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Influence of an AC magnetic field on magnetization tunneling in high-spin molecules
Journal of Magnetism and Magnetic Materials 222 (2000) 375}378
In#uence of an AC magnetic "eld on magnetization tunneling in high-spin molecules I.D. Tokman *, G.A. Vugalter Institute for Physics of Microstructures, Russian Academy of Sciences, Nizhny Novgorod 603600, GSP-105, Russia Department of Physics, Nizhny Novgorod State University, 23 Gagarin Avenue, Nizhny Novgorod 603600, Russia Received 12 April 2000; received in revised form 25 July 2000
Abstract We study theoretically, magnetization (spin) tunneling in high-spin molecules subjected to an AC magnetic "eld. It has been shown that the tunneling time in the presence of an AC magnetic "eld can be essentially less than the tunneling time in the absence of this "eld. The conditions of observation of this e!ect are discussed. 2000 Elsevier Science B.V. All rights reserved. PACS: 75.45.#j; 75.50.Tt Keywords: Tunneling of magnetization; Rabi's oscillations; High-spin molecules
1. Introduction In recent years, great experimental and theoretical e!orts have been made to observe and explain di!erent quantum phenomena in monodomain particles, including quantum coherence and quantum tunneling of magnetization (for an overview, see Ref. [1] and references therein). Experiments on quantum coherence were performed for the "rst time in the antiferromagnetic horse-spleen ferritin [2]. The "rst few experiments giving good indications of tunneling in a single-crystal SmCo Cu have been made in Ref. [3]. Then manifestations of magnetization tunneling have been investigated in * Corresponding author. Tel.: #7-831-2-675185; fax: #7831-2-361972. E-mail addresses: [email protected] (I.D. Tokman), [email protected] (G.A. Vugalter).
molecular magnets of Mn }acetate (Mn }AC) [4}7], in high-spin clusters CrNi , CrMn [8] (however, the last few systems have a rather lowenergy barrier, and it is therefore di$cult to tell that they show tunneling manifestations), and in aluminosilicate glasses doped with di!erent magnetic ions [9}11]. Certainly, the list of interesting works in this "eld, given above, is not complete. Magnetic molecules are characterized by a strong uniaxial anisotropy. In the absence of an external magnetic "eld, the projection of the molecule spin on the anisotropy axis is conserved and all energy levels of the molecule are two-fold degenerate (except the highest level if the total spin is an integer). Each state of any doublet is localized in a corresponding well of a double-well potential with a high-energy barrier (the latter is none other than the anisotropy energy). A weak DC magnetic "eld applied perpendicular to the anisotropy axis
0304-8853/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 5 6 1 - 8
376
I.D. Tokman, G.A. Vugalter / Journal of Magnetism and Magnetic Materials 222 (2000) 375}378
breaks conservation of the spin projection and induces tunneling between two states of any doublet. As a consequence, degenerate levels get splitting. At very low temperature, only tunneling between the states of the ground doublet is important. At higher temperature there exists a regime called thermally assisted tunneling [7]. This regime combines thermal activation to excited levels with tunneling across the barrier. The purpose of our paper is to investigate the in#uence of a weak AC magnetic "eld on the tunneling of magnetization in high-spin molecules (clusters) placed in a DC magnetic "eld perpendicular to the anisotropy axis. We will show that, if the frequency of the AC "eld is resonant, the time of tunneling can be essentially less than the tunneling time in the absence of the AC magnetic "eld.
