Feedback control of noise in spin valves by the spin-transfer torque
Swarnali Bandopadhyay,1,a兲 Arne Brataas,1and Gerrit E. W. Bauer2,3
1Department of Physics, Norwegian University of Science and Technology, 7491 Trondheim, Norway 2Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft,
The Netherlands
3Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
共Received 24 October 2010; accepted 28 January 2011; published online 23 February 2011兲 The miniaturization of magnetic read heads and random access memory elements makes them vulnerable to thermal fluctuations. We demonstrate how current-induced spin-transfer torques can be used to suppress the effects of thermal fluctuations. This enhances the fidelity of perpendicular magnetic spin valves. The simplest realization is a dc to stabilize the free magnetic layers. The power can be significantly reduced without losing fidelity by simple control schemes, in which the stabilizing current-induced spin-transfer torque is controlled by the instantaneous resistance. © 2011 American Institute of Physics. 关doi:10.1063/1.3556270兴
Magnetic spin valves with metal or insulator spacers are used for hard disk read heads as well as for magnetic random access memory elements. The physical principle of operation is the giant magnetoresistance共GMR兲 or tunnel magnetore-sistance共TMR兲, which varies with the relative magnetization between a layer with fixed magnetization共e.g., by exchange biasing兲 and a free layer with a magnetization that is allowed to共re兲orient the magnetization direction. Further miniaturiza-tion is thwarted by the increased thermal fluctuaminiaturiza-tions associ-ated with smaller magnets, however, because the magnetic anisotropy energies scale with the total magnetic moment.1 GMR read heads in the form of nanometer-scale pillars are currently considered as possible replacements for TMR read heads.2 Their higher sensitivity comes at the cost of noise which is enhanced by the spin-transfer-torque-induced cou-pling of electronic and magnetic fluctuations.3,4 Here, we demonstrate that the spin-transfer torque can also be used to suppress the noise in spin valves. The necessary power to achieve a given fidelity can significantly be reduced by feed-back control of the electric current-induced torque by the instantaneous resistance.
An electric current passed through a spin valve interacts with the magnetic order parameter when the magnetizations are noncollinear. This is caused by the spin current compo-nent polarized normal to the magnetization of the free layer, which is absorbed at the interface. The loss of angular mo-mentum in the spin current is transferred to the magnetiza-tion as a spin-transfer torque,5–10 which effectively leads to magnetic damping or antidamping of the magnetization dy-namics, depending on the current direction. Low frequency noise can be suppressed by increasing the magnetic aniso-tropy of the free magnetic layer at the cost of reduced sen-sitivity. Covington3 proposed to reduce noise by applying a spin-transfer torque which opposes the intrinsic damping so that the damping seems to be reduced. According to the fluctuation-dissipation theorem the rms random field that perturbs the magnetization at thermal equilibrium is propor-tional to the square root of the Gilbert damping constant关as Eq.共3兲below兴. The average spin-transfer torque dynamically changes the energy of the ferromagnet 共FM兲 but does not
change the Brownian fluctuating fields the ferromagnet ex-periences, except at very low temperatures through shot noise effects.4Thus, it is not evident that a reduced damping may improve read-head performance. In this letter we ob-serve that in order to suppress the noise we have to increase the damping. Applying a dc of the right direction we are able to increase the fidelity of the read head compared to the zero current situation. The draw back of this approach is the ad-ditional power that has to be invested for a given noise sup-pression. In search of a more energy-conserving method we propose to apply a time-dependent current that is controlled by the instantaneous magnetization configuration as mea-sured by the GMR/TMR. We implement here two physically transparent feedback mechanisms, noting that performance could even be improved by making full use of sophisticated control theory.
Our model corresponds to a realistic11,12 ferromagnet-normal metal-ferromagnet 共FM-NM-FM兲 spin valve as sketched in Fig.1, consisting of a thick FM Permalloy共Py兲 layer with fixed magnetization direction Mˆf and a thin Py layer separated by a nonmagnetic 共Cu兲 layer. The pillar has an elliptical cross section, which defines the easy axis of the
a兲Electronic mail: swarnali.banerjee@gmail.com.
