ISSN 1063-7842, Technical Physics, 2009, Vol. 54, No. 1, pp. 82–88. © Pleiades Publishing, Ltd., 2009. Original Russian Text © M.M. Karpuk, D.A. Kostiuk, Yu.A. Kuzavko, V.G. Shavrov, 2009, published in Zhurnal Tekhnicheskoœ Fiziki, 2009, Vol. 79, No. 1, pp. 86–91.

SOLIDS

Reflection of Elastic Waves in a Crystal of Heusler Alloy Ni2MnGa in the Phase Transition Range M. M. Karpuka, D. A. Kostiukb, Yu. A. Kuzavkoc, and V. G. Shavrovd a

Politechnika Koszalinska, 75-343 ul. Kwiatkowskiego 6E, Koszalin, Poland Brest State Technical University, ul. Moskovskaya 267, Brest, 224017 Belarus c Polessk Agrarian Ecological Institute, National Academy of Sciences of Belarus, ul. Moskovskaya 204, Brest, 224020 Belarus e-mail: [email protected] d Institute of Radio Engineering and Electronics, Russian Academy of Sciences, ul. Mokhovaya 18, Moscow, 103907 Russia b

Received February 4, 2008

Abstract—The reflection of longitudinal and transverse acoustic waves from the free surface of the ferromagnetic shape memory alloy Ni2MnGa that is located in the ranges of the premartensite and martensite phase transformations is considered. The propagation directions and amplitudes of the waves reflected in the (001) plane of the crystal are determined. They acquire the character of substantially quasi-longitudinal and quasitransverse vibrations rather than being pure modes. The angles of wave reflection and conversion are shown to be effectively controlled by temperature and a magnetic field due to the colossal acoustic anisotropy of the crystal over the wide range of its phase transitions. Beginning from a certain critical angle of incidence of a quasitransverse wave, the quasi-longitudinal wave having appeared upon reflection becomes an accompanying surface vibration, and it can be emitted into the bulk of the crystal when the phase transition point is approached. Two angles of full conversion of an incident quasi-longitudinal wave into a quasi-transverse wave are established, and their temperature dependences are found. Trivisonno’s experimental data for the ultrasound velocity and absorption in an Ni2MnGa crystal are used to numerically estimate these acoustic effects. PACS numbers: 75.50.Cc, 72.55.+s DOI: 10.1134/S1063784209010125

INTRODUCTION

materials [1], and shape memory alloys [8–10]. The hierarchy of the reversible deformation achieved due to magnetostriction, the piezoelectric effect, and the shape memory effect is represented by the series 10–3, 10–2, and 10–1, respectively. Shape memory alloys are used for the production of thermosensitive force devices, sensors, microrobot engines, and detachable and permanent joints that do not require mechanical or another fixation [9]. The ferromagnetic Heusler alloy Ni2MnGa exhibits the best properties in this field. Using this unique alloy—specifically, the family of nonstoichiometric Ni2 + x + yMn1 – xGa1 – y alloys—researchers were able to achieve a magnetic field–controlled change of 6% in their linear dimensions, which corresponds to the theoretical limit of deformation upon the transformation from the high-temperature cubic (austenitic) to the low-temperature tetragonal (martensitic) phase.

When an elastic wave is reflected by the free surface of a solid medium, two reflected (longitudinal LA and transverse TA) waves always appear in the plane of incidence [1]. In [2–4], we demonstrated the possibility of effective field-assisted control of the angles of reflection and refraction and wave conversion in an easyplane antiferromagnet in the range of the orientational phase transition (PT) in magnetic field H applied to the basal xy crystal plane. Even in a magnet with isotropic elastic and magnetic properties, a strong anisotropy in dynamic elastic moduli arises near the PT, which changes the velocities of longitudinal and, especially, transverse sound. For example, in hematite α-Fe2O3, this change was experimentally found to be 50% (from 4200 to 2100 m/s) [5]. The induced anisotropy substantially exceeds the intrinsic anisotropy of acoustic crystals [6] and, thus, makes it possible to observe a number of linear and nonlinear acoustic effects [3, 4, 7] in ordinary crystals.

Note that martensitic transformations are structural diffusionless cooperative PTs. Their main specific feature consists in the fact that they proceed in a solid medium at low temperatures, where diffusion processes are frozen. Depending on the stoichiometric composition, the martensite transformation temperature TM can be located near room temperature and below the Curie temperature of the material.

