1. Theory

Definition: Kelvin probe force microscopy measures the contact potential difference between an AFM tip and the sample by applying a DC bias that nulls the electrostatic force or force gradient at a modulation frequency (Bruker; Panchal et al.).
Work function: With a tip calibrated on a reference, the CPD gives the sample’s work function or surface potential (Panchal et al.; Bruker).
AM and FM detection: Amplitude modulation senses the force and averages over the cantilever; frequency modulation senses the force gradient and localizes the signal at the apex (Panchal et al.; Bruker).
Related methods: EFM measures the phase shift from electrostatic force gradients without nulling (Panchal et al.).

2. Working Principle

Modulation: An AC bias at frequency $ω$ creates a force at $ω$ proportional to $V_{D C} - V_{C P D}$ (Bruker).
Feedback: A second loop adjusts $V_{D C}$ to null that component; the null value maps the CPD (Bruker; Panchal et al.).
Lift or single pass: Two-pass lift mode separates topography and potential; single-pass FM and PeakForce modes measure both together (Bruker).
Calibration: Tips are calibrated on Au, HOPG or Au/Al references, often with UPS values (Panchal et al.; Bruker).

3. Applications

Semiconductors and devices: Kelvin-voltage contrast across p-n junctions versus doping, and surface band bending (Polak and Wijngaarden).
2-D materials: Graphene layer thickness from surface potential (Panchal et al.).
Solar cells: Photovoltage and work-function changes under illumination (Bruker).
Metals and alloys: Phase contrast in Sn-Pb and corrosion studies (Bruker).

Electrostatic Detection

Name	Formula	Related Formulae	Variables/Constants
Electrostatic force	$F_{e l} = - \frac{1}{2} \frac{\partial C}{\partial z} Δ V^{2}, Δ V = V_{D C} - V_{C P D} + V_{A C} sin ω t$	$F_{ω} = - \frac{\partial C}{\partial z} (V_{D C} - V_{C P D}) V_{A C} sin ω t, F_{2 ω} = \frac{1}{4} \frac{\partial C}{\partial z} V_{A C}^{2} cos 2 ω t$	$F_{e l} : t i p - s am pl e e l ec t r os t a t i c f or ce$ $C : t i p - s am pl e c a p a c i t an ce$ $z : se p a r a t i o n$ $V_{D C} : a ppl i e d D C bia s$ $V_{C P D} : co n t a c t p o t e n t ia l d i f f er e n ce$ $V_{A C}, ω : m o d u l a t i o n am pl i t u d e an d f r e q u e n cy, F_{ω}$ $F_{2 ω} : f i r s t an d seco n d ha r m o ni cs; B r u k er : f or ce an d ha r m o ni cs (r eco n s t r u c t e d), t h e ω am pl i t u d e n u l l e d w h e n V_{D C} = V_{C P D}; P an c ha l e t a l . : s am e f or ce f or m; ha r m o ni c e x p an s i o n (co m p u t e d)$ $(co m p u t e d : F_{ω} v ani s h es o n l y a t V_{D C} = V_{C P D}, w hi c h i s t h e n u l l in g co n d i t i o n)$
Contact potential and work function	$e V_{C P D} = Φ_{t i p} - Φ_{s am pl e} (one sign convention)$	$Φ_{p r o b e} \approx Φ_{A u} + e Δ V_{C P D}, Φ_{s am pl e} \approx Φ_{p r o b e} - e Δ V_{C P D}$	$Φ : w or k f u n c t i o n$ $Δ V_{C P D} : m e a s u r e d C P D a g ain s t t h e c a l ib r a t i o n r e f er e n ce; P an c ha l e t a l . : t h ese r e l a t i o n s, Φ_{A u} = 4.82 eV f r o m U P S, Φ_{p r o b e} = 4.09 eV, V_{C P D} = 0.20 V (1 L G) an d 0.35 V (2 L G), Φ_{1 L G} = 4.55 an d Φ_{2 L G} = 4.44 eV (see t h e Q / A), U P S c u t - o f f w i t h 21.22 eV p h o t o n s; B r u k er : a cc u r a t e s am pl e w or k f u n c t i o n o n l y a f t er t i p c a l ib r a t i o n, e x t r e m es O s 5.93 an d C s 2.14 eV; P o l ak an d W ij n g aa r d e n : V_{C P D} = (W_{s} - W_{p}) / e w i t h t h e f ee d ba c k v o l t a g e a ppl i e d t o t h e s am pl e; o t h er in s t r u m e n t s ma y d e f in e t h e o pp os i t e s i g n (m y r e a d in g)$ $(co m p u t e d : O s - C s = 3.79 V, t h e ma x im u m p oss ib l e C P D)$
Frequency-modulation KPFM	$f_{0} \pm f_{m o d} \approx f_{0} (1 - \frac{1}{2 k} \frac{d F}{d z})$	$Δ φ = \frac{Q}{k} \frac{d F}{d z} = \frac{Q}{2 k} \frac{d ^{2} C}{d z ^{2}} V^{2} (EFM)$	$f_{m o d} : m o d u l a t i o n f r e q u e n cy$ $k : s p r in g co n s t an t$ $Q : q u a l i t y f a c t or$ $Δ φ : E F M p ha se; P an c ha l e t a l . : t h ese f or m s, f_{0} o f 70 t o 350 kHz, f_{m o d} ab o u t 2 kHz, k ab o u t 0.8 N/m (P F QN E - A L), t i p r a d i u s ab o u t 5 nm; B r u k er : F M - K P F M n u l l in g s i d e ban d s a t ω \pm ω_{m}, m o d u l a t i o n in d e x 0.1 a t 2 kHz g i v in g 200 Hz, se n s i t i v i t y sc a l in g w i t h Q / k (p r in t e d r e l a t i o n g a r b l e d), M E S P - R C d r i v e n a t 133 kHz$ $(co m p u t e d : a m o d u l a t i o n in d e x o f 0.1 a t 2 kHz i s 200 Hz o f f r e q u e n cy d e v ia t i o n)$

