1. Theory

Definition: A piezoresponse force microscope is a contact-mode AFM that applies an AC voltage through a conductive tip and detects the resulting surface vibration from the converse piezoelectric effect, $Δ z = d_{33} V$ (Magonov; Kalinin et al.).
Polarization mapping: The amplitude gives the local electromechanical response and the phase gives the polarization direction, so domains of opposite polarity show 180 degree contrast (Magonov; Kalinin et al.).
Mixed signal: The measured response includes electrostatic and electrochemical contributions besides the piezoelectric one (Kalinin et al.; Seol et al.).
Switching: A larger DC tip bias can locally reverse polarization, so PFM writes domains and measures hysteresis (Magonov; Kalinin et al.).

2. Working Principle

Excitation: A sub-volt to few-volt AC bias is applied between tip and bottom electrode, below or at the contact resonance (Magonov; Kalinin et al.).
Detection: Lock-in amplifiers read the vertical deflection and lateral torsion as amplitude and phase (Magonov).
Resonance tracking: DART and band excitation follow the contact resonance to boost signal-to-noise (Seol et al.).
Spectroscopy: Pulsed DC steps with small AC readout give local hysteresis loops (Seol et al.; Kalinin et al.).

3. Applications

Ferroelectric memory materials: Domain imaging and switching in PZT (Magonov; Seol et al.), and written domains for high-density data storage (Gruverman et al.).
Piezoelectric crystals and films: PPLN, LiNbO3, BiFeO3 and PZT films for transducers and optics (Magonov).
Domain engineering: Writing domain patterns with a biased tip (Magonov).
Electrochemical strain studies: Ionic motion in battery materials such as LiCoO2 (Seol et al.).

Piezoelectric Response

Name	Formula	Related Formulae	Variables/Constants
Converse piezoelectric displacement	$S_{j} = d_{ij} E_{i}, Δ z = d_{33} V$	$S_{3} = \frac{Δ z}{z _{0}}, E_{3} = \frac{V}{z _{0}}$	$S_{j} : s t r ain$ $d_{ij} : p i ez oe l ec t r i c coe f f i c i e n t in m / V$ $E_{i} : f i e l d$ $Δ z : t hi c k n ess c han g e$ $z_{0} : s am pl e t hi c k n ess; M a g o n o v : t h ese r e l a t i o n s, P Z T d 33 o f 100 t o 500 pm/V an d a 1 μ m P Z T f i l m e x am pl e w i t h 3.6 \times 1 0^{- 10} m / V g i v in g b e l o w 1 nm; K a l inin e t a l . : coe f f i c i e n t s f r o m ab o u t 1 pm/V (q u a r t z) t o 1000 pm/V$ $(co m p u t e d : 360 pm/V a t 1 V g i v es 0.36 nm; 1 pm/V g i v es 1 pm)$
AC excitation and lock-in response	$V = V_{d c} + V_{a c} cos ω t, Δ z = d_{33} V_{a c} cos (ω t + ϕ)$	$A = \frac{A _{1 ω}}{V _{a c}}, P R = \frac{a A _{1 ω} c o s ϕ}{V _{a c}}$	$A_{1 ω} : f i r s t - ha r m o ni c am pl i t u d e$ $A : p i ez or es p o n se am pl i t u d e in nm / V$ $ϕ : p ha se$ $P R : mi x e d p i ez or es p o n se s i g na l$ $a : l oc k - in an d p h o t o d i o d e c a l ib r a t i o n; M a g o n o v : d r i v e an d r es p o n se f or m, p ha se 0 f or p o l a r i z a t i o n d o w n an d π f or u p in t ha t se t - u p; K a l inin e t a l . : A an d P R d e f ini t i o n s$ $(co m p u t e d : an t i p a r a l l e l d o main s g i v e P R o f e q u a l ma g ni t u d e an d o pp os i t e s i g n)$
Signal decomposition	$A = A_{e l} + A_{p i ez o} + A_{n l}$	$(P R)_{ω, E S} = k^{- 1} C^{'} (V_{d c} + V_{C P D}) V_{a c}$	$A_{e l} : l oc a l e l ec t r os t a t i c t i p t er m$ $A_{p i ez o} : e l ec t r o m ec hani c a l t er m$ $A_{n l} : n o n - l oc a l c a p a c i t i v e c an t i l e v er t er m$ $k : c an t i l e v er s p r in g co n s t an t, C^{'} : c a p a c i t an ce g r a d i e n t$ $V_{C P D} : co n t a c t p o t e n t ia l d i f f er e n ce; K a l inin e t a l . : d eco m p os i t i o n; S eo l e t a l . : F_{E} S = (1/2) (d C / d z) V^{2} an d t h e f i r s t - ha r m o ni c t er m w i t h V_{C} P D a dd e d (s i g n co n v e n t i o n o f t h e so u r ce)$ $(a ss u m e d : C^{'} = 1 \times 1 0^{- 9} F/m, V_{C} P D = 0.5 V, V_{a} c = 1 V; co m p u t e d : 250 pm o n a 2 N/m l e v er an d 12.5 pm o n a 40 N/m l e v er)$

