- Email: [email protected]
Noise effect on resonant tunneling in the nanoscale molecular magnet (Fe8)
Physica B 329–333 (2003) 1170–1171
Noise effect on resonant tunneling in the nanoscale molecular magnet ðFe8Þ Masamichi Nishino*, Keiji Saito, Seiji Miyashita Department of Applied Physics, Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo Bunkyo-ku, Tokyo 113-0033, Japan
Abstract We study the effect of noise on the resonant tunneling for a realistic model of Fe8 : We discuss both the static and dynamical effect of noise on the estimation of tunnel splitting by the use of the Landau–Zener–Stuckelberg . formula. r 2003 Elsevier Science B.V. All rights reserved. Keywords: Fe8 ; Resonant tunneling; Tunnel splitting; Noise
1. Introduction
2. Estimation of the tunnel splitting
Since nanoscale molecular magnets (Mn12 ; Fe8 ; etc.) were reported to exhibit unprecedented features, much effort has been exerted to the quantum tunneling of magnetization. The tunneling dynamics has been studied from the viewpoint of the nonadiabatic transition [1–3]. The transition probability in the two level system is given by the Landau–Zener–Stuckelberg . (LZS) formula [4]. This formula is useful to pure quantum dynamics. However, in real systems fluctuation and dissipation affect the process of quantum dynamics. At low enough temperatures, effects of the ubiquitous hyperfine field within the molecule and the dipole field of other molecules become significant for the relaxation process. Therefore, we have to take notice of estimating the tunnel splitting by the use of LZS formula. We have already studied the effect of noise in the case S ¼ 12 and clarified that the effect of noise cannot be negligible and the estimated gap deviates from the true tunnel splitting [5]. Here we study the noise effect of a realistic model with S ¼ 10 for Fe8 [6–8]. It is found that the estimated gap coincides with the true gap in the fast sweeping regime, while the effect of noise cannot be negligible in the slow sweeping regime.
We consider the following Hamiltonian which is adopted for the molecular magnet of Fe8 :
*Corresponding author. E-mail address: [email protected] (M. Nishino).
# ¼ DðSz Þ2 þ EððSx Þ2 ðSy Þ2 Þ HðtÞ z * þ CððS þ Þ4 þ ðS Þ4 Þ hðtÞS þ H# noise ðtÞ;
ð1Þ
where we set D ¼ 0:292 K; E ¼ 0:046 K; and C ¼ * 2:9 105 K [6]. The field ðhðtÞÞ in the z-direction is swept linearly with time. We calculate the tunneling probability under the influence of the noise. We adopt H# noise ðtÞ ¼ ha ðtÞS a ; where a denotes a direction of x; y; or z: Here hðtÞ is a random Gaussian noise and has an exponential-decaying autocorrelation. We define B as the amplitude of the noise [5]. Using the LZS formula, the tunneling probability p between the magnetizations m and m0 in the sweeping field is given as a function of the energy gap DEm;m0 and the sweeping velocity c ð¼ dHz =dtÞ: p ¼ 1 expðpðDEm;m0 Þ2 =2_jm m0 jcÞ: In the experiments, p was estimated for given c and the tunnel splitting was estimated as a function of the velocity making use this formula. Here we estimate the effect of noise on estimation of the tunnel splitting by evaluating the following quantity. DEeff D ; DEm;m0
0921-4526/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4526(02)02089-6
DEeff
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2_jm m0 jc lnð1 pÞ ; p
ð2Þ
M. Nishino et al. / Physica B 329–333 (2003) 1170–1171
where DEm;m0 is the true energy gap and DEeff is the estimated energy gap from the obtained value of p: We can evaluate the effect of noise by estimating how much D deviates from 1.
