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[Nobuyuki Ishida](https://orcid.org/0000-0003-0161-0583)

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[Influence of Fermi level pinning on contact potential difference measurements using Kelvin probe force microscopy](https://mdr.nims.go.jp/datasets/aec52709-2acb-4fe7-a0bc-9742174c369b)

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Influence of Fermi level pinning on contact potential difference measurements usingKelvin probe force microscopyNobuyuki Ishida1, ∗1National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan(Dated: March 5, 2025)We theoretically investigated the influence of Fermi-level pinning on contact potential difference(CPD) measurements conducted via Kelvin probe force microscopy (KPFM) on semiconductor sur-faces. To systematically modulate the strength of the surface pinning, virtual surface states wereintroduced within the semiconductor bandgap, and the density of states (DOS) was varied. Thenumerically simulated CPD values varied depending on the DOS of the surface states, reflecting themagnitude of the surface band bending (surface potential). However, we found that under certainconditions, the CPD values obtained by KPFM deviated from the physical quantities directly asso-ciated with the surface potential. Our results provide valuable insights for enhancing the accuracyof KPFM data analysis and interpretation.I. BACKGROUNDKelvin probe force microscopy (KPFM) is a mea-surement technique that combines the Kelvin probe(KP) method, originally developed by Lord Kelvin, withatomic force microscopy (AFM) [1, 2]. This method en-ables local measurement of the contact potential differ-ence (CPD) and has been widely used to characterizevarious electronic [3–9] and ionic devices [10–15]. TheCPD, also known as the Volta potential difference, isdefined as the electric potential difference between onepoint in vacuum close to the surface of M1 and anotherpoint in vacuum close to the surface of M2, where M1and M2 are two uncharged metals brought into contact[16]. This value corresponds to the difference betweenthe work functions of the two metals. In this study, wedenote the work function difference between the probetip and sample as VCPD.In KPFM or KP measurements, the CPD is detectedby identifying the bias voltage (U∗) at which the electro-static force or displacement current between the probeand sample is minimized. KPFM and KP measurementsof pristine metal surfaces using a well-defined metal probeyield U∗ equivalent to VCPD [17, 18]. However, for pris-tine semiconductor surfaces, U∗ does not necessarily co-incide with VCPD because of the variations in the surfacepotential induced by band bending [19]. To accuratelyanalyze and interpret the experimental data, understand-ing the influence of the surface electronic properties onthe U∗ measurements is essential. However, the theoret-ical studies on this topic are limited.In this study, we theoretically investigated the vari-ation in U∗ measured by KPFM on semiconductor sur-faces, focusing on the influence of surface properties, par-ticularly the strength of Fermi-level pinning. The pinningstrength was modulated by introducing virtual surfacestates within the bandgap of the semiconductor sample∗ ishida.nobuyuki@nims.go.jpand altering the density of states (DOS). Our calcula-tions show that U∗ depends not only on the presence orabsence of surface states but also on their DOS. We foundthat when the Fermi-level pinning is either strong or neg-ligibly weak, U∗ corresponds to the difference in workfunctions between the probe tip and the semiconductorat the surface, which is directly related to the surface po-tential. By contrast, under moderate Fermi-level pinningconditions, U∗ deviates from the value expected from thesurface potential. Our findings suggest that U∗ measuredby KPFM on semiconductor surfaces cannot always beinterpreted as a measure of the surface potential, therebyhighlighting the significance of surface electronic proper-ties for the accurate interpretation of KPFM data.II. THEORETICAL COMPUTATIONTo derive the theoretical U∗ measured by KPFM, wenumerically calculated the dependence of the electro-static force (Felec) between the tip and semiconductorsample on the bias voltage (U). For these computations,we employed the Poisson solver (SEMITIP) developedby Feenstra [20–25], which allowed us to calculate theelectrostatic potential distribution in both the vacuumand semiconductor regions, induced by an applied po-tential from a metallic probe tip near the semiconductorsurface. The electrostatic forces acting on the tip werecalculated using the electric field and charge density onthe tip surface, as derived from the potential distributionprovided