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Dynamical response of local magnetic structures induced by inhomogeneities to time-dependent magnetic field
Computer Physics Communications 142 (2001) 382–386 www.elsevier.com/locate/cpc
Dynamical response of local magnetic structures induced by inhomogeneities to time-dependent magnetic field Masamichi Nishino ∗ , Seiji Miyashita Department of Applied Physics, Graduate School of Engineering, The University of Tokyo, Bunkyoku, Tokyo, Japan
Abstract Local magnetic structures induced by the bond impurities in the S = 1/2 bond alternating system are found to be looked on as an effective S = 1/2 spin at low temperatures. The response of such structures to a time dependent magnetic field is an interesting problem to study and it would be important for the analysis of the nanoscale materials. In realistic systems the effect of coupling to the environment causes fluctuation and dissipation and plays a crucial role in such dynamics. We study the tunneling process in noise by solving the time-dependent Schrödinger equation numerically from the viewpoint of nonadiabatic transition in the two level system, and make clear basic properties. We investigate the dependence of the tunneling probability on sweeping velocity of the applied field and also on the direction of the random noise and its relaxation time. 2001 Elsevier Science B.V. All rights reserved. PACS: 75.40.Gb; 75.10.Jm Keywords: Quantum tunneling; Nonadiabatic transition; Noise effect; Local magnetic structures; Tunneling probability; Time-dependent Schrödinger equation
1. Introduction The quantum dynamics in microscale or nanoscale materials is one of topics in the condensed matter physics. At low temperatures quantum dynamical effects are important and these effects have really been observed in the hysteresis phenomena of the nanoscale molecular magnets such as Mn12 , Fe8 , and V15 [1,2]. Among the quantum phenomena in these systems, the quantum tunneling of the magnetization (QTM), which is often called resonant tunneling, has become one of topics of much interest. We have * Corresponding author.
E-mail address: [email protected] (M. Nishino).
studied this tunneling dynamics from the viewpoint of the nonadiabatic transition [3–5]. We have found that local magnetic structures which are induced at bond impurities in the Heisenberg chain (S = 1/2) show similar quantum mechanical dynamics [5,6], as well as the molecular magnets. In the bond-alternating antiferromagnetic Heisenberg chain with defects of the alternation, the local magnetic structures are induced around the defects and they can be looked on as an effective S = 1/2 spin at low temperatures. The response of such structures to magnetic field is also possible to understand in the viewpoint of nonadiabatic transition. In the realistic system the effect of the coupling to the environment plays a crucial role in such phenomena [7]. When we consider control of such a local
0010-4655/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 0 - 4 6 5 5 ( 0 1 ) 0 0 3 8 7 - 3
M. Nishino, S. Miyashita / Computer Physics Communications 142 (2001) 382–386
structure for future application, the noise effect would become one of the main factors. In advance of the study of the noise effect on such local structures, here we focus on the noise effect on just an S = 1/2 spin, i.e. two level system. The transition probability for the nonadiabatic transition is given by the Landau–Zener– Stückelberg (LZS) formula in the pure quantum systems, i.e. without noise. This formula of the probability has been used to estimate the gap at the resonant tunneling [1]. The effect of noise on this formula is studied in this paper. The probability is found not to be affected for large sweeping velocity. We also study how the probability depends on the direction of the noise field.
2. Local magnetic structure Let us first give an overview on the property of the impurity-induced local magnetic structure in the bondalternating antiferromagnetic Heisenberg chain, H = J S · Si+1 . We consider the chain with bond i,i+1 i i alternation · · · J1 J2 J1 J2 · · · , where J1 > J2 . If this system has a defect such as · · · J2 J1 J2 J1 J2 J2 J1 J2 J1 J2 · · · , what magnetic structure appears? We have found that local magnetic structures are induced at the bond impurity or weak edge bonds, and these structures have an effective S = 1/2 spin which behaves almost independently. This results can be understood naturally by the analogy of the VBS picture, i.e. an unpaired spin causes an induced magnetization. In the above example, S = 1/2 moment is allocated in the center site of J2 J2 and a local moment is induced around this site. In the system · · · J1 J2 J1 J2 J1 J1 J2 J1 J2 J1 · · · a local moment of S = 1/2 is also induced at J1 J1 . We show an example of the induced moment {mi } in the Fig. 1. This figure was obtained by using the loop algorithm with continuous time quantum Monte Carlo method (LCQMC) [8]. We adopt a method of specification of the magnetization Mz [6,9] to observe the magnetic profile in low temperatures. There are three local moments and these moments interact with one another very weakly. Under a uniform field the local moment behaves identically as a single spin, while under a field with noise, the behavior of induced moment (a cluster of spins) is not the same as that of isolated single spin [5]. In this paper, we mainly study the basic aspect of noise
383
Fig. 1. Magnetization profiles {mi } of the model with 63 sites at T = 0.01 in the Mz = 1/2 space (106 MCS). mi is the averaged magnetization at ith site. The diamonds denote the strength of bonds {Ji,i+1 }, those at high positions denote J1 = 1.3 and those at the low positions denote J2 = 0.7.