Detection and Contact

Name	Formula	Related Formulae	Variables/Constants
Contact resonance enhancement	$A_{r es} = d_{33} V_{a c} Q$	$f ≪ f_{C R} : A \approx d_{33} V_{a c}$	$A_{r es} : am pl i t u d e a t co n t a c t r eso nan ce$ $Q : co n t a c t - r eso nan ce q u a l i t y f a c t or$ $f_{C R} : co n t a c t r eso nan ce; M a g o n o v : A = d 33 V a c Q, s u b - r eso nan t o p er a t i o n f o l l o w s s am pl e d i s pl a ce m e n t d i r ec tl y, r eso nan t o p er a t i o n m or e se n s i t i v e b u t ha r d er t o q u an t i f y; K a l inin e t a l . : co n t a c t r eso nan ces 407.9 an d 1075.0 kHz an d a 634.1 kHz r o t a t i o na l m o d e, n o n - co n t a c t m o d es 54.5, 346.7 an d 980.5 kHz; S eo l e t a l . : D A R T an d ban d e x c i t a t i o n g i v e p i co m e t r e v er t i c a l r eso l u t i o n$ $(a ss u m e d : Q = 100; co m p u t e d : 1 pm/V a t 1 V b eco m es 100 pm)$
Contact mechanics	$F_{c} = F_{c} (h, V_{t i p})$	$k_{1} \approx 1000 N/m ≫ k \approx 1 to 50 N/m$	$F_{c} : co n t a c t f or ce$ $h : in d e n t a t i o n$ $k_{1} : t i p - s u r f a ce co n t a c t s t i f f n ess; K a l inin e t a l . : t h ese v a l u es, B a T i O 3 co n t a c t p a r am e t er s C 1 * = 403 GPa an d C 3 * = 15.4 N/ (V m) f or R = 50 nm an d P = 100 nN, c an t i l e v er s o f 4.5, 40 an d 0.15 N/m$ $(co m p u t e d : k_{1} / k i s 20 t o 1000, so t h e t i p f o l l o w s t h e s u r f a ce d i s pl a ce m e n t)$
Vertical and lateral PFM	$VPFM \propto d_{33}^{e f f}, LPFM \propto d_{15}^{e f f}$	$in-plane polarization \Rightarrow torsion of the cantilever$	$V P F M, L P F M : v er t i c a l an d l a t er a l s i g na l s; M a g o n o v : b o t h ana l y z e d w i t h l oc k - in am pl i f i er s a s am pl i t u d e an d p ha se; S eo l e t a l . : e l ec t r oc h e mi c a l s t r ain m u c h w e ak er l a t er a l l y t han v er t i c a l l y, p i ez or es p o n se s imi l a r in b o t h; G r uv er man e t a l . : c an t i l e v er m o t i o n r e a d a s v er t i c a l d i s pl a ce m e n t, t or s i o n or b e n d in g; s h e a r coe f f i c i e n t a ss i g nm e n t (m y r e a d in g)$ $(m y r e a d in g : a s t r o n g v er t i c a l b u t n e g l i g ib l e l a t er a l s i g na l o n a s u pp ose d l y i so t r o p i c f i l m hin t s a t n o n - p i ez oe l ec t r i c or i g in)$

