Noncoherent quantum effects in the magnetization reversal of a chemically disordered magnet : SmCo3.5Cu1.5

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Noncoherent quantum effects in the magnetization reversal of a chemically disordered magnet :

SmCo3.5Cu1.5

M. Uehara, B. Barbara

To cite this version:

M. Uehara, B. Barbara. Noncoherent quantum effects in the magnetization reversal of a chem- ically disordered magnet : SmCo3.5Cu1.5. Journal de Physique, 1986, 47 (2), pp.235-238.

�10.1051/jphys:01986004702023500�. �jpa-00210200�

Noncoherent quantum effects in the magnetization ^reversal

of a chemically disordered magnet : SmCo3.5Cu1.5

M. Uehara (*) and B. Barbara Laboratoire Louis Néel, C.N.R.S., 166X, ^{38042 Grenoble} Cedex, ^France

(Reçu ^le 7 juin ^1985, accepté le 3 octobre 1985)

Résumé.

Dans la première partie ^{de cet article nous} justifions théoriquement l’hypothèse d’énergie d’activation moyenne ^{en vue} de décrire le trainage magnétique ^de l’aimantation des systèmes désordonnés en champs ^fixes.

Dans la seconde partie ^{nous mettons en évidence une} importante anomalie de l’énergie d’activation thermique ^de SmCo3,5Cu1,5 en-dessous de 50 K. Cette anomalie est expliquée à l’aide d’un simple ^{modèle de} fluctuations _quan-

tiques ^des parois ^{entre domaines} magnétiques. ^Une température effective de 10 K est obtenue qui ^rend compte ^des fluctuations d’énergie ^de parois à zéro kelvin. Cette température ^{coincide très bien avec la} fréquence caractéristique

03C90 ~ 1012 s-1 des vibrations de parois ^en équilibre métastable.

Abstract.

In the first part of this paper ^we justify theoretically ^the assumption ^{of a mean} activation _energy to describe the time dependent magnetization at constant field in disordered systems. ^In the second part ^{we show the} existence of an important anomaly ^{in the mean} activation _energy of SmCo3.5Cu1.5 ^{below 50 K. This} anomaly ^is explained ^{in terms of a} simple ^{model of} quantum fluctuations of domain walls. An effective temperature ^{of 10 K is} found in order to account for energy fluctuations of domain walls at zero kelvin. This temperature ^coincides very well with the characteristic frequency, 03C90 ~ 1012 s-1, ^{of domain} wall vibrations in metastable equilibrium.

Classification Physics ^Abstracts

75.60L - 05.40J

Recently, ^there has been considerable interest in the

study ^{of the} macroscopic magnetic ^{behaviour of}

disordered systems, ^{such as} spin glasses ^or amorphous magnets. ^{In these} systems ^with spherical symmetry, the characteristic scale for the disorder extends between 10 and 103 A.

Rare earth-based permanent magnets ^{are charac-} terized by ^an extremely large ^uniaxial anisotropy (hexagonal symmetry). ^In Sm(Co-Cu)s alloys ^the

disorder is of chemical origin and extends at the scale of 5-50 _pm [1]. ^The magnetization reversal of such

alloys has been studied above 50 K in terms of an

activation mechanism involving ^local variations of the domain wall surface (kink ^{creation and annihila-}

tion model) [2-7]. ^{Below this} temperature ^{we observed}

an anomalous behaviour indicating ^a possible ^break-

down of the thermal activation mechanism.

In this _paper we show that such a behaviour ^can be attributed to quantum vibrations of domain walls.

Before going further, ^{let us consider some} general

features concerning ^the interpretation of magnetic ^after

effect measurement.

Street and Wooley [8] ^{showed that the rate} of magne-

tization variation is given by :

where A ⁼ I/To e-E(H)/kT. In this expression E(H) =

kT In (1/ À.To) represents ^the activation energy ^{at a}

given field and ro is the characteristic relaxation time of domain walls in critical damping. ^The unspe- cified terms have their usual meaning. ^This expression

can be rewritten :

As the function f(E(H)) ^{is smooth} compared ^to e - It :

where E(H) ^{= kT In} (t/io) ^{is the mean activation}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004702023500

236

energy. The small variations of S(H, T) ^are always negligible compared ^{to :}

therefore, ^{the time rate of} magnetization ^variations I/r ⁼ (1/2 MJ dM/dt ^{can be written}

with 1/z ⁼ (1/2 MJ dM/dt ^and To = 7:o/S(H, ^{T) can}

be considered as a constant.

