Piezoresponse force microscopy (PFM) is a variant of atomic force microscopy (AFM) that allows imaging and manipulation of piezoelectric/ferroelectric materials domains. This is achieved by bringing a sharp conductive probe into contact with a ferroelectric surface (or piezoelectric material) and applying an alternating current (AC) bias to the probe tip in order to excite deformation of the sample through the converse piezoelectric effect (CPE). The resulting deflection of the probe cantilever is detected through standard split photodiode detector methods and then demodulated by use of a lock-in amplifier (LiA). In this way topography and ferroelectric domains can be imaged simultaneously with high resolution. PFM is applied to ferroelectric, semiconductor and biological materials, and has several advanced variants for time-resolved and spectroscopic measurements.
Basic principles
General overview Piezoresponse force microscopy is a technique which since its inception and first implementation by Güthner and Dransfeld has been applied across fields including ferroelectrics, semiconductors and biology. In its most common format PFM allows for identification of domains from relatively large scale e.g. 100×100 μm2 scans right down to the nanoscale with the added advantage of simultaneous imaging of sample surface topography. Also possible is the ability to switch regions of ferroelectric domains with the application of a sufficiently high bias to the probe which opens up the opportunity of investigating domain formation on nanometre length scales with nanosecond time resolution. Several major scanning probe microscope manufacturers now supply systems configured for PFM.
Consider that a static or DC voltage applied to a piezoelectric surface will produce a displacement but as applied fields are quite low and the piezoelectric tensor coefficients are relatively small then the physical displacement will also be small such that it is below the level of possible detection of the system. Take as an example, the d33 piezoelectric tensor coefficient of BaTiO3, it has a value of 85.6 pmV−1 meaning that applying 1 V across the material results in a displacement of 85.6 pm or 0.0856 nm, a minute cantilever displacement even for the high precision of AFM deflection detection. In order to separate this low level signal from random noise a lock-in technique is used wherein a modulated voltage reference signal,
V ( ω ) = V a c cos ( ω t ) {\displaystyle V({\boldsymbol {\omega }})=V_{\mathrm {ac} }\cos({\boldsymbol {\omega }}t)\;}
of frequency ω and amplitude Vac is applied to the tip giving rise to an oscillatory deformation of the sample surface,
d = d 0 + D cos ( ω t + φ ) {\displaystyle d=d_{\mathrm {0} }+D\cos({\boldsymbol {\omega }}t+{\boldsymbol {\varphi }})\;}
from the equilibrium position d0 with amplitude D, and an associated phase difference φ. The resulting movement of the cantilever is detected by the photodiode and so an oscillating surface displacement is converted into an oscillating voltage. A lock-in-amplifier (LiA) is then able to retrieve the amplitude and phase of the CPE induced surface deformation by the process outlined below.
Converse piezoelectric effect The converse piezoelectric effect (CPE) describes how an applied electric field will create a resultant strain which in turn leads to a physical deformation of the material. This effect can be described through the constitutive equations. The CPE can be written as
X i = d k i E k {\displaystyle X_{\mathrm {i} }=d_{\mathrm {ki} }E_{\mathrm {k} }\;}
where Xi is the strain tensor, dki is the piezoelectric tensor, and Ek is the electric field. If the piezoelectric tensor is considered to be that of the tetragonal crystal system (that of BaTiO3) then it is
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