2. Tunneling time We describe a magnetic molecule (cluster) with the total spin S<1 by the Hamiltonian HK "!DSK !gk SK (H #H cos ut), (1) X V where z is the anisotropy axis, D is the anisotropy energy constant, k is the Bohr magneton, H is a DC magnetic "eld perpendicular to the anisotropy axis (we suppose the direction of this "eld to coincide with the x-axis), H and u are, respective ly, the amplitude and angular frequency of an AC magnetic "eld which is parallel to the DC magnetic "eld. In the absence of DC and AC magnetic "elds, the energy spectrum of the molecule is very simple: E "!Dm, m"!S,!S#1,2S, (2) K where m is the magnetic quantum number. The energy levels (2) form S# doublets if S is a half integer, and S doublets and one nondegenerate level if S is an integer. Eigenfunctions t correK sponding to the energy levels (2) are the eigenfunctions of the operator SK , i.e. SK t "mt . X X K K A DC magnetic "eld H directed along the x-axis induces tunneling between two states of any doublet and splits the latter. We assume the magnetic "eld to be small enough (gk H /DS;1). In this
case, the perturbation theory of the 2mth order in the parameter gk H /DS yields the tunneling split ting for two states of the doublet E [12,13] !K
gk H K 2D(S#m)! *E K . K [(2m!1)!](S!m)! 2D
(3)
Here m'0. The lower E\ and upper E> levels of K K the doublet are described by the formula E!KE $*E . K K K
(4)
The quantity *E is much smaller than the unperK turbed level spacing "E !E ""D"2m!1" and K\ K changes essentially with changing m by unity. The eigenfunctions corresponding to the lower and upper levels of the split doublet are symmetric and antisymmetric ones tQK( (t #t ), t?K( (t !t ), \K K \K K K K
(5)
respectively. Now, let us consider the e!ect of a weak AC magnetic "eld on the magnetic molecule. We assume temperature to be low enough, restrict ourselves to comparatively short time (less than the relaxation time of the wavefunction phase) and describe the behaviour of the molecule by a SchroK dinger equation. For brevity we designate e "E\, e "E>, e "E\ , e "E> , u " 1 1 1\ 1\ tQ, u "t?, u "tQ , u "t? . The AC 1 1 1\ 1\ "eld frequency is supposed to be resonant for transitions between the levels e and e , i.e. u"(e !e )/ .
(6)
Such transitions are possible because the matrix element 1u "SK "u 2"(S/2O0. It should be no V ticed that transitions between the levels e and e and between the levels e and e , induced by the AC "eld, are forbidden because the matrix elements 1u "SK "u 2, 1u "SK "u 2 equal zero. Transitions V V between the levels e and e are not forbidden, but we suppose them to be not resonant for the frequency (6). This implies that the condition holds (e !e )! u
(7)
where *u is the linewidth of the source of the AC "eld, q is the relaxation time of the wavefunction
I.D. Tokman, G.A. Vugalter / Journal of Magnetism and Magnetic Materials 222 (2000) 375}378
phase. Since *E <*E , inequality (7) can be 1\ 1 rewritten in the form
*u 1 4(S!1) gk H 1\
(8)
Under this condition the level e (as well as higher levels) is not important and the magnetic molecule can be considered as a three-level system, the transitions between two levels of which are described by the resonance perturbation theory [14]. Let the initial con"guration of the magnetic molecule be t(t"0)"( (u #u ). (9) It corresponds to the state with the z-projection of the spin, equal to S, and with the energy (e #e )/2K!DS. Solving the SchroK dinger equation, we have for t;q 1 1 u e\ C R # u e\ C R cos X t t(t)K 0 (2 (2 i # u e\ C R sin X t, 0 (2
(10)
where gk H gk H (S 1u "SK "u 2" X " V 0 2
2(2
(11)
is the Rabi frequency. Calculating the average zprojection of the spin and taking into account the following relations: 1u "SK "u 2"1u "SK "u 2"0, X X 1u "SK "u 2"1u "SK "u 2"0, X X 1u "SK "u 2"S, X we "nd *E 1S (t)2KS cos X t cos 1 t. X 0
(12)
In the absence of the AC "eld the time required for the molecule to change the direction of its spin (magnetization) is determined by tunneling and equals t "p *E . 1
(13)
377
Certainly, this time should be much less than q. In contrast, if the AC magnetic "eld is not very weak, namely, X <*E / , 0 1
(14)
the time required for the molecule to change the direction of its spin is determined by Rabi's oscillations and equals t "n/X ;t . 0
(15)
Thus, if the amplitude of the resonant AC magnetic "eld satis"es the condition
H 4(2S gk H 1\ < , H (2S!1)! 2D
(16)
