ˆ
−∞
z(t)e−iωtdt (A.1a)
z(t) =
+∞
ˆ
−∞
ˆ
z(ω)e−iωtdω (A.1b)
hz2i =
+∞
ˆ
−∞
ˆ
z(ω)2dω. (A.1c)
In the frequency domain, the cantilever’s displacement can be calculated from the applied force, by using the compliance ˆC(ω) (or response function | ˆC(ω) |2) of the damped oscillator system
ˆ
z(ω) = ˆC(ω) ˆF(ω) (A.2a)
C(ω) =ˆ 1/m
q
(ω2− ω02)2+ (ωω0/Q)2
. (A.2b)
A.2 Transfer function theory
In chapter 3, the basics of contrast formation in magnetic force microscopy have been explained. The underlying theory for the Fourier transfer theory used is presented shortly in this section.
In order to facilitate the discussion of magnetic fields, the region above the sample is current free and contains no time–varying electric fields, i.e. ∇ × H(x) = 0, as discussed in section 3.1.1. On top of this, a scalar magnetic potential can be defined in analogy to the electric scalar potential, in three dimensions x = (x, y, z), as well as the volume and the surface magnetic charge functions:
∇φ(x) ≡ H(x) (A.3a)
ρM(x) ≡ −∇M(x) (A.3b)
σM(x) ≡ (Minside(x) − Moutside(x)) · n. (A.3c) If we apply the Laplace equation to Eq. A.3, we obtain ∆φ = ρM(x), which is valid with the boundary conditions
∂φ
To describe the interaction between the magnetic tip and sample, we use a two–dimensional Fourier transform, which is defined as: becomes ∂/∂x = ikx, ∂/∂y = iky. The Laplace region for the two–dimensional Fourier space in the region above the sample is given by:
−kx2φ(kk, z) − ky2φ(kk, z) +∂2φ(kk, z)
This can also be applied to the components of the magnetic stray field:
Hsample(kk, z) = −∇φ(kk, z) = −
The full solution of the Laplace equation for the stray field in the condition of M(x) = M(xk) has been derived in literature, for example see [52]:
Hsample(kk, z) = −∇
where τsampleis the sample’s thickness. If we now want to convert the conservative force as given by Eq. A.10
F(r) = −∇ 1
we first evaluate the force in Fourier Space, given by
F(kk, z) = −µ0
+∞
ˆ
−∞
Hsample(kk, z′) · ∇kk,z′M∗tip(kk, z′− z) dz, (A.11)
which can be rewritten using the coordinate transformation z′ → z′+ z and Eq. A.8:
F(kk, z) = µ0Hsample(kk, z) ·
σtip is a surface charge distribution located in the plane touching the tip’s apex. If there is no modification, σtip(kk) does not depend on z and an equation similar to equations A.7 and A.8 is valid for the force:
F(kk, z) = −µ0σtip∗ (kk, 0) · e−kkz= F(kk, 0) · e−kkz. (A.13) To evaluate the resulting force in dynamic mode, we have to convert Eq. A.12 to the frequency shift. The frequency shift itself is given by
∆f (x0) = − f0
πkLδ
π
ˆ
0
n· F(x(ϕ)) cos ϕ dϕ, (A.14)
where x = x0+ n · δ cos ϕ, n is the direction of the oscillation of the cantilever and kL the cantilever’s longitudinal spring constant. To calculate the frequency shift, we have to use Eq. A.12 in real space, inserting this into Eq. A.14, we obtain
∆f (x0) = − f0µ0
4π3kLδ
π
ˆ
0
n
+∞
ˆ
−∞
σtip∗ (kk)Hsample(kk, 0) · e−kkz· eikkxkdkxdky cos ϕ dϕ. (A.15)
Taking into account that x = x0+ n · δ cos ϕ, changing the order of integration and changing to Fourier space, Eq. A.15 transforms into:
∆f (kk, z0) =f0µ0
2kL · 2 πδ
π
ˆ
0
σ∗tip(kk)n Hsample(kk, 0) · e−kkz· eikkxkδ cos ϕcos ϕ dϕ
=f0µ0
2kL · σtip∗ (kk)n Hsample(kk, 0) · e−kkz0·2
δℑ1(iδ(kknk− kknz))
=f0µ0
2kL · n F(kk, z0) · 2
Aℑ1(iδ(kknk− kknz)),
(A.16)
where ℑ1(z) = π1
π
´
0
e−z cos ϕcos ϕ dϕ (z ∈ C) is the first order modified Bessel function of the first kind. In the small amplitude limit the Bessel function is proportional to z/2 and the signal is proportional to the force derivative. For large amplitudes and large k this is no longer the case. If the minimal distance to the sample, x0− δn, is kept constant while increasing the oscillation amplitude, the measurement becomes less sensitive to short wavelengths of the sample’s magnetization.
