Current address: ]Advanced Photon Source, Argonne National Laboratory,
Argonne, IL 60439, USA

# Magnetic Structure in Fe/Sm-Co Exchange Spring Bilayers with Intermixed
Interfaces

Yaohua Liu<EMAIL_ADDRESS>S. G. E. te Velthuis<EMAIL_ADDRESS>J. S. Jiang
Y. Choi  [  S. D. Bader  Materials Science Division, Argonne National
Laboratory, Argonne, IL 60439, USA  A. A. Parizzi  H. Ambaye  V. Lauter
Spallation Neutron Source Oak Ridge National Laboratory, Oak Ridge, TN 37831,
USA

###### Abstract

The depth profile of the intrinsic magnetic properties in an Fe/Sm-Co bilayer
fabricated under nearly optimal spring-magnet conditions was determined by
complementary studies of polarized neutron reflectometry and micromagnetic
simulations. We found that at the Fe/Sm-Co interface the magnetic properties
change gradually at the length scale of 8 nm. In this intermixed interfacial
region, the saturation magnetization and magnetic anisotropy are lower and the
exchange stiffness is higher than values estimated from the model based on a
mixture of Fe and Sm-Co phases. Therefore, the intermixed interface yields
superior exchange coupling between the Fe and Sm-Co layers, but at the cost of
average magnetization.

###### pacs:

75.70.Cn, 61.05.fj

##  I Introduction

Exchange-coupled, high-magnetization (soft) and high-anisotropy (hard)
magnetic phases have potential applications as both ultra-strong permanent
magnet KnellerHawigIEEE1991  and ultra-high-density recording media.
VictoraIEEE2005  While the intrinsic properties of the two phases play the
most important role, optimization of the interface properties are also
important to achieve the best performance. For example, the interface
morphology need to be optimized for good exchange coupling between the soft
and hard phases. Micromagnetic simulations suggested that magnetically graded
interfaces, whose magnetic properties are gradually changed over a distance
$\sim 10$ nm, is more effective than sharp interfaces, in order to increase
the nucleation field in the soft phase and decrease the switching barrier of
the hard phase. JiangAPL2004 ; SuessAPL2006  Magnetically graded interfaces
can be fabricated from chemically intermixing phases via nanotechnology.
ChoiAPL2007 ; GollJAP2008 ; KirbyPRB2010  Nanoscale spatial resolution is
needed to experimentally determine the intrinsic magnetic properties of hard-
soft heterostructures. Interestingly, for Fe/Sm-Co spring magnets under
optimized fabrication conditions for the maximum energy product, the Fe layer
and the Sm-Co layer have an intermixed interface over a length scale $\sim
5-10$ nm. ChoiPRB2007 ; LiuAPL2008  Although the intrinsic magnetic properties
of individual Fe films and Sm-Co films have been well studied,
FullertonPRB1998  there lacks quantitative knowledge about the intrinsic
magnetic properties of the intermixed Fe/Sm-Co interface due to its complex
composition. ChoiPRB2007 ; LiuAPL2008

Polarized neutron reflectometry (PNR) is a sensitive tool to study the depth
profile of magnetic structures within multilayers with sub-nanometer depth
resolution, Fitz2005 ; Chatterji2006  and has been used to study exchange
coupling and anisotropy. DonovanPRL2002 ; FitzPRB2006  For example, in a
NiFe/FePt spring magnet, O’Donovan et al. have determined that the twisted
magnetic structure of the spring magnet is not confined to the magnetically
soft layer, but also penetrates into the hard magnetic phase. DonovanPRL2002
Recently, Kirby et al. showed qualitatively that the structural gradation
yields a graded anisotropy in Co/Pd multilayers. KirbyPRB2010

In the present work, we report PNR studies on a Fe/Sm-Co bilayer, which was
fabricated under nearly optimized spring magnet conditions. We first confirmed
that there is a structurally intermixed interface of $\sim 8$-nm wide between
the Fe and Sm-Co layers via combined X-ray reflectometry (XRR) and PNR
studies. We also determined the depth profile of the saturation magnetization
$M_{S}$. Furthermore, the profiles of the magnetic anisotropy $K$ and the
exchange stiffness $A$ were obtained by analyzing the PNR data with the aid of
micromagnetic simulations. The magnetic properties in the intermixed region
were compared to a model based on a mixture of Fe and Sm-Co phases. The
saturation magnetization is slightly lower than the value estimated from the
model, suggesting new compounds formed in the intermixed region. The intrinsic
anisotropy is also lower than the value from the model. However, the exchange
stiffness is higher, so that the interface efficiently couples the Fe and Sm-
Co layers.

