Shaped angular dependence of the spin transfer torque and microwave
  generation without magnetic field
The magnetization of a ferromagnetic body can be manipulated by transfer of spin angular momentum from a spin-polarized curren
Shaped angular dependence of the spin transfer torque and microwave generation without 
magnetic field  
O. Boulle1, V. Cros1, J. Grollier1, L. G. Pereira1,*, C. Deranlot1, F. Petroff1, G. Faini2, J. Barnaś3, 
A. Fert1 
1 Unité Mixte de Physique CNRS/Thales and Université Paris Sud XI, Route départementale 128, 
91767 Palaiseau, France 
2 Laboratoire de Photonique et de Nanostructures LPN-CNRS, Route de Nozay, 91460 
Marcoussis, France 
3 Department of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznań, Poland 
Abstract: The generation of oscillations in the microwave frequency range is one of the most 
important applications expected from spintronics devices exploiting the spin transfer 
phenomenon. We report transport and microwave power measurements on specially designed 
nanopillars for which a non-standard angular dependence of the spin transfer torque (wavy 
variation) is predicted by theoretical models. We observe a new kind of current-induced 
dynamics that is characterized by large angle precessions in the absence of any applied field, as 
this is also predicted by simulation with such a wavy angular dependence of the torque. This type 
of non-standard nanopillars can represent an interesting way for the implementation of spin 
transfer oscillators since they are able to generate microwave oscillations without applied 
magnetic field. We also emphasize the theoretical implications of our results on the angular 
dependence of the torque.  
The magnetization of a ferromagnetic body can be manipulated by transfer of spin angular 
momentum from a spin-polarized current. This is the concept of spin transfer introduced by 
Slonczewski [1] and Berger [2] in 1996. In most experiments, a spin-polarized current is injected 
from a spin polarizer into a “free” magnetic element, for example in pillar-shaped magnetic 
trilayers [3-6]. The phenomenon of spin transfer has a great potential for applications. It can be 
used either to switch a magnetic configuration (the configuration of a magnetic memory for 
example) [3-5] or to generate magnetic precessions and voltage oscillations in the microwave 
frequency range[6-7]. In the most usual situations, such oscillations are observed in the presence 
of a magnetic field. 
From a fundamental point of view, spin transfer effects raise two different types of 
problems [8]. First the spin transfer torque acting on a magnetic element is related to the 
transverse spin polarisation of the current (transverse meaning perpendicular to the magnetization 
axis of the element) and can be derived from spin-dependent transport equations [8-17]. On the 
other hand, the description of the magnetic excitations generated by the spin transfer torque raises 
problems of non-linear dynamics [8,18-20]. For example, in the simple limit where the excitation 
is supposed to be a uniform precession of the magnetization (macrospin approximation), this 
precession can be determined by introducing the spin transfer torque into a Landau-Lifshitz-
Gilbert (LLG) equation for the motion of the magnetic moment. However, the determination of 
the spin transfer torque and the description of the magnetization dynamics cannot be regarded as 
independent problems. In standard trilayered structures with in-plane magnetizations and with the 
usual angular dependence, a switching regime is found at zero and low magnetic field and the 
precession regime with generation of voltage oscillations is mainly observed above some 
threshold field [8]. We will show that a new behavior, characterized by large angle precessions in 
the absence of any magnetic field, can be obtained in specially designed structures presenting a 
non-standard dependence of the spin transfer torque as a function of the angle between the fixed 
magnetization of the polarizer and the magnetization of the free layer. This non-standard angular 
dependence of the torque, that we call “wavy”, is obtained by choosing materials with different 
spin diffusion lengths for the “fixed” and “free” magnetic layers, which changes the distribution 
of the spin currents and spin accumulations in the structure. 
The observation of spin transfer oscillations at zero field in structures with a “wavy” angular 
dependence of the torque can represent a new way to obtain spin transfer oscillators operating 
without any applied field, an other possible way being the use of exchange interactions or 
anisotropy to generate local effective fields or non-collinear equilibrium configurations [21]. In 
addition, the observation of a wavy angular dependence of the torque represents a valuable test of 
the theory and shows that realistic predictions of the spin transfer torque and its angular 
dependence in a given structure are now possible. As we will see, in the models we consider here 
[15-16], the torque is calculated from parameters which, for most of them, can be derived from 
former CPP-GMR experiments [22-23].  