Switching and Interpretation

Name	Formula	Related Formulae	Variables/Constants
Domain switching and hysteresis	$P R = α d_{33} (1 - \frac{η}{V _{d c}})$	$V_{n u c} \approx 0.1 to 10 V$	$α, η : o n e - d im e n s i o na l m o d e l p a r am e t er s$ $V_{n u c} : n u c l e a t i o n t h r es h o l d; K a l inin e t a l . : h y s t er es i s f or m an d t h r es h o l d o f or d er 0.1 t o 10 V; M a g o n o v : d o main s w r i tt e n w i t h 20 V an d er a se d w i t h - 20 V, P F M bia s r an g e - 50 t o + 50 V, s p ec t r osco p y b e l o w 1 V e x c i t a t i o n; S eo l e t a l . : p u l se d D C m o d e r eco mm e n d e d$ $(co m p u t e d : a 20 V w r i t e i s 20 t im es t h e 1 V e x c i t a t i o n l imi t)$
Tip field estimate	$E_{t i p} \approx \frac{V}{R}$	$E_{t i p} ≫ E_{c} \Rightarrow local switching$	$E_{t i p} : f i e l d u n d er t h e a p e x$ $R : t i p r a d i u s$ $E_{c} : coer c i v e f i e l d; or d er - o f - ma g ni t u d e es t ima t e (co m p u t e d); G r uv er man e t a l . : t i p a p e x r a d i u s t y p i c a l l y 20 t o 30 nm, bia se d t i p mani p u l a t in g p o l a r i z a t i o n; K a l inin e t a l . : R = 50 nm in co n t a c t c a l c u l a t i o n s$ $(co m p u t e d : 1 V o n a 50 nm a p e x g i v es ab o u t 200 kV/cm)$
Ferroelectric or not	$hysteresis loop \neq \Rightarrow ferroelectricity$	$checks: frequency, time, lateral response, V_{a c} dependence$	$S eo l e t a l . : b u tt er f l y l oo p s in n o n - f er r oe l ec t r i c S r T i O 3, T i O 2 an d L i C o O 2 f r o m o x y g e n - v a c an cy or l i t hi u m - i o n m o t i o n, e l ec t r oc h e mi c a l s t r ain r i s in g a t l o w er f r e q u e n cy w hi l e A l P O 4 an d P Z T d o n o t c han g e, g r o w in g w i t h l o n g er p u l ses, w e ak er l a t er a l l y, H f O 2 s h o w in g n o s i g ni f i c an t V_{a} c d e p e n d e n ce, co mbina t i o n w i t h P - E l oo p s or S H G$ $(m y r e a d in g : H f O 2 t hin f i l m s u se d in F e F E T an d F e R A M n ee d t h ese c h ec k s)$

Important Questions

Ques: How does PFM turn a voltage into a measurable displacement?

Through the converse piezoelectric effect. An AC voltage $V_{a c} cos ω t$ applied through the tip makes the surface move by $Δ z = d_{33} V_{a c} cos (ω t + ϕ)$ (Magonov). Kalinin et al. define the piezoresponse amplitude as $A = A_{1 ω} / V_{a c}$ in nm/V and the mixed signal as $P R = a A_{1 ω} cos ϕ / V_{a c}$ . Coefficients range from about 1 $pm/V$ in quartz to 1000 $pm/V$ in some ferroelectrics (Kalinin et al.), and Magonov gives 100 to 500 $pm/V$ for PZT. At 1 V, 360 $pm/V$ moves the surface 0.36 $nm$ and 1 $pm/V$ only 1 $pm$ (computed), so lock-in detection is essential. Since contact stiffness, about 1000 $N/m$ , greatly exceeds the cantilever stiffness, the tip tracks the surface (Kalinin et al.). Reference: S. Magonov, ‘Piezoresponse Force Microscopy in Its Applications’, NT-MDT Application Note 083 (fetch-tool summary); S. V. Kalinin, A. Rar and S. Jesse, ‘A Decade of Piezoresponse Force Microscopy: Progress, Challenges and Opportunities’, arXiv:cond-mat/0509009, invited review for IEEE Trans. Ultrason. Ferroelectr. Freq. Control (fetch-tool summary).

