- Email: [email protected]
Phonon confinement in ultrathin nickel films
1 December 2000
Chemical Physics Letters 331 (2000) 115±118
www.elsevier.nl/locate/cplett
Phonon con®nement in ultrathin nickel ®lms A. Melikyan *, H. Minassian State Engineering University of Armenia, 105 Terian Str., 375009 Yerevan, Armenia Received 30 June 2000; in ®nal form 26 September 2000
Abstract We present an interpretation of the recently obtained data related to the measurements of the sound velocity in ultrathin nickel ®lms. The interpretation is based on the phonon frequency spectrum size quantization, which in turn leads to a decreasing of the phonon group velocity in the con®nement direction as the ®lm thickness decreases. A good agreement with the experimental data is obtained for all ®lm thicknesses for which the sound velocity has been measured. Ó 2000 Elsevier Science B.V. All rights reserved.
The ultrafast dynamics of nonequilibrium electrons in metals has recently attracted considerable attention [1±10]. The major reason for conducting femtosecond pump-probe experiments (see e.g., [10]) is the realization, that most modern high-speed electronic contacts are thin metallic ®lms, therefore nonequilibrium dynamics and transport of carriers determine the limiting properties of the contacts. The behavior of electrons, phonons, excitons, etc. in thin solid ®lms diers from that in bulk, because the wave number of quasiparticles in the direction perpendicular to the ®lm surface becomes discrete. This in turn leads to the peculiar dependence of the quasiparticle energy on the ®lm thickness, similar to that which takes place when a particle is localized in quantum well (quantum con®nement). In this case the quasiparticle energy spectrum becomes discrete, and this phenomena is known as size-quantization [11]. As shown in [12], the quantization of the phonons' frequency spectrum in ®lms with thickness less than the *
Corresponding author. Fax: +374-2-151-068. E-mail address: [email protected] (A. Melikyan).
electron's mean-free-path, may lead to a real decrease in the probability of electron±phonon scattering. Thus, the size-quantization of phonons' frequencies lead to the thickness dependence of the nonequilibrium electron relaxation time in thin metallic ®lms. It is well known that in thin semiconductor ®lms the electron±phonon interaction strongly depends on ®lm thickness [11,13]. The mechanism of this in¯uence is related to the size-quantization of the lattice vibrations' frequency spectrum. As a result the interaction matrix elements and energy±momentum conservation relationships (kinematic relationships) are changed. The detailed calculations reveal [11,13] that on decreasing the ®lm thickness, the relaxation rate of electrons will either increase or decrease depending on the character of the process ± interband, intraband, etc. However, to the best of our knowledge, the role of phonon con®nement in electron±phonon interaction in metallic ®lms has not been discussed in publications prior to [12]. The in¯uence of phonon con®nement becomes obvious for example in the case of very low temperatures. Namely, a minimum nonzero phonon frequency xmin equal to
0009-2614/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 0 ) 0 1 1 6 1 - 1
116
A. Melikyan, H. Minassian / Chemical Physics Letters 331 (2000) 115±118
pvs 1 L exists in the ®lm, where vs is the sound velocity, and L is the ®lm thickness. Hence the probability W of electron±phonon scattering at low temperatures will be proportional to Wbulk ; 2 W hx =kT e min ÿ1 xmin
where Wbulk is the scattering probability in bulk. At a sound velocity of 3 103 m/s, ®lm thickness of 10 nm and temperature of 10 K, the exponent will exceed by several orders of magnitude, which will considerably decrease the scattering probability as compared with that of the bulk. Note that, the probability of scattering also depends on the ®lm thickness, see Eqs. (1) and (2). At room temperature, and for the same values of the other parameters, the exponent will be much less than unity. The dependence on ®lm thickness is conditioned by two other eects; ®niteness of electron mean-free-path due to electron±phonon scattering, and quantization of the wave vector of acoustic phonons in the con®nement direction. The numerical calculation [12] indicated that the probability W of electron±phonon scattering considerably decreases as the ®lm thickness decreases (see Fig. 1 in [12]). At L > K, where K is the mean-free-path of electrons, W can be well approximated by an expression of the form