the time of magnetization tunneling is much less than in the case where the AC "eld is absent. Analogously, the same e!ect will occur if the AC "eld is resonant for transitions between the levels e and e . We emphasize that our model neglects the high-order anisotropy terms, the dipolar}dipolar interactions between magnetic molecules, and the coupling between magnetic molecules and nuclear spins. Therefore, we deal with a simple isolated spin. Moreover, the result obtained above is valid if the times t , t are essentially less than the relax ation time of the wavefunction phase, q. Nevertheless, our model seems reasonable due to the following facts. (i) The high-order anisotropy terms are needed in the absence of a DC magnetic "eld perpendicular to the anisotropy axis in order to describe tunneling between the states with opposite projections of the spin. In the presence of the DC "eld H which is perpendicular to the anisotropy axis and is not very weak, tunneling between the states with opposite projections of the spin is induced by the term proportional to H , which pre vails over the high-order anisotropy terms. (ii) The #uctuating part of the dipolar}dipolar interactions, which causes the spin relaxation, may be made ine$cient by diluting the sample. (iii) Despite the dipolar}dipolar interactions between magnetic molecules and the interactions between magnetic molecules and nuclear spins broaden the levels of a magnetic molecule, it is well established that there
378
I.D. Tokman, G.A. Vugalter / Journal of Magnetism and Magnetic Materials 222 (2000) 375}378
exists even the ground doublet (see, for example, Refs. [10,11]). Since we use experimental values of the relaxation time for our estimates, an exact consideration of the dipolar}dipolar interactions and the interactions between magnetic molecules and nuclear spins are not possible. We assume 0.1% disprosium (Dy>) -doped glasses to be appropriate for the observation of the predicted e!ect because of the comparatively long relaxation time and the great distance between magnetic ions. According to the experimental data [11], for such glasses S", DSK10 K, gK, q&10\ s. In this case, the resonant frequency (6) is u/2pK0.55 T Hz. Supposing H "70 kOe, we obtain that inequality (8) is satis"ed if the quality factor of the AC "eld source Q"u/*u<10. According to Eqs. (3), (13) and (16), the time of tunneling in the absence of the AC "eld is about t K7.5;10\ s and the tunneling time in the pres ence of the AC "eld is essentially less than t if H <0.04 Oe. 3. Conclusion We have investigated the in#uence of an AC magnetic "eld on the behaviour of magnetic molecules or magnetic clusters with a high value of spin. It has been shown that, if the AC "eld frequency is resonant for transitions between the lower states of the ground and "rst excited doublets, the time required for spin tunneling between two opposite directions can be much shorter than the time of tunneling in the absence of the AC magnetic "eld. We have pointed out the conditions under which the e!ect can be observed. The e!ect discussed, will also occur in the case where the frequency of the AC "eld is resonant for
transitions between the upper states of two doublets mentioned above.
Acknowledgements This work was supported partly by the Russian Foundation for Basic Research (Grant Nos. 98-0216412 and 97-02-17437), and by the International Center-Foundation for Promising Research in Nizhny Novgorod (Grant No. 99-2-03).
References [1] Quantum Tunneling of Magnetization, L.Gunther, B. Barbara (Eds.), Kluwer Academic Publishers, Dordrecht, 1995. [2] D.D. Awshalom, J.F. Smyth, G. Grinstein, D.P. DiVincenzo, D. Loss, Phys. Rev. Lett. 68 (1992) 3092. [3] M. Uebara, B. Barbara, J. Phys. 47 (1986) 235. [4] B. Barbara, W. Wensdorfer, L.S. Sampaio, J.G. Park, C. Paulsen, M.A. Novak, R. Ferre, D. Mailly, R. Sessoli, A. Caneschi, K. Hasselbach, A. Benoit, L. Thomas, J. Magn. Magn. Mater. 140}144 (1995) 1825. [5] M.A. Novak, R. Sessoli, A. Caneschi, D. Gatteschi, J. Magn. Magn. Mater. 146 (1995) 211. [6] J.R. Friedman, M.P. Sarachik, J. Tejada, R. Ziolo, Phys. Rev. Lett. 76 (1996) 3830. [7] L. Thomas, F. Lionti, R. Ballou, D. Gatteschi, R. Sessoli, B. Barbara, Nature 383 (1996) 145. [8] N. Vernier, G. Bellessa, T. Mallah, M. Verdagner, Phys. Rev. B 56 (1997) 75. [9] N. Vernier, G. Bellessa, Phys. Rev. B 48 (1993) 12842. [10] N. Vernier, G. Bellessa, Phys. Rev. Lett. 71 (1993) 4063. [11] N. Vernier, G. Bellessa, D.A. Parshin, Phys. Rev. Lett. 74 (1995) 3459. [12] D.A. Garanin, J.Phys. A 24 (1991) L61. [13] I.Ya. Korenblit, E.F. Shender, Sov. Phys. JETP 48 (1978) 937. [14] L.D. Landau, E.M. Lifshitz, Quantum Mechanics, 3rd edition. Pergamon Press, New York, 1977.