Resolution and Accuracy

Name	Formula	Related Formulae	Variables/Constants
Lateral resolution	$d_{A M} \approx 50 to 70 nm or worse, d_{F M} < 20 nm$	$d_{F M} (50 nm lift) \approx 200 nm$	$d : l a t er a l r eso l u t i o n; P an c ha l e t a l . : A M - K P F M 50 t o 70 nm, F M - K P F M b e l o w 20 nm; B r u k er : A M s i g na l d o mina t e d b y t h e c an t i l e v er an d co n e w i t h mi cr o m e t r e - sc a l e a v er a g in g, F M ha l f - s i g na l f r o m w i t hin 15 nm o f t h e a p e x a t 5 nm l i f t (12 nm d iam e t er, ab o u t 10 nm r eso l u t i o n p oss ib l e), ab o u t 200 nm a t 50 nm l i f t, P e ak F or ce K P F M a cc u r a t e o n f e a t u r es o f 10 nm or m or e, l i f t a r t e f a c t b o u n d a r y b e tw ee n 25 an d 27 nm o n 5 nm b r u s h p o l y m er s, S C M - P I T p r o b e 225 b y 30 μ m w i t h 10 μ m t i p an d 22.5 d e g r ee ha l f - co n e$ $(co m p u t e d : r eso l u t i o n w or se n s ab o u t 20 - f o l d f r o m 5 t o 50 nm l i f t, f r o m ab o u t 10 t o 200 nm)$
Measured-potential accuracy	$ϵ = \frac{V _{m e a s} - V _{t r u e}}{V _{t r u e}}$	$ϵ_{A M} = - 27.6 per cent, ϵ_{F M} = + 4.4 per cent, ϵ_{E F S} = + 7.8 per cent$	$ϵ : r e l a t i v e er r or in a k n o w n a ppl i e d p o t e n t ia l s t e p; P an c ha l e t a l . : t h ese v a l u es o n a bia se d g r a p h e n e c hann e l (A M w i t h V_{A C} = 2 V m e a s u r in g ab o u t 2.9 o f an e x p ec t e d 4 V, F M w i t h V_{A C} = 8 V m e a s u r in g 4.18), g r a p h e n e 1 L G - 2 L G d i f f er e n ces o f 50 mV (A M), 150 mV (F M) an d 110 mV (E F S) a g ain s t l i t er a t u r e 25 t o 135 meV (see t h e Q / A); B r u k er : S n - P b co n t r a s t 240 mV (P e ak F or ce), 235 mV (F M) an d 97 mV (A M)$ $(co m p u t e d : A M r e a d s 59 p er ce n t b e l o w F M o n S n - P b an d 67 p er ce n t b e l o w F M o n g r a p h e n e l a y er s)$
Repeatability on a reference	$Δ V_{A u - A l} = 0.825 \pm 0.019 V$	$Φ_{A u} - Φ_{A l} \approx 5.1 - 4.26 = 0.84 eV$	$B r u k er : nin e P F QN E - A L p r o b es o n 50 nm A u an d A l f i l m s o n n - S i g i v in g A u 0.639 pl u s or min u s 0.018 V, A l - 0.185 pl u s or min u s 0.020 V an d A u - A l 0.825 pl u s or min u s 0.019 V, p r o b e - t o - p r