Ques: What does the PFM phase tell you?

The direction of polarization. In Magonov’s set-up a phase of 0 means polarization pointing down and $π$ means up, so antiparallel domains appear as 180 degree contrast with equal amplitude (Magonov; computed sign reversal of PR). The amplitude, a measure of the effective coefficient $d_{z z}$ (Gruverman et al.), should drop at domain walls, where opposing responses cancel within the tip footprint (my reading). The absolute phase convention depends on electronics, photodiode wiring and tip-sample geometry, so it is best calibrated on a reference such as PPLN, which Magonov notes is distributed for PFM training (my reading). Electrostatic contributions add a phase offset, so the contrast between domains may differ from exactly 180 degrees (Kalinin et al.; Seol et al.). Lateral PFM reveals in-plane components through cantilever torsion (Magonov). Reference: S. Magonov, ‘Piezoresponse Force Microscopy in Its Applications’, NT-MDT Application Note 083 (fetch-tool summary); A. Gruverman, M. Alexe and D. Meier, ‘Piezoresponse force microscopy and nanoferroic phenomena’, Nat. Commun. 10, 1661 (2019), Paperity copy (abstract, introduction and Box 1 only): cantilever motion as vertical displacement, torsion or bending, amplitude a measure of the effective coefficient d_zz, polarization direction from phase, biased tip manipulating polarization at the nanoscale, tip apex radius typically 20 to 30 nm, sub-10 nm domain resolution, stable written domains below 40 nm in BaTiO3, high-density data storage and domain-wall nanoelectronics (fetch-tool summary); S. V. Kalinin, A. Rar and S. Jesse, ‘A Decade of Piezoresponse Force Microscopy: Progress, Challenges and Opportunities’, arXiv:cond-mat/0509009, invited review for IEEE Trans. Ultrason. Ferroelectr. Freq. Control (fetch-tool summary); D. Seol, B. Kim and Y. Kim, ‘Non-piezoelectric effects in piezoresponse force microscopy’, arXiv:1701.01572 (fetch-tool summary).

Ques: Why do electrostatic forces contaminate PFM and how are they reduced?

The AC bias also drives the capacitor formed by tip, cantilever and sample. Seol et al. write $F_{E S} = \frac{1}{2} (d C / d z) V^{2}$ , giving a first-harmonic response $k^{- 1} C^{'} (V_{d c} + V_{C P D}) V_{a c}$ , and Kalinin et al. split the signal into local electrostatic, piezoelectric and non-local cantilever terms. With assumed C’ = $1 \times 1 0^{- 9}$ $F/m$ , 0.5 V contact potential and 1 V AC, the electrostatic deflection is 250 $pm$ on a 2 $N/m$ lever but 12.5 $pm$ on 40 $N/m$ (computed), compared with 1 $pm$ for quartz. Seol et al. therefore recommend stiff cantilevers, a top electrode, and pulsed DC for hysteresis; Magonov advises excitation below 1 V for spectroscopy. Since the electrostatic term scales with $V_{d c} + V_{C P D}$ , nulling it with a DC offset, as in KPFM, should also help (my reading). Reference: D. Seol, B. Kim and Y. Kim, ‘Non-piezoelectric effects in piezoresponse force microscopy’, arXiv:1701.01572 (fetch-tool summary); S. V. Kalinin, A. Rar and S. Jesse, ‘A Decade of Piezoresponse Force Microscopy: Progress, Challenges and Opportunities’, arXiv:cond-mat/0509009, invited review for IEEE Trans. Ultrason. Ferroelectr. Freq. Control (fetch-tool summary); S. Magonov, ‘Piezoresponse Force Microscopy in Its Applications’, NT-MDT Application Note 083 (fetch-tool summary).

