Tunneling of Dipolar Spin Waves through a Region of Inhomogeneous Magnetic Field
S.O. Demokritov,*A. A. Serga,†A. Andre ´,V. E. Demidov, M. P. Kostylev, and B. Hillebrands
Fachbereich Physik and Focschungsschwerpunkt MINAS, Technische Universita ¨t Kaiserslautern, 67663 Kaiserslautern, Germany
A. N. Slavin
Department of Physics, Oakland University, Rochester, Michigan 48309, USA
(Received 16 November 2003; published 20 July 2004)
We show experimentally and by numerical simulations that spin waves propagating in a magnetic
film can pass through a region of a magnetic field inhomogeneity or they can be reflected by the region
depending on the sign of the inhomogeneity. If the reflecting region is narrow enough, spin-wave
tunneling takes place.We investigate the tunneling mechanism and demonstrate that it has a magnetic
DOI: 10.1103/PhysRevLett.93.047201PACS numbers: 75.30.Ds, 75.40.Gb
The effect of tunneling discovered in 1928 by Gamov
and co-workers in quantum mechanics  is a striking
manifestation of the wave nature of quantum-mechanical
particles. This effect, however, manifests itself for waves
of a different nature, and during the last decade tunneling
of electromagnetic and acoustic waves through spatial
regions where existence of these waves is prohibited
attracted a lot of attention . Spin waves can also
demonstrate a tunneling effect, and since the frequency
of spin waves depends on the applied magnetic field, a
tunneling barrier for propagating spin waves can be cre-
ated by a magnetic field inhomogeneity.
Propagation of spin waves in an inhomogeneous mag-
netic field was discussed for the first time in the 1960s
[3–6]. It was Schlo ¨mann who first noticed a close simi-
larity between propagation of exchange dominated spin
waves and the motion a quantum-mechanical particle .
In fact, neglecting the magnetic dipole interaction and
magnetic anisotropies the Landau-Lifshitz equation de-
scribing magnetic dynamics can be rewritten in the form
of the stationary Schro ¨dinger equation with the dynamic
magnetization m / exp?i!t? being the analog of a wave
function and the magnetic field playing the role of poten-
where A is the exchange stiffness, MSis the saturation
magnetization, and ? is the gyromagnetic ratio of the
medium. The dispersion relation for a plane spin wave
[m / exp?iqz?] can then be written as
! ? ??z? ?2?A
where ??z? ? ?H?z? is the gap of the spectrum, which is
reminiscent to the dispersion of a particle in a potential
E ? U?z? ?? h2
Thus, if a spin wave of frequency ! enters a region
where the field H ? H?z? (and the gap) varies, the wave
m ? 0;
keeps to propagate through the inhomogeneous field,
albeit with changing wave vector, q ? q?z?, to fulfill the
dispersion law Eq. (2). However, if the value of the gap
locally exceeds !, there exists no real wave vector any-
more to fulfill the dispersion law for this frequency. The
wave is reflected from this region, which thus can be
considered as a potential barrier. Recently it was shown
that a strongly inhomogeneous internal field in magnetic
a microstripe can cause such turning points within the
stripe which reflect spin waves and thus create a spin-
wave well [7–9].
A theoretical analysis of spin-wave reflection from a
field inhomogeneity taking into account only the ex-
change interaction and neglecting the magnetic dipole
interaction shows that the dynamic magnetization beyond
the turning point is not zero: it just changes its depen-
dence on z from sinusoidal [m / exp?iqz?] to exponential
[m / exp???z?] . The spin waves tunnel through the
In this Letter we experimentally observe and investi-
gate the effect of spin-wave tunneling. In contrast to
previous studies [3,4,7] the magnetic dipole interaction
dominantly determines the properties of spin waves under
consideration.The character of the magnetic dipole inter-
action is nonlocal, and, as a consequence, the tunneling
transmission coefficient depends nonexponentially on the
The used experimental setup is schematically shown in
Fig. 1. Microwave spin wave packets in an optically trans-
parent yttrium-iron-garnet (YIG) film are generated by a
strip-line antenna and are detected using the time- and
space-resolved Brillouin light scattering (BLS) tech-
nique . Both a homogeneous external field and the
static magnetization are oriented in the plane of the film
parallel to the propagation direction of the spin waves, z.
