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Imaging mechanism of piezoresponse force microscopy of ferroelectric surfaces
Sergei V. Kalinin and Dawn A. Bonnell
Department of Materials Science and Engineering, University of Pennsylvania, 3231 Walnut Street, Philadelphia, Pennsylvania 19104
共Received 10 August 2001; published 11 March 2002兲
In order to determine the origin of image contrast in piezoresponse force microscopy 共PFM兲, analytical
descriptions of the complex interactions between a small tip and ferroelectric surface are derived for several
sets of limiting conditions. Image charge calculations are used to determine potential and field distributions at
the tip-surface junction between a spherical tip and an anisotropic dielectric half plane. Methods of Hertzian
mechanics are used to calculate the response amplitude in the electrostatic regime. In the electromechanical
regime, the limits of strong 共classical兲 and weak 共field-induced兲 indentation are established and the relative
contributions of electroelastic constants are determined. These results are used to construct ‘‘piezoresponse
contrast mechanism maps’’ that correlate the imaging conditions with the PFM contrast mechanisms. Condi-
tions for quantitative PFM imaging are set forth. Variable-temperature PFM imaging of domain structures in
BaTiO
3
and the temperature dependence of the piezoresponse are compared with Ginzburg-Devonshire theory.
An approach to the simultaneous acquisition of piezoresponse and surface potential images is proposed.
DOI: 10.1103/PhysRevB.65.125408 PACS number共s兲: 77.65.⫺j, 77.80.Bh, 77.80.Dj, 73.30.⫹y
I. INTRODUCTION
In recent years, scanning probe microscopy 共SPM兲 based
techniques have been successfully employed in the charac-
terization of ferroelectric surfaces on the micron and nanom-
eter levels.
1
The primary SPM techniques used are variants
of noncontact electrostatic SPM such as electrostatic force
microscopy 共EFM兲, scanning surface potential microscopy
共SSPM兲,
2,3
and contact techniques such as piezoresponse
force microscopy 共PFM兲.
4–7
Both SSPM and PFM are based
on voltage modulation: i.e., during imaging, the actuator
driving the cantilever is disengaged and an ac bias is applied
directly to a conductive tip. In PFM the tip is in contact with
the surface and the electromechanical response of the surface
is detected as the first-harmonic component of the bias-
induced tip deflection. In SSPM the tip is held at a fixed
distance above the surface 共typically 10–100 nm兲 and the
first harmonic of the electrostatic force between the tip and
surface is nullified by adjusting the constant bias on the tip.
An open loop version of SSPM, in which the feedback is
disengaged and the oscillation amplitude in the noncontact
regime is collected as the image, has also been reported.
8
In many cases, the morphological information on domain
structure and orientation obtained from SPM images is suf-
ficient, and numerous observations of local domain dynamics
as related to polarization switching processes,
9–11
ferroelec-
tric fatigue,
12–15
phase transitions,
16–19
mechanical stresses,
20
etc., have been made. However, analysis of local ferroelec-
tric properties including hysteresis measurements,
21
stress ef-
fects in thin films,
22
size dependence of ferroelectric
properties,
23,24
etc., requires quantitative interpretation of the
SPM interaction. A detailed analysis of EFM and SSPM im-
aging on ferroelectric surfaces is given by Kalinin and
Bonnell.
25
Contrast formation mechanism in PFM is less
understood.
26–30
Luo et al.
17
have found that the temperature
dependence of piezoresponse contrast is similar to that of
spontaneous polarization. This behavior was attributed to the
dominance of electrostatic interactions due to the presence of
a polarization bound charge,
31
since the electromechanical
response based on the piezoelectric coefficient d
33
would
diverge in the vicinity of the Curie temperature. The pres-
ence of the electrostatic forces hypothesis is also supported
by observations of nonpiezoelectric surfaces.
32
In contrast,
the existence of a lateral PFM signal
33–35
and the absence of
relaxation behavior in PFM contrast as opposed to SSPM
contrast,
36,25
as well as numerous observations using both
EFM-SSPM and PFM,
37,38
clearly point to a significant elec-
tromechanical contribution to PFM contrast. In order to re-
solve the controversy regarding the origins of PFM contrast,
we analyze the contrast formation mechanism and relative
magnitudes of electrostatic versus electromechanical contri-
butions to PFM interactions for the model case of c
⫹
, c
⫺
domains in tetragonal perovskite ferroelectrics. It is shown
that both electrostatic and electromechanical interactions can
contribute to the PFM image. The relative contributions of
these interactions depend on the experimental conditions.
