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Title Vector piezoresponse force microscopy
Authors(s) Kalinin, S. V.; Rodriguez, Brian J.; Jesse, S.; et al.
Publication date 2006-06
Publication information Microscopy and Microanalysis, 12 (3): 206-220
Publisher Cambridge University Press
Item record/more information http://hdl.handle.net/10197/5514
Publisher's version (DOI) 10.1017/S1431927606060156
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Vector Piezoresponse Force Microscopy
Sergei V. Kalinin,
1,
* Brian J. Rodriguez,
1,2
Stephen Jesse,
1
Junsoo Shin,
1,3
Arthur P. Baddorf,
1
Pradyumna Gupta,
4
Himanshu Jain,
4,5
David B. Williams,
4
and Alexei Gruverman
6
1
Condensed Matter Sciences Division, Oak Ridge National Laboratory, Bldg. 3025, MS 6030, 1 Bethel Valley Rd.,
Oak Ridge, TN 37831, USA
2
Department of Physics, North Carolina State University, 2700 Stinson Drive, Box 8202, Raleigh, NC 27695, USA
3
Department of Physics and Astronomy, University of Tennessee, 1408 Circle Drive, Knoxville, TN 37996, USA
4
Depart ment of Materials Science and Engineering, Lehigh University, 5 E ast Packer Avenue, Bethlehem, PA 18015, USA
5
Center for Optical Technologies, Lehigh University, 5 East Packer Ave., Bethlehem, PA 18015, USA
6
Department of Materials Science and Engineering, North Carolina State University, 2410 Campus Shore Drive,
Raleigh, NC 27695, USA
Abstract: A novel approach for nanoscale imaging and characterization of the orientation dependence of
electromechanical properties—vector piezoresponse forc e microscopy ~ Vector PFM!—is described. The relation-
ship between local electromechanical response, polarization, piezoelectric constants, and crystallographic
orientation is analyzed in detail. The image formation mechanism in vector PFM is discussed. Conditions for
complete three-dimensional ~3D! reconstruction of the electromechanical response vector and evaluation of the
piezoelectric constants from PFM data are set forth. The developed approach can be applied to crystallographic
orientation imaging in piezoelectric materials with a spatial resolution below 10 nm. Several approaches for
data representation in 2D-PFM and 3D-PFM are presented. The potential of vector PFM for molecular
orientation imaging in macroscopically disordered piezoelectric polymers and biolog ical systems is discussed.
Key words: scanning probe microscopy, piezoresponse force microscopy, piezoelectric materials, ferroelectric
materials, domains, orientation imaging
INTRODUCTION
In the last decade, piezoresponse force microscopy ~PFM!
has been established as a pr imary technique for imaging
and nondestructive characterization of piezoelectric and
ferroelectric materials on the nanometer scale ~Eng et al.,
2001; Alexe & Gruverman, 2004; Hong, 2004!. The use of
the electromechanical coupling at the tip–surface junction
for imaging polarized regions in ferroelectric polymers was
originally demonstrated by Güthner and Dransfeld ~1992!.
This approach was later extended by Gruverman et al.
~1996a! to nanoscale domain switching and imaging in
ferroelectric thin films and single crystals. The term piezo-
response, introduced by Gruverman et al. ~1996a! for the
description of this voltage-modulated contact mode of
scanning probe microscopy, comes from the fact that the
measured signal is dominated by the piezoelectric defor ma-
tion of the ferroelectr ic sample. In the original papers on
PFM, only the normal component of the tip displacement
related to the out-of-plane component of polarization vec-
tor has been measured, an approach further referred to as
vertical PFM ~VPFM!~Gruverman et al., 1998!. In 1998,
Eng et al. ~1998, 1999! proposed a lateral PFM ~LPFM!
imaging method for measuring the in-plane component of
polarization by monitoring the angular torsion of the
cantilever. Based on this development, an approach for
three-dimensional ~3D! reconstruction of polarization using
a combination of the VPFM data with two LPFM data sets
obtained at different scanning directions has been devel-
oped ~Eng et al., 1999; Rodriguez et al., 2004!.However,the
lateral and vertical PFM data in these studies were generally
obtained with different sensitivities and therefore could not
be compared directly.
