Gilad Cohen Thesis

TEL AVIV UNIVERSITY

The Iby and Aladar Fleischman Faculty of Engineering

The Zandman-Slaner School of Graduate Studies

Reconstruction of Kelvin Probe Force Microscopy images

A thesis submitted toward the degree of

Master of Science in Electrical and Electronic Engineering

by

Gilad Cohen

November 2012

TEL AVIV UNIVERSITY

The Iby and Aladar Fleischman Faculty of Engineering

The Zandman-Slaner School of Graduate Studies

Reconstruction of Kelvin Probe Force Microscopy images

A thesis submitted toward the degree of

Master of Science in Electrical and Electronic Engineering

by

Gilad Cohen

This research was carried out in the Department of Electrical Engineering -

Physical Electronics, under the supervision of Prof. Yossi Rosenwaks

November 2012

I would like to express my gratitude for the advices and directions provided by Prof.

Yossi Rosenwaks, my supervisor. The discussions and suggestions from him contributed

substantially to this thesis. I also would like to thank Prof. Amir Boag for his guidance

and assistance along the way.

I want to express my appreciations to my colleagues in Tel Aviv University for

supporting me with everything I needed. Special thanks go to: George Elias, Ezer

Halperin, Iddo Amit and Gil Shalev for lots of constructive suggestions and helpful

conversations.

[i]

Table of contents

Abstract ii List of Publications iii List of Figures iv List of symbols and abbreviations ix 1 Introduction 1 1.1 Kelvin Probe Force Microscopy (KPFM) 2 1.2 Frequency Modulation KPFM 5 1.3 Comparison between Amplitude and Frequency Modulations 8 2 Literature Review 9 3 Electrostatic Model 14 3.1 Probe-Sample Electrostatic System 14 3.2 Minimum Force/Force-gradient Condition 18 3.3 Point Spread Function (PSF) Integration 20 3.4 Model Validation 22 4 The Effect of Probe Geometry on PSF 25 4.1 Force/Force-gradient Distributions 25 4.2 Properties of AM-PSF and FM-PSF 28 4.3 The Cantilever Effect 31 5 Tip-Sample Distance Effect 33 6 Image Reconstruction 37 6.1 Outline of the Deconvolution Method 37 6.2 Reconstruction Process 39 6.2.1 PSF Expansion 39 6.2.2 Calibration Sample Measurements 42 6.3 Results and Comparison with Measurements 49 6.3.1 Reconstruction of CdS-PbS Images 49 6.3.2 Reconstruction of Graphene Images 54 7 Summary and Conclusions 58 8 References 59 9 Appendix 62

[ii]

Abstract

The main goal of this thesis is to develop an algorithm for reconstructing the surface

potential from its Kelvin probe force microscopy (KPFM) measurements. The KPFM

measures the surface potential, however due to the long range electrostatic forces the

measured potential is a weighted average and not the real potential under the tip apex. In

addition, we develop a method to calculate the point spread function (PSF) of frequency

modulation KPFM (FM-KPFM) and compare it with amplitude modulation KPFM (AM-

KPFM). In FM-KPFM the probe detects the force-gradient rather than the force and thus

demonstrates better spatial resolution than in AM-KPFM.

In order to reconstruct the real surface potential we estimate the system noise statistics

and calculate the exact PSF for the KPFM measurement. The image reconstruction is

then performed by applying the Wiener filter. The reconstruction algorithm is validated

by measuring a calibration sample under a known bias. We exhibit reconstruction of

surface potential on CdS-PbS nanorods measured with AM-KPFM in argon atmosphere

and surface potential on Graphene layers measured with FM-KPFM in ultra-high

vacuum. We show that in AM-KPFM measurements the averaging effect is very

dominant whereas FM-KPFM measurements demonstrate no averaging effect.

Furthermore, we analyze the effect of the tip-sample distance on the spatial resolution

and on the attenuation factor of the measured potential. By inspecting the full width at

half maximum (FWHM) of the PSFs we show that FM-KPFM demonstrate a superior

spatial resolution than AM-KPFM. We also show that for conventional tip-sample

distances (1nm-50nm) there is almost no attenuation for FM-KPFM, however in AM-

KPFM measurements conducted above 10nm the measured potential is derived mainly

from the substrate and not from the feature beneath the tip apex.

[iii]

List of Publications

Nanayakkara, S., Cohen, G., Jiang. C., Romero, M., Maturova, K., Al-Jassim, M., Lagemaat, J., Rosenwaks, Y., Luther, J. Built-in potential and charge distribution within single heterostructured nanorods measured by scanning Kelvin probe microscopy. Under review (2012).

[iv]

List of Figures

FIG.1: SCHEMATICS OF A TYPICAL KPFM TIP ABOVE A P-N JUNCTION SHOWING THAT THE TIP

DIMENSIONS ARE MUCH BIGGER THAN THE JUNCTION REGION UNDERNEATH IT. 1

FIG. 2: DEFINITION AND BASIC MEASUREMENT SETUP OF CONTACT POTENTIAL DIFFERENCE

(CPD). - WORK FUNCTION OF THE TIP, - WORK FUNCTION OF THE SAMPLE, - FERMI LEVEL, LVL - LOCAL VACUUM ENERGY LEVEL. 3

FIG. 3: PRINCIPLE OF FM-KPFM, THE FREQUENCY SHIFT VS. TIP SAMPLE BIAS. THE KELVIN

CONTROLLER MINIMIZES THE AMPLITUDE OF THE FREQUENCY SHIFT, WHICH IS SMALLEST

WHEN VDC = VCPD. 6 FIG. 4: SCHEMATIC FREQUENCY SPECTRUM OF THE TIP OSCILLATION. THE PEAKS AT fmod AND 2fmod ORIGINATE FROM THE ELECTROSTATIC FORCE, WHEREAS THE PEAKS AT f0 fmod AND f0 2fmod SHOW THE FREQUENCY MODULATION AT f0. 7 FIG. 5: A MODEL IN WHICH THE AFM TIP IS REPRESENTED BY A SERIES OF PARALLEL PLATE

CAPACITORS Ci AT DISTANCES Zi FROM THE SAMPLE SURFACE. ONE PLATE OF A CAPACITOR IS LOCATED ON THE TIP AND THE OTHER ON THE SAMPLE SURFACE. 9

FIG. 6: SCHEMATIC DESCRIPTION OF THE DIHEDRAL CAPACITORS. THE TWO CAPACITOR

PLATES ARE INDICATED TOGETHER WITH LENGTH OF THE ARC, THAT CONNECTS THEM. 11

FIG. 7: SCHEMATIC DESCRIPTION OF THE KPFM SETUP PROPOSED BY JACOBS ET AL. THE

SAMPLE IS DIVIDED INTO A SYSTEM OF IDEAL CONDUCTORS WITH ELECTROSTATIC

INTERACTIONS REPRESENTED BY MUTUAL CAPACITANCES . 12

FIG. 8: CALCULATING THE EQUIPOTENTIAL LINES AND FIELD LINES BY FITTING A

HYPERBOLA TO THE MEASURED TIP FOR EVERY ANGLE . 13

FIG. 9: BOUNDARY CONDITIONS FOR THE PROBE-SAMPLE ELECTROSTATIC SYSTEM. 14

FIG. 10: TWO CONTRIBUTION TO THE PROBE VOLTAGE: (A)FROM THE HOMOGENOUS SYSTEM

AND (B)FROM THE CPD OF THE SAMPLE. 16

FIG. 11: TRANSFORMING THE BOUNDARY CONDITION ON THE SAMPLE SURFACE, VCPD, TO DIPOLES IN FREE SPACE. (A) VCPD IS REPLACED WITH SURFACE CHARGE DENSITY s. (B)THE

[v]

SURFACE CHARGE DENSITY IS DESCRIBED WITH VARYING DIPOLE DENSITY ABOVE A

GROUNDED PLANE. (C)THE GROUNDED PLANE IS REPLACED WITH IMAGE CHARGES TO

FORM AN ADDITIONAL DIPOLE LAYER. (D)THE ORIGINAL DIPOLE LAYER COINCIDES WITH

THE IMAGE DIPOLE LAYER, FORMING A DIPOLE LAYER WITH DOUBLED DENSIT. 17

FIG. 12: TIP-INFINITE FLAT SURFACE SYSTEM AND CHARACTERISTIC DIMENSIONS. 23

FIG. 13: COMPARISON BETWEEN THE FORCE AND FORCE-GRADIENT CALCULATED IN OUR

MODEL TO ANALYTICAL EXPRESSIONS, AS A FUNCTION OF THE TIP-SAMPLE DISTANCE. THE

SOLID LINES (RED) INDICATE ANALYTICAL VALUES AND THE MARKED LINES (GREEN)

INDICATE CALCULATED VALUES FROM OUR MODEL. (A)FORCE VS. TIP-SAMPLE DISTANCE.

(B)GRADIENT-FORCE VS. TIP-SAMPLE DISTANCE. THE INSETS IN (A) AND (B) DESCRIBE THE

PERCENTAGE OF THE ABSOLUTE ERROR FROM THE FORCE AND FORCE-GRADIENT,

RESPECTIVELY. (C)LOGLOG PLOT OF THE FORCE VS. TIP-SAMPLE DISTANCE. (D)LOGLOG

PLOT OF THE FORCE-GRADIENT VS. TIP-SAMPLE DISTANCE. FOR BOTH THE CALCULATION

AND SIMULATIONS WE CONSIDERED: R = 30nm, H = 3.88m AND 0 = 10. 24 FIG. 14: (A)CROSS-SECTION OF A TIP WITH A SPHERICAL APEX RADIUS , CONE LENGTH ,

HALF APERTURE ANGLE 0, CONNECTED TO A CANTILEVER WITH LENGTH, WIDTH AND

THICKNESS OF L, W AND t, RESPECTIVELY. (B)CROSS SECTION OF A PROBE-SAMPLE SYSTEM

COMPOSED FROM A CANTILEVER TILTED IN DEGREES TOWARDS THE SAMPLE AND A TIP-

SAMPLE DISTANCE OF d. ALL THE ANALYSES WERE CARRIED OUT USING THE PARAMETER

VALUES: L = 225m, W = 40m, t = 7m, l = 13m, R = 30nm AND 0 = 17.5. 26 FIG. 15: DISTRIBUTION ALONG THE PROBE OF THE HOMOGENEOUS AND INHOMOGENEOUS

PARTS OF THE (A)FORCE AND (B)FORCE-GRADIENT. LEFT AXES - RELATIVE CONTRIBUTION

OF THE HOMOGENEOUS FORCE/FORCE-GRADIENT ON DIFFERENT SECTION OF THE PROBE.