Ques: Why measure at the contact resonance, and what does it cost?

Resonance multiplies the signal but complicates quantification. Magonov gives $A = d_{33} V_{a c} Q$ at contact resonance, whereas well below it the response follows the sample displacement directly. With an assumed Q of 100, a 1 $pm/V$ material at 1 V gives 100 $pm$ instead of 1 $pm$ (computed). Kalinin et al. report contact resonances at 407.9 and 1075.0 $kHz$ and a 634.1 $kHz$ rotational mode, against non-contact modes of 54.5, 346.7 and 980.5 $kHz$ . The resonance shifts with contact stiffness as the tip crosses regions of different elasticity, which can mimic piezoresponse contrast. Seol et al. note that DART and band excitation track this shift and give picometre vertical resolution. Reference: S. Magonov, ‘Piezoresponse Force Microscopy in Its Applications’, NT-MDT Application Note 083 (fetch-tool summary); S. V. Kalinin, A. Rar and S. Jesse, ‘A Decade of Piezoresponse Force Microscopy: Progress, Challenges and Opportunities’, arXiv:cond-mat/0509009, invited review for IEEE Trans. Ultrason. Ferroelectr. Freq. Control (fetch-tool summary); D. Seol, B. Kim and Y. Kim, ‘Non-piezoelectric effects in piezoresponse force microscopy’, arXiv:1701.01572 (fetch-tool summary).

Ques: Is the PFM response of PZT about 0.2 nm/V or up to 0.5 nm/V?

The values differ because they measure different things. Magonov quotes 100 to 500 $pm/V$ for PZT and uses $3.6 \times 1 0^{- 10}$ m/V, 360 $pm/V$ , for a 1 $μ m$ film, while Kalinin et al. measured 0.2 $nm/V$ on PZT against an average of about 0.3 $nm/V$ . PFM measures an effective coefficient $d_{z z}$ (Gruverman et al.), not the bulk value; substrate clamping and the inhomogeneous tip field plausibly reduce it on films, and electrostatic offsets shift it either way (my reading). The lower 0.2 to 0.3 $nm/V$ range is therefore more likely for a tip on a clamped film, and the 360 $pm/V$ value is better read as a material coefficient (my reading). Kalinin’s BaTiO3 data, falling from 0.1 to 0.04 $nm/V$ after first contact, show how contact conditions also matter. Reference: S. Magonov, ‘Piezoresponse Force Microscopy in Its Applications’, NT-MDT Application Note 083 (fetch-tool summary); S. V. Kalinin, A. Rar and S. Jesse, ‘A Decade of Piezoresponse Force Microscopy: Progress, Challenges and Opportunities’, arXiv:cond-mat/0509009, invited review for IEEE Trans. Ultrason. Ferroelectr. Freq. Control (fetch-tool summary); A. Gruverman, M. Alexe and D. Meier, ‘Piezoresponse force microscopy and nanoferroic phenomena’, Nat. Commun. 10, 1661 (2019), Paperity copy (abstract, introduction and Box 1 only): cantilever motion as vertical displacement, torsion or bending, amplitude a measure of the effective coefficient d_zz, polarization direction from phase, biased tip manipulating polarization at the nanoscale, tip apex radius typically 20 to 30 nm, sub-10 nm domain resolution, stable written domains below 40 nm in BaTiO3, high-density data storage and domain-wall nanoelectronics (fetch-tool summary).

Ques: How does PFM write and switch domains?