effect on the nonadiabatic transition in a single spin system. At the end of summary and discussion we discuss a case of local induced magnetic moments. 3. Dynamics of the two level system under sweeping field with noise 3.1. Method Here we consider the two level system under sweeping field with noise. The Hamiltonian is given by H(t) = 2Γ S x − H (t)S z + Hnoise(t),
(1)
where H (t) is the sweeping field H (t) = (−H0 + ct) and Hnoise(t) is the term of the noise. We investigate the dependence of the transition probability on the direction of the noise. The noise is applied in x-, y-, or z-direction, Hnoise(t) = hx (t)S x , hy (t)S y or hz (t)S z . We solve the time evolution of the time-dependent Schrödinger equation (TDSE) ∂ ih¯ Ψ (t) = HΨ (t) , (2) ∂t where |Ψ (t) denotes the wave function of the spin system at time t. We set h¯ = 1. The present system is a two-level system whose energy levels are shown in Fig. 2 as a function of H when Hnoise(t) = 0. There the avoided level crossing occurs at H = 0. As the initial state, we set H (t = 0) = −H0 < 0, and put the system to be the ground state for this field.
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M. Nishino, S. Miyashita / Computer Physics Communications 142 (2001) 382–386
The random noise h(t) has the properties A2 t exp − , (6) h(t) = 0 and h(0)h(t) = 2γ τ where τ = 1/γ is the relaxation time. Here A2 /2γ is regarded as the amplitude of the noise and we define B ≡ A2 /2γ . In realistic situations, the noise has components in all directions. We also study this case, where we let the noise have the same amplitude as in the above one direction case. We take an average over 1000–4000 samples for estimation of physical properties in this paper.
Fig. 2. Avoided level crossing.
If we set large H0 > 0, the initial state is the down state (Mz = −1/2). Without noise the probability to remain in the ground state after crossing is given by the Landau–Zener–Stückelberg (LZS) [10] formula, −2πΓ 2 . (3) p = 1 − exp c It is a function of the sweeping velocity and the energy gap (E = 2Γ ) at the avoided level crossing point. When the effect of noise exists, the noise disturbs the quantum process and this probability generally changes. We will also investigate how the probability depends on the direction. We define the transition probability under the noise field in the x-, y-, and z-direction as px , py , and pz , respectively. At low enough temperatures phonon-mediated relaxation can be neglected and the effects of the ubiquitous hyperfine filed and the dipole field of the other spins become significant for relaxation process. It is natural assumption that these noise sources (like a white Gaussian noise) independently cause a change of the internal field with some restoring force on the target spin. Therefore, we provide a random noise with an exponential-decaying autocorrelation function by a Langevin equation (Ornstein–Ulenbeck process), ˙ = −γ h(t) + η(t), h(t) (4) where γ is the damping factor. Here h(t) represents hx (t), hy (t), and hz (t). η(t) is a white Gaussian noise, η(t) = 0 and η(0)η(t) = A2 δ(t). (5)
3.2. Dependence of the transition probability on the noise Fig. 3 shows the transition probability under the noise versus the damping factor γ keeping B constant (i.e. B = 0.02) in the case of a slow sweeping velocity c (= 0.0005). There we sweep the field from H (t = 0) = −5.0 to H (t = tfinal ) = 5.0 for the system of Γ = 0.02. For this parameter set, the system shows almost the adiabatic transition, i.e. p 1 when B = 0. In brief, the noise of x- or y-direction has larger influence than the noise in z-direction. This is because the transverse field causes the transition to the other level directly. The probabilities px , py , and pz decrease largely around γ = 0.02. The probabilities px
Fig. 3. γ dependence of the transition probability in the case of slow sweeping velocity.