##  II Experimental Methods

The Fe/Sm-Co thin film was fabricated via _dc_ magnetron sputtering onto an
MgO (110) single-crystal substrate with a nominal structure of (10 nm Cr)/(10
nm Fe)/(20 nm Sm-Co)/(20 nm Cr)/MgO. The sample size is $10\times 10$ mm2. The
Cr (211) buffer layer and the Sm-Co layer were grown at 400 ∘C, and the Fe
layer was grown at 100 ∘C. ChoiAPL2007  The Sm-Co layer, nominally Sm2Co7
composition, has an uniaxial, in-plane magnetic easy axis. BenaissaIEEE1998
Magnetic hysteresis loops were obtained by means of vibrating sample
magnetometer (VSM). XRR studies were performed with a X-ray diffractometer
using Cu $K_{\alpha}$ radiation. PNR experiments were conducted at the SNS at
Oak Ridge National Laboratory, at the Magnetism Reflectometer.
LauterPhysicaB2009  This is a time-of-flight (TOF) instrument with a
wavelength band 2 - 5 Å and the polarization efficiency of $\sim 98\%$. All
experiments were conducted at room temperature.

Figure 1:  (Color online) Easy-axis magnetic hysteresis loop. The film is
saturated at 1 T. The two circles label the magnetization states measured by
PNR, with applied fields of +1.11 and +0.39 T, respectively.

The external field was applied along the magnetic easy-axis for PNR
experiments. For convenience, we define the field direction as $+\hat{x}$ and
the sample surface normal as $-\hat{z}$. Figure 1 shows the easy-axis magnetic
hysteresis loop. The film saturates at fields above $+1$ T. There are two
major reversal processes along the increasing field branch, centered at
$+0.35$ and $+0.75$ T, respectively. A minor soft-phase was also observed,
which reverses below $+0.05$ T. PNR measurements were performed at $+1.11$ T
and in a demagnetized state at $+0.39$ T, after saturation with a $-1.11$ T
field, in order to determine the depth profiles of the saturation
magnetization, the exchange stiffness and the magnetic anisotropy.

Reflectometry is a non-destructive method to determine scattering length
density (SLD) profiles. In the specular condition, XRR yields the depth
profile of the electron density, which can be used to reconstruct the chemical
structures. Neutrons interact with both nuclei and the internal magnetic
field. Spin-up and spin-down neutrons feel the same nuclear scattering
potential, but an opposite magnetic scattering potential. From subsequent
measurement with oppositely polarized neutron beams, the two contributions can
be separated in order to reconstruct both depth profiles of the chemical
structure and the magnetization vector. Fitz2005 ; Chatterji2006  The contrast
of the neutron SLDs between Fe and Sm-Co is high, so that the PNR experiments
are sensitive to the interfacial structure of interest. There are four
reflectivities in PNR, including two non-spin-flip (NSF) reflectivities
$R^{++}$ and $R^{--}$, and two spin-flip (SF) reflectivities $R^{+-}$ and
$R^{-+}$. SF scattering occurs when the sample’s magnetization vector has a
non-zero component perpendicular to both the neutron’s polarization direction
and the momentum transfer direction. During magnetization reversal, the SF
scattering occurred in a sufficiently high magnetic field so that it was
necessary to take into account the Zeeman effect when analyzing the data.
FelcherPB1996  (See Appendix A.)

For PNR, Fredrikze’s formalism FredrikzePB2001  was used to determine the
optics’ polarization efficiency and the direct beam (DB) spectra from the
measured four reflectivities of the DB. The polarization corrections with the
error propagation were made following Wilder’s formalism. WildesNN2006
Simulations of the reflectivities were based on the Parratt formalism. Reflpak
A rough interface was modeled as a sequence of very thin slices whose SLDs
vary, following an error function so as to interpolate between adjacent
layers. Fitz2005  The instrumental resolution was handled by Gaussian
convolution. The GenCurvefit program GenCurvefit  using the genetic algorithm
was employed for the model optimization.