by the SEMITIP, as detailed in our previouswork [26]. Subsequently, the bias voltage at which theelectrostatic force was minimized was identified by fit-ting a ninth-order polynomial function to the calculatedFelec(U) spectra [9, 26]. As a model sample, we usedan n-type GaAs(110) surface with a doping density of5× 1017 cm−3, which has been extensively studied boththeoretically and experimentally [9, 24, 27]. The pris-tine GaAs(110) surface has no surface states within thebandgap, resulting in a flat band from the bulk to thesurface [19].2ENECEFEVEF0.756 eVVirtual surface statesIntrinsic surface states1.150 eV0.25 eV0.25 eV(a) (b)FIG. 1. (a) Schematic illustration of the band structure ofthe GaAs(110) surface used for the electrostatic force com-putation. Virtual surface states with a Gaussian-type energydistribution with a FWHM of 0.25 eV are introduced at thecenter of the bandgap. The CNL aligns with the centroid ofthe Gaussian distribution. In addition, the intrinsic surfacestates originated from the Ga dangling bond are incorporatedabove the conduction band edge. (b) shows the surface bandbending induced by the occupation of the surface states abovethe CNL.To incorporate the effect of Fermi-level pinning intoour simulation, we introduced virtual surface stateswithin the bandgap of GaAs(110). The strength ofFermi-level pinning was controlled by varying the spatialdensity of the surface states ρss. We assumed a Gaussian-type energy distribution with a full width at half max-imum (FWHM) of 0.25 eV and positioned the centroidenergy of the distribution at the center of the bandgap,as illustrated in Fig. 1(a). The charge neutrality level(CNL) EN was aligned with the centroid energy of theGaussian distribution. The CNL represents the energybelow which the states are neutral when filled and pos-itively charged when empty, and above which they arenegatively charged when filled and neutral when empty[19]. We varied ρss from 0 to 4.4 × 1014 cm−2. A ρssof zero corresponds to a pristine GaAs(110) surface. Ata maximum value of 4.4 × 1014 cm−2, which is equiv-alent to the spatial density of the Ga or As atoms onthe GaAs(110) surface, the CNL of the surface states isstrongly pinned at the Fermi level of the semiconductor,as discussed in Section III C.In addition to the virtual surface states, we accountedfor the intrinsic surface states originating from the Gadangling bonds located above the conduction band edge(Fig. 1(a)). We assumed a Gaussian-type energy distri-bution with a FWHM of 0.25 eV. The centroid energyof the distribution was set to 1.150 eV above the CNL(Fig.1(a)) [24]. Although these surface states were in-cluded in our simulation because they have been experi-mentally observed on a pristine GaAs(110) surface [28],their inclusion did not influence the findings of this study.Figure 1(b) illustrates the surface band bending in-duced by the occupation of the virtual surface states.When the band is flat, the virtual surface states are lo-cated below the Fermi level (Fig.1(a)), causing electronsrss = 0rss = 4.4 x 1012 cm-2rss = 4.4 x 1014 cm-2FIG. 2. Felec(U) characteristics under ρss of 0, 4.4 × 1012cm−2, and 4.4× 1014 cm−2. Vertical lines indicate the U∗ foreach ρss.in the conduction band near the surface to transitioninto the surface states. The occupation of the surfacestates above the CNL creates negative surface charges,initiating upward band bending near the surface to forma space charge layer, where dopant atoms are positivelycharged owing to electron depletion. This type of bandbending occurs intrinsically at the surface under equilib-rium conditions, even in the absence of a nearby probetip. This differs from tip-induced band bending (TIBB).In this study, the magnitude of intrinsic band bendingis denoted by φ0surf (Fig.1(b)). Note that the potentialdifference between the surface and far inside the bulk isgenerally referred to as the surface potential.III. SIMULATION RESULTSA. Bias dependence of electrostatic forceFigure 2 shows the Felec(U) characteristics under ρssvalues of 0, 4.4 × 1012, and 4.4 × 1014 cm−2. The di-rection of the force was configured such that a negativeelectrostatic force represents an attractive force. Verti-cal lines indicate U∗ for each ρss. The primary inputparameters of the SEMITIP are the tip radius (Rtip),tip-sample separation (s), and work function differencebetween the tip and sample (VCPD). We used param-eter values of 10 nm, 1.53 nm, and −0.933 V for Rtip,s, and VCPD, respectively. Note that the definition ofVCPD in SEMITIP differs slightly from ours. Thus, theactual input for VCPD has the opposite sign (+0.933 V).These input parameters were determined by fitting theelectrostatic-force simulation to the experimental data,as reported previously [26].The Felec(U) curves in Fig. 2 demonstrates that U∗varies depending not only on the presence or absenceof virtual surface states within the bandgap but also onthe magnitude of ρss. On the pristine GaAs(110) surface(ρss = 0), where the band is flat up to the surface without3Fermi-level pinning, U∗ (−0.931 V) was nearly identicalto the input VCPD (−0.933 V) used in the computations.Therefore, U∗ agrees well with the quantity defined byVCPD =Wsample −Wtipe, (1)where Wsample and Wtip are the work functions of thesample and tip, respectively, and e is the elementarycharge.When ρss was large (4.4 × 1014 cm−2), the intrinsicband bending (φ0surf) was 0.756 eV, indicating that thesurface band was pinned such that the Fermi level andCNL were nearly aligned, as shown in Fig. 1(b). In thissituation, U∗ (−0.175 V) deviates significantly from theinput VCPD, instead closely matching the work functiondifference between the tip and the semiconductor at thesurface (−0.177 V). The quantity V ∗CPD is expressed as:V ∗CPD =(Wsample + φ0surf)−Wtipe, (2)where Wsample+φ0surf represents the work function of thesurface (Wsurf). The relationships between the variablesin Eq. 2 are shown in Fig. 3. This finding suggests thatwhen the surface band is strongly pinned, KPFM is onlysensitive to the potential at the surface, rather than beinga weighted average of the potential near the surface. Thisindicates that KPFM is highly surface sensitive.Although the high surface sensitivity of KPFM hasbeen widely assumed, no theoretical verification has beenperformed for realistic tip-sample configurations, partic-ularly for semiconductor samples. To address this, weexamined the validity of this assumption under moregeneral conditions, including measurements in ambientenvironments, where the values of s and Rtip tend tobe larger. The electrostatic force calculations were per-formed with s varying from 1 to 100 nm and Rtip from5 to 200 nm, as shown in Supplementary Fig. S1 [29].In all cases, the magnitude of the electrostatic force andthe curvature of the Felec(U) spectra exhibited signifi-cant dependence on s and Rtip. However, U∗ remainednearly unchanged, with only minor fluctuations of ap-proximately 1-2 mV. These findings indicate that thehigh surface sensitivity of KPFM is maintained acrossa broader range of conditions typically encountered inAFM/KPFM measurements.For the pristine GaAs(110) surface described above,φ0surf is zero, implying that the values of VCPD and V ∗CPDare identical. Thus, when ρss is both 0 and 4.4 × 1014cm−2, U∗ obtained by KPFM can be considered as ameasure of the physical quantity related to the surfacepotential, expressed in Eq. 2. This explains why KPFMis often described as a method for measuring surface po-tential.By contrast, when ρss was in the middle range (4.4 ×1012 cm−2), corresponding to moderate Fermi-level pin-ning, U∗ (−0.422 V) did not match with either VCPD(−0.933 V) or V ∗CPD (−0.554 V). This finding suggestsEFENWsurfWsampleEFEvacEvacWtipECEVFIG. 3. Band structures of a metallic tip and a semiconductorsample prior to electrical connection. The vacuum level farinside the semiconductor is aligned with the vacuum level inthe metallic tip. Surface band bending is caused by the occu-pation of virtual surface states, which is an intrinsic surfaceproperty irrespective of the presence of a nearby probe tip.that the characterization of KPFM as a tool for measur-ing the surface potential is not entirely accurate. φ0surfwas 0.379 eV; however, φ0surf estimated from U∗ usingEq. 2 is 0.511 eV. This implies that the KPFM overesti-mates the surface band bending compared to the actualvalue.B. Discrepancy between U∗ and V ∗CPDTo analyze the deviation of U∗ from V ∗CPD expressedin Eq. 2, the U∗ and V ∗CPD values were calculated asfunctions of ρss, as shown in Fig. 4(a) as solid circles andtriangles. The dashed horizontal line indicates the valueof V ∗CPD for the pristine GaAs(110) surface. Figure 4(b)illustrates the difference between U∗ and V ∗CPD.The variation in V ∗CPD (solid triangles) arises fromchanges in φ0surf in Eq. 2, which reflects the degree of thesurface band bending. For a small ρss, V ∗CPD is compara-ble to that of pristine GaAs(110) surface, indicating thatthe band is nearly flat up to the surface. As ρss increases,the band gradually bends upward, and the bending satu-rates when the CNL aligns approximately with the Fermilevel.When ρss was below 4.4 × 1011 cm−2 and above4.4 × 1013 cm−2, the difference between U∗ and V ∗CPDwas less than 20 mV. By contrast, when the spatial den-sity of the surface states was in the ranges of 4.4 × 1011cm−2 and 4.4×1013 cm−2, U∗ deviated from V ∗CPD by >20 mV. The largest deviation (132 mV) was observed atapproximately 4.4 × 1012cm−2, with a steep