7 Moment switching in nanotube magnetic force
probes
John R Kirtley1,2,3, Zhifeng Deng4, Lan Luan4, Erhan
Yenilmez1, Hongjie Dai5, and Kathryn A Moler1,4
1 Department of Applied Physics and Geballe Laboratory for Advanced Materials,
Stanford University, Stanford, California 94305 USA
E-mail: jkirtley@stanford.edu
2 IBM Watson Research Center, Route 134 Yorktown Heights, NY 10598 USA
3 Faculty of Science and Technology and MESA+ Institute for Nanotechnology,
University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
4 Department of Physics and Geballe Laboratory for Advanced Materials, Stanford
University, Stanford, California 94305 USA
5 Department of Chemistry, Stanford University, Stanford, California 94305 USA
Abstract.
A recent advance in improving the spatial resolution of magnetic force microscopy
(MFM) uses as sensor tips carbon nanotubes grown at the apex of conventional silicon
cantilever pyramids and coated with a thin ferromagnetic layer [1]. Magnetic images of
high density vertically recorded media using these tips exhibit a doubling of the spatial
frequency under some conditions [1]. Here we demonstrate that this spatial frequency
doubling is due to the switching of the moment direction of the nanotube tip. This
results in a signal which is proportional to the absolute value of the signal normally
observed in MFM. Our modeling indicates that a significant fraction of the tip volume
is involved in the observed switching, and that it should be possible to image very high
bit densities with nanotube magnetic force sensors.
PACS numbers: 75.75.+a,78.67.Ch
Spatial period doubling has been observed for several carbon nanotube tips with
different track widths and bit densities. The MFM images reported here were made
using the nanotube tip shown in the inset of Figure 1. It was approximately 250nm long,
with a ferromagnetic coating to a total tip diameter of 16nm. The means of producing
metal-coated carbon nanotube tips on AFM cantilevers and the techniques for imaging
magnetic media using these tips have been described previously [1, 2]. Briefly, carbon
nanotubes were grown using wafer-scale chemical vapor deposition at the apexes of the
pyramids of commercial silicon tips intended for tapping mode atomic force microscopy.
For the present measurements the nanotubes were shortened to a length of about 250
nm using an electrical cutting method [3], aligned approximately perpendicular to the
cantilever using a focused ion beam [2], and then coated to a total thickness of about
16 nm with a Ti/Co/Ti trilayer by e-beam evaporation from a direction parallel to the
http://arxiv.org/abs/0704.0311v1
Moment switching in nanotube magnetic force probes 2
nanotube long axis (see the inset in Figure 1). The magnetic imaging was done using the
Tapping/LiftTM mode of a Digital Instruments Nanoscope III SPM at room temperature
in air. In this mode, the topography of the sample is first determined for each line with
a scan at low tip-sample spacing z0, then the tip is retracted a specified distance and
a second line scan is made while recording the deviation of the phase angle δ of the
cantilever response with the cantilever driven slightly below its resonance frequency. δ
is proportional to dFz/dz0, the derivative of the force on the cantilever with respect to
The images presented here were made on a vertically polarized magnetic medium.
A collection of phase shift images at 300 kilo-flux changes per inch (kfci) and different
tip-sample spacings (z0’s) are shown in Figure 2(a), and at constant lift height (15nm)
and different bit densities in Figure 2(b). Cross-sections of the data through the center
of the tracks are displayed in Figure 2(c,d). Modeling of this data as described below
is shown in Figure 2(e,f). At high values of z0 and high bit densities the phase images
have the same period as written, but at low z0’s and low bit densities anomalies in
the images gradually develop into sharp features with double the original periodicity.
These anomalies are due to switching of the orientation of the magnetic moment of
the tip. This can be demonstrated most convincingly by inspecting the images and
cross-sections of Figure 2(b,d). In this case a background from the sections of the image
without written bits has been subtracted out, and it can be seen that at low bit densities
the phase shift always stays below the average background level, keeping the tip-sample
force attractive by switching of the tip moment direction just as the force derivative
crosses zero.
This conclusion is supported by detailed modeling: We assume that the magnetic
medium is composed of slabs of length s in the x direction, height h in the z
direction, and width W in the y direction (Figure 1(a)). The slabs have a uniform
magnetization with moment direction alternating between parallel and anti-parallel to
the z-axis direction. The tip is assumed to have a square cross-section with width w
and length L, with the tip end a distance z0 from the upper surface of the medium.
The magnetic fields above the sample (displayed as field lines in Figure 1(b)) were
calculated both analytically and numerically. In our numerical modeling the vector
between the individual medium dipole moments [xm, ym, zm] and a position [x, y, z] is
~r = (x− xm)x̂+ (y − ym)ŷ + (z − zm)ẑ. Then the z and x-component of the field from
the individual dipoles is given by
Bz(~r) =
3(z − zm)2
Bx(~r) =
3µ0mm
(x− xm)(z − zm), (1)
with r = |~r|. By is given by setting x → y and xm → ym in the second equation.