ÿ W Wbulk 1 ÿ eÿL=K :
From (2) and (3) one can see that the electron± phonon coupling constant depends on the ®lm thickness, and this fact should be taken into account in all the theories describing the spatial and temporal evolution of nonequilibrium energy distributions following optical excitation of metallic ®lms (Two-temperature model [2], Fermiliquid theory [6]). Obviously, extremely thin ®lms are most convenient for the experimental veri®cation of phonon con®nement, since in such ®lms the quantization of phonon frequencies in the direction perpendicular to the ®lm plane, is essential. In addition, in ultrathin ®lms the role of eects violating coherence (e.g., diusion) is small, because the thickness is smaller than the mean-free-path of phonons. It is also worth to note that for manifestation of phonon con®nement in electron±phonon interaction a greater time is necessary as compared with the time of ¯ight of phonon from one surface of the ®lm to another ± L=vs ± during which a con®ned phonon is being formed, e.g., at 10-nm thickness and sound velocity 2 103 m/s this time is 5 ps, while at shorter times the eect will not be observed because the phonon has not managed to reach surfaces. However, it comes out that for revealing the phonon con®nement it is not at all necessary to investigate the electron±phonon interaction. An obvious demonstration of this eect is the recent pump-probe experiment [14] with nickel ®lms of 40 to 1 nm thickness on copper substrate, where the dependence of sound velocity on ®lm thickness in the direction (0 0 1) was observed (Fig. 27 in [14]). This phenomenon can be described in terms of phonon con®nement, as it will be shown below. The acoustic pulse propagates with a velocity equal to the phonon group velocity Vg
Fig. 1. Film thickness versus the traversal time of the acoustic signal. Solid line corresponds to the sound velocity in bulk. Bars represent the experimental data. Crosses denote values, calculated according to (11) and (12). Integers neighboring crosses denote the number of monolayers in the sample.
3
qa ox 1 xmax a cos ; oq 2 2
4
where xmax is the maximum phonon frequency, q the wave vector, and a is the interplanar distance in the direction (0 0 1). Under the condition of size quantization expression (4) for group velocity
A. Melikyan, H. Minassian / Chemical Physics Letters 331 (2000) 115±118
must be modi®ed as the phonons' frequency spectrum becomes discrete. Indeed, for the more simple case when in the con®nement direction just two vibrational modes with frequencies x1 and x2 , and wave vectors q1 and q2 exist, the group velocity, instead of (4), obviously becomes Vg
x2 ÿ x1 : q2 ÿ q1
5
In the case of bulk sample frequency spectrum is continuous and the ratio of ®nite dierences in Eq. (5) reduces to the derivative as in (4). Let us ®nd the phonon frequency spectrum in conditions of size quantization when the ®lm is deposited on the substrate. Index 1 will be ascribed to the nearest to the substrate monolayer of the ®lm, and the displacement of monolayer with number m from the equilibrium state will be denoted as um . The equations for the displacements are mum ÿk 2um ÿ um1 ÿ umÿ1 ; mu1 ÿk u1 k u2 ÿ u1
m > 1;
6
with boundary conditions u0 0;
uN 1 uN ;
where N is the number of monolayers, and k is the force constant. Here we neglect the small dierence between the force constants kNi:Ni 37:90 kg=s2 and kNi:Cu 33 kg=s2 [15]. The other important approximation is that we neglect the energy ¯ow into the substrate. As it is shown, in nanostructures [15,16] the leakage leads just to broadening, determined by the imaginary part of an eigenfrequency. The solution of the set (6) is sought in the form um u sin m/ eÿixt :
7
The phase shift / will be found from the second boundary condition sin N / sin N 1/ from where /n
2n ÿ 1 p: 2N 1
8
The frequency will be found by substituting the solution (7) into the initial set (6) and taking into account (8):