Py
Py
2
1
Cu
x
z
m
θ
y
φ
I
M
f
FIG. 1. 共Color online兲 Schematic diagram of a perpendicular spin valve under study. A nonmagnetic spacer layer共Cu兲 is sandwiched between a thick ferromagnetic共Py1兲 layer with “fixed” magnetization along Mˆfand a thin
ferromagnetic layer共Py2兲 with magnetization mˆ which is relatively “free” to
move. A positive electric charge current Iជis defined to flow from the fixed layer to free layer.
APPLIED PHYSICS LETTERS 98, 083110共2011兲
0003-6951/2011/98共8兲/083110/3/$30.00 98, 083110-1 © 2011 American Institute of Physics
free layer to be along x. A charge current I flowing from the thick layer to the thin layer in the perpendicular direction共zˆ兲 is defined to be positive. The device is stable for a dc of up to 1012 A/m2.11
The allowed peak value for a pulsed current is higher.13
We assume a spatially uniform magnetization of the free FM layer共“macrospin” approximation兲. The temporal evolu-tion of the magnetizaevolu-tion vector mˆ = Mជ/Msof the free FM in
the presence of random field and a Slonczewski spin-transfer torque model can be described by the Landau–Lifshitz– Gilbert equation, dmˆ dt = −␥mˆ ⫻ Hជeff+␣mˆ ⫻ dmˆ dt −␥mˆ ⫻ hជ共t兲 +␥aJmˆ ⫻ 共mˆ ⫻ Mˆf兲. 共1兲
The effective field Hជeff= −E共Mជ兲/Mជ depends on the free energy density,
E共Mជ兲 = −1
2HanMs共mˆ · eˆ兲2− Hជext· Mជ + 2Ms
2共mˆ · zˆ兲2, 共2兲 where eˆ is the unit vector along the easy axis. 兩Hជan兩=共Ny
− Nx兲Ms= 0.05 kOe is the uniaxial anisotropic field, 共Nz
− Ny兲Ms= 4Ms共=5 kOe兲 共Ref. 11兲 is the demagnetization
field关last term in the free energy; Eq.共2兲兴 with demagneti-zation factors Nx, Ny, and Nz,
14
and Msis the saturation
mag-netization. A weak 关⬍0.38 kOe 共Ref. 11兲兴 external field 兩Hជext兩=0.1 kOe is chosen to ensure no switching due to an applied in-plane field. The damping torque in Eq.共1兲is pro-portional to the Gilbert constant ␣= 0.01. The next entry on the right-hand side 共rhs兲 of Eq.共1兲 is a stochastic term. The stochastic field hជ共t兲 has zero mean and white noise correlation.15Here,
具h共i兲共t兲h共j兲共t
⬘
兲典 =2kBT␣␥MsV
␦ij␦共t − t
⬘
兲, 共3兲where kBT is the thermal energy, ␥= 0.176⫻108 G−1s−1 is
the gyromagnetic ratio, V is the volume of the free ferromag-net, and具¯典 denotes statistical averaging. The last term on the rhs of Eq.共1兲is the spin-transfer torque parametrized by aJwhich has the dimension of magnetic field and it is
pro-portional to applied current I as aJ=共ប/2e兲共I/4MsV兲,
where共=50%兲 is the spin polarization.
We solve the Landau–Lifshitz–Gilbert equation numeri-cally in the presence of stochastic torques at room tempera-ture by the Heun method.16We use discretized time interval dt = 1 ps. We start from the initial state mx共0兲= 1 and study the temporal evolution of mx. We calculate the fidelity lifetime,
which is defined as the average time it takes until the mag-netization deviates from its initial equilibrium value beyond a cutoff mc= 0.8共arbitrary choice兲. The typical value of
fi-delity lifetime in the absence of applied current0is 500 ps as obtained from our calculations. We average over 10 000 realizations.
Let us first study the dynamics in the presence of a dc and compute the fidelity time enhancement /0 and the average dimensionless power consumption factor Pf
=共aJ/4Ms兲2共the solid curve in Fig.2兲. The dimensionless
fidelity /0 increases exponentially with the square of Pf.
Thus, fidelity increases with magnitude of applied current
共hence torque兲. As a test of our method we repeated the simulations of current-induced magnetization reversal in Ref. 17and found good agreement.