RIGHT AXES - RELATIVE AREA OF EACH SEGMENT OF THE PROBE OUT OF THE TOTAL AREA

OF THE PROBE. THE INSETS IN (A) AND (B) OUTLINE THE DISTRIBUTIONS OF THE

INHOMOGENEOUS FORCE AND INHOMOGENEOUS FORCE-GRADIENT, RESPECTIVELY, ALONG

THE PROBE. THE PROBE IS DIVIDED INTO ELEVEN DISTINCT SEGMENTS DEFINED AS

FOLLOWS (FROM LEFT TO RIGHT): THE TIP APEX, THE TIP SPHERE, THE LOWER PART OF THE

CONE (VERTICAL LENGTH OF 5m), THE UPPER PART OF THE CONE (VERTICAL LENGTH OF 5m) AND SEVEN CANTILEVER SEGMENTS EACH WITH A LATERAL LENGTH OF 26.7m. THE SIMULATIONS WERE CALCULATED FOR = 20 AND PROBE-SAMPLE DISTANCE OF d = 5nm. 27

[vi]

FIG. 16: TWO DIMENSIONAL PSFS OF (A)AM-PSF AND (B)FM-PSF. THE PSFS WERE

CALCULATED FOR = 20 AND TIP-SAMPLE DISTANCE OF d = 5nm. 28 FIG. 17: COMPARISON BETWEEN ONE DIMENSIONAL FM-PSFS(I)(RED) AND AM-PSFS(II)(BLUE).

(A)ONE DIMENSIONAL PSFS ALONG THE Y-AXIS. THE HORIZONTAL LINES REPRESENT THE

WIDTH AT HALF MAXIMUM, THE SOLID AND DASHED LINES CORRESPOND TO THE FM- AND

AM-PSFS, RESPECTIVELY. (B)LOG OF THE PSFS ALONG THE X-AXIS. (C)LOG OF THE PSFS

ALONG THE Y-AXIS. THE SIMULATIONS WERE PERFORMED WITH = 20 AND TIP-SAMPLE DISTANCE OF d = 5nm. 30 FIG. 18: ONE DIMENSIONAL AM- AND FM-PSFS SIMULATED FOR TWO DIFFERENT TIP-SAMPLE

DISTANCES d WITH AND WITHOUT THE CANTILEVER, REPRESENTED BY THE SOLID (RED) AND

DASHED (BLUE) LINES, RESPECTIVELY. (A)AM-PSF FOR d=5nm. (B)AM-PSF FOR d=30nm.

(C)FM-PSF FOR d=5nm. (D)FM-PSF FOR d=30nm. THE HORIZONTAL LINES REPRESENT THE FWHM OF THE PSFS - SOLID LINES FOR THE FULL PROBE (TIP+CANTILEVER) AND DASHED

LINES FOR THE TIP ONLY. THE SIMULATIONS WERE PERFORMED WITH THE PROBE

ILLUSTRATED IN FIG. 14 WITH = 20 FOR THE FULL PROBE. 32 FIG. 19: (A)FULL WIDTH AT HALF MAXIMUM OF BOTH AM-PSF (DASHED LINE) AND FM-PSF

(SOLID LINE) AS A FUNCTION OF THE TIP-SAMPLE DISTANCE. (B)PSF PEAK VALUE (MAXIMAL

PSF VALUE) AS A FUNCTION OF THE TIP SAMPLE DISTANCE FOR AM-PSF (DASHED LINE) AND

FM-PSF (SOLID LINE). ALL SIMULATION WERE PERFORMED WITH THE PROBE IN FIG. 14 FOR

= 15. 33 FIG. 20: HOMOGENOUS FORCE (LEFT AXIS) AND HOMOGENOUS FORCE-GRADIENT (RIGHT

AXIS) VS. THE TIP-SAMPLE DISTANCE. THE FORCE AND FORCE-GRADIENT ARE MARKED WITH

BLUE AND GREEN LINES, RESPECTIVELY. 34

FIG. 21: SUMMATION OVER THE AM-PSF (LEFT AXIS) AND FM-PSF (RIGHT AXIS) VS. THE TIP-

SAMPLE DISTANCE. ALL PSFS WERE SIMULATED FOR 192nm 192nm SAMPLE AREA. 36 FIG. 22: INPUT AND OUTPUT SIGNALS IN THE KPFM SYSTEM. 37

FIG. 23: (A)LOGLOG PLOT OF THE AM-PSF VS. X AND Y LINES. A LINEAR RELATION IS

OBSERVED FAR FROM THE ORIGIN. (B)EXPANDING THE AM-PSF TO AN INFINITE AREA. FIRST,

THE X AND Y LINES ARE EXTRAPOLATED (ILLUSTRATED BY BLUE ARROWS). NEXT,

EXTRAPOLATION IS PERFORMED ON THE DIAGONAL DIRECTION (ILLUSTRATED BY RED

ARROWS). IN THE FINAL STEP, THE REMAINING PIXELS ON THE PLANE (BLACK DOMAINS) ARE

INTERPOLATED USING 2 PIXELS WHICH WERE PREVIOUSLY EXTRAPOLATED. AN EXAMPLE

[vii]

FOR THIS INTERPOLATION IS MARK WITH A GREEN AREA. 40

FIG. 24: (A)CONTACT MODE TOPOGRAPHY OF THE SAMPLE: TWO Ni CONTACTS ON A LAYER

OF SiO2. (B)HEIGHT PROFILE (AS INDICATED IN (A)). (C)CPD IMAGE OF THE SAMPLE. (D)CPD PROFILE (AS INDICATED IN (C)). (E)CPD IMAGE OF THE SAMPLE WHILE THE LEFT Ni

CONTACT WAS BIASED WITH 0.5V AND THE RIGHT Ni CONTACT WAS BIASED WITH -0.5V,

RELATED TO THE SiO2. (F)CPD PROFILE (AS INDICATED IN (E)). FROM THIS PROFILE ONE CAN OBSERVE AN ALMOST LINEAR SLOPE IN THE SURFACE POTENTIAL. A SMALL DEVIATION

UPWARDS IS VISIBLE DUE TO THE HIGHER CPD VALUE ON THE SiO2 COMPARED TO THE Ni. ALL KPFM MEASUREMENTS WERE CONDUCTED WITH AM USING LIFT-MODE WHERE THE LIFT

HEIGHT WAS 5nm. 43

FIG. 25: EVALUATION OF THE STATISTICS OF THE NOISE. (A)SECTIONS CONTAINING ONLY

CLEAN SiO2 WERE FRAMED FOR DISTINGUISHING THE NOISE FROM THE CPD SIGNAL. (B)THE

CPD OF THE SiO2 SUBSTRATE TAKEN FROM (A). (C)HISTOGRAM OUTLINES THE DISTRIBUTION OF THE SUBSTRATE CPD (BARS) AND A NORMAL FIT (DASHED-LINE) WITH MEAN AND

VARIANCE OF 0.5132V AND 3.23 105V2 RESPECTIVELY. (D) AUTOCORRELATION OF THE NOISE EXTRACTED FROM THE MARKED AREA IN (A). 44

FIG. 26: (A)BAND DIAGRAM OF THE UNBIASED SAMPLE. (B)BAND DIAGRAM OF THE SAMPLE

WHERE THE LEFT AND RIGHT Ni ELECTRODES ARE BIASED WITH -0.5V AND 0.5V,

RESPECTIVELY, RELATED TO THE TIP. Ef(Si),Ef(Ni),Ef(tip) REFERS TO THE FERMI LEVELS OF

THE SILICON, NICKEL AND TIP, RESPECTIVELY. LVL(tip) AND LVL(sample) ARE THE LVLS OF

THE TIP AND SAMPLE, RESPECTIVELY, AND = 50meV. THE CPD IS MARKED IN RED. UNDER BIAS, THE CPD INCREASES FROM -0.035V (LEFT ELECTRODE) TO 0.965V (RIGHT

ELECTRODE). 46

FIG. 27: (A)2D IMAGE OF THE THEORETICAL CPD ON THE BIASED SAMPLE. (B)MEASURED CPD

(RED), THEORETICAL CPD (BLUE) AND CONVOLUTION OF THE THEORETICAL CPD WITH PSFS

GENERATED FOR TIP-SAMPLE DISTANCE OF 6nm (BROWN), 8nm (BLACK) AND 10nm(GREEN).

ALL CPD LINESCANS ARE PLOTTED ALONG THE WHITE LINE IN (A). 47

FIG. 28: MEASURED CPD (RED), THEORETICAL CPD (BLACK) AND DECONVOLVED CPD (BLUE).