Recently [8, 9], researchers have paid particular attention to the creation and study of a new class of functional materials the shapes and sizes of which can be reversibly controlled by a mechanical stress and an electric or magnetic field. These materials include magnetoacoustic materials (MAMs) [2–6], piezoelectric 82

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A martensitic PT is sometimes preceded by a premartensitic PT (PMPT) [8], which is characterized by substantial specific features in the velocity and absorption of longitudinal and transverse acoustic waves (see table) [11, 12]. Transverse mode TA2 is especially strongly softened in the vicinity of the PMPT with a wavevector k = (ξ, ξ, 0) at ξ = 0.33 and a polarization C' ------ changes from 2ρ 740 m/s (T = 300 K) to 614 m/s (TPM = 255 K) and 903 m/s (TM = 215 K) at a frequency f = 3.7 MHz [12], where C ' = (C11 – C12)/2 is the elastic modulus responsible for the propagation of this mode; theoretically, this modulus should vanish at ξ = 0.33 at the PT point.

vector e = ( 110 ). Its velocity St2 =

2

The LA velocity along this direction, S l = (C11 + C12 + 2C44)/(2ρ), is almost temperature independent. Thus, theoretically, the anisotropy A = C44 /C ' induced by the magnetic structural phase transition (MSPT) should be infinite, as in the case of MAM at the PT point. The experimentally achieved value for hematite α-Fe2O3 is A = 4, whereas A = 28 in Ni2MnGa upon the PMPT. 2

83

Acoustic parameters of a Ni2MnGa crystal with a density ρ = 8.13 × 103 kg/m3 T, K

Cl

C44

C' 1011

C11 C12

Vl

V44 103

Pa

V'

A

m/s

300 2.50 1.03 0.045 1.52 1.43 5.54 3.56 0.740 23 260 2.43 0.86 0.031 1.59 1.53 5.46 3.26 0.617 28 215 2.48 0.98 0.067 1.61 1.48 5.53 3.47 0.908 15

FORMULATION OF THE PROBLEM To explain this discrepancy, we study the reflection of LA and TA waves from the free (110) surface of the crystal during their oblique incidence in the (100) crystallographic plane, where the ultrasound softening effects are maximal. Let LA be incident at angle of α to the normal to the interface (y = 0) (Fig. 1). The y axis coincides with the [110] direction in the crystal. During reflection, the wave converts into LA specularly and into TA2 at angle of reflection β. Using the expression for the free energy of an Ni2MnGa crystal in [8], we can easily determine the LA and TA2 velocities in the (001) plane, 2

2

2

C 11 + C 44 ± ( C 11 – C 44 ) sin 2α + ( C 11 + C 44 ) cos 2α 2 -, S = -----------------------------------------------------------------------------------------------------------------------------------2ρ where the sign “+” corresponds to LA and the sign “–” corresponds to slow TA2. The rapid TA1 wave, which propagates in the same plane at the velocity S44 = C 44 /ρ and polarization e = [001], is related to LA and TA2 by neither a dispersion equation nor boundary conditions. The LA and TA2 velocities in the low-symmetry tetragonal martensitic phase are also specified by Eq. (1) upon the substitution C44 C66. Given the direction, polarization, and amplitude of the incident wave under certain elastic properties of the medium, we have to determine the propagation directions, polarizations, and amplitudes of the reflected waves. To solve this problem, we set up wave equations for the wave propagation in the medium and boundary conditions on the free crystal surface; in our case, they have the form [1] I

T iy +

∑

R

T iy = 0,

(2)

which represents the continuity of mechanical stresses Tiy (i = x, y). Superscripts I and R correspond to the incident (longitudinal LA or transverse TA2) and reflected (longitudinal LA and transverse TA2) waves. The neglect of the spin system in an explicit form for this consideration is justified for frequencies ω
ωm + ωme, where ωm and ωme are the magnetic and magneVol. 54

toelastic gaps in the spin wave spectrum, respectively [2–4]. The MSPT in a Ni2MnGa crystal is known to be associated with the transition from the cubic into the tetragonal phase, and magnetic moment M0 || [111] deviates toward the [001] direction under these conditions. The presence of the tetragonal phase with M0 || [001] imposes rather stringent conditions on the elastic and magnetoelastic constants of the material and the value and direction of the applied magnetic field. Therefore, the structural PT first occurs, and the orien-

0 4 × 10–4

S t–1 1

2

S l–1 S l–1 α α kl kl β

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S t–1 x

8 × 10–4 12 × 10–4

R

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(1)

kt

16 × 10–4 y Fig. 1. Geometrical construction of the wavevectors of an incident longitudinal wave and the reflected longitudinal and transverse waves at the free surface of an Ni2MnGa crystal. The temperature is T = (1) 260 and (2) 300 K.