Despite this limitation, the 2D and 3D PFM imaging
methods allow for the first time an insight into the domain
arrangement, domain st ructure reconstruction, and mecha-
nism of polarization reversal in microscopic ferroelectric
capacitors. However, there is no universally accepted ap-
proach for 3D PFM imaging and data analysis and even the
qualitative interpretation of 2D and 3D PFM data is subject
to a large number of misconceptions.
Here, we summarize the basic principles of PFM,
illustrate what infor mation can be obtained from PFM
experiments, and delineate the limitations of PFM signal
interpretation. In particular, we demonstrate that the quan-
titative measurement of the local electromechanical re-
Received January 7, 2005; accepted July 12, 2005.
*Corresponding author. E-mail: sergei2@ornl.gov
Microsc. Microanal. 12, 206–220, 2006
DOI: 10.1017/S1431927606060156
MicroscopyAND
Microanalysis
© MICROSCOPY SOCIETY OF AMERICA 2006
sponse vector ~as opposed to measurements of vertical and
lateral PFM maps!, further referred to as vector PFM, is a
unique tool for local crystallographic orientation imaging
in piezoelectric materials, such as micro- and nanocrystal-
line ferroelectric thin films, molecular orientation in ferro-
electric polymers, and biological systems. Several approaches
for PFM signal calibration using external and internal
standards are discussed.
PRINCIPLES OF PIEZORESPONSE
FORCE MICROSCOPY
Piezoresponse force microscopy is based on the detection
of the bias-induced piezoelectric surface deformation. The
tip is brought into contact with the surface, and the
piezoelectric response of the surface is detected as the first
harmonic component, A
1v
, of the tip deflection, A A
0
A
1v
cos~vt w!, induced by the application of the periodic
bias V
tip
V
dc
V
ac
cos~vt ! to the tip. Here, the deflection
amplitude, A
1v
, is determined by the tip motion and is
given in the units of length. When applied to the piezoelec-
tric or ferroelectric materials, the phase of the electrome-
chanical response of the surface, w, yields information on
the polarization direction below the tip. For so-called c
ferroelectric domains ~polarization vector oriented normal
to the surface and is pointing downward! the application of
a positive tip bias results in the expansion of the sample and
surface oscillations are in phase with the tip voltage, w 08,
whereas for opposite c
domains, w 1808. The piezore-
sponse amplitude, A A
1v
/V
ac
, given in the units of nm/V,
defines the local electromechanical activity of the surface.
The difficulty in the acquisition of PFM data stems from
nonnegligible electrostatic interactions between the tip and
the surface, as well as nonlocal interaction between the
cantilever and the surface ~Hong et al., 2001; Kalinin &
Bonnell, 2001, 2004; Huey et al., 2004!. In the general case,
the measured piezoresponse amplitude can be written as
A A
el
A
piezo
A
nl
,whereA
el
is the electrostatic
contribution, A
piezo
is the electromechanical contribution,
and A
nl
is the nonlocal contribution due to capacitive
cantilever–surface interaction ~Hong et al., 2001; Kalinin &
Bonnell, 2002; Harnagea et al., 2003; Huey et al., 2004!.
Quantitative PFM imaging requires A
piezo
to be maximized
to achieve predominantly electromechanical contrast. Pro-
vided that the phase signal varies by 1808 between domains
of opposite polarities, indicative of a small capacitive cross-
talk contribution to the sig nal, PFM images can be conve-
niently represented as A
1v
cos~w!/V
ac
,whereA
1v
is the
amplitude of first harmonic of measured response. Experi-
mentally, the collected signal is the output of the lock-in
amplifier, and we refer to the experimental signal as PR
aA
1v
cos~w!/V
ac
, given in the units of @V#,wherea is a
calibration constant determined by the lock-in settings and
sensitivity of the photodiode. The unique feature of the
scanning probe-based techniques is that, in addition to the
vertical displacement, torsion of the cantilever can be
measured as well, thus allowing measurement of both
VPFM and LPFM signals as illustrated in Figure 1. Note
that in general vertical and lateral sensitivities of PFM are
different, and several approaches for calibration have been
suggested ~Peter et al., 2005!. In general, separate recording
of amplitude, 6PR6, and phase, w, images is a more sensitive
mode of microscope operation.