M. Nishino, S. Miyashita / Computer Physics Communications 142 (2001) 382–386
and py take the minimum values at γ = 0.1–1.0 and these values are near 0.5 while pz has the minimum value at γ = 0.02–0.05 and this value is more than 0.7. As γ becomes large, the noise of Eq. (6) becomes a kind of white noise but its amplitude becomes zero. Therefore the probabilities increase again and reach the probability of LZS [11]. In the region of small γ , namely in the slow fluctuation case, there is a distinguishable difference between px and py , while py and pz approach each other. This is consistent with the results which were obtained by the analytical treatment [11,12]. When γ → 0, we can regard the noise to be static. In this limit the effect of noise on the transition probability can be estimated by simply averaging over the distribution of the noise. Explicitly we have 2 1 2π 2 1 π Γ Γ , (7) − exp px = 1 − D c c 2B 2 D
2πΓ 2 1 − exp py = 1 − , 2B 2 D c 2πΓ 2 , pz = 1 − exp − c
and (8)
where π 1 D≡ + . (9) 2c 2B 2 Generally we find py > pz , py > px . For the present parameters (Γ = 0.02, B = 0.02, c = 0.0005), px = 0.8724, py = 0.9965, and pz = 0.9939. If we extrapolate in Fig. 3, we find these values. 3.3. Estimation of the effective gap Next we investigate the dependence of the transition probability on the sweeping velocity. We estimate the effective gap Eeff from the transition probability by
the LZS formula (Eq. (3)), Eeff = − 2c π ln(1 − p). We show the field sweeping rate dependence of Eeff for various damping factors (Fig. 4), where the noise is applied in all directions. The gap (Eeff ) is normalized by the pure value of the gap without noise (E0 = 2Γ ). As the sweeping velocity is large, the normalized gap becomes constant. This indicates that the effect of noise becomes less important regardless of the damping factor for large sweeping velocity.
385
Fig. 4. Field sweeping velocity dependence of the estimated gap from the transition probability by the LZS formula. The gap is normalized by the gap without noise (E0 ).
This tendency is consistent with the experimental results [1] where the estimated gap is constant for large velocity. In the range of slow sweeping velocity the effective gap roughly linearly depends on the sweeping velocity, and it also depends on the damping factor significantly. When the sweeping velocity is large, i.e. 2πΓ 2 /c 1, the LZS probability is approximated as p 2πΓ 2/c. In this case, the averaged transition probability in the noise of γ = 0 is given by px + py + pz 2πΓ 2 h2
= + , 3 c 3c p¯ Eeff 2 h2
= =1+ . pLZS E0 6Γ 2
p¯ ≡
(10) (11)
Thus we obtain E within a correction of O(h2 /Γ 2 ). In the present case, because h2 /Γ 2 = 1, (Eeff / E0 ) = 1.083. Furthermore we find that the average transition probability gives a close value to pLZS when γ is large. The detailed analyses will be given elsewhere [11].
4. Summary and discussion We have studied the tunneling property in the sweeping field. In particular, we have studied the effect of noise on the adiabatic transition in the
386
M. Nishino, S. Miyashita / Computer Physics Communications 142 (2001) 382–386
sweeping field. The transition probability depends on the amplitude and the relaxation time of noise. The probability also depends on the direction of noise. In the limit γ → 0, we can regard the noise to be static, where we find py > pz = pLZS and py > px . When γ increases, py rapidly decreases and px and py behave similarly. When γ becomes further large, the probabilities increase. When the sweeping field is large, we find that the noise has little effect, and the gap can be obtained from the observed transition probabilities within a correction of O(h2 /Γ 2 ). Here we mainly studied the two level systems with noise. The magnetic structure induced by bond impurity behaves as an effective S = 1/2 spin. For the induced moments at impurity, we also found a similar dependence on the types of noise. It is found that the reduction of magnetization is smaller than the case of the single spin [5]. In the case of local moments there exists a little contribution from the exited states and the situation is rather complicated. Further investigation comparing between them in detail will be made in the future. We hope that this work provides a basic information for the manipulation of microscopic or nanoscale magnetic devices in the future. Acknowledgements The authors would like to thank Keiji Saito for his useful discussion. The present work was supported by
Grant-in-Aid for Scientific Research from Ministry of Education, Science, Sports and Culture of Japan. M.N. is also supported by the Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.