Contrast mechanism maps were constructed to delineate the
regions with dominant electrostatic and electromechanical
interactions. Under some conditions, i.e., those correspond-
ing to a relatively large indentation force and tip radius, the
real piezoelectric coefficient can be determined. This analy-
sis reconciles existing discrepancies in the interpretation of
PFM imaging contrast.
II. PRINCIPLES OF PFM
Piezoresponse force microscopy is based on the detection
of bias-induced 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 of bias-
induced tip deflection d⫽ d
0
⫹ A cos(
t⫹
). The phase
yields information on the polarization direction below the
tip. For c
⫺
domains 共polarization vector pointing downward兲
the application of a positive tip bias results in the expansion
of the sample and surface oscillations are in phase with the
tip voltage,
⫽ 0. For c
⫹
domains,
⫽ 180°. The amplitude
A defines the local electromechanical response and depends
on the geometry of the tip-surface system and material prop-
erties. An additional contribution to PFM contrast originates
from long-range electrostatic tip-surface interactions.
39
This
PHYSICAL REVIEW B, VOLUME 65, 125408
0163-1829/2002/65共12兲/125408共11兲/$20.00 ©2002 The American Physical Society65 125408-1
electrostatic interaction is comprised of a local contribution
due to the tip apex and a nonlocal contribution due to the
cantilever.
40
Distinguishing electrostatic forces in a PFM ex-
periment is problematic; however, it can be achieved in
SSPM. In SSPM, application of an ac bias to the tip located
at 10–100 nm from the surface results in a strong capacitive
interaction. The cantilever deflection is then proportional to
the first harmonic of the force. The amplitude and relative
phase of cantilever oscillations in the noncontact mode can
be well approximated by simple harmonic-oscillator
models.
41
One of the difficulties in a comparison of the relative
magnitudes of electromechanical and electrostatic responses
is the difference in the contrast transfer mechanism. In the
electromechanical case the surface displacement is deter-
mined as a function of applied voltage. In the electrostatic
case the force containing both local and nonlocal compo-
nents is defined. Analysis of contrast formation in PFM
clearly requires reliable estimates of surface displacement
under tip bias for both cases. Given this, frequency-
dependent surface-tip contrast transfer could be constructed.
Analysis of the image formation mechanism requires the
solution of several independent problems. The electrostatic
tip-surface interaction and the magnitude of electrostatic
contrast are analyzed in Sec. III. The mechanism of electro-
mechanical contrast and weak- and strong-indentation limits
are formulated in Sec. IV. PFM contrast mechanism maps
and the temperature dependence of PFM contrast on a
BaTiO
3
surface are analyzed in Sec. V.
III. ELECTROSTATIC REGIME
In the electrostatic regime of piezoresponse imaging the
capacitive and Coulombic tip-surface interactions result in an
attractive force between the tip and surface which cause an
indentation. In some cases, these interactions have been ap-
proximated by a plane-plane capacitor. Obviously, this is in-
applicable in contact because a capacitive force in planar
geometry does not cause a tip deflection. A correct descrip-
tion of the electrostatic tip-surface interaction must take into
account the tip shape.
A. Potential distribution in the tip-surface junction
The potential distribution in the tip-surface junction in
noncontact imaging is often analyzed in the metallization
limit for the surface.
42
In this limit, the tip-surface capaci-
tance C
d
(z,
), where z is the tip-surface separation and
is
the dielectric constant for the sample, is approximated as
C
d
(z,
)⬇C
c
(z), where C
c
(z) is the tip-surface capacitance
for a conductive tip and conductive surface. This approxima-
tion breaks down for small tip-surface separations when the
effect of field penetration in the dielectric sample is non-
negligible. For ferroelectric surfaces the effective dielectric
constant is high,
⬇100–1000, favoring the metallization
limit. However, in contact tip-surface separation z⬇0 leads
to a divergence in the capacitance C
c
(z) and the correspond-
ing force. To avoid this difficulty and, more importantly, take
into account the anisotropy of the ferroelectric medium, we
calculate the tip-surface force using the image charge method
for spherical tip geometry. This approach is applicable when
the tip-surface separation is small, zⰆ R, where R is radius of
curvature of the tip.
The potential in air produced by charge Q at a distance d
above a conductive or dielectric plane located at z⫽ 0 can be
represented as a superposition of potentials produced by the
original charge and the corresponding image charge Q
⬘
lo-
cated at position z⫽ d
⬘
below the plane. The potential in a
dielectric material is equal to that produced by a different
image charge Q
⬙
located at z⫽ d
⬙
.