The usual behaviour observed in pillars with in-plane magnetizations along an anisotropy axis 
corresponds to the standard angular dependence of the inset of Fig.1a, in which the torque starts 
from zero at ϕ = 0 (P equilibrium state with parallel magnetizations of the fixed and free 
magnetic layers) and keeps the same sign till it comes back to zero at ϕ = π (AP antiparallel 
state). At zero field and starting from a P state for example (Fig.1b), a negative current (electrons 
going from the free to the fixed layer in our convention) will destabilize the P state and stabilize 
the AP state, i.e. can switch the system from P to AP. In the presence of a large enough applied 
field favouring the P configuration, the torque cannot stabilize the AP state and leaves the system 
in an intermediate precession state. This is what we call the standard behaviour with irreversible 
switching at low field and precession at high field, as illustrated by Fig.2a (remark: in some low 
field experiments however, the irreversible switching is preceded by precessions in a very narrow 
current range just below the switching current). 
The non-standard behaviour with precession at zero and/or low field presented in this article 
is related to the existence of a wavy angular dependence of the torque acting on the free magnetic 
layer. This oscillatory angular dependence, with an inversion of the torque between ϕ = 0 and ϕ = 
π, is shown in Fig. 1a. We present the results of calculations in the models of Fert et al [15] and 
Barnaś et al [16-17] for a Py(8)/Cu(10)/Co(8) pillar. With respect to standard structures like 
Co/Cu/Co or Py/Cu/Py, the difference we have introduced is a large asymmetry between the spin 
diffusion lengths (SDL) in the magnetic layers, with a long SDL in Co ≈ 38 nm (at room 
temperature) and a short SDL in Py ≈ 4 nm [22-23]. The smaller spin asymmetry of the resistivity 
in Co could also affect the angular dependence but we have checked by additional calculations 
that the wavy variation comes primarily from the shorter SDL in the Py free layer and not from 
the different spin asymmetry coefficients, as this has been mistakenly written in Ref.[24]. The 
solid curves in Fig.1a correspond to the calculation in the model of Barnaś et al [16]. A wavy 
angular dependence is also predicted by the model of Fert et al [15] which gives the terms of first 
order in ϕ and (π-ϕ) in the vicinity of the colinear P or AP states  (the solid straight lines at the 
left and right edges of the graph in Fig.1). Due to the inversion at small values of ϕ, a negative 
current (Fig. 1c) now stabilizes not only the AP state but also the P one and should be ‘inactive’. 
This can be a solution, for example, to reduce the spin-transfer-induced noise that is detrimental 
to read heads. In contrast, an appropriate positive current can destabilize both the P and AP states, 
leading to a precessional solution the motion equation, even at zero field.  
To validate these predictions, we have performed transport and microwave power 
measurements at room temperature on Py(8)/Cu/Co(8) elliptical nanopillars of approximate 
dimensions 100x155 nm². Only the top Py layer (free layer) and the Cu spacer are etched 
through. The unetched Co layer (“fixed” layer) lies directly on the Ta/Cu bottom electrode. Very 
similar results have been obtained on Py(8)/Cu/Co(4)/IrMn nanopillars in which the extended Co 
layer is exchange biased by the IrMn one. We show in Fig. 2b the GMR signal of a 
Py(8)/Cu/Co(8) sample. Starting, for example, from large negative fields, the switching to an AP 
state at about 40 Oe is related to the magnetization reversal of the free layer (Py) to the positive 
direction, as this can be found from subsequent CIMS experiments in which the current-induced 
return to P is made harder by a larger positive field (consistently with a positive orientation of the 
Py magnetization in the switching to the AP state). From the GMR minor cycles of the Py layer 
(see Supplementary Information), we find that the coercive field of the Py layer is 90 Oe and the 
dipolar field acting on it is 43 Oe. 