References [1] Wernsdorfer et al., J. Phys. Soc. Jpn. Suppl. A 69 (2000) 375, Frontiers in Magnetism and references therein. [2] B. Barbara et al., J. Phys. Soc. Jpn. Suppl. A 69 (2000) 383, Frontiers in Magnetism and references therein. [3] S. Miyashita, J. Phys. Soc. Jpn. 64 (1995) 3207. [4] H. De Raedt, S. Miyashita, K. Saito, D. García-Pablos, N. García, Phys. Rev. B 56 (1997) 11 761. [5] M. Nishino, H. Onishi, K. Yamaguchi, S. Miyashita, Phys. Rev. B 62 (2000) 9463. [6] M. Nishino, H. Onishi, P. Roos, K. Yamaguchi, S. Miyashita, Phys. Rev. B 61 (2000) 4033. [7] S. Miyashita, K. Saito, cond-mat/0005013 and references therein. [8] N. Kawashima, J.E. Gubernatis, J. Stat. Phys. 90 (1995) 169 and reference therein; H.G. Evertz, cond-mat/9707221 and reference therein. [9] P. Roos, S. Miyashita, Phys. Rev. B 59 (1999) 13 782. [10] L. Landau, Phys. Z. Sowjetunion 2 (1932) 46; C. Zener, Proc. Roy. Soc. London, Ser. A 137 (1932) 696; E.C.G. Stückelberg, Helv. Phys. Acta 5 (1932) 369. [11] M. Nishino, S. Miyashita, in preparation. [12] Y. Kayanuma, J. Phys. Soc. Jpn. 53 (1984) 108, 118; J. Phys. Soc. Jpn. 54 (1985) 108.
Dynamical response of local magnetic structures induced by inhomogeneities to time-dependent magnetic field Masamichi Nishino ∗ , Seiji Miyashita Department of Applied Physics, Graduate School of Engineering, The University of Tokyo, Bunkyoku, Tokyo, Japan
Abstract Local magnetic structures induced by the bond impurities in the S = 1/2 bond alternating system are found to be looked on as an effective S = 1/2 spin at low temperatures. The response of such structures to a time dependent magnetic field is an interesting problem to study and it would be important for the analysis of the nanoscale materials. In realistic systems the effect of coupling to the environment causes fluctuation and dissipation and plays a crucial role in such dynamics. We study the tunneling process in noise by solving the time-dependent Schrödinger equation numerically from the viewpoint of nonadiabatic transition in the two level system, and make clear basic properties. We investigate the dependence of the tunneling probability on sweeping velocity of the applied field and also on the direction of the random noise and its relaxation time. 2001 Elsevier Science B.V. All rights reserved. PACS: 75.40.Gb; 75.10.Jm Keywords: Quantum tunneling; Nonadiabatic transition; Noise effect; Local magnetic structures; Tunneling probability; Time-dependent Schrödinger equation
1. Introduction The quantum dynamics in microscale or nanoscale materials is one of topics in the condensed matter physics. At low temperatures quantum dynamical effects are important and these effects have really been observed in the hysteresis phenomena of the nanoscale molecular magnets such as Mn12 , Fe8 , and V15 [1,2]. Among the quantum phenomena in these systems, the quantum tunneling of the magnetization (QTM), which is often called resonant tunneling, has become one of topics of much interest. We have * Corresponding author.