##  III Experimental Results and Analysis

###  III.1 Depth profile of Saturated Magnetization

Figure 2:  (Color online) (a) XRR data and (b) PNR data taken at saturation
field $+1.11$ T. (c) Depth profiles of neutron nuclear SLD (thin black line,
left), neutron magnetic SLD (thick red line, left) and X-ray SLD (dashed green
line, right). Only the real parts of the X-ray and the nuclear SLDs are shown.

The XRR data and the specular PNR data taken at a saturation field of $+1.11$
T are displayed in Figs. 2a and 2b, respectively. The reflectivities are shown
as functions of the momentum transfer perpendicular to the film plane
$Q_{z}=4\pi sin\theta/\lambda$, where $\theta$ is the incident angle and
$\lambda$ is the wavelength of the radiation source. Since there is no spin
flip scattering exists at saturation, we measured only $R^{+}$ (=$R^{++}$) and
$R^{-}$ (=$R^{--}$). Because of the sample’s high saturation magnetization,
$R^{+}\gg R^{-}$ for $Q_{z}$ above the critical edges ($Q_{z}>0.018$ Å-1).
There is an interesting feature in the PNR data below the critical edges: both
$R^{+}$ and $R^{-}$ showed frustrated total reflection. This is due to the
enhanced absorption at resonant conditions. MaazaPLA1996  Simulations show
that the $Q$ separation between the two dips strongly depends on the
magnetization along the field direction, especially that of the Sm-Co layer,
which is the locus of absorption. Therefore, this separation, as well as the
splitting between $R^{+}$ and $R^{-}$ above the critical edge, are direct
indications of the sample’s magnetization.

The chemical and the magnetic SLD profiles were determined by fitting the XRR
and the PNR data simultaneously with the same structural model.Model  Since
there are well tabulated values for the neutron and the X-ray scattering
length and absorption length for each element, both the real and imaginary
parts of the X-ray and the neutron’s nuclear SLDs were calculated from the
chemical composition and the film density. Therefore a single parameter was
used to determine both the X-ray and the neutron nuclear SLDs for each layer
that has a well defined composition. This parameter is the mass density for
that particular layer. However, the intermixed interface has a complex
chemical composition that extends $>~{}5$ nm in depth. ChoiPRB2007 ;
LiuAPL2008  The X-ray and the neutron’s nuclear SLDs in the intermixed region
need to be determined separately, which was done by introducing an intermixed
layer between the Fe and Sm-Co layers. Therefore, both the nuclear and the
magnetic SLD profiles can vary in more sophisticated ways and they are less
correlated to each other over the intermixed region. Actually, the peak and
dip positions in $R^{-}$ between 0.025 and 0.045 Å can not be reproduced
without introducing this intermixed layer, which indicates that a single error
function is not sufficient to model the nuclear and/or the magnetic SLD
profiles at the Fe/Sm-Co interface.

Table 1:  The layer thickness, _rms_ interface roughness, the nuclear and magnetic SLDs, saturated magnetization $M_{S}$, exchange stiffness $A$ and uniaxial anisotropy $K$ in the layers of interest from the best fits. Nuclear and magnetic SLDs and $M_{S}$ are from fitting the $+1.11$-T data (Sec. III.1). $A$ and $K$ are from fitting the $+0.39$-T data (Sec. III.2). Typical literature values are $M_{S}=1700~{}(550)$ emu/cc, $A=2.8~{}(1.2)$ ergs/cm, and $K=10^{3}~{}(5\times 10^{7}$) ergs/cc for the Fe (Sm-Co) layer. FullertonPRB1998  (∗Literature value of $K_{Fe}$ was used during the optimization.) | Fe |  | mixed |  | Sm-Co
---|---|---|---|---|---
Thickness (nm) | $7.4\pm 0.3$ |  | $4.2\pm 0.3$ |  | $20.1\pm 0.2$
Roughness (nm) |  | $4.3\pm 0.3$ |  | $2.6\pm 0.2$ |
$\rho_{nuc}$ ($10^{-6}$ Å-2) | $7.9\pm 0.2$ |  | $5.2\pm 0.2$ |  | $1.64\pm 0.04$
$\rho_{mag}$ ($10^{-6}$ Å-2) | $4.9\pm 0.1$ |  | $2.5\pm 0.1$ |  | $1.62\pm 0.03$
$M_{s}$ (emu/cc) | $1700\pm 50$ |  | $890\pm 50$ |  | $570\pm 10$
$M_{+0.39~{}T}$ (emu/cc) | $1540\pm 10$ |  | $730\pm 30$ |  | $420\pm 10$
$A$ (10-6 ergs/cm) | $2.6\pm 0.1$ |  | $2.6\pm 0.1$ |  | $1.2\pm 0.1$
$K$ (106 ergs/cc) | 0.001∗ |  | $3.0\pm 0.2$ |  | $27\pm 1$