decrease inthe difference away from that point (Fig. 4(b)).We also examined the generality of the results pre-sented in this section for larger s and Rtip, as shown inSupplementary Fig. S2[29]. In all cases, the dependence4(a)(b)FIG. 4. (a) Solid circles and triangles represent U∗ and V ∗CPDvalues, respectively, plotted as a function of ρss. (b) showsthe differences between U∗ and V ∗CPD values.of U∗ on the DOS of the surface states and its devia-tions from V ∗CPD closely resembled the trends observedin Fig. 4. These findings suggest that the discussions inthe following sections are applicable to a broader rangeof conditions typically encountered in AFM/KPFM mea-surements.C. Bias dependence of surface potentialTo explore the correlation between U∗ and the strengthof Fermi-level pinning, we analyzed the bias dependenceof the surface potential (φsurf(U)) for each ρss in Fig. 4, asshown in Fig. 5. Surface potential (or surface band bend-ing) is defined as the electric potential difference betweenthe surface and far inside the bulk (see Fig. 1(b)). Onthe pristine GaAs(110) surface, because φ0surf is zero, theobserved variation in φsurf(U) can be attributed solelyto TIBB. By contrast, in the presence of finite virtualsurface states within the bandgap, φsurf(U) encompassesthe contributions from both the TIBB and band bendingcaused by the occupation of the surface states.For ρss above 4.4× 1013 cm−2 (solid circles), φsurf re-mains relatively unchanged with the bias voltage, indi-cating strong Fermi-level pinning at the surface. Strongpinning implies that the surface states can accommodatesufficient charge to compensate for the charge on the tipsurface. In other words, the line of electric force (or elec-tric field) from the tip surface cannot penetrate the in-terior of the semiconductor, which suppresses the TIBB.This implies that the semiconductor surface behaves asa metallic surface. Under these conditions, the electro-static force between the tip and the sample arises mostlyfrom the Coulomb force between the charges on the tipsurface and the charges in the surface states (hereinafterreferred to as surface charge).By contrast, for ρss below 4.4 × 1011 cm−2 (solid�� �� � � ���������������
������������ ����������������������FIG. 5. Change in bias dependence of surface potentials(φsurf(U)) depending on ρss. The ρss values are 0 (orangecrosses), 4.4 × 109 (blue squares), 4.4 × 1010 (cyan squares),4.4×1011 (olive squares), 1.5×1012 (gray squares), 3.0×1012(pink triangles), 4.4×1012 (brown triangles), 7.5×1012 (purpletriangles), 1.5×1013 (red triangles), 4.4×1013 (green circles),1.0× 1014 (orange circles), and 4.4× 1014 (blue circles).squares), the curves exhibit bias dependences similar tothose of pristine GaAs(110) (orange crosses). This sug-gests that surface charges play little role in preventing thepenetration of the line of electric force into the semicon-ductor. In this case, charges generated near the surfaceowing to TIBB (accumulated carriers or charged dopantatoms created by carrier depletion, hereinafter referredto as bulk charge) predominantly contribute to compen-sating for the charges on the tip surface. Therefore, theelectrostatic force between the tip and sample is mostlycaused by charges on the tip surface and bulk chargesnear the surface.When ρss fell within the middle range, ranging from4.4 × 1011 to 4.4 × 1013 cm−2 (solid triangles), φsurf ex-hibited significant variations with the bias voltage, indi-cating relatively weak Fermi-level pinning. Nonetheless,the curve shapes display different bias dependencies com-pared to the one observed for pristine GaAs(110) (orangecrosses). These characteristics suggest that although sur-face charges contribute to shielding the electric field tosome extent, the amount is insufficient to entirely impedethe penetration of the electric field into the semiconduc-tor. In this case, both the surface and bulk charges con-tribute substantially to the electrostatic force acting be-tween the tip and sample. The ratio of the contributionsof these two charge types is considered to vary dependingon ρss.IV. DISCUSSIONThe results discussed thus far indicate that when theelectrostatic force between the tip and the sample arisesprimarily from either surface charges or bulk charges, U∗5measured by KPFM can be interpreted as a measure ofthe physical quantity related to the surface potential, asexpressed in Eq. 2. By contrast, when the electrostaticforce originates substantially from both the surface andbulk charges, strictly speaking, KPFM cannot be de-scribed as a tool for measuring the surface potential.When both surface and bulk charges contribute to theelectrostatic force, one might intuitively expect U∗ (orCPD) to reflect the electric potential at