In#uence of an AC magnetic "eld on magnetization tunneling in high-spin molecules I.D. Tokman *, G.A. Vugalter Institute for Physics of Microstructures, Russian Academy of Sciences, Nizhny Novgorod 603600, GSP-105, Russia Department of Physics, Nizhny Novgorod State University, 23 Gagarin Avenue, Nizhny Novgorod 603600, Russia Received 12 April 2000; received in revised form 25 July 2000
Abstract We study theoretically, magnetization (spin) tunneling in high-spin molecules subjected to an AC magnetic "eld. It has been shown that the tunneling time in the presence of an AC magnetic "eld can be essentially less than the tunneling time in the absence of this "eld. The conditions of observation of this e!ect are discussed. 2000 Elsevier Science B.V. All rights reserved. PACS: 75.45.#j; 75.50.Tt Keywords: Tunneling of magnetization; Rabi's oscillations; High-spin molecules
1. Introduction In recent years, great experimental and theoretical e!orts have been made to observe and explain di!erent quantum phenomena in monodomain particles, including quantum coherence and quantum tunneling of magnetization (for an overview, see Ref. [1] and references therein). Experiments on quantum coherence were performed for the "rst time in the antiferromagnetic horse-spleen ferritin [2]. The "rst few experiments giving good indications of tunneling in a single-crystal SmCo Cu have been made in Ref. [3]. Then manifestations of magnetization tunneling have been investigated in * Corresponding author. Tel.: #7-831-2-675185; fax: #7831-2-361972. E-mail addresses: [email protected] (I.D. Tokman), [email protected] (G.A. Vugalter).
molecular magnets of Mn }acetate (Mn }AC) [4}7], in high-spin clusters CrNi , CrMn [8] (however, the last few systems have a rather lowenergy barrier, and it is therefore di$cult to tell that they show tunneling manifestations), and in aluminosilicate glasses doped with di!erent magnetic ions [9}11]. Certainly, the list of interesting works in this "eld, given above, is not complete. Magnetic molecules are characterized by a strong uniaxial anisotropy. In the absence of an external magnetic "eld, the projection of the molecule spin on the anisotropy axis is conserved and all energy levels of the molecule are two-fold degenerate (except the highest level if the total spin is an integer). Each state of any doublet is localized in a corresponding well of a double-well potential with a high-energy barrier (the latter is none other than the anisotropy energy). A weak DC magnetic "eld applied perpendicular to the anisotropy axis
0304-8853/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 5 6 1 - 8
376
I.D. Tokman, G.A. Vugalter / Journal of Magnetism and Magnetic Materials 222 (2000) 375}378
breaks conservation of the spin projection and induces tunneling between two states of any doublet. As a consequence, degenerate levels get splitting. At very low temperature, only tunneling between the states of the ground doublet is important. At higher temperature there exists a regime called thermally assisted tunneling [7]. This regime combines thermal activation to excited levels with tunneling across the barrier. The purpose of our paper is to investigate the in#uence of a weak AC magnetic "eld on the tunneling of magnetization in high-spin molecules (clusters) placed in a DC magnetic "eld perpendicular to the anisotropy axis. We will show that, if the frequency of the AC "eld is resonant, the time of tunneling can be essentially less than the tunneling time in the absence of the AC magnetic "eld.
