Superparamagnetic dwell times and tuning of switching rates in perpendicular CoFeB/MgO/CoFeB tunnel junctions
TTuning superparamagnetism in perpendicular magnetic tunnel junctions
G. Reiss, J. Ludwig, and K. Rott Center for Spinelectronic Materials and Devices, Department of Physics,Bielefeld University, Universitaetsstrasse 25, 33615 Bielefeld, Germany (Dated: September 23, 2019)Thin electrodes of magnetic tunnel junctions can show superparamagnetism at surprisingly low temperature.We analysed their thermally induced switching for varying temperature, magnetic and electric field. Althoughthe dwell times follow an Arrhenius law, they are orders of magnitude too small compared to a model of singledomain activation. Including entropic effects removes this inconsistency and leads to a magnetic activationvolume much smaller than that of the electrode. Comparing data for varying barrier thickness then allows toseparate the impact of Zeman energy, spin-transfer-torque and voltage induced anisotropy change on the dwelltimes. Based on these results, we demonstrate a tuning of the switching rates by combining magnetic andelectric fields, which opens a path for their application in noisy neural networks.
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Magnetic tunnel junctions (MTJs) with magnetically per-pendicular CoFeB electrodes [1] are key components for harddisk read heads [2] and low power nonvolatile memories [3].For such MTJs with CoFeB electrodes thinner than about ≤ . nm , an unexpectedly low critical current density forspin-transfer-torque switching in the range of 10 A / cm hasbeen found [4], and recent reports demonstrated a superpara-magnetic behavior [5, 6], i.e. a thermally activated switchingof the magnetic electrode that depends on the size, the tem-perature and a variety of external parameters. The observedswitching rates, however, are not compatible with a magneticsingle-domain behavior of the electrode, because the energybarrier for magnetization reversal would be by orders of mag-nitude too large. A granular structure of the CoFeB has beendiscussed [7], but there is no unambiguous explanation up tonow.Such superparamagnetic MTJs (sp-MTJs) can serve tostudy superparamagnetism life and be useful for applications.Recently, we proposed a true random number generator basedon sp-MTJs [8]. Moreover, they can serve in noisy neural-like computing. One precondition is a pronounced maximumof the sp-MTJ’s thermal switching rate in dependence of anexternal input and a shift of these tuning curves by anotherexternal parameter. Mizrahi et al. [9, 10] demonstrated thisby varying the current through sp-MTJs.We investigated magnetically perpendicular sp-MTJs withan exchange biased reference CoFeB electrode [11], an MgObarrier and a 1 . (50)substrate. The layer sequence was Ta(5)/ Ru(30)/ Ta(10)/Pd(2)/ MnIr(8)/ CoFe(1)/ Ta(0.4)/ Co Fe B (0.8)/ MgO(X)/Co Fe B (1.1)/ Ta(3)/ Ru(3) (units in nm), and the thicknessX of the MgO was 1 . . . ◦ C for 30 min in a perpendicularmagnetic field of 0 . . H K ≈
330 kA m − .The coercive field H C measured with an out-of-plane exter-nal field, however, is much smaller than H K and decreasesstrongly with temperature. Similar properties are found forthe quasistatic tunneling magnetoresistance minor loops of thepatterned MTJs. For the example discussed in the supplement, H C is around 0 . − at 50 ◦ C and reaches zero at ≈ ◦ C.The magnetic properties of our film systems and MTJs thusare similar as reported by, e.g., Zhu et al. [12]. To elucidatethis puzzling switching behavior, we analyzed the time de-pendence of the current through MTJs with 140 nm diameterunder varying temperature, magnetic and electric field.From the typical example shown in figure 1 a) it becomesclear, that the MTJ is in a superparamagnetic state with aswitching time in the ms-range at 48 ◦ C. The distributionsof dwell times τ P / AP taken from data as that of figure 1 a) fol-low an exponential law as shown in figure 1 b) with the withmean dwell times ¯ τ P / AP . This property of the MTJs has severeconsequences, because extrinsic quantities such as H C that areevaluated from quasistatic measurements will depend on thetime scale, at which data are taken. If we, e.g., take a mag-netic hysteresis loop with an out-of-plane external magneticfield at a typical measurement time much larger than ¯ τ P / AP ,the apparent H C will be zero, whereas non-zero values will befound otherwise. Nevertheless, the system maintains its pro-nounced out-of-plane anisotropy and switches only betweenthe parallel and the antiparallel state with mean dwell times¯ τ P / AP . Thus the (intrinsic) anisotropy field H K evaluated fromquasistatic characterization is still valid.To obtain quantitative data for the superparamagnetic stateof our MTJs, we have to discuss briefly the statistics of thedwell times. With perpendicular magnetic anisotropy and ina single domain approach, an Arrhenius law describes the a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p FIG. 1: a) Time resolved current through an MTJ with a barrier thickness of 1 . ◦ C, −