ALL CPD PROFILES ARE RELATED TO THE LINESCAN INDICATED IN FIG. 24(E). 48

FIG. 29: SCHEMATIC OF KPFM EXPERIMENTAL APPARATUS. THE AFM TIP USED FOR THIS

SETUP HAS A TIP APEX OF 30nm. THE MEASURED NRS ARE APPROXIMATELY 80nm IN LENGTH

AND 4nm IN DIAMETER. 49

[viii]

FIG. 30: (A)REPRESENTATION OF THE PROBE IN THE Y-Z AXES. IS THE ANGLE BETWEEN THE

CANTILEVER AND THE HORIZONTAL AXIS. DISTANCE d IS THE AVERAGE TIP-SAMPLE

DISTANCE. (B)REPRESENTATION OF THE PROBE IN THE Y'-Z' AXES. A IS THE OSCILLATION

AMPLITUDE. 50

FIG. 31: (A)AM-KPFM MEASUREMENT OF CDS-PBS NRS ON HOPG SUBSTRATE. TWO NRS (ROD1

AND ROD2) ARE SELECTED FOR FURTHER ANALYSIS. (B)THE CPD OF THE HOPG SUBSTRATE

TAKEN FROM (A). (C)HISTOGRAM PRESENTS THE DISTRIBUTION OF THE SUBSTRATE CPD

(BARS) AND A NORMAL FIT (DASHED-LINE) WITH A MEAN AND VARIANCE OF 0.3862V AND 5.4 105V2, RESPECTIVELY. (D)AUTOCORRELATION OF THE NOISE EXTRACTED FROM THE MARKED AREA (BLUE) IN (A). 52

FIG. 32: (A)MEASURED CPD ON CDS-PBS NRS. (B)ACTUAL CPD ON CDS-PBS NRS OBTAINED BY

DECONVOLUTION WITH THE WIENER FILTER. 53

FIG. 33: LINESCANS OF THE MEASURED (DASHED LINE) AND DECONVOLVED (SOLID LINE)

CPD ACROSS THE LONGITUDINAL AXIS OF TWO SYMMETRIC PBS-CDS-PBS NRS (INSET KPFM

IMAGE) USING THE WIENER FILTER, EFFECTIVE PSF AND NOISE STATISTICS. (A)CPD ALONG

ROD1. (B)CPD ALONG ROD2. BOTH NRS ARE HIGHLIGHTED IN FIG. 31(A). 54

FIG. 34: OBTAINING SYSTEM NOISE STATISTICS FROM CPD MEASUREMENTS. (A) AND (B) SHOW

RAW CPD MEASUREMENTS OF SINGLE LAYERS AND DOUBLE LAYERS OF GRAPHENE. (C) AND

(D) PRESENT ONLY PURE AREAS OF SINGLE LAYERS OR DOUBLE LAYERS OF GRAPHENE

WITHOUT THE INTERFACIAL AREA BETWEEN THEM. THE SINGLE LAYERS ARE MARKED WITH

II,IV WHEREAS THE DOUBLE LAYERS ARE MARKED WITH I,III,V. (E) AND (F) SHOW THE CPD

DISTRIBUTION (BLUE BARS) ON THE AREAS PRESENTED IN (C) AND (D), RESPECTIVELY,

ALONG WITH FITTED PDFS OF BIMODAL GAUSSIAN FITS CORRESPONDING TO THEM (DASHED

LINE). 56

FIG. 35: (A) AND (B) SHOW LINESCANS OF THE MEASURED (BLUE LINE) AND DECONVOLVED

(RED LINE) CPD ALONG THE PROFILES ILLUSTRATED IN FIG. 34(A) AND FIG. 34(B),

RESPECTIVELY. THE DECONVOLVED CPD VALUES WERE CALCULATED USING THE WIENER

FILTER, EFFECTIVE PSF AND NOISE STATISTICS OBTAINED BEFOREHAND. 57

[ix]

List of symbols and abbreviations

0 Vacuum permittivity

Work function

q Elementary charge

V Voltage potential difference

Angular frequency

f Frequency

C Capacitance

U Energy

F Force

A Amplitude

m Mass

k Spring constant

Potential

Surface charge density

Standard deviation

Expected value (mean value)

Surface dipole density

R Tip apex radius

H Tip cone length

L Cantilever length

W Cantilever width

[x]

d Tip-sample distance

0 Cone half aperture angle

Cantilever tilt angle

Noise

2 Variance VDC Measured potential, output of Kelvin controller V CPD Actual potential distribution on the sample Vprobe Applied bias voltage on the probe Vsub Substrate voltage VAC Alternating voltage G Probe self-interaction matrix (Greens matrix) D Dipole-probe interaction matrix Ch Probe-surface homogenous capacitance Cinh Probe-surface inhomogeneous capacitance DC Direct current

AC Alternate current

KPFM Kelvin probe force microscopy

AFM Atomic force microscopy

EFM Electrostatic force microscopy

AM-KPFM Amplitude modulation KPFM

FM-KPFM Frequency modulation KPFM

CPD Contact potential difference

LVL Local vacuum level

[xi]

PLL Phase lock loop

SNR Signal-to-noise ratio

LIA Lock-in amplifier

AM-PSF Amplitude modulation PSF

FM-PSF Frequency modulation PSF

BEM Boundary element method

FWHM Full width at half maximum

LPF Low pass filter

AWGN Additive white Gaussian noise

NR Nanorod

HOPG Highly-oriented pyrolytic Graphite

UHV Ultra high vacuum

PDF Probability density function

Chapter 1 - Introduction

[1]

1 Introduction Kelvin Probe Force Microscopy (KPFM) has already been demonstrated as a powerful

tool for measuring electrostatic potential distribution with nanometer spatial resolution.

However, due to the long range electrostatic forces the measured potential is a weighted

average of the real surface potential distribution. This effect is demonstrated in Fig. 1,

showing to scale an atomic force microscopy (AFM) tip above a p-n junction. The figure

clearly demonstrates that the size of the tip apex alone (right) is huge compared to the

typical junction dimensions, which emphasizes further the significance of the averaging

effect in KPFM.

Fig.1: Schematics of a typical KPFM tip above a p-n junction showing that the tip

dimensions are much bigger than the junction region underneath it1.

The actual surface potential can be recovered from the measured surface potential by

using deconvolution. The main goal of this thesis is to develop a reconstruction algorithm

for KPFM measurements.

Chapter 1 introduces the KPFM method and compare between two different scanning

modes: amplitude modulation KPFM (AM-KPFM) and frequency modulation KPFM

(FM-KPFM). In Chapter 2 we review some of the previous works, where a special

attention is paid to those who correlated the measured potential with the real potential on


[2]

the sample. Chapter 3 presents the computational model that was used which is based on

the work of Elias et al.2,3 and Strassburg et al.4,5. In addition, this chapter improves the

model by analyzing the influence of the averaging effect on FM-KPFM and validates it

with comparison to the work of Hudlet et al.6 who obtained an expression for the

electrostatic force acting on a probe above a homogeneous plane. In Chapter 4 and 5 we

examine the dependence of the probe geometry (especially the cantilever) and tip-sample

distance on the averaging effect in AM- and FM-KPFM. Chapter 6 presents a

reconstruction algorithm for the surface potential from a KPFM signal and displays some

results. The thesis is summarized in Chapter 7.

1.1 Kelvin Probe Force Microscopy Kelvin Probe Force Microscopy (KPFM, sometimes KFM or KPM) is a further

development of Electrostatic Force Microscopy (EFM)7, which measures the contact

potential difference (defined below) between tip and sample. It was first developed by

Nonnenmacher et al.8 in 1991. If the work function of the tip is known, the surface

potential of the material under study can be observed.

The work function of a material in vacuum is defined as the minimum energy required

for emitting an electron in the Fermi level to the vacuum level outside the material. The

Contact Potential Difference (CPD) is the difference between the work function of two

materials, defined as:

VCPD = 1 2q (1.1) Where 1 and 2 are the work function of the first and second materials, respectively,

and q is the elementary charge.

The term Kelvin Probe Force Microscopy relates to macroscopic Kelvin probe

techniques, which were invented by William Thomson, known as Lord Kelvin, in 1898

for the measurement of surface potentials using a vibrating parallel plate capacitor

arrangement: The sample constitutes one plate of a parallel plate capacitor, with a known


[3]

metal forming the other plate, which is vibrated at frequency . A DC-voltage applied to

one of the plates is used to minimize the induced current by the vibration9. This voltage

corresponds to the CPD of the two materials.

The KPFM employs the same principle, applying a DC-voltage in order to compensate

the CPD between the Atomic Force Microscope (AFM) probe and the sample10.

However, in a KPFM setup the electric force is used as the controlling parameter instead

of the current.

Fig. 2 illustrates the working principle of Kelvin probe force microscopy. When the

probe and sample are not electrically connected their local vacuum levels (LVL) are

aligned, as shown in Fig. 2(a). When they are electrically wired together, current will

flow from the material with the higher work function to the one with the lower work

function until the Fermi levels are aligned, inducing opposite charges between the probe

and the sample which form a binding electrostatic force, as shown in Fig. 2(b). This force

is nullified by a Kelvin feedback that applies bias between the probe and the

sample11. The magnitude of this bias is the CPD between the probe and the sample, as

shown in Fig. 2(c).

Fig. 2: Definition and basic measurement setup of contact potential difference (CPD).

- work function of the tip, - work function of the sample, - Fermi level, LVL -

local vacuum energy level.


[4]

KPFM measurement is usually conducted with Amplitude Modulation (AM-KPFM). In

AM-KPFM the cantilever is excited electrically by inducing a bias potential of Vprobe = VDC + VACsin (t), where VDC is the output of the Kelvin controller and is the frequency of the Kelvin modulation. Therefore, the potential difference between the

probe and the sample is:

V = VCPD Vprobe = VCPD VDC VACsin (t) (1.2) The electrostatic force can be derived from the energy of a capacitor:

U = 12 CV2 (1.3) Where C is the local capacitance between the probe and the sample. The vertical force is

then the derivative of the energy with respect to the probe-sample separation z:

Fz = dUdz = 12 dCdz V2 (1.4) By substituting Eq. (1.2) in Eq. (1.4) we get that the electrostatic force has spectral

components at DC and at frequencies and 2: Fz = FDC + F + F2 (1.5a) FDC = 12 dCdz (VCPD VDC)2 + 12 VAC2 (1.5b) F = dCdz (VCPD VDC)VACsin (t) (1.5c) F2 = 14 dCdz VAC2 cos (2t) (1.5d)

The Kelvin controller nullifies the force component at the excitation frequency () by


[5]

adjusting = . If the work function of the tip is known, we can infer the work function on the sample.

The KPFM measurement can be conducted simultaneously with the topography with

single-pass technique12, or alternatively by using lift mode technique13.

In single-pass the cantilever is exited in 2 eigenmodes simultaneously. The first

eigenmode is used for distance control and KPFM is performed at the second flexural

eigenmode. To this end, the cantilever is mechanically excited by a dither-piezo in the

first resonance and electrically excited in the second resonance.

In lift-mode technique the measurement of topography and surface potential are

alternated. Each scan line of topography is first recorded in tapping mode; the measured

trajectory is retracted in order to perform AM-KPFM at a constant lift height from the

sample.

1.2 Frequency Modulation KPFM Frequency Modulation KPFM (FM-KPFM) was first proposed by Kitamura et al.14 in

1998. In contrast to AM-KPFM where the force is detected, in FM-KPFM the control

parameter is the force-gradient.