o b e d e v ia t i o n b e l o w 50 mV (t e x t) or 20 mV (l a t er s t a t e m e n t); L ib r e T e x t s t ab l e : A u 5.1 t o 5.47 eV, A l 4.06 t o 4.26 eV; T r an e t a l . : f a ce t d e p e n d e n ce an d co n t amina t i o n er r or s u p t o 0.5 eV$ $(co m p u t e d : t h e 0.825 V d i f f er e n ce i s 2 p er ce n t b e l o w t h e s ma l l es t t ab u l a t e d d i f f er e n ce, 0.84 eV; p r o b e sc a tt er i s 2.8 p er ce n t o f t h e A u v a l u e)$

Applications and Environment

Name	Formula	Related Formulae	Variables/Constants
Illumination and charge	$Δ Φ_{l i g h t} = - 535 mV (300 suns, MDMO-PPV:PCBM)$	$V_{H V} = \pm 200 V (error < 15 per cent), up to \pm 1000 V$	$Δ Φ_{l i g h t} : w or k - f u n c t i o n s hi f t u n d er i l l u mina t i o n$ $V_{H V} : hi g h - v o l t a g e K P F M r an g e; B r u k er : 535 mV d o w n - s hi f t a t 300 s u n s o n an or g ani c so l a r ce l l, s in g l e c a r b o n nan o t u b e 30 mV an d b u n d l e 105 mV ab o v e t h e s u b s t r a t e, K P F M - H V t o 200 V w i t h u n d er 15 p er ce n t er r or an d u p t o 1000 V l imi t e d b y p h o t o d e t ec t or n o n l in e a r i t y, bia s f r e q u e n cy b e l o w ha l f t h e r eso nan ce, P D M S c ha r g e f r o m - 99 t o 156 V (158 in a c a pt i o n), g l o v e - b o x o p er a t i o n a t ab o u t 1 pp m o x y g e n an d w a t er$ $(co m p u t e d : a 30 mV nan o t u b e s i g na l i s 0.8 p er ce n t o f t h e ma x im u m 3.79 V C P D r an g e)$
Humidity and environment	$Δ V_{C P D} increases with humidity (water gating)$	$vacuum or inert gas for absolute values$	$P an c ha l e t a l . : h u mi d i t y g a t in g s h o w n t o in cr e a se Δ V v a l u es, m e a s u r e m e n t s a t ab o u t 35 p er ce n t h u mi d i t y an d 18^{\circ} C, l i t er a t u r e 1 L G - 2 L G v a l u es o f 135 meV in v a c uu m an d 25 meV in ai r; B r u k er : g l o v e - b o x m e a s u r e m e n t s; P o l ak an d W ij n g aa r d e n : s u r f a ce d i p o l es f r o m a d sor b e d m o l ec u l a r l a y er s an d f i x e d c ha r g e f r o m i o n s d e p os i t e d f r o m t h e e n v i r o nm e n t$ $(co m p u t e d : t h e v a c uu m v a l u e i s 5.4 t im es t h e ai r v a l u e in t h e c i t e d l i t er a t u r e)$