2. Tunneling time We describe a magnetic molecule (cluster) with the total spin S<1 by the Hamiltonian HK "!DSK !gk SK (H #H cos ut), (1) X V where z is the anisotropy axis, D is the anisotropy energy constant, k is the Bohr magneton, H is a DC magnetic "eld perpendicular to the anisotropy axis (we suppose the direction of this "eld to coincide with the x-axis), H and u are, respective ly, the amplitude and angular frequency of an AC magnetic "eld which is parallel to the DC magnetic "eld. In the absence of DC and AC magnetic "elds, the energy spectrum of the molecule is very simple: E "!Dm, m"!S,!S#1,2S, (2) K where m is the magnetic quantum number. The energy levels (2) form S# doublets if S is a half integer, and S doublets and one nondegenerate level if S is an integer. Eigenfunctions t correK sponding to the energy levels (2) are the eigenfunctions of the operator SK , i.e. SK t "mt . X X K K A DC magnetic "eld H directed along the x-axis induces tunneling between two states of any doublet and splits the latter. We assume the magnetic "eld to be small enough (gk H /DS;1). In this
case, the perturbation theory of the 2mth order in the parameter gk H /DS yields the tunneling split ting for two states of the doublet E [12,13] !K
gk H K 2D(S#m)! *E K . K [(2m!1)!](S!m)! 2D
(3)
Here m'0. The lower E\ and upper E> levels of K K the doublet are described by the formula E!KE $*E . K K K
(4)
The quantity *E is much smaller than the unperK turbed level spacing "E !E ""D"2m!1" and K\ K changes essentially with changing m by unity. The eigenfunctions corresponding to the lower and upper levels of the split doublet are symmetric and antisymmetric ones tQK( (t #t ), t?K( (t !t ), \K K \K K K K
(5)
respectively. Now, let us consider the e!ect of a weak AC magnetic "eld on the magnetic molecule. We assume temperature to be low enough, restrict ourselves to comparatively short time (less than the relaxation time of the wavefunction phase) and describe the behaviour of the molecule by a SchroK dinger equation. For brevity we designate e "E\, e "E>, e "E\ , e "E> , u " 1 1 1\ 1\ tQ, u "t?, u "tQ , u "t? . The AC 1 1 1\ 1\ "eld frequency is supposed to be resonant for transitions between the levels e and e , i.e. u"(e !e )/ .
(6)
Such transitions are possible because the matrix element 1u "SK "u 2"(S/2O0. It should be no V ticed that transitions between the levels e and e and between the levels e and e , induced by the AC "eld, are forbidden because the matrix elements 1u "SK "u 2, 1u "SK "u 2 equal zero. Transitions V V between the levels e and e are not forbidden, but we suppose them to be not resonant for the frequency (6). This implies that the condition holds (e !e )! u
(7)
where *u is the linewidth of the source of the AC "eld, q is the relaxation time of the wavefunction
I.D. Tokman, G.A. Vugalter / Journal of Magnetism and Magnetic Materials 222 (2000) 375}378
phase. Since *E <*E , inequality (7) can be 1\ 1 rewritten in the form
*u 1 4(S!1) gk H 1\
(8)
Under this condition the level e (as well as higher levels) is not important and the magnetic molecule can be considered as a three-level system, the transitions between two levels of which are described by the resonance perturbation theory [14]. Let the initial con"guration of the magnetic molecule be t(t"0)"( (u #u ). (9) It corresponds to the state with the z-projection of the spin, equal to S, and with the energy (e #e )/2K!DS. Solving the SchroK dinger equation, we have for t;q 1 1 u e\ C R # u e\ C R cos X t t(t)K 0 (2 (2 i # u e\ C R sin X t, 0 (2
(10)
where gk H gk H (S 1u "SK "u 2" X " V 0 2
2(2
(11)
is the Rabi frequency. Calculating the average zprojection of the spin and taking into account the following relations: 1u "SK "u 2"1u "SK "u 2"0, X X 1u "SK "u 2"1u "SK "u 2"0, X X 1u "SK "u 2"S, X we "nd *E 1S (t)2KS cos X t cos 1 t. X 0
(12)
In the absence of the AC "eld the time required for the molecule to change the direction of its spin (magnetization) is determined by tunneling and equals t "p *E . 1
(13)
377
Certainly, this time should be much less than q. In contrast, if the AC magnetic "eld is not very weak, namely, X <*E / , 0 1
(14)
the time required for the molecule to change the direction of its spin is determined by Rabi's oscillations and equals t "n/X ;t . 0
(15)
Thus, if the amplitude of the resonant AC magnetic "eld satis"es the condition
H 4(2S gk H 1\ < , H (2S!1)! 2D
(16)