If we assume that the tip can also be represented by a sum of point dipoles ~mt = mtn̂,
where n̂ is a unit vector in the tip moment direction, the force gradient on the tip is
Moment switching in nanotube magnetic force probes 3
Medium
z0+L1
Figure 1. (a) Tip and sample geometry in model. The inset is a scanning electron
microscope image of the cobalt-coated nanotube tip. (b) Calculated field lines for a
300 kfci track above the center of the domains in the magnetic medium, using the
parameters described in the text. The arrows indicate calculated moment orientations
for a series of tip-sample spacings of 15, 22.5, 28, 33, 39, and 45 nm using the model and
parameters described in the text. At small tip-sample spacings the tip moment flips
to be always anti-aligned with the moment directly below the tip. At large tip-sample
spacings the tip moments are only slightly perturbed by the medium fields.
given by
mt cos(φ)
d2Bz(x, y, z)
tip,medium
3µ0mmmt cos(φ)
− 30(z − zm)
35(z − zm)4
where rtm is the distance between the individual dipole elements in the tip and medium,
and φ is the angle between the tip moment direction and the z-axis.
Moment switching in nanotube magnetic force probes 4
Figure 2. (a) Collection of phase shift images for a 300 kfci track of alternating
vertically recorded media moments at different lift heights. Each line of the image has
an offset such that the average phase shift value for that line is zero. (b) Image of a
section of vertically recorded media with a series of tracks with different bit densities,
at a lift height of 15 nm. A background has been subtracted from this image such
that the regions between the tracks have zero averaged phase shifts. (c) Cross-sections
through the centers of the tracks in (a). (d) Cross-sections through the centers of the
tracks in (b). The zero phase shift levels for each cross-section are indicated by dashed
lines. (e) Modeling of (c) as described in the text. The lines in (c) and (e) have been
offset vertically for clarity. (f) Modeling of the tracks in (d) as described in the text,
with the zero phase shift levels indicated by dashed lines.
Moment switching in nanotube magnetic force probes 5
For positions near the center of the tracks we obtained analytical expressions for
the magnetic fields by assuming that the magnetically oriented slabs have infinite width
W in the y direction. If we take the boundary condition
Hz(~r)|z=0+ −Hz(~r)|z=0− = Hz(~r)|z=−h− −Hz(~r)|z=−h+ = σ(~r), (3)
where σ(~r) is the surface magnetic charge, the magnetic field above the sample can be
written as [4]
Hz(~r, z) =
d2~kA
−i~k·~r, (4)
where
H,z = e
−kz(1− e−kh)
d2~r σ(~r)ei
~k·~r. (5)
Taking the surface magnetic charge σ(~r) to be uniform in y,
σ(~r) = M 2ns < x < (2n+ 1)s
= −M (2n− 1)s < x < 2ns, (6)
We find
tan−1
(2n+ 1)s+ x
+ tan−1
(2n− 1)s+ x
− 2 tan−1 z
2ns+ x
− tan−1 z + h
(2n+ 1)s+ x
− tan−1 z + h
(2n− 1)s+ x
+ 2 tan−1
z + h
2ns+ x
Since Hz = dΦ/dz, Φ a scalar potential, the x-component of the field can be written as
Hx = dΦ/dx, which leads to
Hx = −
((2n+ 1)s+ x)2
+ log
((2n− 1)s+ x)2
−2 log
(2ns+ x)2
− log
(z + h)2
((2n+ 1)s+ x)2
− log
(z + h)2
((2n− 1)s+ x)2
+ 2 log
(z + h)2
(2ns+ x)2
Hy is zero by symmetry. Figure 3 displays the calculated fields in the z and x directions
(a,c), the tip moment orientation angle φ (b), and the switching fields < Hz >c and
< Hx >c (c) for the best fit to the data as described below. It is interesting to note
that the discontinuous switches of φ are controlled predominantly by the size of the x
component of the field. This can be understood by examining the trajectory in field
that the tip takes in moving from one domain to the next. The ovals in Figure 3c are the
calculated trajectories of < Hz > vs. < Hx > for the tip heights listed in the caption.
The “asteroid” is the critical field calculated for the energy functional form of Eq. 1
of the main text and has the form first predicted by Stoner and Wohlfarth[5]. The tip
moment direction is predicted to switch when the field trajectories cross the critical field
asteroid (solid dots in Figure 3c). This happens for relatively large values of | < Hx > |
and small values of | < Hz > |. When the field trajectory crosses the Stoner-Wohlfarth
Moment switching in nanotube magnetic force probes 6
0 1 2 3 4
15 nm
45 nm
0 1 2 3 4
15 nm 30 nm
33 nm
45 nm
–2 –1 0 1 2
<Hx>Mt/K1
Figure 3. (color online) (a) Calculated magnetic fields, averaged over the tip volume
and multiplied by the tip saturation magnetization Mt divided by the anisotropy
parameter K1, above a vertically recorded magnetic medium with infinite extent in
the y-direction, 12 nm thick in the z-direction, with magnetization oriented in the
z-direction and alternating in the x-direction with period 2s = 185 nm, for spacings
between the bottom of the tip and the medium of z0 = 15, 22.5, 26, 28, 30, 33, 39, and
45 nm. The tip has an assumed square cross-section 16nm on a side, and the averaging
is over a tip length L1 = 48nm. The calculated magnetic fields < Hz > normal to
the magnetic medium are offset by 3 units. The best fit values MtMm/K1 = 17.5,
σw/µ0L1K1 = 0 (see main text Figure 3) were used for these calculations. (b)
Calculated variation of the tip moment orientation angle φ relative to the z axis.