The best-fit curves for the reflectivity data are overlaid on the data, and
the SLD depth profiles are shown in Fig. 2c. The SLD profiles show sharp
transitions at the Cr/MgO and the Sm-Co/Cr interfaces with _rms_ roughnesses
of $\sim$ 0.4 nm for both interfaces. However, the SLD profiles gradually
change between the Fe and Sm-Co layers over a distance of $\sim 8$ nm. Such a
large intermixed region is consistent with previous energy-dispersive-
spectroscopy (EDS) results from samples fabricated under the same conditions.
ChoiPRB2007  In the intermixed region, there appears a shoulder in the nuclear
SLD profile (Fig. 2c), showing that it is chemically rich in Fe. However, this
feature is not present in the magnetic SLD profile. This will be discussed
below. The parameters of interest are listed in Table 1, which shows that both
the Fe and the Sm-Co layers have their saturation magnetizations close to the
literature values FullertonPRB1998  despite the large intermixing at the
interface.

###  III.2 Depth Profiles of Exchange Stiffness and Uniaxial Anisotropy

Figure 3:  (Color online) (a) PNR data taken at $+0.39$ T after saturation in
$-1.11$ T. $R^{+-}$ and $R^{-+}$ are offset by a factor of 0.1 for clarity.
(b) The depth profiles of the magnetization vector $\vec{M}$ (obtained from
micromagnetic simulations) that yields the best fit to the PNR data. The field
direction is along $+\hat{x}$.

In order to get insight into the intrinsic magnetic properties, besides
$M_{S}$, in the intermixed region, we also studied the magnetization structure
in a demagnetized state. The PNR data were collected at $+0.39$ T after the
negative saturation in $-1.11$ T. The field is sufficiently high so that the
specular SF reflection showed at off-specular positions on the detector due to
the Zeeman effect, as shown in Fig. 6. Therefore, in the analysis the momentum
transfer $Q_{z}$ was modified according to Eq. 1. FelcherNature1995 ;
FelcherPB1996  It is worth noting that the off-specular scattering in the SF
reflectivity is dominated by the Zeeman effect in our data and possible off-
specular scattering due to in-plane magnetic domains is not distinguishable
from the background, and is therefore not considered in the data.

The PNR data are shown in Fig. 3a. In contrast to the saturation case,
$R^{--}$ is higher than $R^{++}$ for most $Q_{z}$ values above the critical
edge, which indicates that the average $M_{x}$ is still along the negative
applied field direction. Both the splitting between $R^{--}$ and $R^{++}$ and
the separation between the dips below the critical edge are smaller because
the average $M_{x}$ is much lower than $M_{s}$. At the same time, there is
significant SF scattering of the same amplitude as the NSF scattering,
indicating a large in-plane magnetization component perpendicular to the field
direction ($M_{y}$). This is caused by the magnetization spiral structure
observed in exchange-coupled bilayers during demagnetization. DonovanPRL2002
Figure 3b plots the depth profile of the magnetization vector $\vec{M}$ that
yields the best fit, which is obtained from micromagnetic simulations. Our PNR
studies do not reveal the chirality of $\vec{M}$ so that a positive $M_{y}$ is
used for convenience. As expected, the magnetic moment of the Sm-Co layer is
along the negative field direction and the magnetization vector rotates into
the intermixed region and the Fe layer.

Figure 4:  (Color online) Top: The depth profile of the nuclear SLD in gray
scale in the unit of $10^{-6}$ Å-2. Bottom: The depth profile of the exchange
stiffness $A$ (left, red solid line) and the uniaxial anisotropy $K$ (right,
blue dashed line) which gives the best fit to the PNR data.