Important Questions

Ques: How does KPFM measure the contact potential difference?

By nulling an electrostatic force. Tip and sample form a capacitor; with a bias $V_{D C} + V_{A C} sin ω t$ , the force $- \frac{1}{2} (\partial C / \partial z) Δ V^{2}$ contains a component at $ω$ proportional to $(V_{D C} - V_{C P D}) V_{A C}$ and one at $2 ω$ proportional to $V_{A C}^{2}$ (Bruker; harmonics computed). A lock-in detects the $ω$ component and a feedback loop adjusts $V_{D C}$ until it vanishes, at which point $V_{D C} = V_{C P D}$ (Bruker). Since the CPD is the work-function difference divided by e, the map shows local work function or surface potential. The largest possible CPD is the 3.79 $V$ between osmium and caesium (Bruker; computed), so semiconductor signals of tens to hundreds of millivolts require millivolt precision. Reference: Bruker, ‘PeakForce Kelvin Probe Force Microscopy’, application note AN140 (several equations garbled) (fetch-tool summary); L. Polak and R. J. Wijngaarden, ‘Two competing interpretations of Kelvin probe force microscopy on semiconductors put to test’, arXiv:1605.05539: V_CPD = (W_s - W_p)/e with feedback voltage positive when applied to the sample, surface band bending from surface states or fixed charge, Kelvin-voltage difference across Si p-n junctions versus doping, surface dipoles from termination or adsorbed molecular layers, ions from the environment depositing fixed surface charge, measurements in air (fetch-tool summary).

Ques: Why is FM-KPFM more accurate and sharper than AM-KPFM?

Because it senses the force gradient, which is concentrated at the tip apex. AM-KPFM responds to the force on the whole cantilever and cone, so it averages over a wide area: Panchal et al. give 50 to 70 $nm$ resolution for AM against below 20 $nm$ for FM, and Bruker estimates that at 5 $nm$ lift half the FM signal comes from within 15 $nm$ of the apex, about 10 $nm$ resolution. Averaging also reduces contrast: on a known 4 $V$ step, AM read 27.6 per cent low while FM read 4.4 per cent high (Panchal et al.), and on Sn-Pb, AM gave 97 $mV$ against 235 $mV$ for FM (Bruker), 59 per cent lower (computed). Resolution degrades with lift, to about 200 $nm$ at 50 $nm$ (Bruker). Reference: V. Panchal, R. Pearce, R. Yakimova, A. Tzalenchuk and O. Kazakova, ‘Standardization of surface potential measurements of graphene domains’, arXiv:1305.0465 (NPL) (fetch-tool summary); Bruker, ‘PeakForce Kelvin Probe Force Microscopy’, application note AN140 (several equations garbled) (fetch-tool summary).

Ques: What is the potential difference between single- and bilayer graphene?

