Dual-heterodyne Kelvin probe force microscopy

We present a new open-loop implementation of Kelvin probe force microscopy (KPFM) that provides access to the Fourier spectrum of the time-periodic surface electrostatic potential generated under optical (or electrical) pumping with an atomic force microscope. The modulus and phase coefficients are probed by exploiting a double heterodyne frequency mixing effect between the mechanical oscillation of the cantilever, modulated components of the time-periodic electrostatic potential at harmonic frequencies of the pump, and an ac bias modulation signal. Each harmonic can be selectively transferred to the second cantilever eigenmode. We show how phase coherent sideband generation and signal demodulation at the second eigenmode can be achieved by using two numerical lock-in amplifiers configured in cascade. Dual-heterodyne KPFM (DHe-KPFM) can be used to map any harmonic (amplitude/phase) of the time-periodic surface potential at a standard scanning speed. The Fourier spectrum (series of harmonics) can also be recorded in spectroscopic mode (DHe-KPFM spectroscopy), and 2D dynamic images can be acquired in data cube mode. The capabilities of DHe-KPFM in terms of time-resolved measurements, surface photovoltage (SPV) imaging, and detection of weak SPV signals are demonstrated through a series of experiments on difference surfaces: a reference substrate, a bulk organic photovoltaic heterojunction thin film, and an optoelectronic interface obtained by depositing caesium lead bromide perovskite nanosheets on a graphite surface. The conclusion provides perspectives for future improvements and applications.

In this case, F(t) can be defined by: Here W is the pulse width.The duty cycle ratio  is equal to W/.Note that the pulse amplitude (or magnitude), labeled A, should not be confused with the modulus coefficients of the Fourier series.Confusion may arise from the fact that the |cn| together are referred to as the signal amplitude spectrum.Note also that we have chosen to use a function with a zero base level for simplicity.The same Fourier coefficients would be obtained for n1 if the function were shifted by a DC component.Both comments apply to the following.
The set of equations in S2 yields: In this case, F(t) is defined by: Performing an integration by parts yields: Which simplifies in: Finally, one gets: ii) Pulse followed by an exponential decay used to calculate the Fourier coefficients describing the time-periodic surface photovoltage.We make the approximation of an instantaneous sample photo-charging (i.e., τr=0).The waveform amplitude is equal to the steady state SPV, but the base level has been set to zero (this shift does not affect the coefficients for n1).
This waveform is used to adjust the spectroscopic data obtained on the organic heterojunction thin film.The sample is exposed to a periodic light pulse train (periodicity , pulse width W).We assume that the surface photovoltage (SPV) is constant during the pulse duration (approximating an instantaneous photo-charging once the light is turned on, i.e. a SPV rise time constant r equal to zero) and that it follows an exponential decay characterized by a single time constant (d) between light pulses.As above, since we only want to calculate the harmonic coefficients (n1), we are in the simplified case of a signal characterized by a zero baseline and an amplitude (or magnitude) labeled A.
The first integral (cn p for "pulse part") has already been calculated above, it corresponds to the case of a pulse train.The second integral (cn d for "decay part") can be derived as follow: These complex coefficients can be used to calculate "intermediate" and an d and bn d terms, which are added to those already calculated for the pulse part (by the linearity of the Fourier transform).For this purpose, it can be noted that the complex coefficients can be described by a sum of a functions, respectively even and odd in n.This facilitates the calculation of an d and bn d (following Eqs.S2b and S2c).
One gets: Finally, the total coefficients are: These last coefficients are used to generate analytical formulas for the modulus (using Eq.S2d) and phase (Eq.S2e) coefficients that describe the time-periodic SPV signal.In the latter case (phase), the factor A obviously disappears when the ratio bn to an is made.Only one variable parameter (the decay time constant d) is thus needed to adjust the phase data.
Conversely, two parameters (A and d) are used to adjust the amplitude/modulus data (as mentioned above, beware of the possible confusion between the signal amplitude A, and the Fourier amplitude spectrum, i.e. the |cn| coefficients).
Note: to perform the phase adjustment, the function atan2(y,x) must be used instead of atan(y/x).The atan2 function is defined as the argument of the complex number x+jy. to illustrate a situation where the modulation frequency is set such a way that the frequencies of the first harmonics fall well below the first cantilever eigenmode.This would for instance be the case if mod/2=1kHz (while 0/2=80kHz).A "first" frequency mixing effect is symbolized (multiplication symbol within a circle) between the cantilever mechanical oscillation (first eigenmode, 0) and the modulated components at n. mod.Middle panel: a "second" frequency mixing effect has to be taken into account, between the ac bias modulation and the spectral components at 0 n.mod.Here again, an arbitrary choice has been made, the situation is hypothetically depicted with a bias modulation frequency falling below 0.This has been done for pedagogical purposes only.Note also that there is off course no physical meaning to assume that there are two mixing processes that occur "one after the other", all effects take place simultaneously.Bottom panel: 4n side bands result from the dual frequency mixing.

S8
By setting an appropriate value for the modulated bias frequency ac, one can "transfer" a given side band at the second cantilever eigenmode 1.