When applying force F with gradient dFdz

on a cantilever, the fundamental mechanical

resonance frequency of the free cantilever f0 = 12 km changes as follows: f = 1

2k dFdzm (1.6)

Where k is the spring constant of the cantilever and is the effective mass of the

cantilever. For low force-gradient the frequency shift can be approximated by15:

f = f02

dFdz (1.7) For the spectral components of the frequency shift we find:


[6]

fDC dFDCdz = 12 d2Cdz2 (VCPD VDC)2 + 12 VAC2 (1.8a) f dFdz = d2Cdz2 (VCPD VDC)VACsin (2fmodt) (1.8b) f2 dF2dz = 14 d2Cdz2 VAC2 cos (2 2fmodt) (1.8c)

Where fmod is the modulation frequency. The Kelvin controller nullifies the force-gradient component at the modulation frequency (f) by adjusting VDC = VCPD, as shown in Fig. 316.

Fig. 3: Principle of FM-KPFM, the frequency shift vs. tip sample bias. The Kelvin

controller minimizes the amplitude of the frequency shift, which is smallest when VDC = VCPD.


[7]

In contrast to AM-KPFM, in FM-KPFM the cantilever is mechanically excited in its first

resonance and electrically excited in the modulation frequency fmod, simultaneously. The tip oscillation spectrum in FM-KPFM is shown in Fig. 417.

Fig. 4: Schematic frequency spectrum of the tip oscillation. The peaks at fmod and 2fmod originate from the electrostatic force, whereas the peaks at f0 fmod and f0 2fmod show the frequency modulation at f0.

The resonance frequency is measured using a frequency demodulator18, or a phase lock

loop (PLL)13. Then, the Kelvin controller nullifies the sidebands at 0 .


[8]

1.3 Comparison between Amplitude and Frequency Modulations As was shown in the previous two sections, the AM-KPFM detects the electrostatic force,

whereas the FM-KPFM detects the electrostatic force-gradient. Therefore, the properties

of the two measurements are different.

The decay of the electrostatic force is inversely proportional to the square of the distance

1

r2 and the decay of the electrostatic force-gradient is steeper and inversely proportional

to third power of the distance 1r3. Consequently, the detection of electrostatic force is

considered long-ranged detection, whereas the detection of the electrostatic force-

gradient is short-ranged detection. Thus, in AM-KPFM the probe averages over a larger

area on the sample, which leads to more prominent averaging effect in the

measurement13,19-21. On the other hand, in FM-KPFM the short-ranged electrostatic

interactions exist mainly between the tip apex and the sample, therefore averaging takes

place over a much smaller area below the tip. Hence, FM-KPFM has better spatial

resolution than AM-KPFM22.

However, AM-KPFM is considered to have better energy resolution of the measured

CPD than FM-KPFM. The signal-to-noise ratio (SNR) in AM-KPFM is very large

because the lock-in amplifier (LIA) detects the high resonance peak of the oscillating

cantilever23, whereas in FM-KPFM the LIA detects the relatively low sidebands f0 fmod , as portrayed in Fig. 4. In addition, more noise is generated at the output of the frequency demodulator (or PLL) which further degrades the SNR in FM-KPFM.

The FM-KPFM energy resolution can be improved by amplifying the modulation

amplitude, VAC. However, high modulation voltage might generate tip induced sample bend bending in semiconducting surfaces24 or quadratically increase the electrostatic

contribution to the topography signal23, as described in Eq. (1.8a), when scanning in

single-pass technique.

Chapter 2 Literature Review

[9]

2 Literature Review

It is well known that the probe geometry and scan method in KPFM can have a profound

effect on the measured CPD image. Reconstruction of the actual CPD on the sample from

the measured one is therefore of great importance and was carried out by several authors.

In the following we present different models which aim to this target. Hochwitz et al.25 suggested a simple model where the tip was replaced by a series

(staircase) parallel plate capacitors. The model is presented schematically in Fig. 5 which

shows defragmentation of the tip into small capacitors connected in parallel.

Fig. 5: A model in which the AFM tip is represented by a series of parallel plate

capacitors Ci at distances Zi from the sample surface. One plate of a capacitor is located on the tip and the other on the sample surface25.


[10]

After calculating the force acting on each capacitor, the total electrostatic force on the

probe is then nullified by applying a DC voltage of:

VDC = Ciz VCPDii Ci

zi (2.1) Where Ci

z is the capacitance gradient of capacitor i with respect to its distance from the

surface, Zi, and VCPDi is the CPD between the surface and capacitor i. Several groups26 used this model and showed reasonably good agreement to experimental data. However,

it uses two assumptions which are invalid in KPFM setup, as explained below.

The area below the tip is underestimated because the cones aperture angle is

small (usually between 10 15). The expression for the parallel plate capacitor (Ci = 1Zi) is valid only when the

capacitor plates area is much larger than the distance between them and the

sample.

A more precise model that takes into consideration the full area of the probe, and

therefore, a larger interacting sample was presented by Hudlet et al6. The tip was

modeled as a cone with a semi-spherical apex and dihedral capacitors replaced the

parallel plate capacitors (in the previous model), as shown in Fig. 6. The electric field on

each infinitesimal tip surface was assumed to be created by the dihedral capacitance

constituted by two infinite planes in the same relative orientation. This approximated

field determines the infinitesimal force, dFz, and the total tip-surface force is then obtained by summing all these contributions. They formulated analytical expressions for

the electrostatic force of a probe above a homogeneous surface and reported very good

agreement with experimental results.


[11]

Fig. 6: Schematic description of the dihedral capacitors. The two capacitor plates are

indicated together with length of the arc, that connects them6.

This model was validated by Law et al27 against a large set of experimental data. They

also included the contribution of a tilted rectangular cantilever to the electrostatic force.

Jacobs et al.28 introduced a model which correlates the measured potential with the actual

sample potential distribution. They treat the sample as a surface consisting of ideally

conducting electrodes of constant potential i and a tip of potential t, as shown in

Fig. 7.


[12]

Fig. 7: Schematic description of the KPFM setup proposed by Jacobs et al28. The sample

is divided into a system of ideal conductors with electrostatic interactions represented by

mutual capacitances .

The vertical force (Fz) on the probe is calculated by differentiating the expression for the electrostatic field energy with respect to the tip-sample distance. By setting Fz = 0 they obtained the KPFM signal:

VDC = Cit ini=1 Citni=1 (2.2) Where Cit = Cit/ z are the derivatives of the mutual capacitances between element on the sample and the tip. Eq. (2.2) demonstrates that the measured potential is a

weighted average of all potentials i on the surface with Cit being the weighting factors. Finally, they showed that by dividing an infinitely large surface to infinitesimal small

areas (x y) the KPFM signal can be expressed as: VDC(xt, yt) = h(x xt, y yt)(x, y)dxdy

(2.3)

h(x xt, y yt) = limx,y0 C(x xt , y yt)Ctot xy (2.4)

Where (xt, yt) is the tip location and Ctot is the derivative of the total tip-surface capacitance. Eq. (2.3) shows that the measured potential is a two-dimensional

convolution of the actual surface potential (x, y) with the system point spread function (PSF), h(x, y).


[13]

Machleidt et al.29 considered the real AFM tip shape in the determination of the PSF.

They used a prolate spheroidal coordinate system to calculate the electric field between

the charged tip and its mirror charge. The tip apex shape was estimated by measuring it

using a calibration sample, such as (NT-MDT) TGT1. The PSF was then derived by

fitting a hyperbola for every angle and calculating the equipotential lines () and the field lines (), as shown in Fig. 8.

Fig. 8: Calculating the equipotential lines and field lines by fitting a hyperbola to the

measured tip for every angle .

Two assumptions are used by most of the above authors: (a) The bulk material under the

surface does not influence the measurements and (b) the actual CPD on the sample does

not change with the presence or the metallic probe. Shikler et al30,31 validated these

assumptions by simulating a three-dimensional model consisting of both the substrate and

the area above it. These assumptions facilitate the KPFM modeling since we can

disregard the substrate entirely and simulate only a two-dimensional surface.

Chapter 3 Electrostatic Model

[14]

3 Electrostatic Model In this chapter we use a previously developed algorithm2-5 in order to find a relation

between the real CPD of the sample, VCPD, and the measured CPD of the probe, VDC. The model calculates the charge distribution on the probe, which yields the electric force and

force-gradient between the probe and the sample. Minimization of the force and the

force-gradient derives the amplitude modulation PSF (AM-PSF) and frequency

modulation PSF (FM-PSF), respectively.

3.1 Probe-Sample Electrostatic System The model assumes an equipotential probe above a flat sample (Fig. 9). The three inherent

boundary conditions are: (1) a constant probe voltage, Vprobe, (2) zero potential far away from the probe and (3) the surface potential of the measured sample, VCPD.

Fig. 9: Boundary conditions for the probe-sample electrostatic system.


[15]

In the absence of space charge, the potential in the electrostatic system follows the

Laplace equation:

Where V() is the potential in the point . The potential of interest is the potential of the probe. The interior potential in a domain bounded by a surface S is given by the integral

equation:

V() = G(, )()ds + Vs()Sprobe

(3.2)

Where G(,) is Greens function: G(, ) = 1

40| | (3.3)

() is the surface charge density on the probe, and Vs() represents the potential of known charges observed in point . Physically, Greens function G(, ) express the contribution of a unit charge in point to the potential in point .

We define:

Vprobeh () is generated from a homogenous system, consisting the charged probe above a grounded surface. Vprobeinh () is generated exclusively from the CPD of the sample. Therefore:

Vprobe() = Vprobeh () + Vprobeinh () (3.6) For the homogeneous system, the grounded surface is replaced with equivalent image

charges. In this manner we maintain the boundary condition at z = 0, V(x, y, z = 0) = 0 (Fig. 10(a)).

2V() = 0 (3.1)

Vprobeh () = G(,)()dsSprobe

(3.4)

Vprobeinh () = Vs() (3.5)


[16]

Fig. 10: Two contribution to the probe voltage: (a)From the homogenous system and

(b)from the CPD of the sample.