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tational PT is then realized as the temperature decreases in field H || [001]. It is rather difficult to place them near each other. Usually, ωm  ωme in magnetic crystals, and the magnetic gap vanishes only during the orientational PT (ωm = 0). According to [10], the magnetoelastic gap in Ni2MnGa becomes minimal in the austenitic phase (ωme = 20 MHz), since M0 = 500 Oe, the anisotropy field is HA = 300 Oe, the magnetostriction constant is γ = 0.1, the effective magnetostriction field is Hme = ' = (HA Hme)1/2 ≈ 1.2 Oe. Note 0.005 Oe, and, hence, H me that, in Trivisonno’s experiments, the ultrasonic frequencies did not exceed 10 MHz; therefore, the assumption given above was grounded. In the Landau theory of PTs used in this work, the MSPT is a firstorder PT that is caused by second- and third-order elastic anharmonicity, which controls the attenuation of sound waves and their nonlinear interactions. Here, the contribution of elastic anharmonicity to the renormalization of elastic moduli is taken into account with the approximation C' = a(T – TPM), which satisfactorily describes the experimental behavior of the soft mode TA2 velocity in the PT range. As in [2–4], we do not take into account the effect of the dissipation of acoustic vibrations on reflection. If necessary, in the case of the reflection of a plane-parallel and aperture-restricted acoustic beam, we can take into account the attenuation of the reflected signals due to the nonuniform energy dissipation of the beam rays because of their different propagation ways in the material by introducing corresponding exponents into the structures of the obtained solutions [7]. LONGITUDINAL WAVE For the elastic displacements in the case of plane harmonic incident (LA) and reflected (LA, TA2) waves, we have ⎛ I ⎜ U 1l, x ⎜ I ⎝ U 1l, y

⎞ ⎟ = U I ⎛⎜ sin ( α – ∆ l ) ⎞⎟ 1l 0 ⎟ ⎝ – cos ( α – ∆ l ) ⎠ ⎠ I

I

I

× exp [ i ( k 1l x sin α – k 1l y cos α – ω 1l t ) ], ⎛ R ⎜ U 1l, x ⎜ R ⎝ U 1l, y ×

⎞ ⎟ = U R ⎛⎜ sin ( α + ∆ l ) ⎞⎟ 1l 0 ⎟ ⎝ cos ( α + ∆ l ) ⎠ ⎠

R exp [ i ( k 1l x sin α

+

R k 1l y cos α

–

R ω 1l t ) ],

(3)

⎛ R ⎜ U 1t, x ⎜ R ⎝ U 1t, y

⎞ ⎟ = U R ⎛⎜ cos ( β + ∆ t ) ⎞⎟ 1t 0 ⎟ ⎝ – sin ( β + ∆ t ) ⎠ ⎠ R

R

T

× exp [ i ( k 1t x sin β – k 1t y cos β – ω 1t t ) ], where U0 and ω are the wave amplitude and frequency. It follows from boundary conditions (2) that, at any I R R time t, ω 1l = ω 1l = ω 1t = ω and, at any point in the I

R

R

y = 0 plane, k 1l, x = k 1l, x = k 1t, x = kx. As follows from the aforesaid, the propagation direction of the converted TA2 wave is determined from the relation

sin β =

sin β sin α ------------- = --------------, S 2t ( β ) Sl ( α )

(4)

2 ⎛ B 1 – A 1 C 1⎞ B1 ⎜ 1 – ------ + ----------------------------⎟ /2, A1 A1 ⎝ ⎠

(4')

where 2

2

4

A 1, 2 = 4 [ ( C 11 – C 44 ) – ( C 12 + C 44 ) ] sin α 2

+ ( C 11 + C 44 ± D ) , B 1, 2 = ( C 11 + C 44 ± D ) [ ( C 11 + C 44 ) cos 2α ± D ], C 1, 2 = [ ( C 11 + C 44 ) cos 2α ± D ]

2

2

– 4 ( C 11 – C 44 ) sin 4α, D =

2

2

2

( C 11 + C 44 ) sin 2α + ( C 12 + C 44 ) cos 2α .