both the surfaceand the bulk near the surface, that is, a weighted averageof the electric potential in the region where band bendingoccurs. If this were the case, KPFM would underestimatethe magnitude of the band bending (or surface potential).However, as discussed in Section II, U∗ is observed tobe larger than V ∗CPD, indicating that band bending isoverestimated.The underlying cause of this overestimation remainsunclear. However, one possible explanation is as follows.During bias application, the electrostatic force betweenthe charges on the tip surface and surface charges is min-imized when the CNL (or surface Fermi level) aligns withthe the Fermi level of the tip. On the contrary, the elec-trostatic force between the charges on the tip surface andbulk charges reaches its minimum when the Fermi levelof the semiconductor in the bulk aligns with the Fermilevel of the tip. Under moderate Fermi-level pinning con-ditions, the interplay between the two effects determinesU∗ based on the relative amounts of surface and bulkcharges, and the value is expected to correspond to V ∗CPD.However, in KPFM measurements, the surface chargesare much closer to the tip surface than the bulk charges,and their contribution to the electrostatic force is there-fore expected to be significantly larger. Consequently,the surface potential measured by KPFM becomes largerthan the intrinsic surface potential, resulting in an over-estimation of the band bending.Finally, we discuss the relationship between our find-ings and the actual KPFM measurements. KPFM ex-periments are frequently performed on devices and ma-terials under ambient conditions. In such environments,strong Fermi-level pinning frequently occurs because ofthe surface states within the bandgap, which are inducedby factors such as surface reconstruction, oxidation, andcontamination. Therefore, the measured CPD values areexpected to correspond to the quantities represented byEq. 2. However, it is important to note that directly an-alyzing the work function of samples remains challengingbecause the magnitude of band bending (surface poten-tial) is generally unknown. Nonetheless, when measur-ing changes in CPD induced by external stimuli, such asan applied bias voltage or light irradiation, quantitativeanalysis of the variation in the electric potential is fea-sible, provided that the surface pinning position remainsunchanged [5, 11].When performing measurements on surfaces with lowsurface energy that remain clean, even under ambientconditions, or on clean surfaces prepared in ultrahighvacuum, careful analysis of the experimental data maybe required. For example, if surface steps introduce sur-face states within the bandgap, the strength of the Fermi-level pinning gradually weakens as the distance from thesteps increases. In the middle of such regions, the CPDmeasured by KPFM may deviate from V ∗CPD in Eq. 2,because of the interplay between the surface and bulkcharges. This necessitates caution when performing pre-cise quantitative comparisons between experimental dataand theoretical models.V. CONCLUSIONWe utilized a theoretical simulation of the electrostaticforce acting between the probe tip and the semiconduc-tor sample to investigate the influence of Fermi-level pin-ning on CPD measurements via KPFM. The strengthof the Fermi-level pinning was controlled by introducingvirtual surface states within the semiconductor bandgapand varying the DOS. In cases where the Fermi-level pin-ning is negligibly weak or strong, such that the surfaceband bending remains nearly invariant under an appliedbias, KPFM was found to measure quantities directlyrelated to the surface potential. This finding indicatedthat KPFM is highly surface sensitive. By contrast, whenmoderate Fermi-level pinning occurred, the CPD valuesdeviated from those predicted based on the surface poten-tial. Under these conditions, both the surface and bulkcharges in the semiconductor contributed to the electro-static force between the tip and sample. The cause ofthe deviations was inferred to arise from larger contri-bution of the surface charges to the electrostatic forcethan the bulk charges due to their proximity to the tipsurface. Although such situations are not frequently en-countered in typical KPFM experiments, care may be re-quired when performing measurements on clean surfacesprepared under ultrahigh-vacuum conditions and makingdetailed quantitative comparisons with theoretical mod-els.ACKNOWLEDGMENTSThis study was partially supported by JSPS KAK-ENHI Grant Numbers JP17K06366, JP21H01818, and24K01367.[1] M. Nonnenmacher, M. P. O’Boyle, and H. K. Wickramas-inghe, Kelvin probe force microscopy, Appl. Phys. Lett.58, 2921 (1991).6[2] S. 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