E-mail address: [email protected] (M. Nishino).
studied this tunneling dynamics from the viewpoint of the nonadiabatic transition [3–5]. We have found that local magnetic structures which are induced at bond impurities in the Heisenberg chain (S = 1/2) show similar quantum mechanical dynamics [5,6], as well as the molecular magnets. In the bond-alternating antiferromagnetic Heisenberg chain with defects of the alternation, the local magnetic structures are induced around the defects and they can be looked on as an effective S = 1/2 spin at low temperatures. The response of such structures to magnetic field is also possible to understand in the viewpoint of nonadiabatic transition. In the realistic system the effect of the coupling to the environment plays a crucial role in such phenomena [7]. When we consider control of such a local
0010-4655/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 1 0 - 4 6 5 5 ( 0 1 ) 0 0 3 8 7 - 3
M. Nishino, S. Miyashita / Computer Physics Communications 142 (2001) 382–386
structure for future application, the noise effect would become one of the main factors. In advance of the study of the noise effect on such local structures, here we focus on the noise effect on just an S = 1/2 spin, i.e. two level system. The transition probability for the nonadiabatic transition is given by the Landau–Zener– Stückelberg (LZS) formula in the pure quantum systems, i.e. without noise. This formula of the probability has been used to estimate the gap at the resonant tunneling [1]. The effect of noise on this formula is studied in this paper. The probability is found not to be affected for large sweeping velocity. We also study how the probability depends on the direction of the noise field.
2. Local magnetic structure Let us first give an overview on the property of the impurity-induced local magnetic structure in the bondalternating antiferromagnetic Heisenberg chain, H = J S · Si+1 . We consider the chain with bond i,i+1 i i alternation · · · J1 J2 J1 J2 · · · , where J1 > J2 . If this system has a defect such as · · · J2 J1 J2 J1 J2 J2 J1 J2 J1 J2 · · · , what magnetic structure appears? We have found that local magnetic structures are induced at the bond impurity or weak edge bonds, and these structures have an effective S = 1/2 spin which behaves almost independently. This results can be understood naturally by the analogy of the VBS picture, i.e. an unpaired spin causes an induced magnetization. In the above example, S = 1/2 moment is allocated in the center site of J2 J2 and a local moment is induced around this site. In the system · · · J1 J2 J1 J2 J1 J1 J2 J1 J2 J1 · · · a local moment of S = 1/2 is also induced at J1 J1 . We show an example of the induced moment {mi } in the Fig. 1. This figure was obtained by using the loop algorithm with continuous time quantum Monte Carlo method (LCQMC) [8]. We adopt a method of specification of the magnetization Mz [6,9] to observe the magnetic profile in low temperatures. There are three local moments and these moments interact with one another very weakly. Under a uniform field the local moment behaves identically as a single spin, while under a field with noise, the behavior of induced moment (a cluster of spins) is not the same as that of isolated single spin [5]. In this paper, we mainly study the basic aspect of noise
383
Fig. 1. Magnetization profiles {mi } of the model with 63 sites at T = 0.01 in the Mz = 1/2 space (106 MCS). mi is the averaged magnetization at ith site. The diamonds denote the strength of bonds {Ji,i+1 }, those at high positions denote J1 = 1.3 and those at the low positions denote J2 = 0.7.
effect on the nonadiabatic transition in a single spin system. At the end of summary and discussion we discuss a case of local induced magnetic moments. 3. Dynamics of the two level system under sweeping field with noise 3.1. Method Here we consider the two level system under sweeping field with noise. The Hamiltonian is given by H(t) = 2Γ S x − H (t)S z + Hnoise(t),
(1)
where H (t) is the sweeping field H (t) = (−H0 + ct) and Hnoise(t) is the term of the noise. We investigate the dependence of the transition probability on the direction of the noise. The noise is applied in x-, y-, or z-direction, Hnoise(t) = hx (t)S x , hy (t)S y or hz (t)S z . We solve the time evolution of the time-dependent Schrödinger equation (TDSE) ∂ ih¯ Ψ (t) = HΨ (t) , (2) ∂t where |Ψ (t) denotes the wave function of the spin system at time t. We set h¯ = 1. The present system is a two-level system whose energy levels are shown in Fig. 2 as a function of H when Hnoise(t) = 0. There the avoided level crossing occurs at H = 0. As the initial state, we set H (t = 0) = −H0 < 0, and put the system to be the ground state for this field.