This direction can also be graphically found from this relation using the geometrical construction of the surfaces of the inverse phase velocities (see Fig. 1). In the (110) plane of a cubic crystal, LA and TA2 are pure modes only when they propagate along high-symmetry directions, such as [100], [110], and [111]. In an arbitrary direction, which is considered in this work, quasi-longitudinal LA and quasi-transverse TA2 waves take place; that is, their polarizations el and e2t deviate from vector k and the orthogonal vector through small angles ∆l and ∆t, respectively. The deviations of polarizations el and e2t are determined when the solutions for LA and TA2 are substituted into the initial wave equations and are expressed as follows: 1 – A+ B -, tan ∆ l = -----------------A+ – B

(5)

A– – B -, tan ∆ t = -----------------1 + A– B

(6)

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REFLECTION OF ELASTIC WAVES IN A CRYSTAL OF HEUSLER ALLOY

85

where 2

2

2

2

( C 11 – C 44 ) sin 2ϕ ± ( C 11 – C 44 ) sin 2ϕ + ( C 12 + C 44 ) cos 2ϕ -, A = ------------------------------------------------------------------------------------------------------------------------------------------------------2 ( C 12 + C 44 ) cos 2ϕ tan ϕ + 1 B = --------------------- . tan ϕ – 1 Note that, at ϕ = 0, π/4, π/2, etc., ∆l = 0 and ∆t = 0; that is, the modes are pure along these and similar directions. At ϕ
1, the following approximate relations are valid: C 12 – C 11 + 2C 44 2C 44 -ϕ = --------------------- ( 1 – A )ϕ. ∆ l ≈ ∆ t ≈ -------------------------------------C 11 – C 44 C 11 – C 44 Figure 2 shows the dependence of the angle of LA reflection into TA2 β on α at various degrees of closeness of Ni2MnGa to the PMPT point, which was calculated numerically using the experimental data in [12]. The inset to Fig. 2 illustrates the angular dependence of the polarization vectors, and the values of ∆l and ∆t are maximal (∆l = ∆t ≈ 20°) far from the PT and are substantial near the PT. We use boundary conditions (2) after Eqs. (3) are substituted into them and determine the amplitude reflection coefficient of the incident LA wave, R U 1l0

X– – Y – -, - = ----------------R ll = -------I X + + Y+ U 1l0

(7)

ness to the PT point. The full conversion of the longitudinal wave excited by ultrasonic piezoceramic transducers into a transverse wave is important for the creation of measuring devices, in particular, in diagnostics. Moreover, the conversion efficiency can be additionally controlled by temperature and, which is most important, by a magnetic field. Figure 3 shows the calculated α dependences of coefficients All and Alt, which demonstrate that All = 0 at two values of α (α = α1, 2); that is, LA completely converts into TA2. TRANSVERSE WAVE We now consider the reflection of transverse wave TA2 incident on the interface; for the elastic displacements of this wave, we have ⎛ I ⎜ U 1t, x ⎜ I ⎝ U 1t, y

I

and the amplitude coefficient of its conversion into TA2, R

U 1t0 C– -. R lt = --------= ----------------I X + + Y+ U 1l0

(8)

Here,

sin α sin β ------------- = -----------St ( α ) Sl ( β )

(10)

or from the related expression

Y ± = cos ( 2β ± ∆ t ) × [ ( C 11 + C 12 ) cos ∆ l + 2C 44 cos ( 2α ± ∆ l ) ], 2

C – = [ ( C 11 + C 12 ) cos ∆ l sin 2α + C 44 sin 4α ] sin β/ sin α. At certain angles of incidence α0, which are calculated from the expression ( C 11 + C 12 ) [ sin ( 2α – ∆ l ) sin ∆ t – cos ( 2β + ∆ t ) cos ∆ l ] = 2C 44 cos ( 2α + 2β – ∆ l + ∆ t ), LA completely converts into TA2 and α0 is specified by the anisotropy parameter of the crystal, i.e., by its closeNo. 1

I

The reflected LA and TA2 waves are specified by the second and third relations in Eq. (3) with the substitutions β α and α β, respectively. The direction of the converted LA wave is determined from the relationship

× [ ( C 11 + C 12 ) sin ∆ t + 2C 44 sin ( 2β + ∆ t ) ],

Vol. 54

(9)

× exp [ i ( k 1t x sin α – k 1t y cos α – ωt ) ].