(c) The ovals show the calculated trajectories of < Hz > Mt/K1 vs < Hx > Mt/K1
for the various tip heights. The “asteroid” plots the values of the critical fields. The
intersections of these two sets of curves, indicated by solid symbols, are the fields at
which switching occurs in the simulations.
Moment switching in nanotube magnetic force probes 7
asteroid at large values of | < Hz > | the tip has already switched to the low energy
configuration.
For numerical work the medium was assumed to be composed of a collection of
individual dipole moments ~m = mmẑ, where mm = ±Mmv, with Mm the saturation
magnetization and v the volume of the individual medium elements. In what follows
we take both the tip and medium volume elements to be cubes 4 nm on a side. This
results in agreement to within a few percent between our numerical work and analytical
expressions for the medium magnetic fields. Halving the size of the volume elements (to
cubes 2 nm on a side) changes the calculated force derivative curve for z0 = 30 nm (see
Figure 2c) by about 2%.
To model the dynamics of the tip flip process, we conceptually divide the tip into
two domains, one with length L1 close to the medium, the other with length L − L1
further away. Each has sufficiently strong exchange fields that the entire volume within
each domain has the same moment orientation[6]. The section of the tip furthest from
the medium is assumed to have its moment parallel to the z axis; that closest to the
medium has its moment at an angle φ relative to the z-axis (Figure 1). Then the energy
of the tip in an external magnetic field can be written as:
E(φ) = µ0w
2L1K1
µ0L1K1
(1− cos φ) + sin2 φ
(< Hz > cos φ+ < Hx > sin φ)
where we have taken the simplest non-trivial forms for the domain wall energy (first
term) and the anisotropy energy (second term)[5, 7]. The third term in Eq. 9 is the
energy of the dipole moments of the tip in the external magnetic field. Here µ0K1 is
the anisotropy energy density, σw is the domain wall energy per unit area, Mt is the tip
saturation magnetization, and < Hz > and < Hx > are the magnetic fields in the z and
x directions respectively averaged over the tip volume from z = z0 to z = z0 + L1. To
simulate the magnetic force images, the tip moment is at first assumed to be parallel
to the z-axis. The tip is moved to a new position, the local fields are calculated and
averaged over the tip volume, φ is moved to the new local minimum in energy (Eq.
9), the force gradient is calculated, and the process is repeated. This modeling results
in the cross-sections displayed in Figure 2(c,f), which reproduce the absence of tip
switching at high bit densities and high lift heights, and the presence of tip switching
at low bit densities and low lift heights. When tip switching occurs, the modeling also
reproduces the fact that the tip-sample force gradient always stays negative, with the
tip moment reversing as the z-component of the field crosses zero. The quantitative
interpretation of MFM images in the presence of tip switching is straightforward once it
is recognized that the phase shift δ is proportional to the negative of the absolute value
of the tip sample force gradient (-|dFz/dz0|). The experimental phase shift oscillation
amplitudes decrease much more rapidly than the modeling for bit densities above about
500 kfci (Figure 2f). We believe that this is because the as written bits do not have
as abrupt moment orientation reversals as our idealized model. Our modeling indicates
Moment switching in nanotube magnetic force probes 8
that nanotube tips with the geometry of Figure 1 could be used to image bits with sharp
moment direction transitions with densities above 2000 kfci (13 nm/flux reversal).
Figure 4 compares the maximum minus the minimum value for δ along a cross-
section through the center of the bits at a bit density of 300 kfci as a function of
tip height z0. The modeling results in this Figure are labeled by the length L1 of
tip that is allowed to reorient its magnetic moment. There are three parameters
in this analysis - a global multiplicative factor, MtMm/K1, and the reduced domain
wall energy σw/µ0L1K1. Figure 4b plots the best fit values for MtMm/K1 and
(∆δexperimental−∆δmodel)2/(N−1) (N the number of data points). In all cases the
best fit value for σw/µ0L1K1 is 0, and the χ
2 value at σw/µ0L1K1 = 1 is approximately
double that when σw/µ0L1K1 = 0. Increasing the domain wall energy requires larger
switching fields: the best fit value at L1= 64 nm for MtMm/K1 increases from 20.8 to
28.6 when σw/µ0L1K1 increases from 0 to 1. The domain wall energy for Co is reported
to be σw = 25 ± 3J/m2[8]. This leads to σw/µ0L1K1 = 0.85, using L1 = 48nm and
K1=0.25 M
t (neglecting crystalline anisotropy), with Mt = 1.4× 106A/m, so that our
fits are consistent with the calculated wall energy, if one allows for a doubling of the best
χ2 value. The tip end may be magnetically poorly coupled to the rest of the tip because
of an inhomogeneity or grain boundary: it appears (Figure 1 inset) to have granularity
on the scale of a few tens of nm and a kink about 50 nm from its end. We have observed
qualitatively similar spatial frequency doubling using several nanotube tips, particularly
in the smallest diameter tips, where inhomogeneities and weak magnetic coupling are
fundamentally more difficult to avoid.
