IEEE TRANSACTIONS ON MAGNETICS, VOL. 31 , NO. 5 , SEPTEMBER 1995 249 I

Stochastic Resonance in a Bistable Magnetic System A. N. Grigorenko and P. I . Nikitin

Abstruct-Theoretical model of magnetostochastic resonance is developed. It is shown that in the presence of noise the signal- to-noise ratio for a bistable uniaxial ferromagnetic can be a million times greater than that for a one-stable ferromagnetic. The phenomenon can be used for magnetic field measurements, detection of macroscopic quantum tunneling of magnetization and investigation of electron tunneling that depends upon mag- netization.

I. INTRODUCTION ECENTLY investigation of system dynamics in a FL istable potential has attracted a great attention [1]-

[9]. It was shown that the presence of noise can lead to a rise of the signal-to-noise ratio of such a system in con- trast to the widespread opinion. This phenomenon de- scribed in [ l ] was called stochastic resonance (SR). SR was extensively studied theoretically [2]-[6] and was found in different experiments, namely, in laser physics [7], electrical circuits [8], spin dynamics [9], etc.

SR occurs in systems with bistable potentials and can be explained as follows. In the absence of noise a system is situated near one of stable minima. A response of the system to a small periodic force is determined by the stiff- ness of this minimum, which is a local characteristic of the bistable potential. In the presence of noise the system jumps from one minimum to another with a rate that de- pends upon the noise strength. In this case a response to a small periodic force depends upon global characteristics of the bistable potential. This response shows a sharp rise at the noise strength where the transition rate nearly co- incides with the frequency of the external force.

One particular SR application is magnetization dynam- ics. The magnetic energy of a uniaxial ferromagnetic has two minima in respect with the direction of magnetiza- tion. A rise of the signal-to-noise ratio of such ferro- magnetics subjected to a periodic magnetic field and noise was called magnetostochastic resonance (MSR) and was studied theoretically [ 101 and experimentally [ 1 11. MSR has some interesting applications. In addition to detection of small periodic magnetic field MSR can be used for in- vestigation of electron tunneling that depends upon mag- netization and for detection of magnetization tunneling.

In this paper we consider a theoretical model of MSR

Manuscript received August 10, 1993; Revised February 1 , 1995. This work was supported by the Soros International Science Foundation Grant No. M8L000 and by the International Association Grant No. 1720.

The authors are with the General Physics Institute of the Russian Acad- emy of Sciences, 38 Vavilov Street, Moscow, Russia, 119742.

IEEE Log Number 941 179.

in uniaxial ferromagnetics and briefly discuss possible ap- plications.

11. THEORY Let us consider magnetization dynamics of an uniaxial

easy-axis ferromagnetic subjected to an external periodic magnetic field and noise. In the phenomenological ap- proach this dynamics is governed by the Landau-Lifshitz equation

where a is the magnetization, MO = 1 a 1 , He, is the ef- fective field and X is a damping constant. We suppose that the easy axis is along the z-axis and that the external pe- riodic magnetic field Ho(t) is parallel to thez-axis. Noise is produced by a fluctuating magnetic field H in the Lan- gevin approach. Amplitudes of noiselike magnetic fields are chosen to ensure the necessary level of magnetization fluctuations. For the purpose of the paper we consider uniform dynamics of magnetization. Equation (1) written in terms of thespherical coordinates 8, cp of the magne- tization vector M yields the system

d8ly dt = -aHa sin 8 cos 8

- H,(sin cp - CY cos 8 cos cp)

+ H,(cos cp + CY cos 8 sin cp)

- a(HZ + Ho(t)) sin 8

sin 8 dcply dt = Ha sin 8 cos 8

- H,(cos 8 cos cp + CY sin cp)

- Hy(cos 8 sin cp - CY cos cp)

+ (H, + Ho(t)) sin 8, (2b)

where y is the gyromagnetic ratio, CY = XlyM, is the di- mensionless damping constant, Ha is the anisotropy field, Ho(t) = Ho cos (mot) is the external periodic field and Hi are fluctuating magnetic fields, so that (Hi(t)Hj(t’)) = 26,D6(t - t’), i, j = x, y , z . In the previous work [lo] we neglected magnetic fields H,, Hy and studied dynamics of an angle 8. Here we remove this restriction and con- sider full- magnetization dynamics.