the time of magnetization tunneling is much less than in the case where the AC "eld is absent. Analogously, the same e!ect will occur if the AC "eld is resonant for transitions between the levels e and e . We emphasize that our model neglects the high-order anisotropy terms, the dipolar}dipolar interactions between magnetic molecules, and the coupling between magnetic molecules and nuclear spins. Therefore, we deal with a simple isolated spin. Moreover, the result obtained above is valid if the times t , t are essentially less than the relax ation time of the wavefunction phase, q. Nevertheless, our model seems reasonable due to the following facts. (i) The high-order anisotropy terms are needed in the absence of a DC magnetic "eld perpendicular to the anisotropy axis in order to describe tunneling between the states with opposite projections of the spin. In the presence of the DC "eld H which is perpendicular to the anisotropy axis and is not very weak, tunneling between the states with opposite projections of the spin is induced by the term proportional to H , which pre vails over the high-order anisotropy terms. (ii) The #uctuating part of the dipolar}dipolar interactions, which causes the spin relaxation, may be made ine$cient by diluting the sample. (iii) Despite the dipolar}dipolar interactions between magnetic molecules and the interactions between magnetic molecules and nuclear spins broaden the levels of a magnetic molecule, it is well established that there
378
I.D. Tokman, G.A. Vugalter / Journal of Magnetism and Magnetic Materials 222 (2000) 375}378
exists even the ground doublet (see, for example, Refs. [10,11]). Since we use experimental values of the relaxation time for our estimates, an exact consideration of the dipolar}dipolar interactions and the interactions between magnetic molecules and nuclear spins are not possible. We assume 0.1% disprosium (Dy>) -doped glasses to be appropriate for the observation of the predicted e!ect because of the comparatively long relaxation time and the great distance between magnetic ions. According to the experimental data [11], for such glasses S", DSK10 K, gK, q&10\ s. In this case, the resonant frequency (6) is u/2pK0.55 T Hz. Supposing H "70 kOe, we obtain that inequality (8) is satis"ed if the quality factor of the AC "eld source Q"u/*u<10. According to Eqs. (3), (13) and (16), the time of tunneling in the absence of the AC "eld is about t K7.5;10\ s and the tunneling time in the pres ence of the AC "eld is essentially less than t if H <0.04 Oe. 3. Conclusion We have investigated the in#uence of an AC magnetic "eld on the behaviour of magnetic molecules or magnetic clusters with a high value of spin. It has been shown that, if the AC "eld frequency is resonant for transitions between the lower states of the ground and "rst excited doublets, the time required for spin tunneling between two opposite directions can be much shorter than the time of tunneling in the absence of the AC magnetic "eld. We have pointed out the conditions under which the e!ect can be observed. The e!ect discussed, will also occur in the case where the frequency of the AC "eld is resonant for
transitions between the upper states of two doublets mentioned above.
Acknowledgements This work was supported partly by the Russian Foundation for Basic Research (Grant Nos. 98-0216412 and 97-02-17437), and by the International Center-Foundation for Promising Research in Nizhny Novgorod (Grant No. 99-2-03).
References [1] Quantum Tunneling of Magnetization, L.Gunther, B. Barbara (Eds.), Kluwer Academic Publishers, Dordrecht, 1995. [2] D.D. Awshalom, J.F. Smyth, G. Grinstein, D.P. DiVincenzo, D. Loss, Phys. Rev. Lett. 68 (1992) 3092. [3] M. Uebara, B. Barbara, J. Phys. 47 (1986) 235. [4] B. Barbara, W. Wensdorfer, L.S. Sampaio, J.G. Park, C. Paulsen, M.A. Novak, R. Ferre, D. Mailly, R. Sessoli, A. Caneschi, K. Hasselbach, A. Benoit, L. Thomas, J. Magn. Magn. Mater. 140}144 (1995) 1825. [5] M.A. Novak, R. Sessoli, A. Caneschi, D. Gatteschi, J. Magn. Magn. Mater. 146 (1995) 211. [6] J.R. Friedman, M.P. Sarachik, J. Tejada, R. Ziolo, Phys. Rev. Lett. 76 (1996) 3830. [7] L. Thomas, F. Lionti, R. Ballou, D. Gatteschi, R. Sessoli, B. Barbara, Nature 383 (1996) 145. [8] N. Vernier, G. Bellessa, T. Mallah, M. Verdagner, Phys. Rev. B 56 (1997) 75. [9] N. Vernier, G. Bellessa, Phys. Rev. B 48 (1993) 12842. [10] N. Vernier, G. Bellessa, Phys. Rev. Lett. 71 (1993) 4063. [11] N. Vernier, G. Bellessa, D.A. Parshin, Phys. Rev. Lett. 74 (1995) 3459. [12] D.A. Garanin, J.Phys. A 24 (1991) L61. [13] I.Ya. Korenblit, E.F. Shender, Sov. Phys. JETP 48 (1978) 937. [14] L.D. Landau, E.M. Lifshitz, Quantum Mechanics, 3rd edition. Pergamon Press, New York, 1977.