We propose two control methods to reduce power con-sumption, assuming that we can monitor the instantaneous magnetization in real time by the electrical resistance. First, we let the magnetization evolve from its initial equilibrium configuration in the absence of a spin torque and apply a current pulse with constant magnitude when mxdrops below
a chosen lower bound ml共=0.85兲. We switch off the current
pulse once mxreaches a chosen upper bound mu共=0.95兲. We
refer to this method as the SFB scheme. We compute the dimensionless fidelity lifetime /0 and the dimensionless average power consumption factor Pf=共aJ/4Ms兲2共⌬t/兲,
where⌬t is the total duration over which the spin torque is in action. The result is shown as the dashed-dotted line in Fig. 2. The fidelity lifetime as a function of invested power is very similar to the dc case. However, in the large power regime which also refers to large current regime, the SFB leads to significant power savings. A fidelity increased by a factor 1000 compared to the zero-torque situation can be gained by investing approximately three times less power as compared to the dc torque method. Here, our approach is to apply a larger torque for shorter duration as large current ensures enhancement in fidelity and small duration makes the scheme power-saving one. Note that in our numerical simulation the magnitude of current pulses would be such that the magnetization switches slowly in the time scale of picoseconds, the chosen time step. In the small Pf regime,
the smaller enhancement in fidelity 共compare the dashed-dotted curve with the solid one in Fig.2兲 than dc case is due to the initial delay, set by mlon mx. With increasing torque,
the initial delay becomes insignificant.
At this stage we devise an improved control of the cur-rent pulses to save much energy. This case will be called as IFB method. As in the “SFB” case, we allow the magnetiza-tion to evolve from its initial state without applying a cur-rent. When mx drops below mlwe turn on a pulsed current
corresponding to a torque, which now is a function of mx. We
have chosen aJ= aJ
共0兲关共m
l− mc兲/共mx− mc兲兴⑀, where aJ
共0兲 corre-sponds to the initial value of the current pulse when it is turned on. mland mcare two chosen cutoff values for mxas
1 10 100 1000 0.0001 0.001 0.01 0.1 1 τ/ τ0 ,τ (dc) /τ0 100 Pf SFB IFB (ε=0.1) IFB (ε=0.5) dc
FIG. 2.共Color online兲 Fidelity lifetimein the presence of a dc, as well as a pulsed current with constant magnitude 关referred as simple feedback 共SFB兲兴 and varying magnitude 关referred as improved feedback 共IFB兲; larger
⑀indicates stronger variation兴 in units of0, and the fidelity lifetime in the
absence of a spin torque, as a function of the dimensionless average power consumption factor Pf. Results are obtained at room temperature and
aver-aged over 10 000 realizations.
083110-2 Bandopadhyay, Brataas, and Bauer Appl. Phys. Lett. 98, 083110共2011兲
described above. Note that in the SFB case magnitude of the pulse current is aJ= aJ
共0兲 whenever it is in action but in the present case aJincreases monotonically when mxdrops from
ml and then decreases with increasing mx from mc in the
presence of spin torque. Thus, to estimate invested power we accumulate the power consumption factor at each time step, i.e., Pf=兺i=1
N 共a
Ji/4Ms兲2共ti/兲. Since in the simulation we
are using a finite time step, in practice, we do not reach the maximal current limit as mx→mc. As soon as mxreaches mu
we turn off the current again as in the SFB case. We carry out simulations for two values of the exponent ⑀ in Fig.2. For small共⑀= 0.1兲, there is only a slight improvement 共the double dashed curve in Fig.2兲 in the result as compared to the SFB case. For an intermediate value 共⑀= 0.5兲, a significant im-provement 共the dashed curve in Fig.2兲 is visible. A fidelity enhancement by a factor 1000 costs roughly 15 times less power than in the dc limit. Larger⑀requires a smaller size of time step to get reliable convergence in the numerical inte-gration.