It is known [12] that a stochastic

dx,ldt = A,@) + j

equation of the form

B p j W J , (3)

0018-9464/95$04.00 0 1995 IEEE

2492 IEEE TRANSACTIONS ON MAGNETICS, VOL. 31, NO. 5 , SEPTEMBER 1995

where ( Wj Wj ) = 26,D6(t - t ' ) , corresponds to the Fok- ker-Plank equation of the form

apiat = -C a(A,p)/ax, + D C, a2(B,jBji,P)/ax,ax,, P P3V.J

(4) where P(xP, t ) is the probability distribution.

obtain Taking (2) as Ito equation and X = [:I, p = 1, 2, we

It gives us the Fokker-Plank equation for the probability distribution P(t, 8, p):

apiat = ~ ( c ~ Y [ H , sin e COS e + Ho(t) sin elpyae

- a(y[Ha COS e + ~ ~ ( t ) l p ) / a ~

+ Da2((i + a2)P)/ae2

+ Da2((1 + a2)p/sin28)/a+2. (5) This is the basic equation of MSR. Since the matrix B is diagonal, (5) implies that we can search the probability distribution P in the form P = p(t, O)p(p), where p(p) is the uniform distribution. Then ( 5 ) is reduced to the equa- tion proposed in [lo]

a 2 p apiat = - :e [ - + (1 + a 2 ) D ~ , ae where V(0) = ay[(Ha/2)sin20 - Ho(t) cos 191 is the effec- tive bistable potential. Thus, we justify the possibility of neglecting of in-plane noiselike magnetic fields. Equation (6) is analogous to the conventional stochastic resonance equation and has the steady state solution

[ (1 + a2)D p(0) = Nexp - (7)

in case Ho = 0. Here N is a normalizing constant. Note that it is necessary to treat (2) as Ito equation in order to achieve spherical normalization of p(0) . The choice of the paper greatly simplifies calculations and is good for large D.

The most important parameter describing MSR is the Kramers time 7~~ [12] for a transition between the equi- librium positions 8 = 0 and 8 = lr due to the presence of noise. Using (6) , we find

Noise activating system jumps over the barrier can be of different origin. There are following sources of magnetic noise: external magnetic fields, thermal fluctuations of

magnetizaiton, magnetic fields induced by stress, quan- tum fluctuations of magnetization, etc. In the first case the noise strength is determined by extemal parameters. The magnitude of magnetostochastic resonance R M s R is usu- ally defined as the signal-to-noise ratio for narrow band- width detection at the frequency of the periodic extemal field. Following [6], [7], we find

where Afis the detector bandwidth. It is assumed that a signal is a change of the magnetization angle 8 in the pres- ence of the periodic field and noise is a change of the magnetization angle 8 in the absence of the periodic field. Although the analytical solution (9) is obtained for the low frequency wo of the periodic field Ho(t) wo << 1 / ~ ~ ~ , it gives good results in the case where WO = 1 / 7 K r .

To calculate the amplification of signals introduced by the bistability compared to the monostability we replace the bistable potential by a monostable potential. The pa- rabola that makes the best fit to the bistable minimum (say left) was chosen. Using the Fokker-Plank equation (6), we calculate the signal-to-noise ratio for the magnetic sys- tem dynamics in this parabolic potential:

2

RSD = 2(1 + t 2 ) D A f [ 21 Equations (9) and (10) yield the ratio of the signal at the MSR conditions to that of in the absence of MSR for the same level of noise.

To evaluate Q we take typical parameters a 2: 0.1, y =

Then we get Q z 106.This implies that the system re- sponse for the bistable magnetic system is much greater than that for the corresponding one-stable system in the case of selected parameters.

If noise is produced by thermal fluctuations, the noise strength D can be calculated with the aid of the steady state solution (7). Comparison of the fluctuations energy with thermal energy yields

(12)

where U is the volume of the sample. (It is assumed that D < ayH,). Then we find

10~0e-I~-~, H, = 1030e, = io3 HZ, D = 3 - io7.

D = ayrtT/[(l + a2)Mv],

The best signal-to-noise ratio is achieved for magnetic particles with spatial dimensions L 2: dkT/(2HaM0). It gives L 2: cm for Ha = 1030e, MO = 100 G at room temperature. So, MSR can be used for magnetic field de- tection on the nanometer scale. The gain obtained in com- parison with one minimum detection equals Q = (ayHa/

3

GRIGORENKO AND NIKITIN: STOCHASTIC RESONANCE IN A BISTABLE MAGNETIC SYSTEM 2493

= 10” when U,, = 10 kHz. The signal-to-noise ratio as a function of the noise strength and the periodic field frequency can be used to evaluate the level of fluctuations in a bistable system.

111. APPLICATIONS There are two natural ways of effect application. First,

small magnetic fields can be measured by recording of the signal-to-noise ratio for a small ferromagnetic sample ex- posed to magnetic fields. The sample should reveal a suf- ficient level of magnetic fluctuations to ensure MSR con- ditions. The size of the sample could be as small as several nanometers. A sensor based on MSR might be capable to detect magnetic fields on the nanometer scale.

Second, the level of magnetic fluctuations in a sample can be evaluated with the help of MSR. The signal-to- noise ratio for a bistable system depends both on a peri- odic magnetic field and noise. Particularly, the effect dis- cussed might be used for detection of macroscopic quan- tum tunneling of magnetization in a single domain sample of a uniaxial ferromagnetic. Recently, this phenomenon has attracted a lot of interest [13]-[16]. It is assumed that the sample would be immersed in a uniform magnetic field Hx to lower potential barrier for the tunneling and achieve reasonable tunneling rates. Then, the Kramers time for quantum tunneling is

where E = 1 - HJH, [14]. The application of a small periodic magnetic field leads to MSR. Evaluations give the resonance magnitude

Temperature variations provide possibility to change the strength of the total magnetic fluctuations and to detect the curve of the stochastic resonance. The temperature at which the strength of the Langevin quantum source be- comes comparable with the strength of the thermal source is equal Tq = hyHa/k = 1K and could be achieved. For samples smaller than lop2’ cm3 it is not necessary to use an additional field Hx.