Panchal et al. measured 50 $mV$ with AM-KPFM, 150 $mV$ with FM-KPFM and 110 $mV$ with electrostatic force spectroscopy, and cite literature values of 135 $meV$ in vacuum, 25 $meV$ in air, 120 and 68 $meV$ . The AM value is low because of spatial averaging: it is 67 per cent below the FM value (computed), consistent with AM’s 27.6 per cent underestimate of a known step. Humidity also matters, since water gating changes the values and the cited vacuum figure is 5.4 times the air figure (computed). The FM value of about 150 $mV$ , or the vacuum 135 $meV$ , is the more likely intrinsic difference, while air and AM values understate it (my reading). Reports should therefore state mode and environment. Reference: V. Panchal, R. Pearce, R. Yakimova, A. Tzalenchuk and O. Kazakova, ‘Standardization of surface potential measurements of graphene domains’, arXiv:1305.0465 (NPL) (fetch-tool summary).

Ques: How is the tip calibrated, and are Panchal et al.’s graphene work functions consistent?

Calibration measures the CPD on a reference with known work function. Panchal et al. took gold at 4.82 $eV$ from UPS and derived a probe work function of 4.09 $eV$ , then reported 4.55 and 4.44 $eV$ for single and bilayer graphene from CPDs of 0.20 and 0.35 $V$ . These numbers do not all fit one relation: adding the CPDs to 4.09 gives 4.29 and 4.44 $eV$ , matching bilayer but not single layer, while subtracting gives 3.89 and 3.74 (computed). The CPD difference, 0.15 $V$ , matches their FM layer contrast, so the relative values are more likely reliable, and the single-layer absolute value should be treated with caution until checked against the original paper (my reading). Polak and Wijngaarden define $V_{C P D} = (W_{s} - W_{p}) / e$ ; instruments that bias the tip instead reverse the sign (my reading). Reference: V. Panchal, R. Pearce, R. Yakimova, A. Tzalenchuk and O. Kazakova, ‘Standardization of surface potential measurements of graphene domains’, arXiv:1305.0465 (NPL) (fetch-tool summary); L. Polak and R. J. Wijngaarden, ‘Two competing interpretations of Kelvin probe force microscopy on semiconductors put to test’, arXiv:1605.05539: V_CPD = (W_s - W_p)/e with feedback voltage positive when applied to the sample, surface band bending from surface states or fixed charge, Kelvin-voltage difference across Si p-n junctions versus doping, surface dipoles from termination or adsorbed molecular layers, ions from the environment depositing fixed surface charge, measurements in air (fetch-tool summary).

Ques: Does Bruker’s 240 mV Sn-Pb contrast match their work functions?

Not exactly. Bruker reports 240 $mV$ with PeakForce KPFM and 235 $mV$ with FM-KPFM on a Sn-Pb alloy, describing these as consistent with work functions of 4.42 $eV$ for Sn and 4.25 $eV$ for Pb, which differ by only 170 $mV$ ; the measured contrast is 41 per cent larger (computed). The tabulated values, Sn 4.42 and Pb 4.25 $eV$ (LibreTexts), are for clean metals; work functions vary with crystal facet and contamination can shift measured values by up to 0.5 $eV$ (Tran et al.), and both metals oxidize in air (my reading). The agreement between PeakForce and FM, 240 and 235 $mV$ , suggests the measurement is reliable and the reference values are the less likely to apply to this surface (my reading). Absolute work functions need clean, calibrated surfaces. Reference: Bruker, ‘PeakForce Kelvin Probe Force Microscopy’, application note AN140 (several equations garbled) (fetch-tool summary); LibreTexts Chemistry, ‘B1: Workfunction Values (Reference Table)’, no original source named: Au 5.1 to 5.47, Al 4.06 to 4.26, Sn 4.42, Pb 4.25, Os 5.93, Cs 2.1 eV (fetch-tool summary); R. Tran, X.-G. Li, J. H. Montoya, D. Winston, K. A. Persson and S. P. Ong, ‘Anisotropic work function of elemental crystals’, Surf. Sci. 687, 48 (2019): work function varies with crystal facet, surface contamination can cause errors up to 0.5 eV in measured values (fetch-tool summary).