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M. Nishino, S. Miyashita / Computer Physics Communications 142 (2001) 382–386
The random noise h(t) has the properties A2 t exp − , (6) h(t) = 0 and h(0)h(t) = 2γ τ where τ = 1/γ is the relaxation time. Here A2 /2γ is regarded as the amplitude of the noise and we define B ≡ A2 /2γ . In realistic situations, the noise has components in all directions. We also study this case, where we let the noise have the same amplitude as in the above one direction case. We take an average over 1000–4000 samples for estimation of physical properties in this paper.
Fig. 2. Avoided level crossing.
If we set large H0 > 0, the initial state is the down state (Mz = −1/2). Without noise the probability to remain in the ground state after crossing is given by the Landau–Zener–Stückelberg (LZS) [10] formula, −2πΓ 2 . (3) p = 1 − exp c It is a function of the sweeping velocity and the energy gap (E = 2Γ ) at the avoided level crossing point. When the effect of noise exists, the noise disturbs the quantum process and this probability generally changes. We will also investigate how the probability depends on the direction. We define the transition probability under the noise field in the x-, y-, and z-direction as px , py , and pz , respectively. At low enough temperatures phonon-mediated relaxation can be neglected and the effects of the ubiquitous hyperfine filed and the dipole field of the other spins become significant for relaxation process. It is natural assumption that these noise sources (like a white Gaussian noise) independently cause a change of the internal field with some restoring force on the target spin. Therefore, we provide a random noise with an exponential-decaying autocorrelation function by a Langevin equation (Ornstein–Ulenbeck process), ˙ = −γ h(t) + η(t), h(t) (4) where γ is the damping factor. Here h(t) represents hx (t), hy (t), and hz (t). η(t) is a white Gaussian noise, η(t) = 0 and η(0)η(t) = A2 δ(t). (5)
3.2. Dependence of the transition probability on the noise Fig. 3 shows the transition probability under the noise versus the damping factor γ keeping B constant (i.e. B = 0.02) in the case of a slow sweeping velocity c (= 0.0005). There we sweep the field from H (t = 0) = −5.0 to H (t = tfinal ) = 5.0 for the system of Γ = 0.02. For this parameter set, the system shows almost the adiabatic transition, i.e. p 1 when B = 0. In brief, the noise of x- or y-direction has larger influence than the noise in z-direction. This is because the transverse field causes the transition to the other level directly. The probabilities px , py , and pz decrease largely around γ = 0.02. The probabilities px
Fig. 3. γ dependence of the transition probability in the case of slow sweeping velocity.
M. Nishino, S. Miyashita / Computer Physics Communications 142 (2001) 382–386
and py take the minimum values at γ = 0.1–1.0 and these values are near 0.5 while pz has the minimum value at γ = 0.02–0.05 and this value is more than 0.7. As γ becomes large, the noise of Eq. (6) becomes a kind of white noise but its amplitude becomes zero. Therefore the probabilities increase again and reach the probability of LZS [11]. In the region of small γ , namely in the slow fluctuation case, there is a distinguishable difference between px and py , while py and pz approach each other. This is consistent with the results which were obtained by the analytical treatment [11,12]. When γ → 0, we can regard the noise to be static. In this limit the effect of noise on the transition probability can be estimated by simply averaging over the distribution of the noise. Explicitly we have 2 1 2π 2 1 π Γ Γ , (7) − exp px = 1 − D c c 2B 2 D
2πΓ 2 1 − exp py = 1 − , 2B 2 D c 2πΓ 2 , pz = 1 − exp − c
and (8)
where π 1 D≡ + . (9) 2c 2B 2 Generally we find py > pz , py > px . For the present parameters (Γ = 0.02, B = 0.02, c = 0.0005), px = 0.8724, py = 0.9965, and pz = 0.9939. If we extrapolate in Fig. 3, we find these values. 3.3. Estimation of the effective gap Next we investigate the dependence of the transition probability on the sweeping velocity. We estimate the effective gap Eeff from the transition probability by
the LZS formula (Eq. (3)), Eeff = − 2c π ln(1 − p). We show the field sweeping rate dependence of Eeff for various damping factors (Fig. 4), where the noise is applied in all directions. The gap (Eeff ) is normalized by the pure value of the gap without noise (E0 = 2Γ ). As the sweeping velocity is large, the normalized gap becomes constant. This indicates that the effect of noise becomes less important regardless of the damping factor for large sweeping velocity.