_

If the functional dependence E(H) ^is known, expression (4) ^{allows to} determine experimentally ^the

thermal and time evolution ^of E(H). ^Some years ago,

we have shown that E(H) ^oc 1/H ⁱⁿ highly anisotropic alloys [9, ^10, 2]. ^Therefore expression (4) ^{shows that}

the thermal variation of the mean activation _energy

E(H) ^{can be} determined by plotting ^In (dM/dt) ^versus 11H.

The rate of magnetization ^variation 1/7: =(1/2 MJ ^x dM/dt has been measured on a single crystal ^of SMCO3.5Cu1.5 ^{near the coercive field} He ^{for which}

the measured magnetization ^is very small. Consequent- ly ^the demagnetizing ^{field of the} sample (a sphere ^{3 mm}

in diameter) ^is negligible [11]. ^The plot ^{of In} (1/r) ^{as a}

function of the reciprocal applied magnetic ^{field H} gives ^{a set of} convergent straight ^lines (Fig. ^{1, see also}

Fig. ^1.

^{Rate of} magnetization ⁱⁿ SmC03,sCul,s ^for

various temperatures ^{as a} function of the reciprocal magne- tic field.

Refs. [2-5]). ^The intersection point ^{is found} by ^{a linear} extrapolation ^at Ho ^{= 41.7 kOe} (the ^{coercive field at} 0 K) ^and 1/T0 ^{= 1012 s-1} (the ^mean characteristic

frequency ^{of domain} walls).

Such a linear dependence ^{of the} activation _energy with the reciprocal applied field characterizes a wall

propagation involving local surface variations. It has been observed several _{years ago} in the displacement ^of

ferroelectric domain walls [12, 13] ^{and in} Dy3Al2

for ferromagnetic compounds [9, 10].

In our model [4,14] ^{of non} rigid domain wall propa-

gation ^{the mean} activation energy has been evaluated

by considering ^{a small} portion So ^{of domain} wall, which, being ^less trapped ^{than its} surrounding, ^has

an energy par unit surface (y ⁺ &y/2), ^i.e. larger ^than

the _average wall _energy density y. This element can

therefore move. Under the combined influence of an

applied ^{field H} sufficiently high, and of the thermal

activation, the surface of this small element deforms from So ^{to S} by sweeping ^{out a} volume V. The asso-

ciated jump ^{will be} irreversible only if the wall segment

can be trapped ^at points of smaller energy (-y - A7y/2) corresponding ^to other defects. The energy variation associated with this jump ^is

The term - 2 M, HSO 6 ^{where ð is} the wall thickness

accounts for small wall curvatures. The irreversible deformation, ^{assumed to be of} cylindrical symmetry, is possible ^{when its} amplitude ^is larger ^{than or} equal

to a critical value pg rr (V IS): ^obtained by ^maximiza-

tion of equation (5). ^The height ^of the energy barrier to be overcome for an irreversible jump ^{is obtained} by substituting p* ^into equation (5) :

where F is a geometrical factor = 1. The parameter

a ⁼ (AS/S - åyfi) characterizes the local variation of domain wall surface (AS/S) ^and pinning (Ayfi).

The field Ho ⁼ (2 a/n). H., ^where H. ^{is the} anisotropy field, is the kink creation or annihilation field in the absence of activation and h the amplitude ^{of ele-}

mentary forward jumps. ^{We have shown that this} length is of the order of the domain wall thickness and that a rr 0.07 in SmC03., Cul,s ^[2-7].

In order to confirm the validity ^of expression (6), ^we

have compared ^{the variation of} kT(d ^In (’C/to)/d(I/H»)

with 2/Ms ^{= 16} ao K(T) NMs (T), ^where K(T) ^and

NM.I(T) ^{are the} anisotropy ^and exchange ^energy respectively determined on the same single crystal (2-5)

and _{ao the} ^mean lattice parameter (Fig. 2). ^{The linear}

behaviour observed above 50 K and the fact that its

extrapolates ^{to the} origin ^of the coordinates constitute

a good ^{check for} the notion of ^mean activation _energy and for the kink creation annihilation model [4].