Both the probe surface charges and the image charges contribute to the homogenous

potential:

Where G(, )()dsS and G(, )()dsS are the contribution to the homogeneous voltage from the probe surface charge and the image charge, respectively,

and where: = (x, y,z). For evaluating the inhomogeneous potential we model the sample surface with dipoles.

This transforms the boundary condition calculation to free space. The model replaces the

sample surface potential, VCPD (Fig. 10(b)), with a layer of non-uniformed surface charge density, s, above a grounded plane in an infinitesimal distance l (Fig. 11(a)). The

Vprobeh () = [G(, ) G(,)]()dsSprobe

(3.7)


[17]

dipole density is therefore: () = s()l = VCPD0 (Fig. 11(b)). To satisfy the boundary condition for z = 0 the ground plane is replaced with image charges, generating an additional dipole layer (Fig. 11(c)). Both dipole layers coincide to a dipole

layer with doubled density () = 2() = 2VCPD0 (Fig. 11(d)).

Fig. 11: Transforming the boundary condition on the sample surface, VCPD, to dipoles in free space. (a) VCPD is replaced with surface charge density s. (b)The surface charge density is described with varying dipole density above a grounded plane. (c)The

grounded plane is replaced with image charges to form an additional dipole layer. (d)The

original dipole layer coincides with the image dipole layer, forming a dipole layer with

doubled density.

The inhomogeneous potential in point on the probe is calculated by integrating all the

dipole contribution from the sample surface, Ssample:

Where () is a unit vector normal to the sample surface. In our system () = z. In order to solve Eq. (3.7) and Eq. (3.8) numerically the equations are discretized into a

Vprobeinh () = 140 ( ) ()| |3 ()dsSsample = 12

( ) ()| |3 VCPD()dsSsample

(3.8)


[18]

linear system of equations using the boundary element method (BEM)2,4. Thus, the total

voltage on the probe, Vprobeh () + Vprobeinh (), can be represented in matrix form: I is a vector in the length of the number of the probes surface elements whose elements are all ones, and VCPD is a vector of discrete samples of the surface potential, VCPD, in the length of the total number of the image pixels. The matrices and vectors sizes and values

are all defined in the Appendix.

From Eq. (3.9) we obtain the surface charge density on the probe:

Where the vector Ch = G1I describes the capacitance per unit area on the probes surface elements and the matrix Cinh = G1D describes the capacitance per unit area between every probe surface element and sample surface element pair.

3.2 Minimum Force/Force-gradient Condition The CPD measurement in AM-KPFM is derived by minimizing the harmonic vertical

force applied on the probe, whereas the CPD measurement in FM-KPFM is derived by

minimizing the harmonic vertical force-gradient on the probe17. Using these two

conditions, a relation between VDC and VCPD is extracted for each of the scan methods under the assumption that the tip-sample distance is maintained constant during the

measurement.

The vertical force on the probe is calculated using the expression2:

Note that underbars denote vectors and double underbarsmatrices.

VprobeI = G + D VCPD (3.9)

= G1 VprobeI D VCPD = ChVprobe Cinh VCPD (3.10)

Fz = tB (3.11)


[19]

Where B is related to the geometry of the probe, as defined in the Appendix. The potential on the probe, positioned above a point on the sample, contains both DC

and AC components:

Vprobe = VDC() + VACsin (t) (3.12)

Applying Eq. (3.10) and Eq. (3.12) into (3.11) generates the force harmonic components:

In AM-KPFM the Kelvin controller minimizes ,. The minimum force condition therefore compels , = 0, extracting the following relation2: The derivation of the latter expression is followed in the Appendix.

In FM-KPFM the Kelvin controller minimizes the component of the force-gradient, dFzdz

. The force-gradient at frequency of the entire probe is:

Where Ch and Cinh are the capacitances gradients in the z directions (explicitly defined in the Appendix). The minimum force-gradient condition sets dFz

dz = 0, extracting the

relation:

VDC() = Cht B Cinh Cht B Ch VCPD() (3.16)

We define: = Cht B Ch ; = Cht B Cinh. From Eq. (3.14) and Eq. (3.16) we obtain the AM-PSF and FM-PSF of the probe-sample

Fz = Fz,DC + Fz, sin(t) + Fz,2cos (2t) (3.13)

VDC() = Cht B CinhCht B Ch VCPD() (3.14)

dFzdz = 2VAC 2Cht B ChVDC Cht B CinhVCPD Cht B CinhVCPD (3.15)


[20]

system:

PSF(AM) =

; PSF(FM) =

()

(3.17)

3.3 Point Spread Function Integration When scanning an equipotential substrate, the KPFM measurement outputs the exact

surface potential. Therefore, integration of the PSFs over the entire x-y surface must be

equal to one. Thus, the discrete PSFs presented in Eq. (3.17) must be normalized to one.

Elias et al show that AM-PSF is indeed normalized for an infinite surface (However,

numerical limitations yield lower value for the sum of the AM-PSF, as will be discussed

later in Chapter 4). We provide here an appropriate confirmation for the FM-PSF.

Let N and M be the number of probe elements and the number of sample surface

elements, respectively. Then, the mathematical representation of the requirement above

is:

ddz H inht ddz (Hh) I MX1 = 1 (3.18)

Where 1 is a vector in the length of the sample surface elements whose elements are all ones.

By substituting and into Eq. (3.18) we left to show that: ddz Cht B Cinh I MX1 = ddz Cht B Ch (3.19)

By writing Cinh = G1D in the left side and Ch = G1I 1 in the right side, Eq. (3.19) becomes:

Cht B G1

D 1 = Cht B G1

I 1 (3.20)


[21]

Meaning,

D 1 + D 1 = 1 (3.21) Where is defined in Eq. (3.20), and D are the gradients of and D in the z direction, respectively, defined as Qt = dQt

dz and D = dD

dz (explicitly defined in the

Appendix).

The sum of every row in D is the summation of the potential induced by a homogenous dipole layer on a specific probe element. This term is given by2:

Di = Dij = 12 zidxdy((xi x)2 + (yi y)2 + zi2)32 = 1 1R2zi2 + 1SSamplej (3.22)

Where (xi, yi, zi) are the coordinates of the probe element, and is an integrated area of a disc in a radius of R. Thus, by integrating an infinite area we get for each ith

probe element:

limR

D = 1 (3.23) Thus,

D 1 = 1 (3.24) As for the summation of D rows we get:

Di = Dijj

= R2(R2 + zi2)32 (3.25) The derivation of the latter expression is followed in the Appendix.

For each probe element i we then obtain:

limR

D = 0 (3.26) Thus,


[22]

D 1 = 1 (3.27) Where 1 is a vector in the length of the probe elements whose elements are all zeros. Eq. (3.24) and Eq. (3.27) satisfy the condition in Eq. (3.21), hence the FM-PSF in Eq.

(3.17) is normalized.

3.4 Model Validation In order to validate our model and to test its accuracy we compare it to the analytical

model proposed by Hudlet et al.6. Their electrostatic system is constructed of a grounded

sample and a probe composed of tip alone, without the cantilever. They estimate that the

force between the probe and the sample is:

F = 0V2 R2(1 sin0)z[z + R(1 sin0)] + k2 lnz + R(1 sin0)H 1 + Rcos20/sin0z + R(1 sin0) (3.28) Where k2 = 1

lntan(0/2)2. Where V is the voltage between the tip and the sample, R is the tip apex radius, z is the tip-sample distance, H is the length of the tip cone and 0 is the half-aperture angle of the

cone, as described in Fig. 12.


[23]

Fig. 12: Tip-infinite flat surface system and characteristic dimensions6.

In order to evaluate the force-gradient between the tip and the sample the derivative of

the expression in Eq. (3.28) was calculated:

dFdz = 0V2 R2(1 sin0)2z + R(1 sin0)z2[z + R(1 sin0)]2 + k2 1z + R(1 sin0) + Rcos20/sin0[z + R(1 sin0)]2 (3.29) Next, we simulated the above system for different tip-sample distances ranging from 1nm

to 50nm, and for each distance we calculated the force and the force gradient by using

Eq. (A.6) and Eq. (3.15). Fig. 13 compares the force and force-gradient in our model to

the analytical expressions in Eq. (3.28) and Eq. (3.29).


[24]

Fig. 13: Comparison between the force and force-gradient calculated in our model to

analytical expressions, as a function of the tip-sample distance. The solid lines (red)

indicate analytical values and the marked lines (green) indicate calculated values from

our model. (a)Force vs. tip-sample distance. (b)Gradient-force vs. tip-sample distance.

The insets in (a) and (b) describe the percentage of the absolute error from the force and

force-gradient, respectively. (c)Loglog plot of the force vs. tip-sample distance.

(d)Loglog plot of the force-gradient vs. tip-sample distance. For both the calculation and

simulations we considered: R = 30nm, H = 3.88m and 0 = 10.

All the calculations take into consideration only the component of the force and force-

gradient. Therefore, we substitute 2(VCPD VDC)VAC instead of V2 in Eq. (3.28) and Eq. (3.29) and assume: VDC = 0.5V ; VAC = 1V ; VCPD = 1V. Overall, we observe very good agreement between the two models. The maximal error

for the force is 15% and the error for the force-gradient never exceeds 5% for tip-sample

distances over 3nm. The relatively high error for the force-gradient in 1-2[nm] might

arise from coarse meshing of the tip or due to insufficient resolution.

Chapter 4 The Effect of Probe Geometry on PSF

[25]

4 The Effect of Probe Geometry on PSF In this chapter we inspect how the distributions of the force and force-gradient on the

probe affect the AM-PSF and FM-PSF. In addition, we show the influence of the

cantilever on the PSFs.

4.1 Force/Force-gradient Distributions

We separate the expression for the force in Eq. (A.6) to a homogenous part, Fz,h , which in not depended on VCPD, and an inhomogeneous part, Fz,inh, which is depended on VCPD:

Fz,h = 2VACVDCCht B Ch Fz,inh = 2VACCht B CinhVCPD (4.1)

In the same analogy, the homogenous part and inhomogeneous parts of the force-gradient

are:

dFzdz h = 2VACVDC ddz Cht B Ch dFzdz inh = 2VAC ddz Cht B CinhVCPD (4.2)

The probe under study is composed from a conical tip enclosed with a spherical cap and a

tilted cantilever. The geometric parameters of the probe are presented in Fig. 14:


[26]

Fig. 14: (a)Cross-section of a tip with a spherical apex radius , cone length , half

aperture angle 0, connected to a cantilever with length, width and thickness of L, W and

t, respectively. (b)Cross section of a probe-sample system composed from a cantilever

tilted in degrees towards the sample and a tip-sample distance of d. All the analyses were carried out using the parameter values: L = 225m, W = 40m, t = 7m, l = 13m, R = 30nm and 0 = 17.5.