Rohlsberger _et al._ have shown that the depth profile of the magnetization
vector in exchange coupled bilayers agrees with the micromagnetic simulation
from the 1D spin-chain model. RohlsbergerPRL2002  Fitzsimmons _et al._ further
determined the micromagnetic parameters for individual layers in the exchange
coupled DyFe${}_{2}2$/YFe2 superlattice using combined studies of PNR and
micromagnetic simulations. FitzPRB2006  For the Fe/Sm-Co spring magnet studied
here, the Fe and the Sm-Co layers are largely intermixed for $\sim 8$ nm at
the interface and it is naturally to expect that the intrinsic magnetic
properties, i.e. uniaxial anisotropy $K$ and exchange stiffness $A$, vary
gradually due to the gradation of the chemical composition. Hence, we extended
Fitzsimmons’ approach and constructed depth profiles of the intrinsic magnetic
properties. Given a profile of the micromagnetic parameters, the equilibrium
magnetic structure can be simulated by energy minimization via the 1D spin-
chain model, FullertonPRB1998  from which the neutron’s magnetic scattering
potential can be directly calculated followed by the spin-dependent neutron
reflectivities. Therefore, the intrinsic magnetic properties can be optimized
to yield the best fit to the experimental PNR data.

These profiles of $K$ and $A$ are built up of layers identical to those of the
chemical structure as determined from the fits to the XRR and the PNR data in
saturations. The gradual changes between the layers are computed using the
interface roughness in the same way as is done for the SLDs. When fitting, the
chemical structure and the roughnesses are fixed to those already determined.
$M$ is allowed to be lower than $M_{s}$ to account for the effect from the
minor soft phase mentioned above. The model optimization is not sensitive to
the $K_{Fe}$ because the associated magnetic energy is negligible, and
therefore the literature value of $K_{Fe}$ is used. The error bars of the
parameters only reflect the statistical error and the accuracy of the values
depends on the model. Therefore it is important here to have the additional
layer between the Fe and the Sm-Co layers in the model because the exact
mapping function from the chemical structure to the micromagnetic properties
is unknown. The layer has its own free micromagnetic parameters so that $M$,
$A$ and $K$ in the intermixed region are able to vary more independently from
those in the Fe layer and the Sm-Co layer. We checked the robustness of the
model by allowing different interface roughnesses for $M$, $A$, and $K$ and
found that the optimized micromagnetic parameters have similar depth
dependencies.

The depth profiles of $A$ and $K$ from the best fit are shown in Fig. 4 and
the values of $M$, $A$ and $K$ for layers of interest are listed in Table 1.
Both the parameters $A$ and $K$ show monotonic depth dependence as expected.
$M_{Sm-Co}$ is $\sim 20\%$ lower than the saturation value, from which we
estimated that $\sim 10\%$ of the Sm-Co was reversed due to the minor phase.
Therefore, the 1D spin-chain model is approximately valid and yields fairly
good fits, which were overlaid on the data in Fig. 3a. As also listed in Table
1, in the intermixed region, $A$ is close to that of the Fe layer, but $K$ is
much lower than the average value between the Fe and the Sm-Co layers.

##  IV Discussion and Summary

Figure 5:  (Color online) Depth profiles of (a) the saturation magnetization,
and (b) the exchange stiffness (left) and the uniaxial anisotropy (right)
calculated by assuming that the intermixed region is a mixture of the Fe phase
and the Sm-Co phase (dashed lines). The experimentally determined profiles are
plotted as solid lines. Also shown in the top panel is the depth profile of
the nuclear SLD in gray scale. The unit is $10^{-6}$ Å-2.

By assuming that the intermixed region is composed of a mixture of the Fe and
Sm-Co phases, the saturation magnetization can be computed from the relative
volume of both phases via $M=f_{s}M_{s}+f_{h}M_{h}$, where $f_{s}$ ($M_{s}$)
and $f_{h}$ ($M_{h}$) are the relative volume (the saturation magnetization)
of the Fe phase and the Sm-Co phase, respectively. From the depth profile of
the nuclear SLD, the relative volume of the two phases can be determined.
Following this approach, the calculated saturation magnetization profile is
shown in Fig. 5a. Also shown is the saturation magnetization profile
calculated from the magnetic SLD profile. Clearly, the mixture model predicted
slightly larger saturation magnetization in the intermixed region. This agrees
with the suggestion that new compound(s) is (are) formed in the intermixed
region, JiangAPL2004  possibly due to interdiffusion between the Fe and Sm-Co
layers. The estimation shows that the average magnetization of the whole
sample is reduced by $\sim 4\%$ due to the intermixing of the two components.
The exchange stiffness and the anisotropy of a two phase mixture can also be
approximately estimated by the relative volume: $K=f_{s}K_{s}+f_{h}K_{h}$ and
$A=f_{s}A_{s}+f_{h}A_{h}$. SkomskiPRB1993  The results are shown in Fig. 5b.
In comparison to those determined from the PNR experiments, the model predicts
higher anisotropy but lower exchange stiffness in the intermixed region.
Therefore, we conclude that the optimized Fe/Sm-Co interface for a spring
magnet reduces the average magnetization.