Below this temperature ^an important anomaly ^is observed, ^{which cannot be due to the thermal evolution}

of the activation _energy (expression (6)). ^{As a matter of}

fact the magnetic characteristics 6, Ha, M,) ^are nearly

constant at these temperatures (the ^Curie tempera-

ture of SMC03.-iCUl.,5 being ⁷⁰⁰ K). ^{Therefore we}

assign ^the anomaly ^{below 50 K to a} modification in

the thermal activation mechanism itself. In order to

u -2

Fig. ^{2. - Plot of} T U d 111 B. T;/T;O} ^{(11m versus y-2lMs = K(T)}

NMa(T) (see ^{also Ref.} [3]).

show more clearly this effect let us plot ^the apparent elementary ^volume [15]

versus T as shown in figure ^3.

Above 50 K this quantity ^is constant; the apparent volume V* can therefore be identified to the real volume V of the activation process. Furthermore the forward amplitude ^{of the} elementary jump h is found of the order of some interatomic distances i.e. close to the mean wall thickness in SMC03.5cu,.5.

Fig. ^3.

Temperature dependence ^{of the normalized} apparent ^volume V */V ⁱⁿ SmCo3.sCu1.s’

Below 50 K, ^the anomaly ^{shown in} figure ^{2 corres-} ponds ⁱⁿ figure ^{3 to an} abrupt decreasing ^{of the}

apparent ^{volume V} *, suggesting ^{an easier} magnetiza- tion reversal in this temperature range. In particular

a finite probability ^for magnetization ^reversal persists

in the limit of absolute temperature (0 kelvin).

Let us assume that the curve of figure 3 results from two contributions :

- a thermally ^{activated one :}

where g(T) ^and h(H) ^{account for the} functional depen-

dences of the activation _energy E ⁼ V g(T) h(H) ^with temperature ^and magnetic ^{field and Y} is the characte- ristic volume introduced above.

- an athermal one:

where A is practically independent ^of temperature ^and To is the characteristic time of small fluctuations about the metastable equilibrium already introduced above.

The total probability ^{for a} magnetization ^reversal

can be written in terms of an effective activation law :

where V * is the apparent ^volume already ^defined [15].

This volume should be temperature dependent :

- In the limit of high temperatures TT z f and therefore V * = V is independent ^{of the} temperature,

as expected.

- In the limit of low temperatures, zT >> ’tr, and

where A/E ^{is a constant. This limit fits with} the expe- rimental data points ^if E/A ~ ^{10 K as shown in} figure ^4.

Fig. ^4.

^Same experimental points ^{as shown in} figure ^3.

The notion of effective temperature ^{T * is related to the}

one of apparent ^volume by V *IV

TIT *. ^{The continuous}

line represents ^{a fit to} equation (12) ^for E/A

^{10 K and}

E

40 ± 5 K. The low and high temperature ^{limits are}

indicated by ^{the two} straight ^lines intercepting ^{at T rr 10 K.}

238

Now it would be interesting ^{to discuss the} origin ^of

the athermal magnetization process introduced here.

Such a mechanism can be attributed to quantum

phenomena : tunnelling ^of domain walls through ^the potential ^barrier [16, 17, 10] ^or quantum mechanical motion over the potential ^{barrier due to zero} point

energy [2-5]. ^Since quantum tunnelling ^of macroscopic objects ^{such as} domain walls is not so common and needs _very peculiar conditions, ^this phenomenon

should be taken carefully. ^{On the} opposite, ^zero point

motion over potential barriers associated with _energy fluctuations AE = hIAt ^{can be} relatively easily ^consi-

dered.

In our case the motion of a small portion ^{of domain}

wall of effective mass m, equivalent ^{to a} quasi-particle,

can be considered as a simple quantum harmonic oscillator with characteristic frequency ^{ro oc} 1/jm,

and energy levels at nro/2, ³ hcol2, ⁵ hco/2... ^{These loca-}

lized states should interact with the spectrum ^{of lattice} vibrations leading ^{to a} quasi-continuous spectrum ^of wall excitations. At low temperature ^the quantum noise of lattice vibrations must lead to quantum energy fluctuations AEO ^{of the wall} portion. ^These

energy fluctuations of characteristic frequency

a)o = AE01h correspond ^to small wall excitations in the metastable well; roo ^can be considered as the trial

frequency ^{or in other words as the} exponential pre-

factor ; ^{it must also enter in the} argument ^{of the} expo- nential since AEO ^{= kT * can be} considered as an

effective temperature. Therefore when the applied ^field

is such that E(H) = AEO ^{the wall} portion ^should

overcome instantaneously the energy barrier ^{even at}

zero kelvin. Finite relaxation times are simply given by ^{’t =} io elAEo _{with io} ⁼ 2 a/coo ⁼ HIAEO.