1171
1.1 1 0.9
∆
0.8
3. Results and discussion
0.7 0.6
In the beginning, we focus on the regime of the fast sweeping because the estimated gap shows no velocity dependence in the experiments [7,8]. In this regime, the noise could be considered to be static ðcbgDE10;10 Þ [5]. g is the damping factor of the noise. The tunnel splitting DE10;10 is estimated numerically as 0:452 107 K for system (1). We estimate numerically the dependence of the gap on static noise hx and hy ; and approximate them by an algebraic function f ðhÞ ¼ DE10;10 þ ah2 ; where a ¼ 1:356 106 K=T2 for hx and a ¼ 2:056 106 K=T2 for hy : The error bar of a is within 1011 K=T2 : The values of the gap DE in the presence of hy and the fitting function f ðhy Þ are shown in Fig. 1. We find that 0:997pDx p1 and 1pDy p1:005 [9] when qffiffiffiffiffiffiffiffiffiffiffiffiffiffi B ¼ /hy S2 p0:1 101 (T), whose energy scale is 106
∆Ε/ Κ
times as large as the tunnel splitting DE10;10 : This holds true from c ¼ 105 to 100 (T/s), where the experiments were performed. This indicates that the estimated gap is equal to the true gap. This is because the transition from m ¼ 10 to m0 ¼ 10 is the 20th perturbation process and the term of hx Sx or hy Sy can give only little effect on the transition process. Next we investigate the dynamical effect of the noise. . It is difficult to simulate time-dependent Schrodinger equation for system (1) with realistic parameters. However the following parameter set is enough to investigate the dynamical effect of the 20th perturbation: D ¼ 0:1 K; E ¼ 0:07 K; and C ¼ 0 K: The tunnel splitting ðDE10;10 Þ is 0:00113 K and we set B ¼ DE10;10 =40: We show the sweeping rate dependence of Dx in Fig. 2, where _ ¼ 1: Here the arrow of Fig. 2 shows the regime of sweeping rate corresponding to that of the experiment. It is found that the fluctuation of noise has an influence on the 4.58 10
-8
4.56 10
-8
4.54 10
-8
4.52 10
-8
4.5 10
-8
4.48 10
-8
-0.01
-0.005
0 hy / T
0.005
0.01
Fig. 1. DEðhy Þ ðJÞ and f ðhy Þ (solid line).
0.5 0.4 -8 10
-7
10
-6
-5
10 10 11 c/10 T/s
0.0001
Fig. 2. Sweeping rate dependence of Dx : Jðg ¼ 0:01 s1 Þ; Wð0:1Þ; and }ð1:0Þ:
tunneling process and the estimated gap decreases in the slow sweeping region [5] even if B is not so large. This tendency reproduces the experimental results qualitatively. The system S ¼ 12 is sensitive to the effect of noise and the estimated gap should be corrected by OðB2 =DE 2 Þ for the fast sweeping regime. On the other hand, the effect of noise can be ignored in the estimation of the tunnel splitting in the case S ¼ 10: It is mainly due to the 20th perturbation. However the fluctuation of noise gives an influence to the tunneling process in the slow sweeping regime regardless of the amplitude of noise. As a result, reduction of the estimated gap is observed in that regime, which reproduces the corresponding experimental data qualitatively. In the experimental results, isotopically substituted Fe8 s show the different estimated values [7,8]. This difference cannot be explained within our modeling. A treatment from a more microscopic viewpoint may be necessary. References [1] S. Miyashita, J. Phys. Soc. Japan 64 (1995) 3207; S. Miyashita, J. Phys. Soc. Japan 65 (1996) 2734. [2] S. Miyashita, K. Saito, H. De Raedt, Phys. Rev. Lett. 80 (1998) 1525. [3] M. Nishino, H. Onishi, K. Yamaguchi, S. Miyashita, Phys. Rev. B 62 (2000) 9463. [4] L. Landau, Phys. Z. Sowjetunion 2 (1932) 46; C. Zener, Proc. R. Soc. London Ser. A 137 (1932) 696; E.C.G. Stuckelberg, . Helv. Phys. Acta 5 (1932) 369. [5] M. Nishino, K. Saito, S. Miyashita, Phys. Rev. B 65 (2002) 014403. [6] W. Wernsdorfer, R. Sessoli, Science 284 (1999) 133. [7] W. Wernsdorfer, R. Sessoli, A. Caneschi, D. Gatteschi, A. Cornia, EuroPhys. Lett. 50 (2000) 552; W. Wernsdorfer, R. Sessoli, A. Caneschi, D. Gatteschi, A. Cornia, J. Phys. Soc. Japan 69 (Suppl. A) (2000) 375 (Frontiers in Magnetism). [8] Wernsdorfer, cond-mat/0101104, Adv. Chem. Phys. and references therein, in preparation. [9] M. Nishino, K. Saito, S. Miyashita, Progr. Theo. Phys. 145 (Suppl.) (2002) 399.