Ques: How repeatable is KPFM between probes?

To about 20 $mV$ on a good reference. Bruker measured an Au/Al standard, 50 $nm$ films on n-type silicon, with nine PFQNE-AL probes: Au at 0.639 plus or minus 0.018 $V$ , Al at -0.185 plus or minus 0.020 $V$ , and the Au-Al difference at 0.825 plus or minus 0.019 $V$ . Tabulated work functions, Au 5.1 to 5.47 and Al 4.06 to 4.26 $eV$ (LibreTexts), give at least 0.84 $eV$ , so the measurement sits 2 per cent below the smallest tabulated difference (computed). Probe-to-probe scatter of 0.018 $V$ is 2.8 per cent of the Au value (computed). Bruker’s text gives deviations below 50 $mV$ in one place and below 20 $mV$ in another; the table supports the 20 $mV$ figure. Differences measured with one tip are more repeatable than absolute values. Reference: Bruker, ‘PeakForce Kelvin Probe Force Microscopy’, application note AN140 (several equations garbled) (fetch-tool summary); LibreTexts Chemistry, ‘B1: Workfunction Values (Reference Table)’, no original source named: Au 5.1 to 5.47, Al 4.06 to 4.26, Sn 4.42, Pb 4.25, Os 5.93, Cs 2.1 eV (fetch-tool summary).

Ques: What else can KPFM measure?

Photovoltages, high surface potentials and nanoscale charge. Bruker reports a work-function down-shift of 535 $mV$ on an organic solar cell under 300 suns, a single carbon nanotube 30 $mV$ above its substrate and a bundle 105 $mV$ , high-voltage KPFM to plus or minus 200 $V$ with under 15 per cent error and up to 1000 $V$ , and charge on PDMS from -99 to 156 $V$ . Bias frequencies must stay below half the cantilever resonance in the high-voltage mode. A 30 $mV$ nanotube signal is only 0.8 per cent of the maximum possible CPD (computed), showing the sensitivity required. Bruker also uses a glove box with about 1 ppm oxygen and water to avoid adsorbates, which Panchal et al. show alter measured potentials. Reference: Bruker, ‘PeakForce Kelvin Probe Force Microscopy’, application note AN140 (several equations garbled) (fetch-tool summary); V. Panchal, R. Pearce, R. Yakimova, A. Tzalenchuk and O. Kazakova, ‘Standardization of surface potential measurements of graphene domains’, arXiv:1305.0465 (NPL) (fetch-tool summary).

References

Bruker, “PeakForce Kelvin Probe Force Microscopy,” application note AN140, hosted by Boise State University. Available: https://www.boisestate.edu/wp-content/uploads/sites/551/2019/07/Bruker-AN-140-PeakForce-KPFM.pdf
L. Polak and R. J. Wijngaarden, “Two competing interpretations of Kelvin probe force microscopy on semiconductors put to test,” arXiv:1605.05539, 2016. Available: https://arxiv.org/pdf/1605.05539
LibreTexts Chemistry, “B1: Workfunction Values (Reference Table).” Available: https://chem.libretexts.org/Ancillary_Materials/Reference/Reference_Tables/Bulk_Properties/B1:_Workfunction_Values_(Reference_Table)
R. Tran, X.-G. Li, J. H. Montoya, D. Winston, K. A. Persson and S. P. Ong, “Anisotropic work function of elemental crystals,” Surface Science, vol. 687, pp. 48-55, 2019. Available: https://perssongroup.lbl.gov/papers/tran-2019-work-function.pdf
V. Panchal, R. Pearce, R. Yakimova, A. Tzalenchuk and O. Kazakova, “Standardization of surface potential measurements of graphene domains,” arXiv:1305.0465. Available: https://arxiv.org/pdf/1305.0465

Quartz 5

Explorer

Kelvin Probe Force Microscope (KPFM)