We now estimate the typical power consumption of our feedback loop in a practical situation. The average power consumption is P = I2R共⌬t/兲=C
SPf, with CS= IS2R and IS =共2eV/ប兲4Ms2are system specific constant terms. For our system IS= 45.5⫻107 esu/s共=0.15 A兲. For a resistivity18 = 70 ⍀ cm of 2-nm-thick Py at room temperature or resis-tance R = 0.75 ⍀ the constant CS= 17 mW. We consider the data from Fig.2for the fidelity time= 10000. For the “dc” case, the enhancement is achieved by investing Pf= 1.8
⫻10−2 or average power P = 0.31 m W. The same order of enhancement in fidelity is obtained by consuming Pf= 0.76
⫻10−2or P = 0.13 mW in SFB scheme by applying a current of pulse of equivalent magnetic field aJ/4Ms= 0.275. For a
valve with a cross-section 100⫻50 nm2 this requires a cur-rent of magnitude I = 12.51⫻107 esu/s共=0.04 A兲 which corresponds to a current density 8⫻1012 A/m2. Clearly, at such current densities our approach will break down due to Joule heating. For the IFB scheme the same enhancement in fidelity is achieved by investing Pf= 0.13⫻10−2 or P
= 0.02 mW when ⑀= 0.5. This strong reduction in the in-vested power will alleviate the aforementioned heating prob-lem in SFB case. The efficiency 共fidelity兲 of the operation depends not only on the different system parameters 共spin-torque aJ/I, external field, anisotropy and demagnetization
field, etc.兲 but also on the chosen cutoff value mc and the
boundary values ml and mu. However, the qualitative
mes-sage appears to be valid for all sensible parameter sets. In conclusion, we propose to suppress noise in metallic magnetic spin valves making use of the current-induced spin-transfer torque. With moderate power consumption it is possible to increase the fidelity of the device, especially when a feedback loop is implemented which controls the spin-transfer torque depending on the instantaneous magne-tization configuration. The latter can be measured by the electric resistance. The correcting current pulses should have sharp rise and decay times which can be generated by state-of-the-art electronics.19Our method can be easily adopted to magnetic tunneling junctions. The method can be used to suppress noise in read heads and increase reliability of memory elements.
This work was supported by EC Contract No. IST-033749 DynaMax and by EU FP7 ICT Grant No. 251759 MACALO. S.B. acknowledges hospitality of TU Delft.
1N. Smith and P. Arnett,Appl. Phys. Lett. 78, 1448共2001兲.
2S. Maat, N. Smith, M. J. Carey, and J. R. Childress,Appl. Phys. Lett. 93,
103506共2008兲.
3M. Covington, U.S. Patent No. 7,042,685共9 May 2006兲.
4J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer,Phys. Rev. Lett.
95, 016601共2005兲.
5J. C. Slonczewski,J. Magn. Magn. Mater. 159, L1共1996兲. 6L. Berger,Phys. Rev. B 54, 9353共1996兲.
7D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320, 1190共2008兲. 8A. Brataas, Yu. V. Nazarov, and G. E. W. Bauer,Phys. Rev. Lett. 84, 2481
共2000兲.
9V. Brataas, Yu. Nazarov, and G. E. W. Bauer, Eur. Phys. J. B 22, 99
共2001兲.
10J. Z. Sun,Phys. Rev. B 62, 570共2000兲.
11S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, A. G. F. Garcia,
R. A. Buhrman, and D. C. Ralph,Phys. Rev. B 72, 064430共2005兲. 12S. Urazhdin, N. O. Birge, W. P. Pratt, Jr., and J. Bass,Phys. Rev. Lett. 91,
146803共2003兲.
13T. Devolder, J. Hayakawa, K. Ito, H. Takahashi, S. Ikeda, J. A. Katine, M.
J. Carey, P. Crozat, J. V. Kim, and C. Chappert, J. Appl. Phys. 103,
07A723共2008兲.
14J. A. Osborn,Phys. Rev. 67, 351共1945兲. 15W. F. Brown, Jr.,Phys. Rev. 130, 1677共1963兲.
16J. L. García-Palacios and F. J. Lázaro,Phys. Rev. B 58, 14937共1998兲. 17N. C. Emley, I. N. Krivorotov, O. Ozatay, A. G. F. Garcia, J. C. Sankey,
D. C. Ralph, and R. A. Buhrman,Phys. Rev. Lett. 96, 247204共2006兲. 18G. Counil, T. Devolder, J.-V. Kim, P. Crozat, C. Chappert, S. Zoll, and R.
Fournel,IEEE Trans. Magn. 42, 3323共2006兲.
19O. J. Lee, V. S. Pribiag, P. M. Braganca, P. G. Gowtham, D. C. Ralph, R.
A. Buhrman, and F. J. Lazaro,Appl. Phys. Lett. 95, 012506共2009兲.
083110-3 Bandopadhyay, Brataas, and Bauer Appl. Phys. Lett. 98, 083110共2011兲