In this case the dynamic magnetization components are
mxand my. Such an orientation of the field and the
magnetization corresponds to the backward volume mag-
netostatic wave (BVMSW) geometry, characterized by a
negative group velocity of the waves . The microwave
PH YSICA LR EVI EW L ET T ERS
23 JULY 2004
VOLUME 93, NUMBER 4
2004 The American Physical Society 047201-1
excitation part consists of a microwave generator and a
modulator, which is controlled by a pulse generator (pulse
length 10–30 ns) and connected to the antenna situated
on the YIG film for spin-wave generation. BVMSW
packets are generated with a carrier frequency ! ? 2? ?
7:095 GHz and a carrier wave vector q ? 210–220 cm?1,
thevalue of q being determined by the dispersion relation
of the spin wave in an external field of H0? 1840 Oe. A
narrow conductor of 50 ?m diameter mounted across the
film carries a dc current. It is used to create a local
inhomogeneous field, Hj?z?. Depending on the direction
of the dc current, the total field (and, thus, the gap of the
spin-wave spectrum) is locally either enhanced or re-
ducedby the oerstedfield of the current up to a maximum
field inhomogeneity of about 200 Oe.
The idea of the experiment is further illustrated in
Fig. 2. As mentioned above, the frequency of the
BVMSW decreases with increasing wave vector, and
the allowed states are situated below the zero-wave-
vector gap, ?0. Thus, to realize a scenario of spin-
wave reflection from a field inhomogeneity, one should
rather decrease, than increase the local field (gap). The
inset of Fig. 2 illustrates the geometry of the field, the
profile of the gap, and the creation of turning points,
which are determined by the following condition: ! ?
??H?z? ? H0? Hj?z??.Asisseen fromtheinset ofFig. 2,
this equation has two solutions, z1and z2. The interval
between the turning points is a prohibited region with no
spin-wave state with !. The width of the interval w ?
z2? z1is considered as the barrier width. On the other
hand, an enhanced magnetic field does not essentially
disturb the propagation: spin waves propagate through
the inhomogeneity while increasing the wave vector ac-
cording to the local field.
The propagation of spin-wave packets through the field
inhomogeneity, measured using the space- and time-
resolved Brillouin light scattering technique, is illus-
trated by Fig. 3 and the movies . Two sequences of
snapshots for different delay times are shown. Figure 3(a)
 corresponds to the enhancement of the local field by
the dc current, whereas the images shown in Fig. 3(b) 
were obtained when the local field was reduced. Each
snapshot displays the distribution of the spin-wave inten-
sity (normalized to its maximum) as a function of the z
Figure 3(a)  demonstrates no significant reflection
of the spin waves. As is discussed above, a region with a
slightly enhanced local field cannot contain a turning
point for a BVMSW packet. The wave accommodates its
wave vector according to the dispersion law and passes
through the region of the field inhomogeneity almost
On the other hand, the region of the reduced field
inhibits propagation of spin waves as seen in Fig. 3(b)
. The spin-wave packet, however, is only partially
reflected, and a certain part of it is transmitted through
the barrier. The transmission and reflection probabilities
depend on the barrier width and height, and the carrier
wavevector of thewave. Indeed, this effect is reminiscent
of the quasiclassical problem of a particle reflection and
tunneling in quantum mechanics.
To understand the physics of the observed spin-wave
tunneling the transmission coefficient, T, defined for the
intensity of the wave, was measured as a function of the
dc current. In agreement with the above consideration, T
is close to unity until the current reaches a certain value,
jc, which corresponds to the creation of a turning point
just below the dc conductor. Figure 4(a) demonstrates
the normalized barrier transmission coefficient, Tb?
