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Simultaneous quantitative imaging of surface and magnetic forces

We demonstrate quantitative force imaging of long-range magnetic forces simultaneously with near-surface van-der-Waals and contact-mechanics forces using intermodulation atomic force microscopy. Magnetic forces at the 200 pN level are separated from near-surface forces at the 30 nN level. Imaging of these forces is performed in both the contact and non-contact regimes of near-surface interactions.

Separating and identifying the various forces acting between a sharp tip and a sample surface has been a long standing challenge for the interpretation of contrast in atomic force microscopy (AFM). One typical example is the separation of near-surface forces, such as the attractive van-der-Waals and repulsive contact forces which dominate a few nanometers from the surface, from magnetic 1,2 or electrostatic 3 forces which dominate at larger distances from the surface. To measure these long-range forces a two-pass method is typically used 4,5 , which has the disadvantage of doubled scan time, loss of resolution and limited ability to measure very close to the surface during the second lifted pass, when the feedback is turned off. Recently multifrequency AFM modes have emerged which provide more measurement channels while scanning 6 , allowing single-pass imaging of magnetic forces 7,8 . Some of these methods have demonstrated the ability to rapidly capture the entire force-distance curve [9][10][11][12][13][14][15] allowing for simultaneous imaging of topography and material properties. In this letter we perform Intermodulation AFM (ImAFM) 16 with a magnetically coated tip and analyze the data so as to separate the long-range magnetic force from near-surface forces, thus obtaining simultaneous imaging of topography, mechanical and magnetic properties in one scan, with a standard cantilever, at a typical scan speed for dynamic AFM.
In narrow-band ImAFM the cantilever is excited simultaneously with two drive tones closely separated in frequency and centered around a resonance frequency of the cantilever.
Upon interaction with the surface the nonlinear tip-surface force creates a spectrum of many intermodulation products of the two drive tones (figure 1a), which can be measured by a phase-sensitive lockin technique with good signal-to-noise ratio. We have previously shown that this spectrum contains the information necessary to accurately reconstruct the tipsurface force curve and we have demonstrated several methods of force reconstruction from the raw spectral data [13][14][15] . One of these methods involves assuming a parametrized force model for the interaction where the force parameters are determined from the measured spectrum with the help of a numerical solver 13 . Here we performed this model-based force reconstruction method using a force model which takes into account both near-surface and long-range forces. We image a hard disk sample with perpendicular magnetization and observe that the parameters of the long-range force capture the expected magnetic image.
It is difficult to derive a simple and exact analytic expression describing the force between the magnetic AFM tip and the hard disk surface 17,18 . A good approximate model should have few parameters and the correct asymptotic behavior: The force should go to zero as 2 the distance between the tip and the sample is increased, and the characteristic length for this decay should be the order of the separation of magnetic poles on the surface. We choose a power-law decay, where F 0 is the magnitude of the magnetic force at the position of the surface z = z s and λ 0 the characteristic decay length. We note that the power p = 2 corresponds to a monopol-monopol while p = 4 corresponds to dipole-dipole interaction 19,20 . An estimate of the characteristic decay length is the size of the individual magnetic bits, 170 nm × 23 nm 21 . Thus we expect the magnetic force to extend significantly beyond the range of the van-der-Waals force which is typically less than 2 nm.
To account for the surface forces we used the Derjaguin-Muller-Toporov (DMT) model 22,23 .
The DMT model describes the attractive force of van-der-Waals interactions and the repulsive force due to Hertzian contact mechanics. The qualitative shape of the DMT force-distance curve coincides with that observed when a polynomial 15 or model-free force reconstruction 14 is performed on the data. The net force is reconstructed as a function of where h is the probe height and z the instantaneous position of the tip, both being measured in the laboratory frame. In terms of the cantilever deflection the combined force model reads, Where H is the Hamaker constant, a 0 is the intermolecular distance in the van-der-Waals model, E * is the effective Young's modulus, R the tip radius and d s the distance to the surface measured from the probe height h.  One can also visualize the data as slices of the 3-dimensional force volume 29 . Figure 3a shows In conclusion, we demonstrate a novel method to separately image the long-range magnetic forces and near-surface forces in atomic force microscopy. We simultaneously measured the repulsive contact-mechanics force, attractive van-der-Waals force and magnetic force with a single-pass scan, which, to our knowledge, has not been previously demonstrated.
The method can also be applied to other long-range forces such as electrostatic forces, and it provides a calibrated means of quantitatively determining of the separate forces. A long term goal of magnetic force imaging is to obtain quantitative measurements of 3-dimensional magnetic fields and magnetization at the sample surface. Achieving this goal requires not only quantitative measurement of force, but also well-characterized magnetic tips and suitable magnetic force models.