X ± = sin ( 2α ± ∆ l )

TECHNICAL PHYSICS

⎞ ⎟ = U I ⎛⎜ – cos ( α – ∆ t ) ⎞⎟ 1t 0 ⎟ ⎝ – sin ( α – ∆ t ) ⎠ ⎠

2009

sin β =

2 ⎛ B 1 – A 1 C 1⎞ B1 – ----– ---------------------------1 ⎜ ⎟ /2. A1 A1 ⎝ ⎠

(10')

This direction can also be graphically found using the geometrical construction of the surfaces of the inverse phase velocities for all waves, similar to the construction in Fig. 1. Figure 2 shows the dependence of the angle of conversion β on the angle of TA2 incidence α. It is seen that β = 90° at α = αcr. Beginning from α > αcr, sin2β > 1, and we only considered the real part of sinβ when plotting the β(α) curves. In all cases where β > α at a certain

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KARPUK et al. β, deg

R 0.4

∆l, ∆t, deg 20

80

4 3

0 60 5 4 40

0

1 0 20 40 60 80 100 α, deg

–0.4

3 2

–0.8

TA2 → LA

20

LA → TA2 20

Rll

1 –1.2

0

2

Rlt

–20 6

6

5

40

60

0

20

40

60

80 α, deg

80 α, deg

Fig. 2. Angle of conversion β of a transverse wave vs. the angle of incidence α of a longitudinal wave and the angle of conversion β of a longitudinal wave vs. the angle of incidence α of a transverse wave. The inset shows the angular dependences of the wave polarizations. The temperature is T = (1, 4) 260, (2, 5) 215, and (3, 6) 300 K. These temperatures are used in all following figures.

Fig. 3. Reflection (Rll) and conversion (Rlt) coefficients of a longitudinal wave vs. the angle of its incidence α.

|Rtt| (a) 4

C 11 C 44 -⎞ α cr = arccos ⎛ ---------------------------------------------------⎝ 2C ( C – C ) + C 2 ⎠ 44 11 12 12

1/2

(11)

following from Eq. (10), LA becomes an inhomogeneous wave glancing along the interface; that is, it becomes an accompanying surface vibration (ASV). Its velocity S l' and penetration depth Λl (Ul ≈ exp[–y/Λl ]) begin to depend on the angle of incidence as Sl -, S l' = ---------sin β

(12)

Sl -, Λ = ----------------------------2 ω sin β – 1

(13)

3

3 2 2 1 1

0

20

40

60

ψtt, deg 200 (b)

4

where Sl = C 11 /ρ . The amplitude reflection coefficient of the incident TA2 wave is written as

80

5

100

6

R

U 1t0 U– + V – -, R tt = --------= – -----------------I U+ + V + U 1t0

(14)

0

and the amplitude coefficient of its conversion into LA is

–100

R U 1l0

C+ -, R tl = --------= – -----------------I U+ + V + U 1t0

(15) –200

0

20

40

60

80 α, deg

where U ± = ( C 11 + C 12 ) [ cos ( 2α ± ∆ t ) + sin ( 2β + ∆ l ) sin ∆ t ], V ± = 2 [ C 44 cos ( 2α − + 2β ± ∆ t − + ∆ l ) ],

Fig. 4. Effect of the angle of incidence α of a transverse wave on (a) the modulus of the reflection coefficient |Rtt | and (b) its phase ψtt. TECHNICAL PHYSICS

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REFLECTION OF ELASTIC WAVES IN A CRYSTAL OF HEUSLER ALLOY |Rtl|

87

LA, which is absorbed much more weakly than TA2 in the crystal under study; as a result, the effect can be detected.

(a)

6

CONCLUSIONS 4 3 2

2 1 0

20

ψtl, deg 200

40

60

80

(b)

100

For an incident longitudinal wave, its reflection is always specular due to the symmetry of the surface of the inverse phase velocities, and the appearing transverse wave always has a limiting angle. The full conversion of a longitudinal into a transverse wave takes place at two angles of incidence, which depend on the closeness of the crystal to its PT points.