43,44
Values of Q
⬘
, Q
⬙
,
d
⬘
, and d
⬙
for metal and isotropic or anisotropic dielectric
materials are summarized in Table I. Note that the potential
in air above an anisotropic dielectric material is similar to the
isotropic case with an effective dielectric constant
eff
⫽
冑
x
z
, where
x
and
z
are the principal values of the
dielectric constant tensor. Potential and field distributions in-
side the dielectric material are more complex
45
and are out of
the scope of the present paper.
To address tip-surface interactions and taking the effect of
the dielectric medium into account, the image charge distri-
bution in the tip can be represented by charges Q
i
located at
distances r
i
from the center of the sphere such that
Q
i⫹1
⫽
⫺ 1
⫹ 1
R
2
共
R⫹ d
兲
⫺ r
i
Q
i
, 共1a兲
r
i⫹1
⫽
R
2
2
共
R⫹ d
兲
⫺ r
i
, 共1b兲
where R is tip radius, d is tip-surface separation, Q
0
⫽ 4
0
RV, r
0
⫽ 0, and V is the tip bias. The tip-surface
capacitance is
C
d
共
d,
兲
⫽
1
V
兺
i⫽0
⬁
Q
i
, 共2兲
from which the force can be found. The rotationally invariant
potential distribution in air can be found from Eqs. 共1a兲 and
共1b兲. Specifically, the potential on the surface directly below
the tip is
V
共
0,0
兲
⫽
1
4
0
2
⫹ 1
兺
i⫽0
⬁
Q
i
R⫹ d⫺ r
i
. 共3兲
In the conductive surface limit,
⫽ ⬁ and Eq. 共2兲 is sim-
plified to
46
TABLE I. Image charges for conductive, dielectric, and aniso-
tropic dielectric planes.
Conductive Isotropic dielectric Anisotropic dielectric
Q
⬘
⫺ Q
⫺
⫺1
k⫹1
Q
⫺
冑
z
x
⫺ 1
冑
z
x
⫹ 1
Q
d
⬘
⫺ d ⫺ d ⫺ d
Q
⬙
0
2
k⫹1
Q
2
冑
z
x
冑
z
x
⫹ 1
Q
d
⬙
d
d
冑
z
/
x
SERGEI V. KALININ AND DAWN A. BONNELL PHYSICAL REVIEW B 65 125408
125408-2
C
c
⫽ 4
0
R sinh

0
兺
n⫽1
共
sinh n

0
兲
⫺ 1
, 共4兲
where

0
⫽ arccosh
关
(R⫹ d)/R
兴
. For the conductive tip-
dielectric surface,
C
d
⫽ 4
0
R sinh

0
兺
n⫽1
冉
⫺ 1
⫹ 1
冊
n⫺1
共
sinh n

0
兲
⫺ 1
. 共5兲
While in the limit of small tip-surface separation C
c
diverges
logarithmically, C
d
converges to the universal ‘‘dielectric’’
limit
47
C
d
共
兲
z⫽0
⫽ 4
0
R
⫺ 1
⫹ 1
ln
冉
⫹ 1
2
冊
. 共6兲
The distance dependence of the tip-surface capacitance
and surface potential directly below the tip are shown in
Figs. 1共a兲 and 1共b兲. For relatively large tip-surface separa-
tions, C
d
(z,
)⬇C
c
(z), which is the usual assumption in
noncontact SPM imaging. The most prominent feature of this
solution is that, while for low-
dielectric materials the tip-
surface capacitance achieves the dielectric limit in contact
and hence surface potential is equal to the tip potential, this
is not the case for high-
materials. The tip-surface capaci-
tance, capacitive force, and electric field can be significantly
smaller than in the dielectric limit. The surface potential be-
low the tip is smaller than the tip potential and is inversely
proportional to dielectric constant 关Fig. 1共b兲兴. This is equiva-
lent to the presence of an apparent dielectric gap between the
tip and surface that attenuates the potential, which is often
the explanation for experimental observations.