Fig. 15(a) and Fig. 15(b) show the distribution of the force and force-gradient,

respectively, along the probe presented in Fig. 14 with a tip-sample distance of d = 5[nm] and cantilever tilt angle of = 20. The probe was divided into eleven segments where each bar corresponds to a distinct segment of the probe specified as

follows (from left to right): The tip apex, the tip sphere, the lower and upper parts of the

cone (each has vertical length of 5nm) and seven cantilever segments with an equal

length. The left axes in Fig. 15 outline the contribution of each probe section to the total

homogenous force/force-gradient and the right axes describe the relative mesh area of

each segment out of the total surface area on the probe. The insets in Fig. 15(a) and

Fig. 15(b) present the inhomogeneous force and inhomogeneous force-gradient

distributions, respectively, along the probe using the same segments. The force and force-

gradients were calculated for a 192nm by 192nm square sample, with equipotential

surface forming VCPD = 1V between the probe and the sample, Kelvin voltage of VDC = 1V and modulation voltage of VAC = 1V.


[27]

Fig. 15: Distribution along the probe of the homogeneous and inhomogeneous parts of

the (a)force and (b)force-gradient. Left axes - relative contribution of the homogeneous

force/force-gradient on different section of the probe. Right axes - relative area of each

segment of the probe out of the total area of the probe. The insets in (a) and (b) outline

the distributions of the inhomogeneous force and inhomogeneous force-gradient,

respectively, along the probe. The probe is divided into eleven distinct segments defined

as follows (from left to right): the tip apex, the tip sphere, the lower part of the cone

(vertical length of 5m), the upper part of the cone (vertical length of 5m) and seven cantilever segments each with a lateral length of 26.7m. The simulations were calculated for = 20 and probe-sample distance of d = 5nm.


[28]

One can observe from Fig. 15(a) that the cantilever contributes 22% of the homogeneous

force and therefore affects greatly the absolute measured CPD. In addition, almost all the

inhomogeneous force (99%) stems from the tip sphere and the cone. Thus, the effect of

the cantilever of the resolution in AM-KPFM can be neglected2. On the other hand, it can

be observed from Fig. 15(b) that the vast majority of the force-gradient (homogenous and

inhomogeneous) resides within the tip sphere (98%), the rest of the force-gradient stems

from the bottom cone (2%), and we observe zero contribution from the upper cone and

the cantilever. Therefore, it indicates that in FM-KPFM the cantilever has no influence on

the measured CPD. Since most of the homogeneous and inhomogeneous force-gradients

stem from the tip apex (81%), whose vertical length is 5nm, very high resolution is

expected in FM-KPFM with measured CPD close to the real CPD. The significant role of

the tip apex in FM-KPFM measurements is consistent with the short-ranged behavior of

the force-gradient, which decays faster than the force.

4.2 Properties of AM-PSF And FM-PSF The differences between the minimum force/force-gradient conditions presented in the

previous chapter and the force and force-gradient distributions yield two distinct PSFs

with different characteristics. In this subsection we distinguish between the properties of

the AM- and FM-PSF.

The two dimensional AM-PSF and FM-PSF for the probe presented in Fig. 14 are shown

in Fig. 16.

Fig. 16: Two dimensional PSFs of (a)AM-PSF and (b)FM-PSF. The PSFs were

calculated for = 20 and tip-sample distance of d = 5nm.


[29]

The sums of the AM-PSF and the FM-PSF components are 0.5856 and 0.9999,

respectively. The FM-PSF converges to a total sum of 1, as we prove in the previous

chapter. However, integration over the AM-PSF yields a smaller value since the

simulations cannot be performed on an infinite sample due to computational limitations.

Instead, we considered a finite square sample (192nm by 192nm) which is not sufficient

to satisfy Eq. (3.23). It should be noted that the sum of the AM-PSF components

approaches 1 as the tip-sample distance is shortened, since most of the inhomogeneous

force stems from the tip-sphere and by lowering the tip-sample distance the tip-sphere

sees more of the sample and less from the truncated area. In Chapter 6 we present a

method to extrapolate the AM-PSF into an infinite size.

Next, we inspect the influence of the force and force-gradient distributions on the KPFM

measurements. Fig. 17 presents a comparison between the one dimensional FM-PSFs () and AM-PSFs (). The FM-PSF peak is greater than the AM-PSF by a factor of 3.5 (Fig. 17(a)) which indicates higher CPD contrast in FM-KPFM measurements than in

AM-KPFM. Moreover, the full width at half maximum (FWHM) of the FM-PSF is

smaller than the AM-PSF by a factor of 1.25, indicating better CPD resolution when

conducting FM-KPFM. The probe is symmetric along the x-axis and therefore both the

AM-PSF and FM-PSF are symmetric along the x-axis (Fig. 17(b)). In the y-axis the

symmetry does not exist, both the AM-PSF and FM-PSF exhibit higher components at

negative y values (Fig. 17(c)). The asymmetry of the PSFs is due to the inhomogeneous

force and force-gradient which stem from the tip-sphere, as shown in Fig. 15. Since the

cantilever is tilted and clamped at positive y, the tip-sphere and the tip cone are mostly

located in negative y values, thus contributing more to the PSF components at these

values. Log scale was used in Fig. 17(b),(c) to highlight the change of the PSFs in the

positive and negative x,y axes, which is unnoticeable from Fig. 17(a) . The log of the

FM-PSF is truncated in order to prevent divergence where the FM-PSF reaches zero.


[30]

Fig. 17: Comparison between one dimensional FM-PSFs(i)(red) and AM-PSFs(ii)(blue).

(a)One dimensional PSFs along the y-axis. The horizontal lines represent the width at

half maximum, the solid and dashed lines correspond to the FM- and AM-PSFs,

respectively. (b)Log of the PSFs along the x-axis. (c)Log of the PSFs along the y-axis.

The simulations were performed with = 20 and tip-sample distance of d = 5nm.


[31]

4.3 The Cantilever Effect Elias et al.2 examined the AM-PSFs generated from probes with and without the

cantilever. They observed that the cantilever reduces the PSF values but has no influence

on the FWHM. Consequently, they concluded that the cantilever alters the absolute CPD

values but does not affect the AM-KPFM resolution.

In this section we inspect the influence of the cantilever on FM-KPFM measurements.

The effect of the cantilever on the AM- and FM-PSFs is demonstrated in Fig. 18 for two

different tip-sample distances with and without the cantilever, represented by the solid

and dashed lines, respectively. For the probe with the cantilever we simulated a tilt angle

of = 20. At a tip-sample distance of 5nm the peak value of the full probe AM-PSF is decreases by a factor of 1.65 compared to the tip AM-PSF (Fig. 18(a)), whereas the FM-PSF is unchanged in the presence of the cantilever (Fig. 18(c)). For a tip-sample distance

of 30nm the peak is attenuated by a factor of 4.12 for the AM-PSF(Fig. 18(b)), whereas the FM-PSF peak value is slightly decreased by 15% (Fig. 18()); this small attenuation of the full probe (tip+cantilever) FM-PSF is resulted from a higher portion of

homogeneous force-gradient (25%) stems on the tilted tip cone and not from the

cantilever. The horizontal arrows represent the FWHM for the four cases demonstrate the

conclusion that the cantilever hardly affects the measurement resolution.


[32]

Fig. 18: One dimensional AM- and FM-PSFs simulated for two different tip-sample

distances d with and without the cantilever, represented by the solid (red) and dashed (blue) lines, respectively. (a)AM-PSF for d=5nm. (b)AM-PSF for d=30nm. (c)FM-PSF

for d=5nm. (d)FM-PSF for d=30nm. The horizontal lines represent the FWHM of the

PSFs - solid lines for the full probe (tip+cantilever) and dashed lines for the tip only. The

simulations were performed with the probe illustrated in Fig. 14 with = 20 for the full probe.

Chapter 5 Tip-Sample Distance Effect

[33]

5 Tip-Sample Distance Effect In this chapter we analyze the effect of the tip-sample distance on the measurement

parameters in AM- and FM-KPFM. To this end, we calculate PSFs for the probe in

Fig. 14 for tip-sample distances ranging from 1nm to 100nm. Tilt angle of = 15 was set for all simulations. Fig. 19(a) shows the FWHM dependence on the tip-sample

distance for both AM-PSF (dashed line) and FM-PSF (solid line). One can observe that

FM-PSF has smaller FWHM values than AM-PSF for all tip-sample distances, which

indicates a superior resolution of FM-KPFM over AM-KPFM. Fig. 19(2) shows the PSF

peak value (maximal PSF value) as a function of the tip-sample distance. It is observed

that the FM-PSF (solid line) demonstrates much higher peak value than the AM-PSF

(dashed line). For example, at tip-sample distance of 30nm, which is frequently used in

ambient KPFM, the FM-PSF peak is higher by an order of magnitude than the AM-PSF

peak. Since contrast of CPD images correlates with the PSF peak value, we conclude a

superior contrast in FM-KPFM compared to AM-KPFM.

Fig. 19: (a)Full width at half maximum of both AM-PSF (dashed line) and FM-PSF

(solid line) as a function of the tip-sample distance. (b)PSF peak value (maximal PSF

value) as a function of the tip sample distance for AM-PSF (dashed line) and FM-PSF

(solid line). All simulation were performed with the probe in Fig. 14 for = 15.


[34]

Fig. 20 presents the homogenous force (blue line) and homogenous force-gradient (green

line) as a function of the tip-sample distance. The left axis corresponds to the force

whereas the right axis corresponds to the force-gradient. The force-gradient demonstrates

much steeper descent than the force; from 1nm to 100nm the force-gradient decreases by

4 orders of magnitude whereas the force decreases merely by 1 order of magnitude.