190 Oe magnetic bias field and 100 mV biasvoltage. The red lines indicate individual dwell times in the parallel (P) and the antiparallel (AP) state. The green line separates the two states.b) Histogram of the dwell times (P: blue, AP: red) of the complete data set with an exponential fit and the mean well times ¯ τ P / AP . mean dwell time ¯ τ P / AP in the Neél-Brown model [13, 14]:¯ τ P / AP = τ · exp ( ∆ E P / AP ( (cid:126) H , (cid:126) E ) / k B T ) . ∆ E P / AP ( (cid:126) H , (cid:126) E ) is theenergy barrier depending on the magnetic and electric field (cid:126) H and (cid:126) E , k B the Boltzmann constant. τ is the attempt time,which is the inverse of the ferromagnetic resonance frequency[15]. For ferromagnetic out-of-plane systems the attempt timeat zero external field is of the order of 10 − s [16]. In real su-perparamagnetic systems with possibly granular substructure[7], however, the entropy S = k B · Ln ( w ) will play a signif-icant role, because the system has plenty of possible path-ways w for magnetization switching. Using the free energy F = E − T S = E − T k B Ln ( w ) results in [15]:¯ τ P / AP = τ w exp (cid:32) ∆ E P / AP ( (cid:126) H , (cid:126) E ) k B T (cid:33) . (1)To correspondingly analyze the behavior in more detail, wethus have three handles: the temperature T, the magnetic andthe electric field.If we apply a magnetic field H perpendicular to the filmplane, the Zeman energy E Z = µ V E / A (cid:126) M · (cid:126) H ( µ : magneticvacuum permeability) will prefer an alignement of the mag-netization and H and thus change the mean dwell times corre-spondingly. V E / A is either the electrode’s volume V E [17, 18]or in case of granularity the magnetic activation volume V A [19]. The dependence of the energy barrier in equation 1 on (cid:126) H in absence of spin torque is then given by ∆ E P / AP ( (cid:126) H ) = ∆ E + E Z . In figure 2, we show exemplarily the dwell timesas a function of the perpendicular magnetic field for an MTJwith 1 . τ P and ¯ τ AP are equal at an external field H comp that compensates the cou-pling and adds an energy of µ V E / A H comp M to the basic en-ergy barrier ∆ E = K · V E / A (K: effective anisotropy). We thusevaluated the data by shifting the field axis to H (cid:48) = H − H comp .With this and the collinearity of (cid:126) M and (cid:126) H , ¯ τ P / AP are given by:¯ τ ( H , T ) = τ w exp (cid:18) ∆ E + µ V E / A M S H (cid:48) k B T (cid:19) . (2) FIG. 2: Mean dwell times ¯ τ P / AP as a function of the external mag-netic field for varying temperature for an MTJ with t MgO = . The energy barrier ∆ E and the product V E / A M S can be de-termined from taking the derivative of Ln ( ¯ τ / τ ) with respectto 1 / k B T or µ H , respectively. The results of this evaluationare summarized in table I. t MgO V E / A M S ∆ E K ∗ K / K ∗ M S nm Anm eV kJ / m kA / m V E / A M S ,the activation energy ∆ E , the apparent effective anisotropy K ∗ = ∆ E / V E , the ratio K / K ∗ for three MgO thicknesses with the MTJ’selectrode volume V E = . · − m , and the saturation magneti-zation M S evaluated with the magnetic activation volume V A . If we compare the measured effective anisotropy ( K ≈ kJ / m ) with the apparent value K ∗ deduced from K ∗ = ∆ E / V E , there is a large discrepancy ranging from K / K ∗ = . . . ±
7. Taking, however,granularity of the electrode into account, then the magnetic ac-tivation volume V A = ∆ E / K is correspondingly smaller. Witha radius of the electrode of r E =