Contrast modeling
In chapter 3, we have outlined the basics of contrast formation and modeling. In this Appendix we will extend this discussion to three different cases, i.e. perpendicular and longitudional media and thin film contrast modeling. For this, a short abbreviation of the work of Hartman [25] is adapted. As it turns out, combining the classical potential theory with the introduction of free magnetic charges, is a convenient concept to understand the contrast produced by MFM. In the case where one has an arbitrary two–dimensional periodic magnetic charge distribution at the sample’s surface, Fourier expansion of the charge density is given by
σ(x, y) = the stray field produced by the periodically charged sample. Directly at the surface, i.e. z = 0, one obtains for the vertical stray field component
Hz(x, y, 0) = σ(x, y)/2. (B.3)
The exterior solution for the Fourier coefficients of the magnetic potential are thus
φmn= − (σmn/2νmn) exp (−νmnz), (B.4)
with the ‘spatial frequencies’
νmn= q
(m/Lx)2+ (n/Ly)2. (B.5)
The complete exterior Laplace solution is therefore given by
φ(r) = − (1/2) In general, the stray field from the sample, HS(r), can be calculated from the sample’s magnetization, MS(r) by the following formula; where R = r − r′ and ˆns is the outward unit normal vector from the sample’s surface. Using the component form for the force force derivative
Fd= µ0
and integrating over the tip’s volume, we get the total magnetic force derivative for an arbitrary cantilever’s orientation. For the special case, where the cantilever is parallel to the sample’s surface, (θc= 0), then ˆn= ˆz and Eq. B.8 transforms into;
where rm the position of this resultant moment inside a cantilever. Thus, to analyze the force gradient of a tip–sample system using equations B.8 and
Fd′(r) = µ0
as already seen in chapter 3, a model of the tip’s shape and magnetization must be assumed. The simplest geometry has been discussed in chapter 3, i.e. the point–dipole geometry. More complex geometries are discussed in the following sections.
Figure B.1: Schematic of a thin–film recording longitudinal recording medium. t denotes the film thickness, w denotes the spacing between the individual transitions. δ denotes the effective transition width, z denotes the working distance and M denotes the spontaneous magnetization.
An important area of MFM applications, is the study of thin–film structures, e.g. of recording media. If the probe–sample separation becomes comparable with or even exceeds the film thickness, the stray fields of both top and bottom sample’s surface contribute to the contrast. Thus, if t is the film thickness,
φ(r) = − where z is the vertical distance measured from the center of the film. In longitudinal magnetic recording, the recording head is flown over the medium with a spacing of a few hundred nanometers or less. Upon writing, oppositely magnetized regions with head–to–head and tail–to–tail transitions are created. Since the transitions of width δ involve free magnetic charges, a stray field is generated which transmits the stored bit configuration
to the recording head. MFM is thus a particularly useful method of analysis, since it detects the generated stray field profile, which is detected by the recording head upon reading operation. Since the stray field is produced right at the transitions between the antiparallel magnetic regions, the detailed internal structure of the transition regimes is of great importance. The latter is determined by demagnetized effects of the medium under investigation. The charge approximation used in earlier approaches to MFM contrast formation may thus be an inexpedient approximation. An approximation commonly used in recording physics is [89]
Mx(x) = −2M
π arctan x δ
, (B.12)
where Mx denotes the in–plane magnetization component near the transition which is centered at x = 0, as shown in Fig. B.1, δ denotes the characteristic transition width and M is the spontaneous magnetization in the uniformly magnetized regions. An estimate of δ may be obtained, for example, from the Williams–Comstock model [90]. If we use the formula for the magnetic potential as given in chapter 3
φs(r) = 1
and substitute Eq. B.12 into this equation, we obtain the stray field for an isolated transition
Hx(x, z) = M
for the vertical component. The total field is obtained by a linear superposition procedure
H(x, z) = X∞ n=−∞
(−1)nH(x − nw, z), (B.16)
where w is the spacing between the transitions. The stray field components together with the quantity ∂Mx/∂x derived from Eq. B.12 are shown in Fig. B.2.
d/w=0.1, t/w=0.05
(a) Contrast formation at the surface medium
H/M
d/w=0.1, t/w=0.05 z/w=0.1
Hz Hx
(b) Contrast formation at working distance w Figure B.2: Contribution to the magnetic contrast produced by a longitudinal recording medium. The field components are considered with respect to the in–plane spontaneous magnetization and are plotted as a function of lateral position. w denotes the spacing between the individual transitions. δ denotes the effective transition width for which a representative value has been chosen. (a) Contrast contributions directly at the surface of the medium, together with the magnetization divergence. (b) The working distance has been increased and is now equal to one–tenth of the transition spacing. (Adapted from [25])
d/w=0.1, t/w=0.05
(a) Contrast formation at the surface medium
d/w=0.1, t/w=0.05
(b) Contrast formation at working distance w Figure B.3: Same as in Fig. B.2, but for the second order derivatives of the stray field components. These quantities become relevant if the force microscope is operated in dynamic mode. (Adapted from [25])
According to Eq. B.8, these stray field components predominantly produce the MFM contrast, if the probe mainly exhibits a monopole moment. Apart from the two field components, their first derivatives with respect to x and z also contribute to the contrast formation if the MFM is operated in static mode. Because of the constraint given in chapter 3
only three out of four derivatives are required to model the MFM contrast. Finally, the second derivatives, which according to Eq. B.9 are relevant in dynamic mode operation, are shown in Fig. B.3. When we once again apply the constraint of Eq. B.17 and combine this with symmetry considerations reflected by ∂2Hx/∂z2 =
−∂2Hz/∂x2, we only have to calculate three out of six possible second derivatives.