B. Tip-surface interaction in the electrostatic regime
From Eqs. 共2兲, 共4兲, and 共5兲, the magnitudes of capacitive
and Coulombic forces between the cantilever-tip assembly
and the surface can be estimated. The capacitive force is
2F
cap
⫽ C
loc
⬘
共
V
tip
⫺ V
loc
兲
2
⫹ C
nl
⬘
共
V
tip
⫺ V
s
兲
2
, 共7兲
where V
tip
is the tip potential, V
loc
is the domain-related local
potential directly below the tip, V
s
is the surface potential
averaged over the distance comparable to the cantilever
length, C
loc
⬘
is the local part of tip-surface capacitance gradi-
ent, and C
nl
⬘
is the nonlocal part due to the cantilever. Typi-
cally, the cantilever length is significantly larger than the
characteristic size of ferroelectric domains; therefore, the
nonlocal part results in a constant background on the image
that does not preclude qualitative domain imaging. The non-
local capacitance gradient can be estimated using plane-
plane geometry as C
nl
⬘
⫽
0
S(z⫹ L)
⫺ 2
, where S is the effec-
tive cantilever area and L is the tip length. For a typical tip
with L⬇10
m and S⬇2⫻ 10
3
m
2
, the nonlocal contribu-
tion is C
nl
⬘
⬇1.8⫻ 10
⫺ 10
F/m and is independent of the tip
radius. The force for a tip-surface potential difference of 1 V
is F
nl
⬇0.9⫻ 10
⫺ 10
N. 共The nonlocal contribution rigorously
should also contain a term describing an effect of the conical
part of the tip.
25
兲 The local capacitive contribution due to
the tip apex is F
loc
⫽ 1.4⫻ 10
⫺ 8
N for z⫽ 0.1 nm, R
⫽ 50 nm, i.e., two orders of magnitude larger. However, C
loc
⬘
scales linearly with tip radius and, therefore, for the sharp
tips capable of high-resolution nonlocal contributions to the
signal increase. Similar behavior is found for noncontact
SPMs.
48
The Coulombic tip-surface interaction due to the
polarization charge can be estimated using the expression for
the electric field above a partially screened ferroelectric sur-
face, E
u
⫽ (1⫺
␣
)P
0
⫺ 1
(1⫹
冑
x
z
)
⫺ 1
, where
␣
is the de-
gree of screening and P is spontaneous polarization 共P
⫽ 0.26 C/m
2
for BaTiO
3
兲. For unscreened surfaces,
␣
⫽ 0, so
this Coulombic contribution in the limit F
Coul
Ⰶ F
cap
is
F
Coul
⫽ C
loc
(V
tip
⫺ V
loc
)E
u
and for the same tip parameters as
above F
Coul
⫽ 2.2⫻ 10
⫺ 9
N. However, polarization charge is
almost completely screened in air, typically 1⫺
␣
Ⰶ 10
⫺ 3
,
and under these conditions the Coulombic contribution can
be excluded from the electrostatic tip-surface interaction.
Capacitive force results in an indentation of the surface.
In the Hertzian approximation the relationship between the
indentation depth h, tip radius of curvature R, and load P is
49
h⫽
冉
3P
4E
*
冊
2/3
R
⫺ 1/3
, 共8兲
where E
*
is the effective Young’s modulus of the tip-surface
system defined as
1
E
*
⫽
1⫺
1
2
E
1
⫹
1⫺
2
2
E
2
. 共9兲
E
1
, E
2
and
1
,
2
are Young’s moduli and Poisson ratios of
tip and surface materials 共Fig. 2兲. For ferroelectric perovs-
kites Young’s modulus is of the order of E
*
⬇100 GPa. The
contact radius a is related to the indentation depth as a
⫽
冑
hR. Hertzian contact does not account for adhesion, and
capillary forces in a tip-surface junction and a number of
more complex models for nanoindentation processes are
known.
50
Under typical PFM operating conditions the total force
acting on the tip is F⫽ F
0
⫹ F
el
, where F
0
⫽ kd
0
is the elas-
tic force exerted by the cantilever of the spring constant k at
set point deflection d
0
and F
el
is the electrostatic force. Since
FIG. 1. 共a兲 Tip-dielectric surface capacitance for
⫽ 10 共dotted
line兲,
⫽ 100 共dashed line兲, and
⫽ 1000 共dot-dashed line兲, com-
pared to the metallic limit 共solid line兲. Vertical lines delineate the
region of characteristic tip-surface separations 共0.1–1 nm兲 in con-
tact mode for tip radius R⫽ 50 nm. 共b兲 Surface potential below the
tip for tip-surface separations z⫽ 0.1 R 共dot-dashed line兲, z⫽ 0.01 R
共dashed line兲, and z⫽ 0.001 R 共solid line兲 as a function of the di-
electric constant of the surface.
IMAGING MECHANISM OF PIEZORESPONSE FORCE... PHYSICAL REVIEW B 65 125408
125408-3
