experimental time-resolved tunnel current obtained by
PPX-STM diﬃcult, in particular, to extract the ultrafast
decay process of the sample. Namely, the exponential decayof a sample parameter does not appear in the time-resolved
tunnel current as an exponential function. Furthermore, theapparent time scale of the obtained decaying curve can be
diﬀerent from that of the actual sample parameter. A
detailed understanding of the nonlinear dependence isneeded for the analysis. In addition, in the experimental
results, there can exist multiple components in a timeresolved tunnel current signal. Currently, separating thesecomponents to investigate their light intensity dependence
remains to be solved in future studies.
In order to apply PPX-STM for a wide variety of
specimens other than GaNAs and to obtain more precise
experimental results for detailed analysis, experiments witha more intense light source that is capable of varying
wavelength and has higher stability for output angle against
the ﬂuctuation in the room temperature, a wider scan range
of delay time and an ultrahigh vacuum environment aredesired. A new experimental system that meets theserequirements is under development.
Acknowledgement
This work was supported in part by a Grant-in-Aid for
Scientiﬁc Research from the Ministry of Education, Culture,
Sports, Science and Technology, Japan.

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