Noise effect on resonant tunneling in the nanoscale molecular magnet ðFe8Þ Masamichi Nishino*, Keiji Saito, Seiji Miyashita Department of Applied Physics, Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo Bunkyo-ku, Tokyo 113-0033, Japan
Abstract We study the effect of noise on the resonant tunneling for a realistic model of Fe8 : We discuss both the static and dynamical effect of noise on the estimation of tunnel splitting by the use of the Landau–Zener–Stuckelberg . formula. r 2003 Elsevier Science B.V. All rights reserved. Keywords: Fe8 ; Resonant tunneling; Tunnel splitting; Noise
1. Introduction
2. Estimation of the tunnel splitting
Since nanoscale molecular magnets (Mn12 ; Fe8 ; etc.) were reported to exhibit unprecedented features, much effort has been exerted to the quantum tunneling of magnetization. The tunneling dynamics has been studied from the viewpoint of the nonadiabatic transition [1–3]. The transition probability in the two level system is given by the Landau–Zener–Stuckelberg . (LZS) formula [4]. This formula is useful to pure quantum dynamics. However, in real systems fluctuation and dissipation affect the process of quantum dynamics. At low enough temperatures, effects of the ubiquitous hyperfine field within the molecule and the dipole field of other molecules become significant for the relaxation process. Therefore, we have to take notice of estimating the tunnel splitting by the use of LZS formula. We have already studied the effect of noise in the case S ¼ 12 and clarified that the effect of noise cannot be negligible and the estimated gap deviates from the true tunnel splitting [5]. Here we study the noise effect of a realistic model with S ¼ 10 for Fe8 [6–8]. It is found that the estimated gap coincides with the true gap in the fast sweeping regime, while the effect of noise cannot be negligible in the slow sweeping regime.
We consider the following Hamiltonian which is adopted for the molecular magnet of Fe8 :
*Corresponding author. E-mail address: [email protected] (M. Nishino).
# ¼ DðSz Þ2 þ EððSx Þ2 ðSy Þ2 Þ HðtÞ z * þ CððS þ Þ4 þ ðS Þ4 Þ hðtÞS þ H# noise ðtÞ;
ð1Þ
where we set D ¼ 0:292 K; E ¼ 0:046 K; and C ¼ * 2:9 105 K [6]. The field ðhðtÞÞ in the z-direction is swept linearly with time. We calculate the tunneling probability under the influence of the noise. We adopt H# noise ðtÞ ¼ ha ðtÞS a ; where a denotes a direction of x; y; or z: Here hðtÞ is a random Gaussian noise and has an exponential-decaying autocorrelation. We define B as the amplitude of the noise [5]. Using the LZS formula, the tunneling probability p between the magnetizations m and m0 in the sweeping field is given as a function of the energy gap DEm;m0 and the sweeping velocity c ð¼ dHz =dtÞ: p ¼ 1 expðpðDEm;m0 Þ2 =2_jm m0 jcÞ: In the experiments, p was estimated for given c and the tunnel splitting was estimated as a function of the velocity making use this formula. Here we estimate the effect of noise on estimation of the tunnel splitting by evaluating the following quantity. DEeff D ; DEm;m0
0921-4526/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0921-4526(02)02089-6
DEeff
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2_jm m0 jc lnð1 pÞ ; p
ð2Þ
M. Nishino et al. / Physica B 329–333 (2003) 1170–1171
where DEm;m0 is the true energy gap and DEeff is the estimated energy gap from the obtained value of p: We can evaluate the effect of noise by estimating how much D deviates from 1.