These curves are in good agreement with the long-range behavior of the force (Fz 1r2) and the short-range behavior of the force-gradient (dFz

dz

1

r3).

Fig. 20: Homogenous force (left axis) and homogenous force-gradient (right axis) vs. the tip-sample distance. The force and force-gradient are marked with blue and green lines, respectively.


[35]

From Eq. (4.1) and Eq. (4.2) we derive:

Fz,inhFz,h = PSF(AM) VCPDVDC (5.1)

dFzdz inhdFzdz h = PSF(FM) VCPDVDC (5.2)

The force and force-gradient were calculated for VDC = 1V , VAC = 1V and an equipotential sample with VCPD = 1V, therefore the sums over the PSFs equals:

PSF(AM) = Fz,inhFz,h (5.3) () =

(5.4) In AM-KPFM and FM-KPFM we should theoretically obtain Fz,inh = Fz,h and dFzdz

inh = dFzdz

h, respectively. However, due to computational limitations mentioned in

Section 4.2 we cannot calculate the inhomogeneous force and force-gradient for infinite

surfaces. Therefore, a 192nm 192nm sample area was simulated beneath the tip. Hence, summation over the PSFs gives the portion of the inhomogeneous force/force-

gradient component in the simulated area. Fig. 21 presents the summation over the PSFs

as a function of the tip-sample distance. The left axis corresponds to the AM-PSF

whereas the right axis corresponds to the FM-PSF. It is observed that the summation over

the AM-PSF decreases rapidly whereas the summation over the FM-KPFM maintains

above 90% for conventional tip-sample distances (1nm to 50nm). Thus, less attenuation

of the CPD signal is expected with FM-KPFM since most of the detected force-gradient

stems from the area beneath the tip.


[36]

Fig. 21: Summation over the AM-PSF (left axis) and FM-PSF (right axis) vs. the tip-

sample distance. All PSFs were simulated for 192nm 192nm sample area.

Summation on the simulated PSF also provides quantitative information about the

attenuation factor. If we assume a square feature of 192nm 192nm with a constant CPD of VCPD surrounded by an infinite substrate with a constant CPD of Vsub, then the KPFM measures:

VDC = VCPD PSF + Vsub 1 PSF (5.5) Where is the simulated PSF on a 192 192 sample. Consequently, the real CPD on the feature can be extracted with:

VCPD = VDC Vsub 1 PSF PSF (5.6) The next chapter demonstrates a more rigorous method to reconstruct the real CPD on a

feature.

Chapter 6 Image Reconstruction

[37]

6 Image Reconstruction The following chapter describes an algorithm to reconstruct the actual CPD of a sample

from the KPFM measurements. In the first section we overview the CPD image

degradation resulting from the KPFM measurement and explain the deconvolution

process. In the second section we validate our reconstruction algorithm with a calibration

sample. The last section shows several CPD reconstruction results.

6.1 Outline of the Deconvolution Method The effect of the measuring AFM probe in KPFM is very large, especially in AM-KPFM

where the measured forces are long range. The probe averaging effect degrades the

spatial resolution but more important largely change the magnitude of the measured

surface potential. The CPD measured on flat surfaces is modeled by a linear shift-

invariant system where the impulse response is the Point Spread Function (PSF). Fig. 22

illustrates the KPFM measurement:

Fig. 22: Input and Output signals in the KPFM system


[38]

The measured surface potential, VDC, is the sum of the convolution VCPD PSF with the system noise, :

VDC = VCPD PSF + (6.1) Where * denotes convolution. Once the PSF and the noise statistics are obtained,

reconstruction of the KPFM measurements is possible using deconvolution. One might

consider using the straight-forward inversion:

VCPD = PSF1(VDC ) (6.2) Because the PSF is a low pass filter (LPF), the noise is amplified greatly at high

frequencies. Therefore, Eq. (6.2) must be replaced with an optimized filter. We choose to

deconvolve the KPFM data using the Wiener filter, since it is straightforward, easy to

implement and in most cases gives good results. The Wiener filter is an optimal estimator

in term of minimum square error (MSE). Wiener deconvolution is performed in the

frequency domain by32:

VCPD(u, v) = PSF(u, v)|PSF(u, v)|2 + 1SNR(u, v)Wiener FilterVDC(u, v) (6.3)

Where (u, v) are the coordinates in the frequency domain, SNR(u, v) = Sf(u,v)Sn(u,v) is the

signal-to-noise ratio with Sf(u, v) = |VCPD(u, v)|2 and Sn(u, v) = |N(u, v)|2 as the power spectrums of VCPD and the noise, respectively. We estimate the power spectrum of VCPD by the power spectrum of VDC33. PSF(u, v) is the complex conjugate of PSF(u, v).


[39]

6.2 Reconstruction Process

6.2.1 PSF Expansion

As was shown in Section 4.2, summation over the AM-PSF components yields result

smaller than 1 due to the long-ranged electrostatic force. Therefore, we must extrapolate

the AM-PSF beyond the simulated surface.

Away from the origin, the two dimensional AM-PSF illustrated in Fig. 16(a) converges to

the expression:

|x| 50nm PSF(x, y) = A1 xP1 |y| 50nm PSF(x, y) = A2 yP2 , y > 0A3 yP3 , y < 0

(6.4)

Where A1 = A1(y) , P1 = P1(y) are positive numbers which vary for each y line and A2 = A2(x) , A3 = A3(x) , P2 = P2(x) , P3 = P3(x) are positive numbers which vary for each x line. 2 and 2 are related to the cantilever side whereas 3 and 3 are related to

the opposite side.

Considering the above expressions, we calculated all the parameters in Eq. (6.4) by

observing log(PSF) vs. log (x) and log(PSF) vs. log (y), as shown in Fig. 23(a). Once all the parameters are obtained, we extrapolated the PSF for all x and y lines, as

illustrated by the blue arrows in Fig. 23(b). Next, we observe that near the corners the PSF

converges to the expression:

|r| 70nm PSF(r) = Ac rPc , y > 0At rPt , y < 0 Where r = x2 + y2 s.t. |x| = |y|

(6.5)

, , , are positive numbers. r is the distance from the origin in a diagonal direction, as illustrated by the red squares in Fig. 23(b).


[40]

Fig. 23: (a)Loglog plot of the AM-PSF vs. x and y lines. A linear relation is observed far

from the origin. (b)Expanding the AM-PSF to an infinite area. First, the x and y lines are

extrapolated (illustrated by blue arrows). Next, extrapolation is performed on the

diagonal direction (illustrated by red arrows). In the final step, the remaining pixels on

the plane (black domains) are interpolated using 2 pixels which were previously

extrapolated. An example for this interpolation is mark with a green area.


[41]

By calculating the parameters in Eq. (6.5) as demonstrated earlier, we extrapolate the PSF

to the corners. The red arrows delineate the direction of this extrapolation. The PSF

components on the remaining areas of the plane (black domains) are interpolated by

including two pixels which were extrapolated in the preceding steps. An example of such

interpolation is illustrated by the green area in Fig. 23(b). The two pixels used for this

interpolation are marked with red and blue filled squares.

The above algorithm was tested on 2 AM-PSFs generated from a full probe

(tip+cantilever) and a probe composed from only a tip, in both cases the tip-sample

distance was d = 2[nm], the tilt of the cantilever was = 15, all additional geometric parameters are given in Fig. 14(a). By expanding the PSFs, the summation over the PSF

of the tip was increased from 0.98132 (for 192nm X 192nm area) to 1.00107 (for 37.5m X 37.5m area) and the summation over the PSF of the full probe was increased from 0.83189 (for 5m X 5m area) to 0.99824 (for 0.3mm X 0.3mm area).

Alternatively, deconvolution can be performed by subtracting the average CPD on the

sample, Vsub, which is usually the CPD value of the substrate outside the scan area. Then, reconstruction is performed using the following relation between VCPD Vsub and VDC Vsub (Mathematical formation can be found in the Appendix):

VDC VsubDeconvolution input = (VCPD Vsub) PSF

Deconvolution output PSF PSF (6.6)

Where PSF denotes summation over the PSF.


[42]

6.2.2 Calibration Sample Measurements

We validate our reconstruction algorithm by scanning a calibration sample assembled of Ni electrodes under predefined bias, and then deconvoluting by using an effective PSF and acquiring the system noise statistics.

Thin lines of were patterned on an isolating layer of 2 using e-beam lithography.

Next, we mount the sample onto an AFM (NTMDT, Eindhoven, Netherlands) for

measuring topography and CPD in ambient. The topography of the sample was measured

via contact mode, and is shown in Fig. 24(a). From a cross section of the topography

(white line) we observe a height difference of 50nm (Fig. 24 (b)). The CPD of the sample

was measured via AM-KPFM twice, once where the substrate and the Ni electrodes share common ground (Fig. 24 (c)) and second where the Ni contacts were biased (Fig. 24 (e)). The left and right Ni contacts were biased with a voltage of 0.5V and 0.5V, respectively. All KPFM measurements were conducted in lift-mode with 5nm lift height.

Conductive TiN coated tips (NTMDT) with a 1st resonance frequency of ~160kHz were used. Cross sections of the surface potentials with and without external bias are shown in

Fig. 24(f) and Fig. 24(d), respectively. One can observe that due to the averaging effect

of the probe, the measured CPD contrast in the biased sample is 0.58V instead of 1V. It should be noted that the actual CPD values are in opposite sign to our measured CPD

values since all the KPFM measurements were conducted by applying Kelvin voltage (VDC) on the probe34. We assume that the measured CPD is a convolution result of the actual CPD and a PSF despite the existence of the topography observed in Fig. 24(b). Our

assumption was validated by Baier et al35 and Sadewasser et al36 who measured AM-

KPFM on flat surfaces and on surfaces with topography step. They observed almost no

change in CPD distributions for both cases. Simulations on these surfaces also confirmed

very little topography influence on the averaging effect.


[43]

Fig. 24: (a)Contact mode topography of the sample: two Ni contacts on a layer of SiO2. (b)Height profile (as indicated in (a)). (c)CPD image of the sample. (d)CPD profile (as

indicated in (c)). (e)CPD image of the sample while the left Ni contact was biased with

0.5V and the right Ni contact was biased with -0.5V, related to the SiO2. (f)CPD profile (as indicated in (e)). From this profile one can observe an almost linear slope in the

surface potential. A small deviation upwards is visible due to the higher CPD value on

the SiO2 compared to the Ni. All KPFM measurements were conducted with AM using

Lift-Mode where the lift height was 5nm.