The different behaviours observed for standard and wavy angular dependences are first 
illustrated in Fig.2a and 2c. In Fig. 2a, we show the standard variation of differential resistance 
(dV/dI) versus I measured on a  Py(4 nm)/Cu(10 nm)/Py(15 nm) pillar: starting from a P state, a 
negative current  induces an irreversible switching from P to AP at low field and a reversible 
variation with the characteristic peak of steady precessions at high field. In contrast, starting 
again from a P magnetic configuration with magnetizations in the positive field direction but now 
with a Py(8 nm)/Cu(10 nm)/Co(8 nm) pillar for which a wavy angular dependence is expected, 
we detect (Fig. 2c) reversible peaks of dV/dI for positive currents and at very small fields on both 
sides of Happ= 0. The peak current increases with increasing applied positive field as expected 
since the P state becomes more stable. We have also performed experiments with an AP initial 
state. We find that dV/dI first drops to the level of a P state at some positive current and then, at 
higher current, exhibits the same characteristic precession peak we observe in measurements with 
a P initial state (data not presented).  
In Fig. 3, we present microwave power spectra recorded with the same P initial state and for 
several values of the current. Fig.3a is for zero applied field (actually, Happ ≈ 2 Oe) and Fig.3b for 
zero effective in-plane field (after subtracting the dipolar field). Coloured dots in the insets 
indicate the values of the current on the corresponding dV/dI vs I curves. A peak in the 
microwave power spectrum turns out approximately in some current range above the maximum 
of dV/dI. The frequency f of the microwave peak increases with the current (blue shift), in 
contrast with the red shift generally observed in standard pillars with in-plane magnetization. 
Actually, with the standard angular dependence of the torque, the theoretical prediction is a 
succession of red and blue shift regimes at increasing current but, in experiments with in-plane 
applied fields, the crossover to a blue shift regime has been seldom observed [25]. In macrospin 
simulations, a blue shift in f is predicted for the regime of out-of-plane (OP) precessions and is 
also associated with a decrease of f with increasing in-plane field. As shown in Fig.3 c, we 
observe this decrease of f with Happ. In Fig.4a, we present the current-field diagram of the 
microwave power. Microwave signals are emitted only in the top left corner of the diagram, i.e. 
at low field and in a zone which is also a region of increased resistance (Fig.4b). No excitation is 
observed at higher field. 
We can therefore put forward two main results from our microwave power data: i) Pillars in 
which a wavy angular dependence of the spin transfer torque is expected, generate microwave 
oscillations, but, in contrast with the standard behavior, when there are excited by  positive 
currents and at zero field; ii) These microwave oscillations present a blue shift of their frequency 
with current, a behaviour generally associated with out-of-plane precessions.  
We first want to exclude that the effects described in the preceding paragraphs could arise 
from other origins than the wavy angular dependence of the STT. Could they arise from 
excitations of the Co “fixed” layer? We can first argue that the same behaviour is also observed 
when the 4nm thick and extended Co layer is pinned by an IrMn layer and that an excitation of a 
thin Co layer in the presence of such a strong pinning is quite improbable. We can also point out 
that, for un-pinned continuous magnetic layers, the switching current densities obtained by Chen 
et al. [26] are about one order of magnitude larger than ours. In addition, whereas a reduction of 
the thickness of the Co layer to 4 nm for the same 8nm Py thickness should make the excitation 
of Co easier (smaller current), our experimental results are in the opposite direction.  
The sample of Fig.2-3 exhibits the relatively simple behaviour predicted for a wavy angular 
dependence of the torque in a macrospin picture, i.e. precessions at zero and low field in positive 
current. However, in a series of five similar samples (with or without pinning by IrMn), we have 
also observed additional features in transport measurements. For example, in some samples and 
with an initial P state, we see not only peaks in dV/dI in positive current at zero or low field but 
also partial or total switchings in negative current. These excitations can be ascribed to a non-
uniform distribution of the magnetization [27]. For a part of the sample, the angle ϕ between the 
magnetizations of the two layers is above ϕc, the angle of torque inversion, and can be excited by 
a negative current. However, we emphasize that these additional excitations observed in transport 
measurements are never associated with peaks in the emitted power in the Gigahertz range. All 
the samples share the same main features with microwave emission only at low field in positive 
current.  