70 nm, the radius of the acti-vation volume is r A = r E / (cid:112) K / K ∗ = ( ± ) nm . One test ofthis model is the resulting saturation magnetization M S , whichwas reported to be between between 500 kA / m and 1 kA / m[7, 11] depending on the preparation conditions and if deadmagnetic layers at the interfaces are considered or not. Usingthe full electrode’s volume, we obtain from V E M S in table Ian M ∗ S smaller than 250 kA m − , i.e. unphysically low values.Using the determined magnetic activation volume V A , the re-sulting values for M S (see table I) are in agreement with the lit-erature. Moreover, the lateral size of V A is very well compara-ble to the typical domain wall width found for similar CoFeBfilms with perpendicular magnetic anisotropy [20], and abouttwice the value of the typical lateral grain size in our samples(see supplement appendix C). Thus modelling the propertieswith this activation volume leads to physical consistency.Using these results, we can now estimate, how much en-tropic effects impact the dwell times. The effect of the en-tropy S = k B Ln ( w ) can be evaluated for H (cid:48) = Ln ( ¯ τ P / AP / τ ) versus ( k B T ) − . This gives values for Ln ( w ) of the order of35, which means that w is of the order of 10 ! This numberseems to be extraordinarily large. Similar values have, how-ever, already been found for the decay of skyrmionic magne-tization patterns [15] and their magnitude was related with thevast amount of pathes w S for skyrmion decay. For our MTJs,we can again estimate the magnetic activation volume fromthe number w of the entropic pathways that can lead to theirthermally activated magnetization switching. If we assume,that the switching can start at either of N sub-volumes of thefree electrode, then we have around w ≈ N ! possible pathes.Thus ln w ≈ = Ln ( N ! ) = ∑ Ni = Ln ( i ) resulting in N ≈
17 forour sp-MTJs with 70 nm radius. The radius of a single activa-tion volume is then r A = nm / √ ≈ nm . It is remarkable,that the two approaches to evaluate V A lead to almost the sameradii, although the underlying physics is in the first case theevaluation of the energetics of the switching process, whilethermodynamic considerations are used in the second.We now turn to the influence of the bias voltage on theswitching process and the dwell time. If a bias voltage U B is applied to the MTJ, two additional effects act on the mag-netization: first, the spin torque due to the spin polarizationof the current I = U B / R · exp ( − B · t MgO ) with the MTJ’scontact resistance R and the inverse decay length B. Sec-ond, the electric field E = U B / t MgO at the interfaces modi-fies the interfacial magnetic anisotropy energy density [21–23] and leads to a linear change of the anisotropy with U B [24–26]: ∆ K ( U B ) = β | (cid:126) E | = β U B / t MgO , where β character-izes the strength of the dependence of K on E . Spin torqueand anisotropy change, however, have fundamentally differ-ent impact on ¯ τ : While spin torque will stabilize one of the (P,AP) states and destabilize the other one, the anisotropy changewill either de- or increase both ¯ τ P and ¯ τ AP , depending on thepolarity of U B . In addition, for fixed U B , the influence of thespin torque on the magnetization decreases exponentially withincreasing MgO barrier thickness. In contrast, the anisotropy FIG. 3: a) Ln ( τ P / AP / s ) with linearized fit in dependence of U B ,each for five different magnetic fields, 65 ◦ C, on a 140 nm struc-ture and MgO thicknesses of 1 . | ∆ − | in dependence ofthe MgO-thickness and numerical fit of the data with | ∆ − | = A · exp (cid:0) − B · t MgO (cid:1) . change decreases only with t − MgO . Thus for thin barriers, thespin torque can be expected to be dominant, while for thickbarriers the change of the anisotropy can have larger impact.At fixed t MgO , the dependence of ∆ E on the bias voltageis taken into account by multiplying equation 2 with the fac-tor exp (( β (cid:48) V ± A ) U B / ( k B T )) , where β (cid:48) and A describe thestrength of the anisotropy change and the spin torque, respec-tively. By convention, we use the plus sign for the P- andthe minus sign for the AP-state. Then, the influences of theanisotropy change and the spin torque can be separated byevaluating ∆ ± = k B T · (cid:18) ddU ( Ln ( τ P / s ) ± Ln ( τ AP / s )) (cid:19) (3)where ∆ + gives the anisotropy change and ∆ − the spintorque influence (see section Data evaluation).In figure 3 a), we show the results for the dwell times as afunction of U B for t MgO = . nm at 65 ◦ C, and in b) the resultsof fitting the spin torque part ∆ − as a function of t MgO withan exponential function. As expected, the spin torque term isfor 1 . ≈ .