A DC tip bias above the local nucleation threshold reverses polarization under the apex. Kalinin et al. give a threshold of about 0.1 to 10 V for realistic tips and model hysteresis as $P R = α d_{33} (1 - η / V_{d c})$ . Magonov wrote a domain in PZT by scanning at 20 V and switched it back at -20 V, with a bias range of -50 to +50 V; 20 V is 20 times the 1 V read limit he advises for spectroscopy (computed). Concentrating 1 V on a 50 $nm$ apex gives roughly 200 $kV/cm$ (computed estimate), and Gruverman et al. report stable written domains below 40 $nm$ in BaTiO3 films, so switching is local. Seol et al. recommend pulsed rather than continuous DC for loops, reading the response between pulses to suppress electrostatic contributions. Reference: S. V. Kalinin, A. Rar and S. Jesse, ‘A Decade of Piezoresponse Force Microscopy: Progress, Challenges and Opportunities’, arXiv:cond-mat/0509009, invited review for IEEE Trans. Ultrason. Ferroelectr. Freq. Control (fetch-tool summary); S. Magonov, ‘Piezoresponse Force Microscopy in Its Applications’, NT-MDT Application Note 083 (fetch-tool summary); D. Seol, B. Kim and Y. Kim, ‘Non-piezoelectric effects in piezoresponse force microscopy’, arXiv:1701.01572 (fetch-tool summary); A. Gruverman, M. Alexe and D. Meier, ‘Piezoresponse force microscopy and nanoferroic phenomena’, Nat. Commun. 10, 1661 (2019), Paperity copy (abstract, introduction and Box 1 only): cantilever motion as vertical displacement, torsion or bending, amplitude a measure of the effective coefficient d_zz, polarization direction from phase, biased tip manipulating polarization at the nanoscale, tip apex radius typically 20 to 30 nm, sub-10 nm domain resolution, stable written domains below 40 nm in BaTiO3, high-density data storage and domain-wall nanoelectronics (fetch-tool summary).

Ques: Does a PFM hysteresis loop prove ferroelectricity?

No. Seol et al. state that a PFM hysteresis loop cannot be used as concrete evidence for ferroelectricity: butterfly-like loops appear in non-ferroelectric SrTiO3, TiO2 and LiCoO2, attributed to oxygen-vacancy or lithium-ion motion. They list checks: electrochemical strain grows at lower frequency and with longer pulses, while PZT and AlPO4 responses did not change with frequency; electrochemical strain is much weaker laterally; and ferroelectric loop shapes depend on AC amplitude relative to the coercive voltage, although their HfO2 example showed no significant dependence. Combining PFM with macroscopic P-E loops or optical second-harmonic generation is recommended. This matters for the doped HfO2 films used in FeFET and FeRAM, where thin films and vacancies are common (my reading). Reference: D. Seol, B. Kim and Y. Kim, ‘Non-piezoelectric effects in piezoresponse force microscopy’, arXiv:1701.01572 (fetch-tool summary).

References

S. Magonov, “Piezoresponse Force Microscopy in Its Applications,” NT-MDT Spectrum Instruments Application Note 083. Available: https://www.ntmdt-si.com/data/media/files/applications/application_notes/083_PFM_in_its_Applications__A4_en.pdf
S. V. Kalinin, A. Rar and S. Jesse, “A Decade of Piezoresponse Force Microscopy: Progress, Challenges and Opportunities,” invited review for IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, preprint arXiv:cond-mat/0509009. Available: https://arxiv.org/pdf/cond-mat/0509009
A. Gruverman, M. Alexe and D. Meier, “Piezoresponse force microscopy and nanoferroic phenomena,” Nature Communications, vol. 10, 1661, 2019 (abstract, introduction and Box 1 read). Available: https://paperity.org/p/192710883/piezoresponse-force-microscopy-and-nanoferroic-phenomena
D. Seol, B. Kim and Y. Kim, “Non-piezoelectric effects in piezoresponse force microscopy,” preprint arXiv:1701.01572. Available: https://arxiv.org/pdf/1701.01572

Quartz 5

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Piezoresponse Force Microscope (PFM)