2n ÿ 1 p xn xmax sin ; 2N 1 2
117
9
n 1; 2; 3; . . . ; N : From (9) one can see that the group velocity depends on the thickness and at N n, we obtain from (4) p2 n 2 a 2 ; 10 Vg L vs 1 ÿ 8L2 where vs is the group velocity in bulk. From this it follows that the group velocity decreases at the decrease of ®lm thickness. It is interesting to note, that the frequency which corresponds to the maximum wavelength in the case of four monolayers, i.e., n 1 and N 4, is very close to the value of 1.4 THz, observed in [14]. Indeed, adopting for vs the value 4200 m/s, and 0.17 nm for interplanar distance ± a (both used in [14]), one can see from (4) and (9) that x=2p 1:39 THz. In [14], Fig. 27, the ®lm thickness versus appearance time of the ®rst peak of second harmonic generation in nickel ®lm on Cu (1 0 0) was measured. For a thickness of more than 4 nm, all points are in excellent agreement with an estimate of the longitudinal sound velocity, i.e., 4200 m/s. For a thickness less than 4 nm, the plotted curve shows a trend toward smaller values of velocity which is in qualitative agreement with (10). This regularity clearly demonstrates that the group velocity of phonons was in fact measured. Thus, the experimental values on Fig. 27 of [14] should correspond to the curve described by equation L s; Vg L
11
where s is the time interval during which the acoustic pulse travels from the back wall to the front one. Indeed, the ratio L=s, according to Fig. 27 of [14] decreases at the decrease of thickness. For the interpretation of experimental results we come from (9) and (11). The best agreement is achieved when the group velocity is calculated on the assumption that phonon modes with n 4 and n 5 are the most essential ones, i.e.,
118
Vg L
A. Melikyan, H. Minassian / Chemical Physics Letters 331 (2000) 115±118
x5 ÿ x4 2vs 2N 1 q5 ÿ q4 p p 4p cos : sin 4N 2 2N 1
Prof. E. Ivchenko for helpful discussions. This work was supported by CRDF Award AP1-376. 12
The values, calculated according to (11) and (12) are in excellent agreement with the data obtained in [14], as shown in Fig. 1. Thus, the phonon con®nement in metals manifests itself not only through electron±phonon interaction [12], but also has a considerable impact on the propagation of sound waves in ultrathin ®lms. The fact that in the case under consideration the main contribution is due to phonon modes with n 4 and n 5, and is probably conditioned by the excitation mechanism accompanying the second harmonic generation, and the materials of ®lm and substrate as well. It is possible that in experiments with ®lms of heavier metals, e.g. gold, other phonon modes will be essential. One can see from Fig. 1, that with increasing the number of monolayers, N , crosses approach the straight line, i.e., the group velocity becomes insensitive to N for n 6 5. Acknowledgements We express our appreciation to Prof. E. Matthias for putting at our disposal the manuscript of his paper before it has been published and
References [1] G.L. Esley, Phys. Rev. Lett. 51 (1983) 2140. [2] J.G. Fujimoto, J.M. Liu, E.P. Ippen, N. Bloembergen, Phys. Rev. Lett 53 (1984) 1837. [3] G.L. Esley, Phys. Rev. B 33 (1986) 2144. [4] H.E. Elsayed-Ali, T.B. Norris, M.A. Pessot, G.A. Mourou, Phys. Rev. Lett. 58 (1987) 1212. [5] S.D. Brorson, J.G. Fujimoto, E.P. Ippen, Phys. Rev. Lett. 59 (1987) 1962. [6] H.E. Elsayed-Ali, T. Juhasz, G.O. Smith, W.E. Bron, Phys. Rev. B 43 (1991) 4488. [7] W.S. Fann, R. Storz, H.W.K. Tom, Phys. Rev. B 46 (1992) 13592. [8] T. Juhasz, H.E. Elsayed-Ali, G.O. Smith, C. Suarez, W.E. Bron, Phys. Rev. B 48 (1993) 15488. [9] C.K. Sun, F. Vallee, L.H. Aciolo, E.P. Ippen, J.G. Fujimoto, Phys. Rev. B 50 (1994) 15337. [10] C. Suarez, W.E. Bron, T. Juhasz, Phys. Rev. Lett. 75 (1995) 4536. [11] B.K. Ridly, Rep. Prog. Phys. 54 (1991) 169. [12] A. Melikyan, H. Minassian, A. Guerra III, W. Wu, Appl. Phys. B, Lasers and Optics 5 (1999) 411. [13] J. Shah, Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures, Springer, Berlin, 1996. [14] J. Hohlfeld, S.-S. Wellershof, J. Guedde, U. Conrad, V. Jaehnke, E. Matthias, Chem. Phys. 251 (2000) 237. [15] S.Y. Tong, Y. Chen, J.M. Yao, Z.Q. Wu, Phys. Rev. B 39 (1989) 5611. [16] S.V. Gupalov, I.A. Merkulov, Physics of Solid State 41 (1999) 1349.