Interfaces with graded magnetic properties are desirable for superior
performances in exchange coupling systems, such as spring magnets and magnetic
media. JiangAPL2004 ; VictoraIEEE2005  A straightforward approach to achieve
magnetically graded interfaces is to make structurally graded interfaces.
ChoiAPL2007 ; GollJAP2008 ; KirbyPRB2010  The micromagnetic properties could
be predicted from the composition. However, it is challenging if multiple
phases coexist even when the micromagnetic properties of each individual phase
are known, due to lacking of prior knowledge of the interphase coupling. As
shown above, the intrinsic magnetic properties in the intermixed Fe/Sm-Co
interface deviates from the results predicted by the mixture model. Therefore,
a complex correlation between the magnetic properties and the nominal
composition may exist at the chemically graded interfaces.

In summary, we determined the intrinsic magnetic properties of a Fe/Sm-Co
bilayer under the nearly optimized fabrication condition for spring magnets.
We determined that the intermixed interface between the Fe and Sm-Co layers
extends $\sim 8$ nm, where the magnetic properties changes gradually, as
expected. We compared the magnetic properties with the prediction of a mixture
model and found, in the intermixed region, the saturation magnetization is
slightly lower than that estimated from the model, but the exchange stiffness
is higher. This observation indicates that the intermixed interface is
efficient for magnetically coupling the Fe and Sm-Co layers but at the cost of
the average magnetization. The intrinsic anisotropy is also lower than the
value from the model. Overall, the intrinsic magnetic properties in the
structurally intermixed region may not be predicted correctly by the mixture
model of nominal compositions, which is worth keeping in mind when designing
the magnetically graded interfaces.

###### Acknowledgements.

We thank Gian P. Felcher for helpful discussions. Research at Argonne was
supported by the U.S. Department of Energy, Office of Basic Energy Sciences,
Division of Materials Sciences and Engineering under Award No.DE-
AC02-06CH11357. Research at Oak Ridge National Laboratory’s Spallation Neutron
Source was sponsored by the Scientific User Facilities Division, Office of
Basic Energy Sciences, U. S. Department of Energy.

##  Appendix A Zeeman effect and off-specular scattering

The Zeeman effect in PNR was firstly clarified by Felcher  _et. al_.
FelcherNature1995 ; FelcherPB1996  The effect shows up in the SF
reflectivities when the difference of the Zeeman energy for spin-up and spin-
down neutrons is not negligible in a sufficiently high magnetic field. Let
$k_{\perp}$ and $k_{//}$ label the perpendicular and the parallel components
of the wavevector with respect to the sample surface in the _vacuum_ region,
_i.e._ , before neutrons enter the magnetic-field region of interest.
$k_{\perp}=2\pi sin\theta/\lambda\ll k_{//}$ at glancing angles. There is
negligible energy exchange between the sample and the neutrons for elastic
scattering, therefore the Zeeman energy change after SF scattering accompanies
a change of the kinetic energy, which is associated with $\vec{k}$. The energy
change after spin-flip is twice the Zeeman energy. If the sample is
magnetically homogenous in the film plane as seen by the neutrons, the energy
change is totally associated with $k_{\perp}$. Since $k_{\perp}$’s are small
quantities, the outgoing angles of the spin-flipped neutrons are different
from the incoming angles. Therefore the SF reflections appear at off-specular
positions, which have a characteristic field dependence. FelcherNature1995 ;
FelcherPB1996  The Zeeman effect on the SF reflections becomes clear in a
magnetic field on $\sim 0.1$ T or higher. Figure 6 shows an example. It is
clear that the locus of the SF reflections deviates away from the specular
position, but are described by the prediction after considering the Zeeman
effect. Beside the reflection due to the Zeeman effect, there is no noticeable
off-specular scattering in the SF channels; therefore, the scattering from the
in-plane magnetic domains are not considered during the data analysis.
Consequently, the momentum transfer $Q_{z}$ are no longer equal to
$2k_{\perp}$ for spin-flipped neutrons, but follows