385
Fig. 4. Field sweeping velocity dependence of the estimated gap from the transition probability by the LZS formula. The gap is normalized by the gap without noise (E0 ).
This tendency is consistent with the experimental results [1] where the estimated gap is constant for large velocity. In the range of slow sweeping velocity the effective gap roughly linearly depends on the sweeping velocity, and it also depends on the damping factor significantly. When the sweeping velocity is large, i.e. 2πΓ 2 /c 1, the LZS probability is approximated as p 2πΓ 2/c. In this case, the averaged transition probability in the noise of γ = 0 is given by px + py + pz 2πΓ 2 h2
= + , 3 c 3c p¯ Eeff 2 h2
= =1+ . pLZS E0 6Γ 2
p¯ ≡
(10) (11)
Thus we obtain E within a correction of O(h2 /Γ 2 ). In the present case, because h2 /Γ 2 = 1, (Eeff / E0 ) = 1.083. Furthermore we find that the average transition probability gives a close value to pLZS when γ is large. The detailed analyses will be given elsewhere [11].
4. Summary and discussion We have studied the tunneling property in the sweeping field. In particular, we have studied the effect of noise on the adiabatic transition in the
386
M. Nishino, S. Miyashita / Computer Physics Communications 142 (2001) 382–386
sweeping field. The transition probability depends on the amplitude and the relaxation time of noise. The probability also depends on the direction of noise. In the limit γ → 0, we can regard the noise to be static, where we find py > pz = pLZS and py > px . When γ increases, py rapidly decreases and px and py behave similarly. When γ becomes further large, the probabilities increase. When the sweeping field is large, we find that the noise has little effect, and the gap can be obtained from the observed transition probabilities within a correction of O(h2 /Γ 2 ). Here we mainly studied the two level systems with noise. The magnetic structure induced by bond impurity behaves as an effective S = 1/2 spin. For the induced moments at impurity, we also found a similar dependence on the types of noise. It is found that the reduction of magnetization is smaller than the case of the single spin [5]. In the case of local moments there exists a little contribution from the exited states and the situation is rather complicated. Further investigation comparing between them in detail will be made in the future. We hope that this work provides a basic information for the manipulation of microscopic or nanoscale magnetic devices in the future. Acknowledgements The authors would like to thank Keiji Saito for his useful discussion. The present work was supported by
Grant-in-Aid for Scientific Research from Ministry of Education, Science, Sports and Culture of Japan. M.N. is also supported by the Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists.
References [1] Wernsdorfer et al., J. Phys. Soc. Jpn. Suppl. A 69 (2000) 375, Frontiers in Magnetism and references therein. [2] B. Barbara et al., J. Phys. Soc. Jpn. Suppl. A 69 (2000) 383, Frontiers in Magnetism and references therein. [3] S. Miyashita, J. Phys. Soc. Jpn. 64 (1995) 3207. [4] H. De Raedt, S. Miyashita, K. Saito, D. García-Pablos, N. García, Phys. Rev. B 56 (1997) 11 761. [5] M. Nishino, H. Onishi, K. Yamaguchi, S. Miyashita, Phys. Rev. B 62 (2000) 9463. [6] M. Nishino, H. Onishi, P. Roos, K. Yamaguchi, S. Miyashita, Phys. Rev. B 61 (2000) 4033. [7] S. Miyashita, K. Saito, cond-mat/0005013 and references therein. [8] N. Kawashima, J.E. Gubernatis, J. Stat. Phys. 90 (1995) 169 and reference therein; H.G. Evertz, cond-mat/9707221 and reference therein. [9] P. Roos, S. Miyashita, Phys. Rev. B 59 (1999) 13 782. [10] L. Landau, Phys. Z. Sowjetunion 2 (1932) 46; C. Zener, Proc. Roy. Soc. London, Ser. A 137 (1932) 696; E.C.G. Stückelberg, Helv. Phys. Acta 5 (1932) 369. [11] M. Nishino, S. Miyashita, in preparation. [12] Y. Kayanuma, J. Phys. Soc. Jpn. 53 (1984) 108, 118; J. Phys. Soc. Jpn. 54 (1985) 108.