T?j?=T?jc?. As is seen in the inset of Fig. 2, it is possible
to increase the distance between the turning points (i.e.,
the barrier width) by increasing the value of the dc
current. The profile of the inhomogeneous field, the posi-
tions of the turning points and, accordingly, the width
- ∆ ∆ ∆ ∆
∆ ∆ ∆ ∆0
∆ ∆ ∆ ∆
ω ω ω ω
ω ω ω ω
∆ ∆ ∆ ∆
∆ ∆ ∆ ∆
gap of the spectrum. The inset illustrates the profile of the gap,
the creation of the turning points, and the forbidden interval
due to the inhomogeneous field, caused by the dc conductor.
Spectrum of BVMSW in a magnetic film. ??z? is the
setup in the forward scattering geometry with space and time
resolution for the study of spin-wave propagation through an
inhomogeneous magnetic field. For a discussion of the compo-
nents, see the main text.
Schematic layout of the Brillouin light scattering
PH YSICA LR EVI EW L ET T ERS
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barrier, w, can be easily calculated for a given j in the
experimental geometry. For the sake of clarity Tbis
shown as a function of w. With increasing w the trans-
mission coefficient decreases. From a comparison of the
measured dependency Tb?w? with the predictions of dif-
ferent theoretical models, the origin of the spin-wave
tunneling can be clarified.
Despite the obvious similarity to the quantum me-
chanic problem, the physics of the spin-wave tunneling
effect is more complex since the long-range magnetic
dipole interaction important for magnetic systems must
be taken into account.
If both magnetic dipole and exchange interactions are
taken into account, the Landau-Lifshitz equation of mo-
tion for the variable magnetization (which, in particular,
describes spin-wave processes in a magnetic film) can be
reduced to an integro-differential equation (see, e.g.,
), where a differential operator of second order de-
scribes the inhomogeneous exchange interaction [see
Eq. (1)], while the integral operator (having the dipolar
Green’s function as a kernel) describes the nonlocal di-
pole-dipole interaction. For long-wavelength spin waves
(q ? 102–103cm?1) we neglect the small contribution of
inhomogeneous exchange interaction and assume har-
monic time dependence for the variable magnetization /
exp?i!t?. The resulting equation for dipolar spin waves
turns out to be a purely integral equation. This equation
written for the transverse component mxof the variable
magnetization in a magnetic film has the form
mx?z? ? 0;
where Gxx?z;z0? is a component of the magnetostatic
Green’s function in the coordinate representation (z is
the coordinate along the direction of wave propagation)
derived for a magnetic film of a finite thickness in our
previous work [see Eq. (3) in ].
The analytical solution of the integral Eq. (4) for any
realistic profile of the inhomogeneous external magnetic
field H?z? is rather complex. Therefore it has been solved
numerically for the experimental field profile shown in
the inset of the Fig. 2 using the iterational convergence
method. As a zero-order approximation for mxthe ana-
lytical solution of Eq. (4) for the spatially homogeneous
static magnetic field H?z? ? H0with an excitation source
0?P?q? ? 1?g ? i?00
f1 ? ?0
0?P?q? ? 1?dq;
where ?0? 4?MSH0=?H2
ceptibility at H ? H0, P?q? is the dipole matrix element
calculated in , and hhsxiqis the spatial Fourier compo-
nent of hsx?z?, the magnetic microwave field of the antenna
which excites the propagating waves in the film. Wave
0? ?!=??2? is the dynamic sus-
the barrier width. Full circles: experimental values. Solid line:
results of the numerical simulations using Eq. (5). (b) Profiles
of the dynamic magnetization, mx, (full circles) and dipole
field, hx, (open squares) in the wave, normalized to the values
of the incident wave. The dashed line is a guide to the eye.
Incident, reflected, and tunnelled waves are indicated by the
arrows. The barrier is shown by the hatched area.