For future discussion of the PFM image for mation
mechanism, we recall the basic principles of atomic force
microscopy ~AFM!~Sarid, 1991!. The vertical displacement
of the tip, w
3
t
, results in the bending of the cantilever by
angle u
d
; w
3
t
/L,whereL is the cantilever length ~Fig. 2a!.
The deflection of the laser beam reflected from the cantile-
ver is detected by the split photodiode. Under normal con-
tact mode imaging conditions, the AFM employs a feedback
loop to keep the cantilever deflection at a set-point value by
adjusting the height of the cantilever base while scanning.
This feedback signal thus provides a topographic image of
the surface, which is relatively insensitive to the exact details
of cantilever and tip motion. Howev e r, PFM imaging is based
on the direct measurement of the changes in cantilever an-
gle. For purely vertical motion of the tip ~Fig. 2a!, the mea-
sured photodiode signal can be directly related to the tip
displacement using a suitable calibration procedure, because
A w
3
t
. However, an additional contribution to the deflec-
tion signal will originate from the buckling oscillations of
the cantilever, as illustrated in Figure 2b. Although these
oscillations do not change the position of the tip, the change
in the deflection angle will be recorded as apparent height
contrast. Buckling of the cantilever is the primary source of
the nonlocal electrostatic signal contribution to the PFM
contrast and can be reduced by using sufficiently stiff canti-
levers ~Kalinin & Bonnell, 2004!. Similarly, the longitudinal
motion of the tip end, w
1
t
, will also result in the cantilever
bending u
d
; w
1
t
/H,whereH is the tip height. This nontriv-
ial detection mechanism must be taken into account in the
interpretation of the PFM data as discussed below.
ELECTROMECHANICAL MEASUREMENTS
BY
PFM
In the general case of a piezoelectric sample with arbitrary
crystallographic orientation, application of the bias to the
tip results in the surface displacement, w, with both normal
and in-plane components, w ~w
1
,w
2
,w
3
!. The usual
assumption in the interpretation of PFM data is that the
displacement of the tip apex in contact with the surface is
equal to the surface displacement, w
t
w. It has been
shown that this is generally true for the normal component
of the tip displacement, w
3
t
w
3
~Karapetian et al., 2005!,
because the effective spring constant of the tip–surface
junction is typically 2–3 orders of magnitude higher than
Vector Piezoresponse Force Microscopy 207
the cantilever spring constant. This approximation breaks
down in the cases when the spring constant of the tip–
surface junction becomes comparable to the spr ing con-
stant of the cantilever, when, for example ~a! PFM imaging
of soft materials, such as ferroelectric polymers or biological
systems, with large spring constant cantilevers or ~b! PFM
imaging at frequencies well above the first resonance fre-
quency of the cantilever when the dynamic stiffening effects
are important.
Alternatively, amplification of the effective PFM signal
is possible due to the resonance enhancement or the onset
of the regime when the tip loses contact with the surface,
when, for example ~c! PFM imaging at the cantilever reso-
nances or ~d! using high modulation amplitudes.
These phenomena can be adequately described using
dynamic models that take into account the frequency-
dependent oscillatory behavior of the cantilever. The linear
model ~tip oscillation magnitude is proportional, rather
then equal, to surface displacement! then can be still ap-
plied using a proper calibration approach, provided that
dynamic characteristics of the system ~e.g., spring constant
of the tip–surface junction! do not vary.
The origins of the in-plane component of electrome-
chanical response are much less understood. It is generally
agreed that the use of a conventional four-quadrant photo-
detector allows the lateral piezoresponse component in the
direction normal to the cantilever axis ~lateral transversal
displacement! to be determined as torque of the cantilever.