The shift ∆ω in the resonance frequency ω0 of the cantilever due to a force gradient
between the tip and sample dFz/dz0 is given by ∆(ω)/ω0 = −(dFz/dz0)/2k, where k
is the spring constant of the cantilever. The phase shift δ of the cantilever response is
then given by
tan δ =
/ω − ω/ω′
, (10)
where Q is the quality factor, ω is the driving frequency and ω
is the perturbed
resonance frequency of the cantilever. At z0 = 15nm our model predicts an excursion of
∆(dF/dz)/µ0MtMm ≈ 2 × 10−9m. Using a driving frequency at optimal sensitivity
ω = ω
(1 − 1/
8Q), k = 2.8N/m, Q=350, ω0 = 168kHz, and estimating Mt =
1.4 × 106A/m [9] and Mm = 2 × 105A/m [10], this corresponds to an excursion
in the phase shift ∆δ = 0.7o, in reasonable agreement with the experimental value
of ∆δ = 0.4o given the uncertainties in the values for the saturation magnetizations.
Using Mt| < Hx >c |/K1 <∼ 2 [5], the best fit value MtMm/K1 ∼ 17 (Figure 4)
implies a critical field of approximately 2.4×104 A/m, much smaller than the saturation
magnetization of cobalt of 1.4×106 A/m, but comparable to a switching field of 3.2×104
A/m reported for 30 nm thick, 0.34 µm wide, 2.04 µm long ellipsoidal amorphous
cobalt nanodots [11]. This reduction in switching field could result from competition
between the crystalline and shape anisotropies [9] if, for example, the uniaxial crystalline
anisotropy favors moment alignment along the tip radial direction, while the shape
Moment switching in nanotube magnetic force probes 9
10 20 30 40 50
z0 (nm)
Experiment
L1= 0
L1= 4 nm
L1= 16 nm
L1= 32 nm
L1= 48 nm
L1= 256 nm
0 100 200 300
L1 (nm)
0 100 200 300
0.0000
0.0025
0.0050
0.0075
0.0100
Figure 4. (color online) (a) Full-scale variation in phase angle along cross-sections
through the centers of the recorded tracks in Figure 2b. The + symbols represent
experiment. The other symbols represent modeling as described in the text. The
modeling curves are labeled by the length L1 of tip that switches magnetic moment
orientation. (b) Best fit values for MtMm/K1, and χ
(∆δexp −∆δmod)2/(N − 1)
as a function of the tip switching length L1, with σw/µ0L1K1 = 0.
anisotropy favors the axial direction. The tip material could also have a complicated
structure incorporating grain and domain boundaries, reducing the anisotropy energy.
The highly non-uniform fields in our case could also play a role in reducing the switching
field.
Although our analysis has been presented using a specific model for the tip
dynamics, the conclusion that the tip moment flips at relatively small fields can be
presented simply: The magnetic field at the tip must be less than the magnetic field
at the medium surface, which is given by the saturation magnetization of the medium.
The fields required to flip the tip are expected, using for example the Stoner-Wohlfarth
model [5], to be about the saturation magnetization of the tip, which is for epitaxial
cobalt much larger than the saturation magnetization of the medium. The switching
fields of our nanotube, just as for amorphous nanodots, are much smaller than those
of epitaxial Co nanodots [7, 9], which are comparable to the saturation magnetization
Moment switching in nanotube magnetic force probes 10
of cobalt. It might be possible to avoid switching and the attendant spatial frequency
doubling by developing processes to epitaxially coat the nanotube or by using single-
crystal nanorods. However, such tips would also generate larger local magnetic fields
at the sample, increasing the possibility of changing the magnetic state of the sample,
particularly for the smallest samples. Our results show that reliable information on the
moment orientations of the media can be inferred even in the presence of tip switching
if it is realised that the MFM signal is proportional to the absolute magnitude of the
tip-sample force gradient.
Acknowledgments
We would like to thank Dennis Adderton of First Nano and Dr. Steve Minne of Veeco
Instruments for the AFM probes, and Dr. David Guarisco of Maxtor Corporation for
the recorded disks. This work was supported by the Center for Probing the Nanoscale
(CPN), an NSF NSEC, NSF Grant No. PHY-0425897, by NSF Grant No. DMR
0103548, by the Dutch Foundation for Research on Matter (FOM), the Netherlands
Organization for Scientific Research (NWO), and the Dutch STW NanoNed program.
References
[1] Deng Z, Yenilmez E, Leu J, Hoffman J E, Straver E W J, Dai H, and Moler K A 2004 Appl. Phys.
Lett. 85 6263
[2] Deng Z, Yenilmez E, Reilein A, Leu J, Dai H and Moler K A 2006 Appl. Phys. Lett. 88 023119
[3] Yenilmez E, Wang Q, Chen R J, Wang D W, and Dai H J 2002 Appl. Phys. Lett. 80 2225
[4] Steifel B, “Magnetic Force Microscopy at Low Temperatures and in Ultra High Vacuum -
Application on High Temperature Superconductors”, Inauguraldissertation, University of Basel,
1998.
[5] Stoner E C and Wohlfarth E P 1948 Phil. Trans. Royal Soc. London A 240 599
[6] Kittel C, 1946 Physical Review 70 965
[7] Bonet E, Wernsdorfer W, Barbara B, Benoit A, Mailly D and Thiaville A, 1999 Phys. Rev. Lett.