1171
1.1 1 0.9
∆
0.8
3. Results and discussion
0.7 0.6
In the beginning, we focus on the regime of the fast sweeping because the estimated gap shows no velocity dependence in the experiments [7,8]. In this regime, the noise could be considered to be static ðcbgDE10;10 Þ [5]. g is the damping factor of the noise. The tunnel splitting DE10;10 is estimated numerically as 0:452 107 K for system (1). We estimate numerically the dependence of the gap on static noise hx and hy ; and approximate them by an algebraic function f ðhÞ ¼ DE10;10 þ ah2 ; where a ¼ 1:356 106 K=T2 for hx and a ¼ 2:056 106 K=T2 for hy : The error bar of a is within 1011 K=T2 : The values of the gap DE in the presence of hy and the fitting function f ðhy Þ are shown in Fig. 1. We find that 0:997pDx p1 and 1pDy p1:005 [9] when qffiffiffiffiffiffiffiffiffiffiffiffiffiffi B ¼ /hy S2 p0:1 101 (T), whose energy scale is 106
∆Ε/ Κ
times as large as the tunnel splitting DE10;10 : This holds true from c ¼ 105 to 100 (T/s), where the experiments were performed. This indicates that the estimated gap is equal to the true gap. This is because the transition from m ¼ 10 to m0 ¼ 10 is the 20th perturbation process and the term of hx Sx or hy Sy can give only little effect on the transition process. Next we investigate the dynamical effect of the noise. . It is difficult to simulate time-dependent Schrodinger equation for system (1) with realistic parameters. However the following parameter set is enough to investigate the dynamical effect of the 20th perturbation: D ¼ 0:1 K; E ¼ 0:07 K; and C ¼ 0 K: The tunnel splitting ðDE10;10 Þ is 0:00113 K and we set B ¼ DE10;10 =40: We show the sweeping rate dependence of Dx in Fig. 2, where _ ¼ 1: Here the arrow of Fig. 2 shows the regime of sweeping rate corresponding to that of the experiment. It is found that the fluctuation of noise has an influence on the 4.58 10
-8
4.56 10
-8
4.54 10
-8
4.52 10
-8
4.5 10
-8
4.48 10
-8
-0.01
-0.005
0 hy / T
0.005
0.01
Fig. 1. DEðhy Þ ðJÞ and f ðhy Þ (solid line).
0.5 0.4 -8 10
-7
10
-6
-5
10 10 11 c/10 T/s
0.0001
Fig. 2. Sweeping rate dependence of Dx : Jðg ¼ 0:01 s1 Þ; Wð0:1Þ; and }ð1:0Þ:
tunneling process and the estimated gap decreases in the slow sweeping region [5] even if B is not so large. This tendency reproduces the experimental results qualitatively. The system S ¼ 12 is sensitive to the effect of noise and the estimated gap should be corrected by OðB2 =DE 2 Þ for the fast sweeping regime. On the other hand, the effect of noise can be ignored in the estimation of the tunnel splitting in the case S ¼ 10: It is mainly due to the 20th perturbation. However the fluctuation of noise gives an influence to the tunneling process in the slow sweeping regime regardless of the amplitude of noise. As a result, reduction of the estimated gap is observed in that regime, which reproduces the corresponding experimental data qualitatively. In the experimental results, isotopically substituted Fe8 s show the different estimated values [7,8]. This difference cannot be explained within our modeling. A treatment from a more microscopic viewpoint may be necessary. References [1] S. Miyashita, J. Phys. Soc. Japan 64 (1995) 3207; S. Miyashita, J. Phys. Soc. Japan 65 (1996) 2734. [2] S. Miyashita, K. Saito, H. De Raedt, Phys. Rev. Lett. 80 (1998) 1525. [3] M. Nishino, H. Onishi, K. Yamaguchi, S. Miyashita, Phys. Rev. B 62 (2000) 9463. [4] L. Landau, Phys. Z. Sowjetunion 2 (1932) 46; C. Zener, Proc. R. Soc. London Ser. A 137 (1932) 696; E.C.G. Stuckelberg, . Helv. Phys. Acta 5 (1932) 369. [5] M. Nishino, K. Saito, S. Miyashita, Phys. Rev. B 65 (2002) 014403. [6] W. Wernsdorfer, R. Sessoli, Science 284 (1999) 133. [7] W. Wernsdorfer, R. Sessoli, A. Caneschi, D. Gatteschi, A. Cornia, EuroPhys. Lett. 50 (2000) 552; W. Wernsdorfer, R. Sessoli, A. Caneschi, D. Gatteschi, A. Cornia, J. Phys. Soc. Japan 69 (Suppl. A) (2000) 375 (Frontiers in Magnetism). [8] Wernsdorfer, cond-mat/0101104, Adv. Chem. Phys. and references therein, in preparation. [9] M. Nishino, K. Saito, S. Miyashita, Progr. Theo. Phys. 145 (Suppl.) (2002) 399.