Thus, if the cantilever orientation is given by the vector n
~cos u
c
,sin u
c
,0!,whereu
c
is the angle between the long axis
of the cantilever and x-axis of the laboratory coordinate
system, the lateral PFM signal is proportional to the projec-
tion of the surface displacement on the vector perpendicu-
lar to the cantilever axis, PR
p
b~w
1
sin u
c
w
2
cos u
c
!
~Fig. 2c!. For a low-symmetry surface, the relationship be-
tween the piezoresponse and surface displacement can be
more complex and b can be a second rank tensor; however,
due to the difficulty in determining the relevant parameters,
such description is unlikely to be practical.
The fundamental difference between VPFM and LPFM
is that in the latter case the displacement of the tip apex can
be significantly smaller than that of the surface, for exam-
ple, due to the onset of sliding friction ~Bdikin et al., 2004!.
Therefore, whereas in VPFM the response amplitude is
expected to scale linearly with the modulation amplitude, in
LPFM the response amplitude will eventually saturate.
Another issue in the LPFM imaging , which, to our
knowledge, has not yet been reported,
1
is the presence of the
piezoresponse component along the cantilever axis ~longitu-
dinal displacement!, PR
l
c~w
1
cos u
c
w
2
sin u
c
!, as illus-
trated in Figure 2d. As discussed above, the vertical
displacement of the cantilever is measured through the deflec-
tion angle of the cantilever, u
d
; h/L,whereL is cantilever
length. Surface displacement along the cantilever axis will
also result in the change of deflection angle. If the surface
1
This effect is partially considered by Abplanalp ~2001!.
Figure 1. a: Vector nature of the electromechanical response. Schematics of vertical ~b! and lateral ~c! PFM detection.
208 S.V. Kalinin et al.
displacement and the tip displacement are equal, the deflec-
tion angle is u
d
; h/H,whereH is tip height. Given that
typically H
L, this implies that the “vertical” PFM signal is
more sensitive to the lateral longitudinal surface displace-
ment than the vertical surface displacement! This difference
in signal transduction mechanism is also observed for lateral
PFM signal ~Peter et al., 2005!; however, unlike longitudinal
and vertical modes, the coupling between normal and tor-
sional modes of rectangular cantilever oscillations is gener-
ally weak ~Jeon et al., 2004! and can be ignored.
On close inspection of existing experimental data on
model s ystems ~Abplanalp, 2001; Ganpule, 2001; Harnagea
2001; Kalinin 2002!, such as a and c domains on BaTiO
3
~100! surface, a nonzero VPFM signal can in some cases be
observed over a-domain regions with in-plane polarization.
However, this signal is much weaker than that in c-domain
regions. Although a complete description of this behavior is
unavailable, it can be argued that the longitudinal surface
displacement is not effectively t ransmitted to the cantilever
due to the onset of sliding. This contribution of the lateral
longitudinal surface displacement to the vertical signal can
be determined from comparison of VPFM images obtained
for different cantilever ori entations in the X–Y plane as
discussed below.
It is important to emphasize that the simple combina-
tion of VPFM and LPFM measurements is insufficient to
unambiguously determine the 3D piezoresponse vector for
an arbitrarily oriented sample. Indeed, in the general case,
all three component of w ~w
1
,w
2
,w
3
! are unknown. The
VPFM signal is PR
v
aw
3
c~w
1
cos u
c
w
2
sin u
c
!, whereas
the LPFM signal gives information on the linear combina-
tion of in-plane components, PR
l
b~w
1
sin u
c
w
2
cos u
c
!.
Figure 2. a: Vert ical tip displacement in beam-deflection AFM is achieved through the detection of cantilever deflection
angle. b: Distributed force results in cantilever buckling and change in the deflection angle, detected as apparent
tip-height change. c: Contributions to cantilever deflection. In-plane cantilever orientation in the arbitrary laboratory
coordinate system including lateral ~perpendicular to the cantilever axis! and longitudinal ~along the cantilever axis!
components. d: Longitudinal surface displacement along the long cantilever axis results in the change in deflection
angle, providing contribution to the VPFM signal.
Vector Piezoresponse Force Microscopy
209