83 4188
[8] Hehn M, Padovani S, Ounadjela K and Bucher J P, 1996 Phys. Rev. B 54 3428
[9] Otani Y, Kohda T, Novosad V, Fukamichi K, Yuasa S and Katayama T, 2000 J. Appl. Phys. 87
[10] Ross C A 2001 Annu. Rev. Mater. Res. 203
[11] Johnson J A, Grimsditch M, Metlushko V, Vavassori P, Illic B, Neuzil P and Kumar R 2000 Appl.
Phys. Lett. 77 4410

  A recent advance in improving the spatial resolution of magnetic force
microscopy (MFM) uses as sensor tips carbon nanotubes grown at the apex of
conventional silicon cantilever pyramids and coated with a thin ferromagnetic
layer. Magnetic images of high density vertically recorded media using these
tips exhibit a doubling of the spatial frequency under some conditions. Here we
demonstrate that this spatial frequency doubling is due to the switching of the
moment direction of the nanotube tip. This results in a signal which is
proportional to the absolute value of the signal normally observed in MFM. Our
modeling indicates that a significant fraction of the tip volume is involved in
the observed switching, and that it should be possible to image very high bit
densities with nanotube magnetic force sensors.

7 Moment switching in nanotube magnetic force
probes
John R Kirtley1,2,3, Zhifeng Deng4, Lan Luan4, Erhan
Yenilmez1, Hongjie Dai5, and Kathryn A Moler1,4
1 Department of Applied Physics and Geballe Laboratory for Advanced Materials,
Stanford University, Stanford, California 94305 USA
E-mail: jkirtley@stanford.edu
2 IBM Watson Research Center, Route 134 Yorktown Heights, NY 10598 USA
3 Faculty of Science and Technology and MESA+ Institute for Nanotechnology,
University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
4 Department of Physics and Geballe Laboratory for Advanced Materials, Stanford
University, Stanford, California 94305 USA
5 Department of Chemistry, Stanford University, Stanford, California 94305 USA
Abstract.
A recent advance in improving the spatial resolution of magnetic force microscopy
(MFM) uses as sensor tips carbon nanotubes grown at the apex of conventional silicon
cantilever pyramids and coated with a thin ferromagnetic layer [1]. Magnetic images of
high density vertically recorded media using these tips exhibit a doubling of the spatial
frequency under some conditions [1]. Here we demonstrate that this spatial frequency
doubling is due to the switching of the moment direction of the nanotube tip. This
results in a signal which is proportional to the absolute value of the signal normally
observed in MFM. Our modeling indicates that a significant fraction of the tip volume
is involved in the observed switching, and that it should be possible to image very high
bit densities with nanotube magnetic force sensors.
PACS numbers: 75.75.+a,78.67.Ch
Spatial period doubling has been observed for several carbon nanotube tips with
different track widths and bit densities. The MFM images reported here were made
using the nanotube tip shown in the inset of Figure 1. It was approximately 250nm long,
with a ferromagnetic coating to a total tip diameter of 16nm. The means of producing
metal-coated carbon nanotube tips on AFM cantilevers and the techniques for imaging
magnetic media using these tips have been described previously [1, 2]. Briefly, carbon
nanotubes were grown using wafer-scale chemical vapor deposition at the apexes of the
pyramids of commercial silicon tips intended for tapping mode atomic force microscopy.
For the present measurements the nanotubes were shortened to a length of about 250
nm using an electrical cutting method [3], aligned approximately perpendicular to the
cantilever using a focused ion beam [2], and then coated to a total thickness of about
16 nm with a Ti/Co/Ti trilayer by e-beam evaporation from a direction parallel to the
http://arxiv.org/abs/0704.0311v1
Moment switching in nanotube magnetic force probes 2
nanotube long axis (see the inset in Figure 1). The magnetic imaging was done using the
Tapping/LiftTM mode of a Digital Instruments Nanoscope III SPM at room temperature
in air. In this mode, the topography of the sample is first determined for each line with
a scan at low tip-sample spacing z0, then the tip is retracted a specified distance and
a second line scan is made while recording the deviation of the phase angle δ of the
cantilever response with the cantilever driven slightly below its resonance frequency. δ
is proportional to dFz/dz0, the derivative of the force on the cantilever with respect to
The images presented here were made on a vertically polarized magnetic medium.
A collection of phase shift images at 300 kilo-flux changes per inch (kfci) and different
tip-sample spacings (z0’s) are shown in Figure 2(a), and at constant lift height (15nm)
and different bit densities in Figure 2(b). Cross-sections of the data through the center
of the tracks are displayed in Figure 2(c,d). Modeling of this data as described below
is shown in Figure 2(e,f). At high values of z0 and high bit densities the phase images
have the same period as written, but at low z0’s and low bit densities anomalies in
the images gradually develop into sharp features with double the original periodicity.
These anomalies are due to switching of the orientation of the magnetic moment of
the tip. This can be demonstrated most convincingly by inspecting the images and
cross-sections of Figure 2(b,d). In this case a background from the sections of the image
without written bits has been subtracted out, and it can be seen that at low bit densities
the phase shift always stays below the average background level, keeping the tip-sample
force attractive by switching of the tip moment direction just as the force derivative
crosses zero.
