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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

dipole origin.

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 [1] 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 [2]. 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 [3].

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-

tial energy:

?2A

MS

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

field U?z?,

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

@2m

@z2?

?

H?z? ?!

?

?

m ? 0;

(1)

MS

q2;

(2)

2mq2:

(3)

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?] [3]. The spin waves tunnel through the

barrier.

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

barrier width.

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 [10]. 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 [11]. The microwave

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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 [12]. Two sequences of

snapshots for different delay times are shown. Figure 3(a)

[12] corresponds to the enhancement of the local field by

the dc current, whereas the images shown in Fig. 3(b) [12]

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

coordinate.

Figure 3(a) [12] 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

unaffected [13].

On the other hand, the region of the reduced field

inhibits propagation of spin waves as seen in Fig. 3(b)

[12]. 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

j

0+ H

j

0

- ∆ ∆ ∆ ∆

∆ ∆ ∆ ∆0

∆ ∆ ∆ ∆

j

- H

ω ω ω ω

0

0

H

H

H

Frequency

Wave vector

ω ω ω ω

0

∆ ∆ ∆ ∆

(z)

∆ ∆ ∆ ∆

w

21

z

z

z

H0

Hj

FIG. 2.

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

H H

MW antenna

MW antenna

dc conductorconductor

Laser

beam

beam

Laser

MWMW

modulator

modulator

Pulse

generator

generator

PulseMW MW

generator

generator

dc current

source

source

current

x

y

z

FIG. 1.

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

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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.,

[14]), 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

4?MS

Z1

?1Gxx?z;z0?mx?z0?dz0?

?

?2H?z?

H?z? ?

!2

?

mx?z? ? 0;

(4)

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 [15]].

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

is used:

mx?z? ?

Z1

?1

??0

0?P?q? ? 1?g ? i?00

0? i?00

0?exp?iqz?hhs

xiq

f1 ? ?0

0?P?q? ? 1?dq;

(5)

where ?0? 4?MSH0=?H2

ceptibility at H ? H0, P?q? is the dipole matrix element

calculated in [16], 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-

FIG. 4.

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

300

250

200

Time (ns)

150

100

50

0

1

2

3

4

5

6

7

300

250

200

Time (ns)

150

100

50

0

1

2

3

4

5

6

7

z (mm)

Field

z (mm)

Field

a)

b)

FIG. 3.

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

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damping was taken into account in the usual way [14],

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? [11]. 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

[12], 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

dipole interaction.

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: demokrit@physik.uni-kl.de

†Also at Department of Radiophysics, Taras Shevchenko

National University of Kiev, Kiev, Ukraine.

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[12] See EPAPS Document No. E-PRLTAO-93-007424 for

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The document may also be reached via the EPAPS home-

page (http://www.aip.org/pubservs/epaps.html) or from

ftp.aip.org in the directory /epaps/. See the EPAPS home-

page for more information.

[13] An observed weak reflection is due to the nonadiabatical

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implies that the wavelength of the wave is much smaller

than the lateral scale of the field inhomogeneity. In the

experiment, however, these two values are of the same

order of magnitude.

[14] A.G.GurevichandG. A.

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