(a) Barrier transmission coefficient as a function of
with local field inhomogeneity, observed by space- and time-
resolved BLS. (a) Field/gap has a local maximum, creating a
potential dip for the wave; (b) field/gap has a local minimum,
creating a potential barrier for the wave. The maximum
absolute value of Hjis the same for both cases and is 56 Oe.
An animated version of this figure can be viewed in the
online/PDF version of this Letter.
Propagation of a spin-wave packet across a YIG film
PH YSICA LR EVI EWL ET T ERS
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VOLUME 93, NUMBER 4
damping was taken into account in the usual way ,
i.e., by adding an imaginary part to the static magnetic
field: H0! H0? i?H, where ?H is the linewidth of
ferromagnetic resonance of the film.
The condition f1 ? ?0
provides the dispersion equation for BVMSW. Since for
large fields it can be satisfied for real values of q, the
amplitude of the variable magnetization is relatively
large. Contrary to that, for small H, where the above
condition cannot be satisfied, the denominator in Eq. (5)
becomes large, which leads to the decrease of magnitude
of the dynamic magnetization.
A numerical algorithm providing fast convergence of
the solution of Eq. (4) with the initial function mx?z?
defined by Eq. (5) has been constructed. We used the
fast Fourier transform algorithm for discrete values of z
and q and then employed by the Runge-Kutta technique
to solve the resulting system of differential equations.
The calculated dependence of the barrier transmission
coefficient on the barrier width, Tb?w?, is shown in
Fig. 4(a) by the solid line. Obviously, the obtained depen-
dence is not exponential (contrary to the exchange case),
due to the nonlocal character of the long-range magnetic
dipole interaction. The quantitative agreement between
the theory and the experiment is striking, especially if
one takes into account that no fitting parameters were
used in the numerical calculations. Such an agreement
allows us to conclude that the observed effect can be
interpreted as a tunneling effect of dipolar spin waves
through the field inhomogeneity. It is important to men-
tion that not only the dipole field, but also the dynamic
magnetization tunnels through the barrier. This is illus-
trated by Fig. 4(b), where the calculated profiles of the
dynamic magnetization and of the dipole field are pre-
sented. Both profiles are normalized by the values of the
incident wave. It is clearly seen that the observed tunnel-
ing effect is, indeed, a tunneling of the dynamic magne-
tization and the dipole field. In fact, both values are
connected by the local value of the dynamic susceptibil-
ity ? ? 4?MSH=?H2? ?!=??2? . Since as it was
mentioned above the value of the static field is just
slightly changed by the field inhomogeneity, the relations
between mxand hxare almost the same inside the barrier
and far from it. A standing wave on the left side of the
barrier is due to the interference between the incident and
reflected waves. This interference is observed in the ex-
periment if longer spin-wave pulses, as shown in Fig. 3
, are used.
In conclusion, we have demonstrated, both experimen-
tally and by numerical simulation, that tunneling of di-
polar spin-wave pulses takes place when it encounters a
‘‘well’’ -type localized inhomogeneity in the course of its
propagation in a magnetic film. The dependence of the
dynamic magnetization on the propagation coordinate,
m?z?, inside the prohibited region is nonexponential,
due to the nonlocal character of the dominating magnetic
0?P?q? ? 1?g ? 0 [see Eq. (5)]
Support by the Deutsche Forschungsgemeinschaft,
by the European Communities Human Potential pro-
gramme under Contract No. HRPN-CT-2002-00318
ULTRASWITCH, by European Commission under
Contract No. IST-2001-37334-NEXT, by NSF Grant
No. DMR-0072017, and by the joint NSF-DAAD Grant
No. INT-0128823 is gratefully acknowledged. M.K. is
indebted to the DAAD for financial support.
*Electronic address: email@example.com
†Also at Department of Radiophysics, Taras Shevchenko
National University of Kiev, Kiev, Ukraine.
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