We now present the theoretical implications of our experimental results and first comment briefly 
on the origin of the wavy angular dependence of the spin transfer torque in our samples. The 
physics governing this angular dependence can be discussed simply by considering that, in all the 
models [8,13-17] based on interfacial absorption of the transverse spin component and boundary 
conditions of the mixing conductance type (the language can be different in different 
formalisms), the spin transfer torque is proportional to the transverse component of the spin 
accumulation in the spacer layer. The key point is that the spin accumulation in a nonmagnetic 
conductor is directly related to the gradient of the spin current along the current axis z, 
dzjdm m /)(−∝  [28]. In configurations close to the P state of a standard pillar, with a thick 
fixed layer and a thin free layer made of the same material, the spin polarization of the current in 
the spacer decreases from the fixed layer to the free layer.  This corresponds to a given sign of 
the spin accumulation. But an opposite sign is expected if, in the same configuration, the spin 
polarization of the current increases from the fixed layer to the free layer. This is what occurs for 
our Py(8nm)/Cu(10nm)/Co(8nm) pillars in an angular range close to the P configuration, as this 
can be seen from the spin accumulation calculated in the Section Methods. As shown in Fig 1a, 
calculations of STT based on two different models reflect this inversion of the spin accumulation 
by an inversion of the torque on the left part of the figure with respect to the standards case. 
However, as shown the figure, the inversion is a little less pronounced (less steep slope) in the 
model of Ref.[15] which goes beyond the simple mixing conductance approximation of Ref.[16].  
For a further understanding, we have performed additional macrospin simulations of the 
current-induced precessions by solving a Landau-Lifschitz Gilbert equation including a spin 
transfer term using parameters compatible with the actual structure of the measured samples (see 
Methods, the simulations have been performed by two of the co-authors, O.B. and J.G., 
independently of those published in Ref.[24]). The simulated current-field diagram at T = 0 K is 
presented on Fig 4.d with a colour scale corresponding to the change of resistance. At high field 
(Happ larger than the anisotropy field) and in the current range we have considered, the only 
excitations are in-plane (IP) precessions occurring above a threshold current Ic1 and associated 
with a small change of resistance (which also corresponds to a small microwave power). At low 
field, the IP precessions above Ic1 (black and blue trajectories in Fig. 4c) are followed by out-of-
plane (OP) precessions (orange and red trajectories) above a second current threshold Ic2.  
There is a general good agreement between the main features of the experimental and 
calculated phase diagrams. In particular, the zone of OP precessions in the top left corner of the 
diagram of Fig. 4d turns out to be also the zone where we measure the larger DC resistance 
increase (Fig. 4b) and also detect microwave excitations (Fig.4a). Quantitatively, if one compares 
the colours in Fig.4 b and c, one can see that the distribution of the resistance change in the 
diagram is well reproduced and that the experimental ΔR in the OP zone is only somewhat 
smaller than the calculated one (by about 20% in average). The simulations also give a 
distribution of microwave power (not shown) concentrated in OP top-left zone as in the 
experimental plot of Fig.4a but with a power which is about 80 times larger than the experimental 
one. This could be due to several reasons. First, there are certainly technical factors, like a large 
impedance mismatch in the detection circuit. Second, for the OP excitations, the limits of a 
macrospin approach for a quantitative prediction [6,30], have been put forward by several 
publications. Finally, for the IP precessions we could not detect in the microwave spectra, it can 
be pointed out that a very small variation of GMR is expected for angles between P and an angle 
similar to our ϕc in structures with our type of torque angular dependence [29]. This has also led 
us probably to overestimate the resistance change and the microwave power, since our 
calculation is based on a standard angular dependence of the GMR as sin2(ϕ/2). 
A confirmation that the zone of maximum resistance and microwave excitations in the top-
left corner (positive currents and low fields) of the diagrams in Fig.4a-b can be identified with the 
zone of Out-of-Plane precessions in the calculated diagram (Fig. 4c) comes from the current and 
field dependence of the frequency. As shown in the inset of Fig. 4c, the simulations predict that a 
decrease of the frequency at increasing current for IP precessions is followed by an increase at 
the transition to OP precessions. This is in agreement with the frequency blue shift of the 
microwave excitations detected in the same zone of the phase diagram. The simulations also 
predict correctly the red shift for the variation with the field. Our simulations therefore support 
the picture of a non-standard behavior induced by a wavy angular dependence of the STT torque 
and characterized by out-of-plane precessions excited by positive current at zero and low field. 