S9
Python routine for switching the demodulation configuration The following script can be used to switch the HF2LI configuration during the spectroscopic ramps.
import daq.setint('/dev18005/mods/0/enable', 0): toggles the HF2LI modulation unit such a way, that it will not combine the reference signal at 1-0 with the reference signal at n.mod.Instead, the signal at the output of the HF2LI modulation unit simply "follows" the reference signal at 1-0 provided by the MFLI."normalized" (i.e. by fixing its value to 5.2 for n=1) to ease the comparison with the experimental data.

S14
Concerning the SPV decay time-constant, the fit error is in average slightly smaller in the case of the data calculated by adjusting the phase spectra.Nevertheless, a certain fraction of the phase spectra could not be properly adjusted.
For these "bad" pixels, the error diverges.The upper limit of the color scale has thus been fixed to 100%.Errors above that threshold appear as magenta pixels in g).These points correspond to an "anomalous" distribution of time-constant values in the histogram h), highlighted by an arrow.functions [S1].b and d stand for the time-constants that characterize the SPV rise (or build-up) and decay dynamics, respectively.A detailed description of the pp-KPFM setup and data analysis protocol can be found in our previous report [S1].

S17
CsPbBr 3 nanosheets: synthesis protocol The nanosheets were synthesized following the same route as the one described in the work by Shamsi et al. [S2] Preparation of Cesium-oleate (Cs-OA) precursor: Cs-OA was obtained by loading 0.032 g Cs2CO3 with 10 mL oleic acid (OlAc) into a 25 mL 3neck flask This mixture was dried for 1h at 120 ºC under vacuum, and then heated to 140 ºC under Ar for 30 minutes until all Cs2CO3 reacted with OlAc.The major change in this way of preparing Cs-OA precursor is that it is dissolved in OlAc without using ODE, which alone leads to the formation of nanoplatelets (NPls) but not nanocubes (NCs), over a wide temperature range from 50 to 150 °C according to Shamsi et al. [S2] Synthesis of CsPbBr3 nanosheets (NSs): 0.013 g PbBr2 and 10 ml ODE were loaded with 250 µl OA, 250 µl oleylamine (OAm), 250 µl octanoic acid (OctA), and 250 µl octylamine (OctAm) into a 25 ml 3-neck flask and dried under vacuum for 20 minutes at 100 °C.After complete solubilization of the PbBr2 salt, the temperature was increased to 145 °C under Ar and 1 ml of the prepared Cs-OA (heated again at 100 °C) was swiftly injected.Accordingly, we used a volumetric ratio of 0.33 which represents the ratio of short to long ligands.After 5 minutes, the reaction mixture was quenched using a cold water bath [1].

Isolation of CsPbBr3 NSs:
To collect the NSs, 10 ml of anhydrous hexane was added to the crude solution and then the mixture was centrifuged at 700 RPM for 5 minutes.The supernatant was discarded and the NSs were dispersed in 3 ml of hexane.

Purification of CsPbBr3 NSs:
After depositing the solution on HOPG or SiO2 by spin coating (done at 2000 RPM for 45 seconds), the substrate was gently dipped in anhydrous ethyl acetate (EtOAc) for few seconds to remove the excess of organics and thus be compatible for AFM characterization.A=30mV.The 7 th harmonic is clearly visible, demonstrating that modulated components of a few mV only can be detected (with n=7, the amplitude should be equal to 2.7mV)

Figure S1
Figure S1 Periodic pulse train of period T, consisting of rectangular pulses of duration W and amplitude A. The duty cycle  is defined by W/T.

Figure S2
Figure S2 Sawtooth waveform of period T, and amplitude A.

Figure
Figure S3 a) Time evolution of the surface potential (SP) under pulsed illumination.The time intervals corresponding to the pump signal are marked by semi-transparent green rectangles.Illumination is performed for 0 ≤ t ≤ W. T: Pump signal period.VD: surface potential in the dark.VI: maximum value of the surface potential during illumination.If the pulse duration exceeds the photovoltage build-up or rise time constant (τb or τr), VI is equal to the SP value that would be measured under continuous wave illumination (Vcw).The "static" (or steady state) surface photovoltage is equal to VI-VD.τd: surface photovoltage decay time constant.b) Time function F(t) (or waveform)

Figure S4 -
Figure S4 -illustration of the dual frequency mixing effect.Top panel: modulated components of the electrostatic potential at harmonic (n1) frequencies of the optical pump (mod) are depicted in red.NB: here, we made the choice Figure S5 -a) Comparison of DHe-KPFM amplitude/phase spectra acquired by rejecting at the second eigenmode the n_4 side bands (ac=1-(0+0)+n.mod) or the n_1 side bands (ac=1-(0+0)-n.mod).The data are the same that the ones shown in Figure 2. The phase has been inverted (minus sign) for the first series of data (n_4 side bands).b) Plot of the Fourier coefficients obtained by an analytic calculus.The amplitude has been

Figure S9 -
Figure S9 -Images of the first ten harmonic phase signals recorded on the PTB7:PC71BM blend, reconstructed from the matrix of spectroscopic curves (same measurement as the one discussed in Figure 4)

Figure S11 -S22Figure S15 -
Figure S11 -CsPbBr3 nanosheets : photoluminescence spectroscopy.The main peak is consistent with the existence of nanosheets with a thickness equal or higher than 4-5nm [S2].The left doublet indicates the existence in the solution of objects (nanosheets) characterized by a smaller thickness.These were not clearly visible in our AFM measurements.