[44]

In order to evaluate the system noise, we mask features in Fig. 24(c) to analyze the CPD

statistics of the pure SiO2 substrate (Fig. 25(a)). The measured CPD on the substrate is shown in Fig. 25(b). The histogram in Fig. 25(c) shows the distribution of the SiO2 CPD which is a Gaussian distribution with an expected value and variance of 0.5132V and 3.23 105V2, respectively. The noise is obtained by subtracting the mean value. Fig. 25(d) presents the autocorrelation of the noise in the marked area of Fig. 25(a). The

delta-function-like feature in the center indicates that white noise can accurately describe

the system; therefore we have used an Additive White Gaussian Noise (AWGN) in the

deconvolution process.

Fig. 25: Evaluation of the statistics of the noise. (a)Sections containing only clean SiO2

were framed for distinguishing the noise from the CPD signal. (b)The CPD of the SiO2

substrate taken from (a). (c)Histogram outlines the distribution of the substrate CPD

(bars) and a normal fit (dashed-line) with mean and variance of 0.5132V and 3.23 105V2, respectively. (d) Autocorrelation of the noise extracted from the marked area in (a).


[45]

The theoretical peak of the autocorrelation function of the AWGN is given by32:

= 2 (6.7) Where and are the number of pixels in the x and y axes, respectively, and 2 is the

variance of the noise. By substituting the values of , and 2 we obtain a theoretical

= 0.344272, which is in excellent agreement with = 0.344322 obtained from Fig. 25(d).

Deconvolution of Fig. 24(e) requires the PSF of the probe in the exact tip-sample

distance that was used in the KPFM measurement, which is d = dtopo + dlift where dtopo is the averaged tip-sample distance used for recording the topography trajectory and dlift = 5nm is the lift height distance for the KPFM measurement. Since dtopo varies for every scan, more analytical approach is used to determine the exact tip-sample distance.

Since the averaging effect is not prominent when the Ni electrodes share common ground (Fig. 24(c)), we approximate the actual CPD values on the SiO2 and Ni to be 0.513V and 0.465V, respectively. The KPFM is suitable for measuring only semiconductor or metallic surfaces18; In practice, the measured CPD on the SiO2 is the CPD of the Si substrate beneath the oxide layer37.

Fig. 26(a) shows the band diagram of the unbiased sample where Ef(Ni),Ef(Si) and Ef(tip) indicate the Fermi levels of the Ni, Si and tip, respectively. LVL(tip) and LVL(sample) indicate the LVLs of the tip and sample, respectively. The band diagram of the sample under bias is presented in Fig. 26(b). When biased, the CPD of the left and

right Ni electrodes are 0.035[] and 0.965[], respectively. is defined as the CPD between the Ni and Si when they are unbiased. We measured = Ef(Si) Ef(Ni) = 50mV which is different from the theoretical value (= 220), the deviation might be related to absorption of molecules from the ambient on the sample

surface or due to the small averaging effect of the probe, which is not entirely negligible

in Fig. 24(c). The voltage drop along the is not linear24, however, for simplicity we

assume constant electric field between the two electrodes.


[46]

Fig. 26: (a)Band diagram of the unbiased sample. (b)Band diagram of the sample where

the left and right Ni electrodes are biased with 0.5V and -0.5V, respectively, related to

the tip. Ef(Si),Ef(Ni),Ef(tip) refers to the Fermi levels of the silicon, Nickel and tip,

respectively. LVL(tip) and LVL(sample) are the LVLs of the tip and sample,

respectively, and = 50meV. The CPD is marked in red. Under bias, the CPD increases from -0.035V (left electrode) to 0.965V (right electrode).


[47]

From the theoretical CPD shown in Fig. 26(b) we guess the actual CPD on the surface, VCPD (Fig. 27 (a)). Due to a voltage drift between the first (unbiased) and second (biased) scan, we shift all CPD values by 16.5[mV].

Fig. 27: (a)2D image of the theoretical CPD on the biased sample. (b)Measured CPD

(red), theoretical CPD (blue) and convolution of the theoretical CPD with PSFs generated

for tip-sample distance of 6nm (brown), 8nm (black) and 10nm(green). All CPD

linescans are plotted along the white line in (a).


[48]

The guessed VCPD is then convolved with PSFs at several tip-sample distances (6nm 10nm) by using Eq. (A.22) in the Appendix:

VDC = (VCPD Vsub) PSF + Vsub (6.8)

Fig. 27(b) shows the different convolution results with the theoretical and measured CPD

along the cross-section (white line) marked in Fig. 27(a).

The PSF generated for tip-sample distance of 8nm is our best estimate since the convolution result at 8nm bears the lowest error from VDC (by L2 norm). A good indication for our estimate is the contrast of the measured CPD (0.557) which is very similar to the summation over the PSF at 8nm (0.544).

Finally, after obtaining the PSF and the noise statistics, deconvolution is performed on VDC (Fig. 24(e)) using Eq. (6.6). Fig. 28 shows the profiles of the measured and deconvolved CPDs.

Fig. 28: Measured CPD (red), theoretical CPD (black) and deconvolved CPD (blue). All

CPD profiles are related to the linescan indicated in Fig. 24(e).

The deconvolved CPD resembles the measured CPD in shape and preserves a contrast of 1.03, as expected from the actual CPD on the surface.


[49]

6.3 Results and Comparison with Measurements

6.3.1 Reconstruction of CdS-PbS Images

CdS-PbS nanorods (NRs) were fabricated on freshly-peeled highly-oriented pyrolytic

Graphite (HOPG) and topography and KPFM images were simultaneously recorded

using single-pass technique in an argon glovebox. The topography was measured in

tapping mode with ~50 (1 resonance frequency) while an AC bias modulation of 300 400 (2 resonance frequency) was added to measure KPFM38. A schematic of the KPFM tip and sample characteristics is shown in Fig. 29:

Fig. 29: Schematic of KPFM experimental apparatus. The AFM tip used for this setup

has a tip apex of 30nm. The measured NRs are approximately 80nm in length and 4nm in

diameter.

In single pass technique, the Kelvin probe controller minimizes the oscillations of the

cantilevers second resonance, leaving it to oscillate only in the first mechanical

resonance. We define d as the average tip-sample distance and A as the amplitude of the cantilever oscillation, as illustrated in Fig. 30(a). Since the cantilever in inclined towards


[50]

the surface by degrees, the movement of the probe is better comprehended in the

rotated axis system , as shown in Fig. 30(b).

Fig. 30: (a)Representation of the probe in the y-z axes. is the angle between the

cantilever and the horizontal axis. Distance d is the average tip-sample distance.

(b)Representation of the probe in the y'-z' axes. A is the oscillation amplitude.

Neglecting the influence of the tip on the cantilever oscillation, the vertical deformation

along the y axis as a function of time is given by39:

Z(y, t) = A2 cos ByL coshByL + sinByL sinhByL cos 2tT0 (6.9) Where B = 1.875, = 0.7341, L is the cantilever length and T0 is the oscillation period. Thus, the deflection of the tip is:

Z(y = L, t) Acos 2tT0 Considering the tip position of the cantilever unaffected by the sample (free cantilever) in

Fig. 30(a) to be at (y, z) = (0, d), the tip position in the rotated system will be: (y, z) = (dsin(), dcos()) Therefore, when oscillating the tip position over time is:


[51]

(y(t), z(t)) = dsin() , dcos() Acos 2tT0 Using the rotation matrix, we describe the oscillations in the y z system: y(t)z(t) = cos () sin ()sin () cos () dsin()dcos() Acos 2tT0 =

Acos 2tT0 sin ()d Acos 2tT0 cos () (6.10)

The KPFM measurement was performed using tapping-mode, therefore we bound min{ z} = 0 for the tip bottom position. Meaning, A = dcos(). To take into account the

cantilever oscillations we sampled the position of the oscillating tip 25 times in one

oscillation period:

y(n)z(n) = dtan()cos 2nN d 1 cos 2nN

Where N = 25 and n increases from 1 to 25. In the measurement = 12 and d = 16.7nm were used. Since this first mechanical resonance frequency is around 50kHz, whereas the time constant of the Kelvin controller is 1ms, the effective PSF is calculated by minimizing the averaged electrostatic force on the probe rather than

minimizing the force at each tip-sample distance3.

Fig. 31(a) shows AM-KPFM measurement of CdS-PbS NRs on the HOPG substrate. The

CPD on the substrate is obtained by masking all the NRs in the images, similar to the

method used in Fig. 25(a) for the calibration sample. The measured CPD on the substrate

is presented in Fig. 31(b). The histogram in Fig. 31(c) shows the CPD distribution on the

substrate, which is a Gaussian distribution with an expected value and variance of

0.3862V and 5.4 105V2, respectively; the noise is obtained by subtracting the mean value. Fig. 31(d) presents the autocorrelation function of the noise in the marked

rectangle area in Fig. 31(a). The peak of the delta-function-like feature in the center

satisfies Eq. (6.7), therefore we deduce that the noise in the system can be considered as

an AWGN.


[52]

Fig. 31: (a)AM-KPFM measurement of CdS-PbS NRs on HOPG substrate. Two NRs (rod1 and rod2) are selected for further analysis. (b)The CPD of the HOPG substrate taken from (a). (c)Histogram presents the distribution of the substrate CPD (bars) and a normal fit (dashed-line) with a mean and variance of 0.3862V and 5.4 105V2, respectively. (d)Autocorrelation of the noise extracted from the marked area (blue) in (a).


[53]

Fig. 32(b) shows the actual CPD image, compared to the measured raw data (Fig. 32(a)).

Fig. 32: (a)Measured CPD on CdS-PbS NRs. (b)Actual CPD on CdS-PbS NRs obtained

by deconvolution with the Wiener filter.

The color scale of the Y-axis is indicative of the magnitude of CPD, and is thus increased

from Fig. 32(a) to Fig. 32(b)

Documents

Gilad Cohen Thesis