83 eV / V more than fourtimes larger than that of the anisotropy change ( ≈ .
17 eV / V),while for t MgO = . nm this ratio reduces to two ( | ∆ | − ≈ .
045 eV / V and | ∆ | + ≈ .
021 eV / V). The numerical eval-uation of the data for the three MgO thicknesses gives a spintorque term ∆ − = − ( ± ) pJV × exp ( − ( ± ) / nm · t MgO ) ,which mirrors the exponential dependence of the current den-sity on t MgO . The anisotropy contribution results in β = β (cid:48) · t MgO = ( ± ) f J / V m . The separation of the smallanisotropy contribution from the spin torque term is, however,not very reliable. The value obtained here is, however, verysimilar to the result of − f J / V M obtained by Endo et al.for MgO/CoFeB/Ta thin films [24], which corresponds to thefree layer system used in our MTJs.Now, we can combine the impacts of the magnetic field andthe electric field to realize tuning curves. If an sp-MTJ is onlyeither in the P or the AP state such as the 140nm diameterdevices used in this work, the tuning curve is given by theswitching frequency ν ( H , U B ) = / ( ¯ τ P ( H , U B )+ ¯ τ AP ( H , U B )) .If the barrier thickness is too large, the application of U B would change only the anisotropy. This would give a simul-taneous de- or increase of both dwell times and thus tuningof the switching rates, i.e. a shift of the gaussian dependenceon one parameter by the other one would not be possible. If,however, spin torque is dominating the impact of U B , one ofthe dwell times is driven exponentially to zero and the otherone to infinity by application of either H (figure 2), or U B (fig-ure 3). In the corresponding range of barrier thickness, wecan, therefore, use the combination of both to shift the depen-dence of the switching rate on one parameter by varying theother one.The switching rate ν ( H , U B ) was evaluated exemplarily forthe sample with 1 . a) ) and for constant bias voltage de-pending on the magnetic field (figure 4 b) ). FIG. 4: Tuning curves for a 1 . ◦ C (circles) and fit with a Gaussian function (lines).a) Bias dependence with constant magnetic field. b) Magnetic fielddependence with constant bias voltage.
The results in figure 4 are in agreement with our finding,that spin torque dominates the impact of U B for a the 1 . b) comes from the change of theanisotropy, which de- or increases the switching rates depend-ing on the sign of the applied voltage U B .The tuning curves can be very well described with a Gaus-sian dependence on both input parameters H and U B : ν = ν · exp (cid:32) (cid:18) C − C σ (cid:19) (cid:33) (4)where C is either the magnetic field or the bias voltage, C is the peak position and σ the full width at half maximum.For, e.g., figure 4 b, we find H = . Oe , σ = . Oe and H = . Oe , σ = . Oe for for U B = mV and for U B = − mV , respectively. This tuning of the switching rates matches exactly the requirements for the firing rates of neu-rons in population coding networks [27] and thus can emulatetuning curves for noisy neural-like computing[9, 10, 28].The authors gratefully acknowledge the support of the workby the Deutsche Forschungsgemeinschaft under contract RE1052/22-1 and -2. We also thank Hans-Werner Schumacher(PTB Braunschweig) for Kerr-microscopy. [1] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan,M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno.A perpendicular-anisotropy CoFeB-MgO magnetic tunnel junc-tion. Nature materials , 9(9):721–724, 2010.[2] Stefan Maat and Arley C. Marley. Physics and design of harddisk drive magnetic recording read heads. In Yongbing Xu,David D. Awschalom, and Junsaku Nitta, editors,
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