This conclusion is supported by detailed modeling: We assume that the magnetic
medium is composed of slabs of length s in the x direction, height h in the z
direction, and width W in the y direction (Figure 1(a)). The slabs have a uniform
magnetization with moment direction alternating between parallel and anti-parallel to
the z-axis direction. The tip is assumed to have a square cross-section with width w
and length L, with the tip end a distance z0 from the upper surface of the medium.
The magnetic fields above the sample (displayed as field lines in Figure 1(b)) were
calculated both analytically and numerically. In our numerical modeling the vector
between the individual medium dipole moments [xm, ym, zm] and a position [x, y, z] is
~r = (x− xm)x̂+ (y − ym)ŷ + (z − zm)ẑ. Then the z and x-component of the field from
the individual dipoles is given by
Bz(~r) =
3(z − zm)2
Bx(~r) =
3µ0mm
(x− xm)(z − zm), (1)
with r = |~r|. By is given by setting x → y and xm → ym in the second equation.
If we assume that the tip can also be represented by a sum of point dipoles ~mt = mtn̂,
where n̂ is a unit vector in the tip moment direction, the force gradient on the tip is
Moment switching in nanotube magnetic force probes 3
Medium
z0+L1
Figure 1. (a) Tip and sample geometry in model. The inset is a scanning electron
microscope image of the cobalt-coated nanotube tip. (b) Calculated field lines for a
300 kfci track above the center of the domains in the magnetic medium, using the
parameters described in the text. The arrows indicate calculated moment orientations
for a series of tip-sample spacings of 15, 22.5, 28, 33, 39, and 45 nm using the model and
parameters described in the text. At small tip-sample spacings the tip moment flips
to be always anti-aligned with the moment directly below the tip. At large tip-sample
spacings the tip moments are only slightly perturbed by the medium fields.
given by
mt cos(φ)
d2Bz(x, y, z)
tip,medium
3µ0mmmt cos(φ)
− 30(z − zm)
35(z − zm)4
where rtm is the distance between the individual dipole elements in the tip and medium,
and φ is the angle between the tip moment direction and the z-axis.
Moment switching in nanotube magnetic force probes 4
Figure 2. (a) Collection of phase shift images for a 300 kfci track of alternating
vertically recorded media moments at different lift heights. Each line of the image has
an offset such that the average phase shift value for that line is zero. (b) Image of a
section of vertically recorded media with a series of tracks with different bit densities,
at a lift height of 15 nm. A background has been subtracted from this image such
that the regions between the tracks have zero averaged phase shifts. (c) Cross-sections
through the centers of the tracks in (a). (d) Cross-sections through the centers of the
tracks in (b). The zero phase shift levels for each cross-section are indicated by dashed
lines. (e) Modeling of (c) as described in the text. The lines in (c) and (e) have been
offset vertically for clarity. (f) Modeling of the tracks in (d) as described in the text,
with the zero phase shift levels indicated by dashed lines.
Moment switching in nanotube magnetic force probes 5
For positions near the center of the tracks we obtained analytical expressions for
the magnetic fields by assuming that the magnetically oriented slabs have infinite width
W in the y direction. If we take the boundary condition
Hz(~r)|z=0+ −Hz(~r)|z=0− = Hz(~r)|z=−h− −Hz(~r)|z=−h+ = σ(~r), (3)
where σ(~r) is the surface magnetic charge, the magnetic field above the sample can be
written as [4]
Hz(~r, z) =
d2~kA
−i~k·~r, (4)
where
H,z = e
−kz(1− e−kh)
d2~r σ(~r)ei
~k·~r. (5)
Taking the surface magnetic charge σ(~r) to be uniform in y,
σ(~r) = M 2ns < x < (2n+ 1)s
= −M (2n− 1)s < x < 2ns, (6)
We find
tan−1
(2n+ 1)s+ x
+ tan−1
(2n− 1)s+ x
− 2 tan−1 z
2ns+ x
− tan−1 z + h
(2n+ 1)s+ x
− tan−1 z + h
(2n− 1)s+ x
+ 2 tan−1
z + h
2ns+ x
Since Hz = dΦ/dz, Φ a scalar potential, the x-component of the field can be written as
Hx = dΦ/dx, which leads to
Hx = −
((2n+ 1)s+ x)2
+ log
((2n− 1)s+ x)2
−2 log
(2ns+ x)2
− log
(z + h)2
((2n+ 1)s+ x)2
− log
(z + h)2
((2n− 1)s+ x)2
+ 2 log
(z + h)2
(2ns+ x)2
Hy is zero by symmetry. Figure 3 displays the calculated fields in the z and x directions
(a,c), the tip moment orientation angle φ (b), and the switching fields < Hz >c and
< Hx >c (c) for the best fit to the data as described below. It is interesting to note
that the discontinuous switches of φ are controlled predominantly by the size of the x
component of the field. This can be understood by examining the trajectory in field
that the tip takes in moving from one domain to the next. The ovals in Figure 3c are the
calculated trajectories of < Hz > vs. < Hx > for the tip heights listed in the caption.
