Natural logarithm of 2: Difference between revisions

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[[File:Bulk BTO PFM scan.png|thumb|alt=Piezoresponse Force Microscopy image of BaTiO3 domains|PFM of [[barium titanate|BaTiO<sub>3</sub>]] single crystal showing simultaneously acquired topography (top) and domain structure (bottom). The scale bar is 10 μm]]
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'''Piezoresponse force microscopy''' (PFM) is a variant of [[atomic force microscopy]] (AFM) that allows imaging and manipulation of [[ferroelectric]] 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.
 
==Basic principles==
 
===General overview===
Piezoresponse force microscopy is a technique which since its inception and first implementation<ref>{{cite journal|last1=Gruverman|first1=A.|doi=10.1116/1.587909|title=Domain structure and polarization reversal in ferroelectrics studied by atomic force microscopy|year=1995|pages=1095|volume=13|journal=Journal of Vacuum Science and Technology B|url=http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1034&context=physicsgruverman&sei-redir=1#search=%22Domain+structure+and+polarization+reversal+in+ferroelectrics+studied+by+atomic+force+microscopy%22|bibcode = 1995JVSTB..13.1095G }}</ref> has steadily attracted more and more interest. This is due in large part to the many benefits and few drawbacks that PFM offers researchers in varying fields from ferroelectrics, semiconductors and even biology.<ref>{{cite journal|last1=Rodriguez|first1=B.J.|last2=Kalinin|first2=S.V.|last3=Shin|first3=J.|last4=Jesse|first4=S.|last5=Grichko|first5=V.|last6=Thundat|first6=T.|last7=Baddorf|first7=A.P.|last8=Gruverman|first8=A.|doi=10.1016/j.jsb.2005.10.008|title=Electromechanical imaging of biomaterials by scanning probe microscopy|pmid=16403652|year=2006|pages=151|issue=2|volume=153|journal=Journal of Structural Biology|url=http://physics.unl.edu/~agruverman/pubs/AG%207%20%28JSB%29.pdf}}</ref> In its most common format PFM allows for identification of domains from relatively large scale e.g. 100×100&nbsp;µm<sup>2</sup> 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.<ref>{{cite journal|last1=Kalinin|first1=Sergei V|last2=Morozovska|first2=Anna N|last3=Chen|first3=Long Qing|last4=Rodriguez|first4=Brian J|title=Local polarization dynamics in ferroelectric materials|journal=Reports on Progress in Physics|volume=73|pages=056502|year=2010|doi=10.1088/0034-4885/73/5/056502|bibcode = 2010RPPh...73e6502K }}</ref> Many recent advances have expanded the list of applications for PFM and further increased this powerful technique. Indeed what started as a user modified AFM has now attracted the attention of the major SPM manufacturers so much so that in fact many now supply ‘ready-made’ systems specifically for PFM each with novel features for research. This is testament to the growth of the field and reflects the numbers of users throughout the scientific world who are at the forefront of scientific research.
 
[[File:Piezoresponse of parallel and antiparallel domains.png|thumb|550px|left|alt=Diagram showing cantilever movements from mechanical deformation of piezoelectric domains|Top line shows an in-phase piezoresponse to the driving voltage and the bottom line shows a 180° out-of-phase piezoresponse to driving voltage. Alignment of electric field and polarisation orientation (top right) results in an expansion of the domain, giving a positive deflection as measured by the photodiode. When the bias is negative the domain contracts giving a negative deflection as measured by the photodiode meaning that the piezoresponse will always be in-phase with the driving voltage. For anti-alignment of electric field and polarisation orientation (bottom right) a positive bias results in a contraction of the domain and so gives a negative deflection as measured by the photodiode therefore the piezoresponse is 180° out-of-phase with the driving voltage. In this way the orientation of polarisation within a domain can be observed.]]
 
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 ''d<sub>33</sub>'' piezoelectric tensor coefficient of BaTiO<sub>3</sub>, it has a value of 85.6 [[picometre|pm]][[Volt|V<sup>−1</sup>]] meaning that applying 1 V across the material results in a displacement of 85.6 pm or 0.0856 [[nanometre|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,
 
: <math>V(\boldsymbol\omega)=V_{\mathrm{ac}}\cos(\boldsymbol\omega t) \;</math>
 
of frequency ''ω'' and amplitude ''V<sub>ac</sub>'' is applied to the tip giving rise to an oscillatory deformation of the sample surface,
 