Another possible application of MSR is electron tun- neling that depends upon magnetization. This effect has been discussed in [17]. Consider two magnetic metals separated by a vacuum gap. In the absence of external fields the electron energy has two minima separated by a potential barrier. Tunneling of electrons through the bar- rier gives a fluctuating current. The application of a pe- riodic magnetic or electric field changes the barrier height and leads to MSR. Then, the component of an electron current at the frequency of the applied magnetic field will depend upon the energy gap and will be essentially larger than that in the absence of magnetostochastic resonance. Again, we can change gap size to achieve the maximum value of stochastic resonance.

IV. CONCLUSION To conclude, we have developed theory of magneto-

stochastic resonance in the phenomenological approach for bistable magnetic systems. It is shown that MSR pro- vides new possibilities for investigation of magnetization fluctuations and for detection of weak magnetic fields. It is proposed to use the effect for registration of macro- scopic magnetization tunneling and for observation of electron tunneling that depends upon magnetization.

REFERENCES

[ l ] R. Benzi, A. Sutera, and A. Vulpiani, “The mechanism of stochastic resonance,” J. Phys. A, vol. 14, p. L453, 1981.

[2] S. Fauve and F. Heslot, “Stochastic resonance,” Phys. Lett A, vol. 97, no. 5 , 1983.

[3] P. Jung and P. Hanggi, “Resonantly driven Brownian motion: Basic concepts and exact results,” Phys. Rev. A, vol. 41, p. 2977, 1990.

[4] C. Presilla, F. Marchesoni, and L. Gammaitoni, “Periodically time- modulated bistable systems: Stochastic resonance,” Phys. Rev. A, vol. 40, p. 2105, 1989. R. F. Fox, “Stochastic resonance in a double well,” Phys. Rev. A, vol. 39, p. 4148, 1989. M. I. Dykman, P. V. E. McClintok, R. Manella, and N. Stokes, “Stochastic resonance,” JETP Lett., vol. 52, p. 141, 1990. B. McNamara, K. Weienfeld, and R. Roy, “Observation of stochas- tic resonance in a ring laser,” Phys. Rev. Lett., vol. 60, p. 2626, 1988. L. Gammaitoni, F. Marchesoni, E. Menichella-Saeta, and S. San- tucci, “Stochastic resonance in bistable systems,” Phys. Rev. Lett., vol. 62, p. 349, 1989. L. Gammaitoni, M. Martinelli, L. Pardi, and S. Santucci, “Stochas- tic resonance in bistable paramagnetic system,” Phys. Rev. Lett., vol. 67, 1799, 1991. A. N. Grigorenko, V. I. Konov, and P. I. Nikitin, “Magnetosto- chastic resonance,” JETP Lett., vol. 52, p. 593, 1990. A. N. Grigorenko et a l . , J . Appl. Phys., vol. 76, p. 6335, 1994. C. W. Gardiner, Handbook of Stochastic Methods. Berlin: Springer- Verlag, 1985. A. J . Legget et al., “Dynamics of the dissipative two-state system,” Rev. Mod. Phys., vol. 59, no. 1, 1987. E. M. Chudnovsky and L. Gunter, “Quantum tunneling of magne- tization in small ferromagnetic particles,” Phys. Rev. Lett., vol. 60, p. 661, 1988. A. Gary and Kim Gwan-Hee, “Dissipation in macroscopic quantum tunneling.” Phvs. Rev. Len. . vol. 63, p. 2512, 1989.

V I <

[16] D. D. Awschalom, M. A. McCord, and G. Grinstein, “Observation of macroscopic spin phenomena in nanometer-scale magnets,” Phys. Rev. Lett., vol. 65, p. 783, 1990.

[17] J. C. Slonczewski, “Conductance and exchange coupling of two fer- romagnets separated by a tunneling barrier,” Phys. Rev. B , vol. 39, p. 6995, 1989.

A. N. Grigorenko was born in 1963. He received the B.S. and Ph.D. degrees in physics from the Moscow Physical Technical Institute in 1986 and 1989, respectively.

Since 1989 he has been employed as Senior Scientific Researcher at the General Physics Institute. He is the head of the group of magnetic field measurements.

P. I. Nikitin was born in 1956. He received the B.S. and Ph.D. degrees in physics from the Moscow Physical Technical Institute, in 1979 and 1983, respectively. Since 1976 he has worked at the P. N. Lebedev Physical Institute. He also has been employed since 1983 as a Senior Scientific Re- searcher at the General Physics Institute. He is the head of the laboratory of low temperature plasma.