The “asteroid” is the critical field calculated for the energy functional form of Eq. 1
of the main text and has the form first predicted by Stoner and Wohlfarth[5]. The tip
moment direction is predicted to switch when the field trajectories cross the critical field
asteroid (solid dots in Figure 3c). This happens for relatively large values of | < Hx > |
and small values of | < Hz > |. When the field trajectory crosses the Stoner-Wohlfarth
Moment switching in nanotube magnetic force probes 6
0 1 2 3 4
15 nm
45 nm
0 1 2 3 4
15 nm 30 nm
33 nm
45 nm
–2 –1 0 1 2
<Hx>Mt/K1
Figure 3. (color online) (a) Calculated magnetic fields, averaged over the tip volume
and multiplied by the tip saturation magnetization Mt divided by the anisotropy
parameter K1, above a vertically recorded magnetic medium with infinite extent in
the y-direction, 12 nm thick in the z-direction, with magnetization oriented in the
z-direction and alternating in the x-direction with period 2s = 185 nm, for spacings
between the bottom of the tip and the medium of z0 = 15, 22.5, 26, 28, 30, 33, 39, and
45 nm. The tip has an assumed square cross-section 16nm on a side, and the averaging
is over a tip length L1 = 48nm. The calculated magnetic fields < Hz > normal to
the magnetic medium are offset by 3 units. The best fit values MtMm/K1 = 17.5,
σw/µ0L1K1 = 0 (see main text Figure 3) were used for these calculations. (b)
Calculated variation of the tip moment orientation angle φ relative to the z axis.
(c) The ovals show the calculated trajectories of < Hz > Mt/K1 vs < Hx > Mt/K1
for the various tip heights. The “asteroid” plots the values of the critical fields. The
intersections of these two sets of curves, indicated by solid symbols, are the fields at
which switching occurs in the simulations.
Moment switching in nanotube magnetic force probes 7
asteroid at large values of | < Hz > | the tip has already switched to the low energy
configuration.
For numerical work the medium was assumed to be composed of a collection of
individual dipole moments ~m = mmẑ, where mm = ±Mmv, with Mm the saturation
magnetization and v the volume of the individual medium elements. In what follows
we take both the tip and medium volume elements to be cubes 4 nm on a side. This
results in agreement to within a few percent between our numerical work and analytical
expressions for the medium magnetic fields. Halving the size of the volume elements (to
cubes 2 nm on a side) changes the calculated force derivative curve for z0 = 30 nm (see
Figure 2c) by about 2%.
To model the dynamics of the tip flip process, we conceptually divide the tip into
two domains, one with length L1 close to the medium, the other with length L − L1
further away. Each has sufficiently strong exchange fields that the entire volume within
each domain has the same moment orientation[6]. The section of the tip furthest from
the medium is assumed to have its moment parallel to the z axis; that closest to the
medium has its moment at an angle φ relative to the z-axis (Figure 1). Then the energy
of the tip in an external magnetic field can be written as:
E(φ) = µ0w
2L1K1
µ0L1K1
(1− cos φ) + sin2 φ
(< Hz > cos φ+ < Hx > sin φ)
where we have taken the simplest non-trivial forms for the domain wall energy (first
term) and the anisotropy energy (second term)[5, 7]. The third term in Eq. 9 is the
energy of the dipole moments of the tip in the external magnetic field. Here µ0K1 is
the anisotropy energy density, σw is the domain wall energy per unit area, Mt is the tip
saturation magnetization, and < Hz > and < Hx > are the magnetic fields in the z and
x directions respectively averaged over the tip volume from z = z0 to z = z0 + L1. To
simulate the magnetic force images, the tip moment is at first assumed to be parallel
to the z-axis. The tip is moved to a new position, the local fields are calculated and
averaged over the tip volume, φ is moved to the new local minimum in energy (Eq.
9), the force gradient is calculated, and the process is repeated. This modeling results
in the cross-sections displayed in Figure 2(c,f), which reproduce the absence of tip
switching at high bit densities and high lift heights, and the presence of tip switching
at low bit densities and low lift heights. When tip switching occurs, the modeling also
reproduces the fact that the tip-sample force gradient always stays negative, with the
tip moment reversing as the z-component of the field crosses zero. The quantitative
interpretation of MFM images in the presence of tip switching is straightforward once it
is recognized that the phase shift δ is proportional to the negative of the absolute value
of the tip sample force gradient (-|dFz/dz0|). The experimental phase shift oscillation
amplitudes decrease much more rapidly than the modeling for bit densities above about
500 kfci (Figure 2f). We believe that this is because the as written bits do not have
as abrupt moment orientation reversals as our idealized model. Our modeling indicates
Moment switching in nanotube magnetic force probes 8
that nanotube tips with the geometry of Figure 1 could be used to image bits with sharp
moment direction transitions with densities above 2000 kfci (13 nm/flux reversal).