: <math>d=d_{\mathrm{0}}+D\cos(\boldsymbol\omega t + \boldsymbol\varphi) \;</math>
 
from the equilibrium position ''d<sub>0</sub>'' 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 [[Deformation (mechanics)|strain]] which in turn leads to a physical deformation of the material. This effect can be described through the constitutive equations.<ref>Rosen, C.Z., Hiremath, B.V., Newnham, R. (ed) "Piezoelectricity" American Institute of Physics, Key Papers in Physics, No 5, 227–283 (1992)</ref> The CPE can be written as
 
: <math>X_{\mathrm{i}}=d_{\mathrm{ki}}E_{\mathrm{k}} \;</math>
 
where ''X<sub>i</sub>'' is the strain tensor, ''d<sub>ki</sub>'' is the piezoelectric tensor, and ''E<sub>k</sub>'' is the electric field. If the piezoelectric tensor is considered to be that of the tetragonal crystal system (that of BaTiO<sub>3</sub>) then it is
 
:<math>
\begin{bmatrix} X_1 \\ X_2 \\ X_3 \\ X_4 \\ X_5 \\ X_6 \end{bmatrix}
=
 
\begin{bmatrix} 0 & 0 & d_{31} \\
0 & 0 & d_{31} \\
0 & 0 & d_{33} \\
0 & d_{15} & 0 \\
d_{15} & 0 & 0 \\
0 & 0 & 0 \end{bmatrix}
\begin{bmatrix} E_1 \\ E_2 \\ E_3 \end{bmatrix}
</math>
 
such that the equation will lead to the strain components for an applied field. If the field is applied exclusively in one direction i.e. ''E<sub>3</sub>'' for example, then the resulting strain components are: ''d<sub>31</sub>E<sub>3</sub>'', ''d<sub>31</sub>E<sub>3</sub>'', ''d<sub>33</sub>E<sub>3</sub>''
 
Thus for an electric field applied along the c-axis of BaTiO<sub>3</sub> i.e. ''E<sub>3</sub>'', then the resulting deformation of the crystal will be an elongation along the c-axis and an axially symmetric contraction along the other orthogonal directions. PFM uses the effect of this deformation to detect domains and also to determine their orientation.
 
===Conductive probe===
The most important property of the probe for use in PFM is that it should be conducting. This is generally required in order to provide a means of applying a bias to the sample, and can be achieved through manufacturing standard silicon probes and coating them in a conductive material. Common coatings are [[platinum]], [[gold]], [[tungsten]] and even [[diamond|conductive diamond]].
 
[[File:Conductive probe SEM images.png|thumb|center|700px|alt=Scanning electron microscope images of increasing magnification of a conductive coated scanning probe| [[Scanning electron microscopy]] images of a PtIr<sub>5</sub> coated scanning probe. From left to right shows images of increasing magnification where the scale bar in the first image is 50 μm and in the third is 200 nm. The first image shows the substrate, cantilever and the tip whereas the second image shows the tip geometry whilst the last image shows the tip apex and demonstrates the fine point that is achieved e.g. radius of curvature of less than 40 nm.]]
 
===Lock-in amplifier===
In the general case a LiA ‘compares’ an ''input signal'' against that of a ''reference signal'' (either generated internally or supplied by an external function generator) in order to separate the information contained in the input signal at the frequency of the reference signal. This is called ''[[demodulation]]'' and is done in a number of easy steps. The reference signal <math>\scriptstyle V_\mathrm{ref}=B\cos(\boldsymbol\omega t)</math>, and input signal, <math>\scriptstyle V_\mathrm{in}=A\cos(\boldsymbol\omega t + \boldsymbol\varphi)</math>, are multiplied together to give the ''demodulator output'',
 
<math>V_\mathrm{out}=\frac{1}{2}AB\cos(\boldsymbol\varphi)+ \frac{1}{2}AB\cos(2\boldsymbol\omega t +\boldsymbol\varphi)</math>
 
where ''A'' is the input signal Amplitude and ''B'' is the reference signal Amplitude, ''ω'' is the frequency of both the reference and input signals, and ''φ'' is any phase shift between the two signals.
 