Chemical Physics Letters 331 (2000) 115±118
www.elsevier.nl/locate/cplett
Phonon con®nement in ultrathin nickel ®lms A. Melikyan *, H. Minassian State Engineering University of Armenia, 105 Terian Str., 375009 Yerevan, Armenia Received 30 June 2000; in ®nal form 26 September 2000
Abstract We present an interpretation of the recently obtained data related to the measurements of the sound velocity in ultrathin nickel ®lms. The interpretation is based on the phonon frequency spectrum size quantization, which in turn leads to a decreasing of the phonon group velocity in the con®nement direction as the ®lm thickness decreases. A good agreement with the experimental data is obtained for all ®lm thicknesses for which the sound velocity has been measured. Ó 2000 Elsevier Science B.V. All rights reserved.
The ultrafast dynamics of nonequilibrium electrons in metals has recently attracted considerable attention [1±10]. The major reason for conducting femtosecond pump-probe experiments (see e.g., [10]) is the realization, that most modern high-speed electronic contacts are thin metallic ®lms, therefore nonequilibrium dynamics and transport of carriers determine the limiting properties of the contacts. The behavior of electrons, phonons, excitons, etc. in thin solid ®lms diers from that in bulk, because the wave number of quasiparticles in the direction perpendicular to the ®lm surface becomes discrete. This in turn leads to the peculiar dependence of the quasiparticle energy on the ®lm thickness, similar to that which takes place when a particle is localized in quantum well (quantum con®nement). In this case the quasiparticle energy spectrum becomes discrete, and this phenomena is known as size-quantization [11]. As shown in [12], the quantization of the phonons' frequency spectrum in ®lms with thickness less than the *
Corresponding author. Fax: +374-2-151-068. E-mail address: [email protected] (A. Melikyan).
electron's mean-free-path, may lead to a real decrease in the probability of electron±phonon scattering. Thus, the size-quantization of phonons' frequencies lead to the thickness dependence of the nonequilibrium electron relaxation time in thin metallic ®lms. It is well known that in thin semiconductor ®lms the electron±phonon interaction strongly depends on ®lm thickness [11,13]. The mechanism of this in¯uence is related to the size-quantization of the lattice vibrations' frequency spectrum. As a result the interaction matrix elements and energy±momentum conservation relationships (kinematic relationships) are changed. The detailed calculations reveal [11,13] that on decreasing the ®lm thickness, the relaxation rate of electrons will either increase or decrease depending on the character of the process ± interband, intraband, etc. However, to the best of our knowledge, the role of phonon con®nement in electron±phonon interaction in metallic ®lms has not been discussed in publications prior to [12]. The in¯uence of phonon con®nement becomes obvious for example in the case of very low temperatures. Namely, a minimum nonzero phonon frequency xmin equal to
0009-2614/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 0 ) 0 1 1 6 1 - 1
116
A. Melikyan, H. Minassian / Chemical Physics Letters 331 (2000) 115±118
pvs 1 L exists in the ®lm, where vs is the sound velocity, and L is the ®lm thickness. Hence the probability W of electron±phonon scattering at low temperatures will be proportional to Wbulk ; 2 W hx =kT e min ÿ1 xmin
where Wbulk is the scattering probability in bulk. At a sound velocity of 3 103 m/s, ®lm thickness of 10 nm and temperature of 10 K, the exponent will exceed by several orders of magnitude, which will considerably decrease the scattering probability as compared with that of the bulk. Note that, the probability of scattering also depends on the ®lm thickness, see Eqs. (1) and (2). At room temperature, and for the same values of the other parameters, the exponent will be much less than unity. The dependence on ®lm thickness is conditioned by two other eects; ®niteness of electron mean-free-path due to electron±phonon scattering, and quantization of the wave vector of acoustic phonons in the con®nement direction. The numerical calculation [12] indicated that the probability W of electron±phonon scattering considerably decreases as the ®lm thickness decreases (see Fig. 1 in [12]). At L > K, where K is the mean-free-path of electrons, W can be well approximated by an expression of the form