In the framework of this physical model, ^the argu- ment of the exponential ⁱⁿ equation (9) ^{is A =} E/AEO

and therefore

where T * is an effective temperature accounting ^for

the zero point motion of domain walls; AE 0 ^{= kT* =} hcoo. ^{It is} interesting ^{to note} that the value E/A ⁼

kT* -- 10 K corresponds ^{well to the} frequency ^of

wo ⁼ 1.3 x 1012 s-1 determined independently ⁱⁿ figure ^{1 :} the effect of zero point energy motion is observed at temperatures ^{lower than or} equal ^to hwo.

In conclusion ^we have interpreted ^{the crossover}

between thermal hopping ^and quantum regime ^{in the}

reversal of the magnetization ^{of a} disordered ferroma- gnet ^{in terms of the} gradual transition from thermal to

microscopic quantum fluctuations. This transition

might ^be qualitatively ^{described in terms of an effec-}

tive temperature equal ^{to the mean} energy of ^a quan- tum harmonic oscillator AE ⁼ /KU/2 ^coth (hco/2 kT),

at finite temperatures (AE -+ hw/2 ^{when T -+ 0 and}

AE - kT when T -+ oo). ^{In our} phenomenological

model we take also into account interactions between modes of frequency ^with the lattice vibrations _spec-

trum. This leads to the two limiting exponential ^rates

~ e-E/AEu at low temperature ^{and e-E/k1’ at} high temperature. ^{It is at the moment difficult to find} any difference between this model of quantum fluctuations

over energy barriers and tunnelling through energy barriers. Therefore we do not exclude the possibility ^of quantum tunnelling ^{of a} domain wall in disordered systems. ^{Our model} may provide ^a simple physical picture ^{for the crossover} between thermal hopping

and quantum mechanical tunnelling ^assisted by phonons [18-20].

The high ^field magnetization measurements were

performed ^{at Service National des} Champs Intenses in Grenoble.

References

[1] UEHARA, M., ^J. Appl. Phys. ⁴⁸ (1977) ^5197.

[2] BARBARA, ^{B. and} UEHARA, M., I.E.E.E. Trans. Mag.

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[3] BARBARA, ^B. and UEHARA, M., Physica ^86-88 (1977)

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[10] BARBARA, B., ^J. Physique ³⁴ (1973) ^1039.

[11] During ^a typical after effect measurement the range of variation of the demagnetizing ^{field is}

± 100 Oe to be compared ^{with the} applied magnetic ^fields

of 40 kOe at low temperature ^and 20 kOe at 50 K.

[12] MERZ, ^W. J., Phys. ^{Rev. 95} (1954) ^690.

[13] MILLER, ^{R. C. and} WEINREICH, G., Phys. ^{Rev. 117} (1960) ^1460.

[14] BARBARA, B., MAGNIN, ^{J. and} JOUVE, H., Appl. Phys.

Lett. 31 (1977) ^133.

[15] ^{This is the} activation volume measured on the basis of

expression (4). ^This expression accounting ^for

thermal activation is not necessarily ^{valid in the}

whole range of temperature ^and especially ^{at low} temperature. ^{This is the reason} why ^{we have}

defined for _any temperature ^an apparent ^activa- tion volume V*. This volume has a physical meaning only in the range of temperature ^where the thermal activation mechanism is preponderant.

[16] EGAMI, T., Phys. ^{Status Solidi} (a) ²⁰ (1973) ^157.

[17] EGAMI, T., Phys. ^{Status Solidi} (b) ⁵⁷ (1973) ^211.

[18] CALDEIRA, ^{A. O. and} LEGGETT, ^A. T., Phys. ^{Rev. Lett.}

46 (1981) ^211.

[19] LARKIN, ^{A. I. and} OVCHINNIKOV, ^Yu. N., ^Soviet Phys.

JETP 86 (1984) ^719.

[20] GRABERT, ^{H. and} WEISS, U., Phys. Rev. Lett. 53 (1984)

1787.