The above equation has an AC component at twice the frequency of the original signals (second term) and a DC component (first term) whose value is related to both the amplitude and phase of the input signal. The demodulator output is sent through a low-pass filter to remove the 2''ω'' component and leave the DC component then the signal is integrated over a period of time defined as the ''Time Constant'', ''τ<sub>LiA</sub>'' which is a user-definable parameter. Several different outputs are commonly available from a LiA: ''X'' output is the demodulator output and ''Y'' is the second demodulator output which is shifted by 90° in reference to the first output, together they hold both the phase, ''θ'', and magnitude, ''R'', information and are given by
 
<math>X=\frac{1}{2}AB\cos(\boldsymbol\theta)</math> and
 
<math>Y=\frac{1}{2}AB\cos(\boldsymbol\theta +\frac{\pi}{2})=\frac{1}{2}AB\sin(\boldsymbol\theta) </math>
 
However, phase and amplitude of the input signal can also be calculated and outputted from the LiA if desired, so that the full amount of information is available. The phase output can be determined from the following equation:
 
<math>\boldsymbol\theta=\tan^{-1}\frac{Y}{X}</math>
 
The magnitude is then given by:
 
<math>R=\sqrt{X^2+Y^2}</math>
 
This allows ''R'' to be calculated even if the input signal differs in phase from the reference signal.
 
===Differentiating vertical and lateral PFM signals===
[[File:Cantilever movements and optical deflections.png|thumb|300px|alt=Diagram showing cantilever dynamics and the optical detection through AFM split photodiode detector|Diagrams showing the effect of cantilever movement with the photodetector represented by the square with quadrants labelled A, B, C and D. Torsional bending of the cantilever (left) leads to a change in lateral deflection and (right) vertical displacement of the cantilever leads to a change in vertical deflection]]
 
A basic interpretation of PFM (which is generally accepted) identifies that two modes of imaging are possible, one that is sensitive to out-of-plane and one to in-plane piezoresponse, termed, vertical and lateral PFM (VPFM and LPFM) respectively.<ref>{{cite journal|last1=Kalinin|first1=SV|last2=Rodriguez|first2=BJ|last3=Jesse|first3=S|last4=Shin|first4=J|last5=Baddorf|first5=AP|last6=Gupta|first6=P|last7=Jain|first7=H|last8=Williams|first8=DB|last9=Gruverman|first9=A|title=Vector piezoresponse force microscopy|journal=Microscopy and microanalysis : the official journal of Microscopy Society of America, Microbeam Analysis Society, Microscopical Society of Canada|volume=12|issue=3|pages=206–20|year=2006|pmid=17481357|doi=10.1017/S1431927606060156|bibcode = 2006MiMic..12..206K }}</ref> The separation of these components is possible through the use of a split photodiode detector, standard to all optical detection AFM systems. In this setup the detector is split into quadrants, nominally A, B, C and D. The centre of the entire detector outputs 0 V but as the laser spot moves a radial distance from this centre point the magnitude of the outputted voltage will increase linearly. A vertical deflection can be defined as {(A+B)-(C+D)}/(ABCD) so that now positive and negative voltages are ascribed to positive and negative cantilever vertical displacements. Similarly a lateral deflection is defined as {(B+D)-(A+C)}/(ABCD) to describe positive and negative torsional movements of the cantilever. So VPFM will utilise the vertical deflection signal from the photodiode detector so will only be sensitive to out-of-plane polar components and LPFM will utilise the lateral deflection signal from the photodiode and will only be sensitive to in-plane polar components.
 
For polar components orientated such that they are parallel to the electric field the resulting oscillating movement will be entirely in-phase with the modulated electric field but for an anti-parallel alignment the motion will be 180° out-of-phase. In this way it is possible to determine the orientation of the vertical components of polarisation from analysis of the phase information, ''φ'', contained in the input signal, readily available after demodulation in the LiA, when using the VPFM mode. In a similar sense the orientations of in-plane polar components can also be determined from the phase difference when using the LPFM mode. The amplitude of the piezoresponse of either VPFM or LPFM is also given by the LiA, in the form of the magnitude, ''R''.
 
==Examples of PFM imaging==
[[File:180 degree ferroelectric domains with profiles.png|thumb|left|alt=180° ferroelectric domains as imaged by PFM|500px|180° ferroelectric domains in [[potassium titanyl phosphate|KTP]] as imaged by PFM. Below are the associated line profiles across the domains]]
 
The image shows periodically poled 180° domains in [[potassium titanyl phosphate]] (KTP) as imaged by VPFM. In the image piezoresponse amplitude can be seen where dark areas represent the zero amplitude that is expected at domain boundaries where the unit cell is cubic i.e. centrosymmetric and so therefore not ferroelectric. On the left hand side piezoresponse phase can be seen where the measured phase changes to show the out-of-plane components that are pointing out of the screen, white areas, and into the screen, dark areas. The scan area is 20×10&nbsp;µm<sup>2</sup>. Below each scan is the relevant cross-section that shows in arbitrary units the PR amplitude and phase.
 