ÿ W Wbulk 1 ÿ eÿL=K :
From (2) and (3) one can see that the electron± phonon coupling constant depends on the ®lm thickness, and this fact should be taken into account in all the theories describing the spatial and temporal evolution of nonequilibrium energy distributions following optical excitation of metallic ®lms (Two-temperature model [2], Fermiliquid theory [6]). Obviously, extremely thin ®lms are most convenient for the experimental veri®cation of phonon con®nement, since in such ®lms the quantization of phonon frequencies in the direction perpendicular to the ®lm plane, is essential. In addition, in ultrathin ®lms the role of eects violating coherence (e.g., diusion) is small, because the thickness is smaller than the mean-free-path of phonons. It is also worth to note that for manifestation of phonon con®nement in electron±phonon interaction a greater time is necessary as compared with the time of ¯ight of phonon from one surface of the ®lm to another ± L=vs ± during which a con®ned phonon is being formed, e.g., at 10-nm thickness and sound velocity 2 103 m/s this time is 5 ps, while at shorter times the eect will not be observed because the phonon has not managed to reach surfaces. However, it comes out that for revealing the phonon con®nement it is not at all necessary to investigate the electron±phonon interaction. An obvious demonstration of this eect is the recent pump-probe experiment [14] with nickel ®lms of 40 to 1 nm thickness on copper substrate, where the dependence of sound velocity on ®lm thickness in the direction (0 0 1) was observed (Fig. 27 in [14]). This phenomenon can be described in terms of phonon con®nement, as it will be shown below. The acoustic pulse propagates with a velocity equal to the phonon group velocity Vg
Fig. 1. Film thickness versus the traversal time of the acoustic signal. Solid line corresponds to the sound velocity in bulk. Bars represent the experimental data. Crosses denote values, calculated according to (11) and (12). Integers neighboring crosses denote the number of monolayers in the sample.
3
qa ox 1 xmax a cos ; oq 2 2
4
where xmax is the maximum phonon frequency, q the wave vector, and a is the interplanar distance in the direction (0 0 1). Under the condition of size quantization expression (4) for group velocity
A. Melikyan, H. Minassian / Chemical Physics Letters 331 (2000) 115±118
must be modi®ed as the phonons' frequency spectrum becomes discrete. Indeed, for the more simple case when in the con®nement direction just two vibrational modes with frequencies x1 and x2 , and wave vectors q1 and q2 exist, the group velocity, instead of (4), obviously becomes Vg
x2 ÿ x1 : q2 ÿ q1
5
In the case of bulk sample frequency spectrum is continuous and the ratio of ®nite dierences in Eq. (5) reduces to the derivative as in (4). Let us ®nd the phonon frequency spectrum in conditions of size quantization when the ®lm is deposited on the substrate. Index 1 will be ascribed to the nearest to the substrate monolayer of the ®lm, and the displacement of monolayer with number m from the equilibrium state will be denoted as um . The equations for the displacements are mum ÿk 2um ÿ um1 ÿ umÿ1 ; mu1 ÿk u1 k u2 ÿ u1
m > 1;
6
with boundary conditions u0 0;
uN 1 uN ;
where N is the number of monolayers, and k is the force constant. Here we neglect the small dierence between the force constants kNi:Ni 37:90 kg=s2 and kNi:Cu 33 kg=s2 [15]. The other important approximation is that we neglect the energy ¯ow into the substrate. As it is shown, in nanostructures [15,16] the leakage leads just to broadening, determined by the imaginary part of an eigenfrequency. The solution of the set (6) is sought in the form um u sin m/ eÿixt :
7
The phase shift / will be found from the second boundary condition sin N / sin N 1/ from where /n
2n ÿ 1 p: 2N 1
8
The frequency will be found by substituting the solution (7) into the initial set (6) and taking into account (8):
2n ÿ 1 p xn xmax sin ; 2N 1 2
117
9