0 6

–100

We theoretically analyzed the reflection of longitudinal and transverse waves by the free surface of a ferromagnetic shape memory alloy Ni2MnGa crystal that was in the ranges of the premartensitic and martensitic phase transitions. We determined the propagation directions, polarizations, amplitudes, and character of the waves reflected in the (001) plane in the crystal when [110] is normal to this surface. When the crystal transforms from the austenitic phase through the intermediate premartensitic phase into the martensitic phase, the acoustic waves become substantially quasilongitudinal and quasi-transverse because of the colossal acoustic anisotropy, which increases as the PT points are approached.

5 4

–200

ACKNOWLEDGMENTS 0

20

40

60

80 α, deg

Fig. 5. Effect of the angle of incidence α of a transverse wave on (a) the modulus of the conversion coefficient |Rtl | and (b) its phase ψtl . 2

C + = 2 [ ( C 11 + C 12 ) sin ∆ t sin 2α + C 44 sin 4α ] sin β/ sin α. Figures 4 and 5 show the numerically calculated dependences of coefficients Att and Atl. Numerical analysis of Eq. (10') demonstrates that the determinant of Eq. (10') becomes negative at certain elastic moduli C11, C12, and C44 and certain angles of TA2 incidence α. This finding indicates that, formally, sin2β becomes complex and the ASV is broken; that is, at certain angle β' dependent on the parameters given above, it begins to be emitted into the bulk of the crystal. Note that these requirements are rather stringent and cannot be fulfilled for Trivisonno’s experimental data used in this work. It should be noted that no propagation of TA2 in the immediate vicinity of the PMPT (∆T = |T – TPM | < 5 K) has been experimentally detected because of its significant attenuation. On the other hand, the appearance of an ASV and its emission into the bulk are induced by TECHNICAL PHYSICS

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This work was supported by the Belarussian and Russian Foundations for Basic Research (project nos. 04-02-81058-Bel, 05-02-19935-YaF, 05-08-50341, 0602-16266, 06-02-16984, and T06M-227), the Council for Grants of the President of the Russian Federation for the Support of Leading Scientific Schools (project nos. NSh-8269.2006.2, MK-5658.2006.2), the Ministry of Education of the Russian Federation, and CRDF (Y2-P-05-19). REFERENCES 1. M. K. Balakirev and I. A. Gilinskii, Waves in Piezoelectric Crystals (Nauka, Novosibirsk, 1982) [in Russian]. 2. Yu. A. Kuzavko and V. G. Shavrov, Akust. Zh. 39, 170 (1993) [Acoust. Phys. 39, 572 (1993)]. 3. M. M. Karpuk, Yu. A. Kuzavko, D. A. Kostyuk, and V. G. Shavrov, Pis’ma Zh. Tekh. Fiz. 20 (17), 86 (2003) [Tech. Phys. Lett. 20, 743 (2003)]. 4. M. M. Karpuk, Yu. A. Kuzavko, D. A. Kostyuk, and V. G. Shavrov, Zh. Tekh. Fiz. 73 (7), 97 (2003) [Tech. Phys. 48, 893 (2003)]. 5. E. A. Andryushchak, N. N. Evtikhiev, S. A. Pogozhev, and V. L. Preobrazhenskii, Akust. Zh. 27, 170 (1981) [Sov. Phys. Acoust. 27, 93 (1981)].

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6. E. A. Turov and V. G. Shavrov, Usp. Fiz. Nauk 140, 429 (1983) [Sov. Phys. Usp. 26, 593 (1983)]. 7. M. M. Karpuk and Yu. A. Kuzavko, in Proceedings of the 7th International Conference on Intermolecular and Magnetic Interactions in Matter, Miendzyzdroje, Poland, 2002, p. 44. 8. A. N. Vasil’ev, V. D. Buchel’nikov, T. Takagi, et al., Usp. Fiz. Nauk 173, 577 (2003) [Phys. Usp. 46, 559 (2003)]. 9. K. Ullakko, I. Aaltrio, P. Yakovenko, et al., J. Phys. (France) 11, 8 (2001).

10. F. Albertini, L. Morellon, P. A. Algarabel, et al., J. Appl. Phys. 89, 5614 (2001). 11. J. Worgull, E. Petti, and J. Trivisonno, Phys. Rev. B 54, 15695 (1996). 12. T. E. Stenger and J. Trivisonno, Phys. Rev. B 57, 2735 (1998).

Translated by K. Shakhlevich

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