$\begin{array}[]{rcl}Q_{z}^{+-}&=&k_{\perp}+\sqrt{k_{\perp}^{2}+2CB}\\\
Q_{z}^{-+}&=&k_{\perp}+\sqrt{k_{\perp}^{2}-2CB},\end{array}$ (1)

where $C=|2m_{n}\mu_{n}/\hbar^{2}|=2.906\times 10^{-5}$ Å-2/T and $B=\mu_{0}H$
is the applied magnetic field. $R^{-+}$ is forbidden when
$k_{\perp}<\sqrt{2CB}$ because there is no enough kinetic energy to compensate
the Zeeman energy change. vandeKruijsPB2000  At the same time, the minimum
$Q_{z}^{+-}$ is also $\sqrt{2CB}$ for finite reflectivity. Therefore, there is
a cutoff $Q_{z}$ for non-zero SF scattering, $Q_{SFcutoff}=\sqrt{2CB}$.

Figure 6:  (Color online) The coutour map of the neutron intensities measured
with the polarization analysis, which are presented as functions of the
neutron wavelengths $\lambda$. The data were collected from a Fe/Sm-Co spring
magnet sample at an incident angle $\theta_{i}=0.24^{o}$ in an external field
of +0.39 T after a negative saturation. The dashed lines indicate the specular
reflection positions without considering the Zeeman effect, i.e.
$\theta_{f}=\theta_{i}$. The solid lines in the SF channels indicate the off-
specular scattering following the the Zeeman effect, i.e.
$\sin^{2}\theta_{f}=\sin^{2}\theta_{i}\pm\frac{cB\lambda^{2}}{2\pi^{2}}$.

Actually, the kinetic energy also changes when neutrons enter and leave the
magnetic-field region of interest, but the change of $\vec{k}$ is essentially
in $k_{//}$ in these cases, Chatterji2006  which changes the time-of-flight.
However, the relative change of $k_{//}$ is so small in laboratory fields, and
TOF only changes $\sim 10$ ns, while the width of the neutron pulse is $\sim
0.2$ ms. Therefore, it does not result in any observable affect.

SF reflection is typically weak when the Zeeman effect matters, therefore the
Zeeman effect is usually not considered during data reduction and analysis.
Reflpak ; Fitz2005  However, we observed significant SF scattering at $+0.39$
T, which is sufficiently high so that the Zeeman effect needs to be
considered. We adopted the generalized algorithm of the GEPORE Chatterji2006
for the PNR simulations. $\vec{B}$ rather than $\vec{M}$ is used to calculate
the magnetic scattering potential since $H$ is now comparable to $M$ and the
spin eigenstates are aligned along $\vec{B}$ rather than $\vec{M}$.
Considering the problem along the normal direction of the sample surface (let
it be $\hat{z}$), it is reduced to a pair of coupled 1D differential wave
equations, Chatterji2006

$\begin{array}[]{rcl}\left[-\frac{\hbar^{2}}{2m}k_{0}^{2}+V_{++}(z)-E\right]\Psi_{+}(z)+V_{+-}\Psi_{-}(z)&=&0\\\
\left[-\frac{\hbar^{2}}{2m}k_{0}^{2}+V_{--}(z)-E\right]\Psi_{-}(z)+V_{-+}\Psi_{+}(z)&=&0.\end{array}$
(2)

$V$ is the potential operator. $\Psi_{\pm}(z)$’s are the wave functions for
spin-up and spin down states. $k_{0}$ is the wavevector in vacuum, which can
be either a real number or a pure imaginary number. States with
${k_{0}}^{2}\pm CB<0$ correspond to evanescent waves outside the sample. Due
to the Zeeman energy, the coupled eigenstates of $\Psi_{+}$ and $\Psi_{-}$
have different wavevectors outside the sample, namely $\sqrt{{k_{0}}^{2}-CB}$
and $\sqrt{{k_{0}}^{2}+CB}$, which has been considered to set up the boundary
conditions at the sample surface in the modified algorithm to take account the
Zeeman effect.

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