n 1; 2; 3; . . . ; N : From (9) one can see that the group velocity depends on the thickness and at N n, we obtain from (4) p2 n 2 a 2 ; 10 Vg L vs 1 ÿ 8L2 where vs is the group velocity in bulk. From this it follows that the group velocity decreases at the decrease of ®lm thickness. It is interesting to note, that the frequency which corresponds to the maximum wavelength in the case of four monolayers, i.e., n 1 and N 4, is very close to the value of 1.4 THz, observed in [14]. Indeed, adopting for vs the value 4200 m/s, and 0.17 nm for interplanar distance ± a (both used in [14]), one can see from (4) and (9) that x=2p 1:39 THz. In [14], Fig. 27, the ®lm thickness versus appearance time of the ®rst peak of second harmonic generation in nickel ®lm on Cu (1 0 0) was measured. For a thickness of more than 4 nm, all points are in excellent agreement with an estimate of the longitudinal sound velocity, i.e., 4200 m/s. For a thickness less than 4 nm, the plotted curve shows a trend toward smaller values of velocity which is in qualitative agreement with (10). This regularity clearly demonstrates that the group velocity of phonons was in fact measured. Thus, the experimental values on Fig. 27 of [14] should correspond to the curve described by equation L s; Vg L
11
where s is the time interval during which the acoustic pulse travels from the back wall to the front one. Indeed, the ratio L=s, according to Fig. 27 of [14] decreases at the decrease of thickness. For the interpretation of experimental results we come from (9) and (11). The best agreement is achieved when the group velocity is calculated on the assumption that phonon modes with n 4 and n 5 are the most essential ones, i.e.,
118
Vg L
A. Melikyan, H. Minassian / Chemical Physics Letters 331 (2000) 115±118
x5 ÿ x4 2vs 2N 1 q5 ÿ q4 p p 4p cos : sin 4N 2 2N 1
Prof. E. Ivchenko for helpful discussions. This work was supported by CRDF Award AP1-376. 12
The values, calculated according to (11) and (12) are in excellent agreement with the data obtained in [14], as shown in Fig. 1. Thus, the phonon con®nement in metals manifests itself not only through electron±phonon interaction [12], but also has a considerable impact on the propagation of sound waves in ultrathin ®lms. The fact that in the case under consideration the main contribution is due to phonon modes with n 4 and n 5, and is probably conditioned by the excitation mechanism accompanying the second harmonic generation, and the materials of ®lm and substrate as well. It is possible that in experiments with ®lms of heavier metals, e.g. gold, other phonon modes will be essential. One can see from Fig. 1, that with increasing the number of monolayers, N , crosses approach the straight line, i.e., the group velocity becomes insensitive to N for n 6 5. Acknowledgements We express our appreciation to Prof. E. Matthias for putting at our disposal the manuscript of his paper before it has been published and
References [1] G.L. Esley, Phys. Rev. Lett. 51 (1983) 2140. [2] J.G. Fujimoto, J.M. Liu, E.P. Ippen, N. Bloembergen, Phys. Rev. Lett 53 (1984) 1837. [3] G.L. Esley, Phys. Rev. B 33 (1986) 2144. [4] H.E. Elsayed-Ali, T.B. Norris, M.A. Pessot, G.A. Mourou, Phys. Rev. Lett. 58 (1987) 1212. [5] S.D. Brorson, J.G. Fujimoto, E.P. Ippen, Phys. Rev. Lett. 59 (1987) 1962. [6] H.E. Elsayed-Ali, T. Juhasz, G.O. Smith, W.E. Bron, Phys. Rev. B 43 (1991) 4488. [7] W.S. Fann, R. Storz, H.W.K. Tom, Phys. Rev. B 46 (1992) 13592. [8] T. Juhasz, H.E. Elsayed-Ali, G.O. Smith, C. Suarez, W.E. Bron, Phys. Rev. B 48 (1993) 15488. [9] C.K. Sun, F. Vallee, L.H. Aciolo, E.P. Ippen, J.G. Fujimoto, Phys. Rev. B 50 (1994) 15337. [10] C. Suarez, W.E. Bron, T. Juhasz, Phys. Rev. Lett. 75 (1995) 4536. [11] B.K. Ridly, Rep. Prog. Phys. 54 (1991) 169. [12] A. Melikyan, H. Minassian, A. Guerra III, W. Wu, Appl. Phys. B, Lasers and Optics 5 (1999) 411. [13] J. Shah, Ultrafast Spectroscopy of Semiconductors and Semiconductor Nanostructures, Springer, Berlin, 1996. [14] J. Hohlfeld, S.-S. Wellershof, J. Guedde, U. Conrad, V. Jaehnke, E. Matthias, Chem. Phys. 251 (2000) 237. [15] S.Y. Tong, Y. Chen, J.M. Yao, Z.Q. Wu, Phys. Rev. B 39 (1989) 5611. [16] S.V. Gupalov, I.A. Merkulov, Physics of Solid State 41 (1999) 1349.