===PFM applied to biological materials===
 
PFM has been successfully applied to a range of biological materials such as teeth,<ref>{{cite journal|last1=Kalinin|first1=Sergei V.|last2=Rodriguez|first2=B. J.|last3=Jesse|first3=S.|last4=Thundat|first4=T.|last5=Gruverman|first5=A.|title=Electromechanical imaging of biological systems with sub-10 nm resolution|doi=10.1063/1.2006984|year=2005|pages=053901|volume=87|journal=Applied Physics Letters|url=http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1017&context=physicsgruverman&sei-redir=1#search=%22Electromechanical+Imaging+of+Biological+Systems+with+Sub-10+nm+Resolution%22|arxiv = cond-mat/0504232 |bibcode = 2005ApPhL..87e3901K }}</ref> bone, and single collagen fibrils.<ref>{{cite journal|last1=Minary-Jolandan|first1=Majid|last2=Yu|first2=Min-Feng|doi=10.1021/nn900472n|title=Uncovering Nanoscale Electromechanical Heterogeneity in the Subfibrillar Structure of Collagen Fibrils Responsible for the Piezoelectricity of Bone|year=2009|pages=1859|volume=3|journal=ACS Nano|url=https://netfiles.uiuc.edu/mfyu/www/Documents/Publications/ACSnano-1-2009.pdf}}</ref> It has been hypothesized that the endogenous piezoelectricity in these materials may be relevant in their mechanobiology. For example, using PFM it has been shown that a single collagen fibril as small as 100&nbsp;nm behaves predominantly as a shear piezoelectric materials with an effective piezoelectric constant of ~1 pm/V.
 
==Advanced PFM modes==
Several additions have been made to PFM that substantially increase the flexibility of the technique to probe nanoscale features.
 
===Stroboscopic PFM===
Stroboscopic PFM allows for time resolved imaging of switching in pseudo real-time.<ref>{{cite journal|last1=Gruverman|first1=A.|last2=Rodriguez|first2=B. J.|last3=Dehoff|first3=C.|last4=Waldrep|first4=J. D.|last5=Kingon|first5=A. I.|last6=Nemanich|first6=R. J.|last7=Cross|first7=J. S.|doi=10.1063/1.2010605|title=Direct studies of domain switching dynamics in thin film ferroelectric capacitors|year=2005|pages=082902|volume=87|journal=Applied Physics Letters|url=http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1015&context=physicsgruverman&sei-redir=1#search=%22Direct+studies+of+domain+switching+dynamics+in+thin+film+ferroelectric+capacitors%22|bibcode = 2005ApPhL..87h2902G }}</ref> A voltage pulse of amplitude much higher than the coercive voltage of the sample but shorter in duration than the characteristic switching time is applied to the sample and subsequently imaged. Further pulses with the same amplitude but longer in time are then applied with regular PFM imaging at the intervals. In this way a series of images showing the switching of the sample can be obtained. Typical pulses are of tens of [[nanoseconds]] in duration and are therefore capable of resolving the first nucleation sites of domain reversal and then observing how these sites evolve.
 
===Contact resonance PFM===
Remembering that in PFM an AC bias of a certain frequency causes a deformation of the sample material at that same frequency the system can be considered as a [[harmonic oscillator|driven harmonic oscillator]]. As such there exists a resonance as a function of driving frequency. This effect has been exploited in PFM to provide an enhancement in the PR signal, thus allowing for a higher [[signal-to-noise]] or similar signal-to-noise at lower driving bias amplitude.<ref>{{cite journal|last1=Harnagea|first1=C.|last2=Alexe|first2=M.|last3=Hesse|first3=D.|last4=Pignolet|first4=A.|doi=10.1063/1.1592307|title=Contact resonances in voltage-modulated force microscopy|year=2003|pages=338|volume=83|journal=Applied Physics Letters|url=http://www.mpi-halle.mpg.de/mpi/publi/pdf/4367_03.pdf|bibcode = 2003ApPhL..83..338H }}</ref> Typically this contact resonance is in the kilo- to mega-[[hertz]] range which is several times higher in frequency than the first free harmonic in air of the cantilever used. However a drawback is that the contact resonance is dependent not only on the dynamic response of the cantilever but also on the [[elastic modulus]] of the sample material immediately in contact with the probe tip and so therefore can change during scanning over different areas. This leads to a change in the measured PR amplitude and so is undesirable. One method of bypassing the inherent disadvantages of contact resonance PFM is to change the driving frequency in order to shadow or track the changes in the frequency of the contact resonance. This feature as developed by Asylum Research called Dual AC™ Resonance Tracking (DART) uses two limit frequencies on either side of the contact resonance peak and so can sense changes in the peak position. It is then possible to adapt the AC bias driving frequency correspondingly in order to maintain the signal boost that results from the contact resonance.
 