Figure 4 compares the maximum minus the minimum value for δ along a cross-
section through the center of the bits at a bit density of 300 kfci as a function of
tip height z0. The modeling results in this Figure are labeled by the length L1 of
tip that is allowed to reorient its magnetic moment. There are three parameters
in this analysis - a global multiplicative factor, MtMm/K1, and the reduced domain
wall energy σw/µ0L1K1. Figure 4b plots the best fit values for MtMm/K1 and
(∆δexperimental−∆δmodel)2/(N−1) (N the number of data points). In all cases the
best fit value for σw/µ0L1K1 is 0, and the χ
2 value at σw/µ0L1K1 = 1 is approximately
double that when σw/µ0L1K1 = 0. Increasing the domain wall energy requires larger
switching fields: the best fit value at L1= 64 nm for MtMm/K1 increases from 20.8 to
28.6 when σw/µ0L1K1 increases from 0 to 1. The domain wall energy for Co is reported
to be σw = 25 ± 3J/m2[8]. This leads to σw/µ0L1K1 = 0.85, using L1 = 48nm and
K1=0.25 M
t (neglecting crystalline anisotropy), with Mt = 1.4× 106A/m, so that our
fits are consistent with the calculated wall energy, if one allows for a doubling of the best
χ2 value. The tip end may be magnetically poorly coupled to the rest of the tip because
of an inhomogeneity or grain boundary: it appears (Figure 1 inset) to have granularity
on the scale of a few tens of nm and a kink about 50 nm from its end. We have observed
qualitatively similar spatial frequency doubling using several nanotube tips, particularly
in the smallest diameter tips, where inhomogeneities and weak magnetic coupling are
fundamentally more difficult to avoid.
The shift ∆ω in the resonance frequency ω0 of the cantilever due to a force gradient
between the tip and sample dFz/dz0 is given by ∆(ω)/ω0 = −(dFz/dz0)/2k, where k
is the spring constant of the cantilever. The phase shift δ of the cantilever response is
then given by
tan δ =
/ω − ω/ω′
, (10)
where Q is the quality factor, ω is the driving frequency and ω
is the perturbed
resonance frequency of the cantilever. At z0 = 15nm our model predicts an excursion of
∆(dF/dz)/µ0MtMm ≈ 2 × 10−9m. Using a driving frequency at optimal sensitivity
ω = ω
(1 − 1/
8Q), k = 2.8N/m, Q=350, ω0 = 168kHz, and estimating Mt =
1.4 × 106A/m [9] and Mm = 2 × 105A/m [10], this corresponds to an excursion
in the phase shift ∆δ = 0.7o, in reasonable agreement with the experimental value
of ∆δ = 0.4o given the uncertainties in the values for the saturation magnetizations.
Using Mt| < Hx >c |/K1 <∼ 2 [5], the best fit value MtMm/K1 ∼ 17 (Figure 4)
implies a critical field of approximately 2.4×104 A/m, much smaller than the saturation
magnetization of cobalt of 1.4×106 A/m, but comparable to a switching field of 3.2×104
A/m reported for 30 nm thick, 0.34 µm wide, 2.04 µm long ellipsoidal amorphous
cobalt nanodots [11]. This reduction in switching field could result from competition
between the crystalline and shape anisotropies [9] if, for example, the uniaxial crystalline
anisotropy favors moment alignment along the tip radial direction, while the shape
Moment switching in nanotube magnetic force probes 9
10 20 30 40 50
z0 (nm)
Experiment
L1= 0
L1= 4 nm
L1= 16 nm
L1= 32 nm
L1= 48 nm
L1= 256 nm
0 100 200 300
L1 (nm)
0 100 200 300
0.0000
0.0025
0.0050
0.0075
0.0100
Figure 4. (color online) (a) Full-scale variation in phase angle along cross-sections
through the centers of the recorded tracks in Figure 2b. The + symbols represent
experiment. The other symbols represent modeling as described in the text. The
modeling curves are labeled by the length L1 of tip that switches magnetic moment
orientation. (b) Best fit values for MtMm/K1, and χ
(∆δexp −∆δmod)2/(N − 1)
as a function of the tip switching length L1, with σw/µ0L1K1 = 0.
anisotropy favors the axial direction. The tip material could also have a complicated
structure incorporating grain and domain boundaries, reducing the anisotropy energy.
The highly non-uniform fields in our case could also play a role in reducing the switching
field.
Although our analysis has been presented using a specific model for the tip
dynamics, the conclusion that the tip moment flips at relatively small fields can be
presented simply: The magnetic field at the tip must be less than the magnetic field
at the medium surface, which is given by the saturation magnetization of the medium.
The fields required to flip the tip are expected, using for example the Stoner-Wohlfarth
model [5], to be about the saturation magnetization of the tip, which is for epitaxial
cobalt much larger than the saturation magnetization of the medium. The switching
fields of our nanotube, just as for amorphous nanodots, are much smaller than those
of epitaxial Co nanodots [7, 9], which are comparable to the saturation magnetization
Moment switching in nanotube magnetic force probes 10
of cobalt. It might be possible to avoid switching and the attendant spatial frequency
doubling by developing processes to epitaxially coat the nanotube or by using single-
crystal nanorods. However, such tips would also generate larger local magnetic fields
at the sample, increasing the possibility of changing the magnetic state of the sample,
particularly for the smallest samples. Our results show that reliable information on the
moment orientations of the media can be inferred even in the presence of tip switching
if it is realised that the MFM signal is proportional to the absolute magnitude of the
tip-sample force gradient.
Acknowledgments
We would like to thank Dennis Adderton of First Nano and Dr. Steve Minne of Veeco
Instruments for the AFM probes, and Dr. David Guarisco of Maxtor Corporation for
the recorded disks. This work was supported by the Center for Probing the Nanoscale
(CPN), an NSF NSEC, NSF Grant No. PHY-0425897, by NSF Grant No. DMR
0103548, by the Dutch Foundation for Research on Matter (FOM), the Netherlands
Organization for Scientific Research (NWO), and the Dutch STW NanoNed program.
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