===Switching spectroscopy (SS) PFM===
In this technique the area underneath the PFM tip is switched with simultaneous acquisition of a [[hysteresis]] loop that can be analysed to obtain information about the sample properties.<ref>{{cite journal|last1=Rodriguez|first1=Brian J|last2=Jesse|first2=Stephen|last3=Baddorf|first3=Arthur P|last4=Zhao|first4=T|last5=Chu|first5=Y H|last6=Ramesh|first6=R|last7=Eliseev|first7=Eugene A|last8=Morozovska|first8=Anna N|last9=Kalinin|first9=Sergei V|title=Spatially resolved mapping of ferroelectric switching behavior in self-assembled multiferroic nanostructures: strain, size, and interface effects|journal=Nanotechnology|volume=18|pages=405701|year=2007|doi=10.1088/0957-4484/18/40/405701|bibcode = 2007Nanot..18N5701R }}</ref> A series of hysteresis loops are acquired across the sample surface in order to map the switching characteristics as a function of position. In this way an image representing switching properties such as coercive voltage, remnant polarisation, imprint and work of switching amongst others can be displayed in which each pixel displays the desired data from the hysteresis loop acquired at that point. This allows spatial analysis of switching properties to be compared with sample topography.
 
==Advantages and disadvantages==
 
===Advantages===
* High resolution on the nanometre scale
* Simultaneous acquisition of topography and piezoresponse
* Allows manipulation of ferroelectric domains
* Non-destructive
* Little sample preparation required
 
===Disadvantages===
* Scans can be slow, e.g. tens of minutes
* Tip wear changes surface interaction and can affect contrast
* Limited to lateral range of AFM i.e. approximately 100×100&nbsp;µm<sup>2</sup>
 
==References==
<!--- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes on how to create references using <ref></ref> tags which will then appear here automatically -->
{{Reflist|30em}}
 
==External links==
* [http://www.asylumresearch.com/Applications/PFMAppNote/PFMAppNote.shtml Asylum Research PFM Application Note]
[http://www.asylumresearch.com/Applications/ContactResonance/ContactResonance.shtml Contact Resonance Application Note]
* [http://www.ntmdt.com/spm-principles/view/piezoresponse-force-microscopy NT-MDT website with animation]
* [http://cp.literature.agilent.com/litweb/pdf/5989-7611EN.pdf Agilent Technologies PFM Application Note]
* [http://www.jpk.com/jpk-tech-piezoresponse-force-microscopy.download.c09f1c29f9ab908fa15897f36f87c510 JPK Instruments PFM Technical Report]
* [http://www.veeco.com Veeco website]
 
{{DEFAULTSORT:Piezoresponse Force Microscopy}}
[[Category:Scanning probe microscopy]]

Revision as of 06:44, 15 February 2014

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You don’t simply ship folks to struggle without providing missionspecific training. One of the frequent observationsin mock drills and precise urban fight is you’re getting into through doors while otherpeople try to return out the identical door. Following these collisions you knives for sale may end up on the ground wrestling withsomeone who desires to kill you. Firearmsare not all the time useful as your buddies can’t shoot them off you and your weaponis typically trapped between you and the unhealthy guy. You need a knife. I stole this from the web, but the actual CQB Device and sheath is manner cooler!

I do know I shouldn't need to get into this however choose a knife that matches the dimensions of your hand! I do know someone who carries around a type of obscenely long Spydercos! It's best to have the ability to wrap your fist around the blade without it poking out. Don't decide up one thing that does not suit your body, you will not carry it, and should wikipedia.orgthebestpocketknifereviews.com/best-combat-knife-fightingtactical-pocket-knives-reviews/">best tactical knives in the World</a> you do, you get into aggressive look points (examine the legislation part under). Knives are short for a purpose, their power is just not in the how hard you may swing the blade like a sword, it's how solidly you'll be able to manipulate it together with your arm and hand strength.

Administrator John S. Pistole believes that is a part of an over all threat-primarily based safety strategy, which allows TSA officers to better focus their efforts on finding increased risk items such as explosives. I'm glad that they are going to have extra time to verify I take my footwear off and throw away that bottle of water That I simply purchased for $three.00. I am additionally glad that I nonetheless can solely have 3oz. Of shampoo, conditioner & lotions in a plastic bag. God only is aware of what would occur if I had more in my keep on and what could occur if I mixed them together.