During the submission process, we learned that oscillations of vortex structures in thick Py 
layers excited by STT have been observed at relatively low field [31,32]. However this leads to 
oscillations at relatively low frequency, below 1 GHz for layers in our aspect ratio [33], and the 
oscillations above 3 GHz we observe cannot be explained by this mechanism. 
Leaning on recent theoretical models of spin transfer torque, our experimental results 
should help designing more efficient spin transfer oscillators operating in a very small or even 
without an applied magnetic field. This is a necessary step (among others) on the implementation 
of these new spintronics-based oscillators in a microwave receiver system for telecommunication 
applications. 
Methods : 
The multilayers are grown by sputtering onto oxidized Si substrates. Two types of stacks 
were deposited : structure 1 = Au(20 nm)/Cu(5 nm)/Py(8 nm)/Cu(10 nm)/Co(8 nm)/Ta(10 
nm)/Cu(80 nm)/Ta(10 nm) and structure 2 = Au(25 nm)/Py(8 nm)/ Cu(8 nm)/Co(4 nm)/IrMn(15 
nm)/Ru(15 nm)/Cu(35 nm). Py stands for Permalloy. The results we present, are on a nanopillar 
with structure 1, but very similar results are observed with structure 2 when the fixed Co layer is 
pinned with an IrMn layer. This indicates that, even without an IrMn pinning layer, the 
magnetization of the extended Co layer is similarly fixed. 
For the nanofabrication process, we first defined (by e-beam lithography, evaporation 
deposition and lift-off) a Ti(15 nm)/Au (55 nm) elliptical mask on the magnetic multilayer. Then, 
the magnetic pillar is etched by ion milling with a real-time monitoring by mass spectroscopy 
down to the Cu/Co interface. The bottom electrode is defined by optical lithography and ion 
milling. The next step is a planarization of the pillar with a Su-8 resist layer that is also used to 
electrically isolate the bottom and the top electrode. The Su-8 layer on the top of the pillar is 
removed by reactive ion etching. Finally, the top Ti/Au electrode is defined by optical 
lithography, evaporation deposition and lift-off.  
We measured both the dc resistance and the differential resistance dV/dI using an additional 
20µA ac current modulated at 5kHz. For the frequency-domain measurements, we applied a dc 
current on the sample through a bias-T. The high frequency voltage signal is then amplified (68 
dB) and analysed on a commercial spectrum analyzer. The power spectra we show are extracted  
from the spectrum analyser (we do not correct them from a calibration done for quantities like the 
frequency-dependent amplifier gain, the attenuation in the transmission lines, and the impedance 
mismatches). They are only obtained by subtracting a reference spectra measured at Idc = 0 in the 
same magnetic field conditions. Note that the measured emitted power is therefore only a fraction 
of the actual emitted power from the pillars. Both transport and frequency measurements have 
been performed at room temperature and with in-plane magnetic field. 
The torques of Fig.1a have been calculated by introducing in the models of Refs.[16] and 
[17] parameters mostly derived from CPP-GMR experimental data [22-23]. For respectively Au, 
Py, Cu, Co and Ta, these parameters are: bulk resistivity ρ (μΩ.cm) = 2, 15, 2.9, 24, 170; bulk 
spin asymmetry coefficient β = 0, 0.76, 0, 0.46, 0; spin diffusion length lsf (nm) = 35, 4, 350, 38, 
10. For the interface parameters, respectively Au/Cu, Cu/Py, Cu/Co, Co/Ta, Ta/Cu, the 
parameters are : interfacial resistance rb (fΩ.m²) = 0.17, 0.5, 0.51, 0.5, 0.5; interfacial spin 
asymmetry coefficient γ : 0, 0.7, 0.77, 0.7, 0; interfacial spin memory loss coefficient δ : 0.13, 
0.25, 0.25, 0.25, 0.1. Note that the values of the Co and Ta resistivity have been measured on thin 
film we had grown in the same conditions. The unknown value of lsf in Ta has been estimated by 
fitting the calculated and experimental variations of resistance ΔR. We have also used the same 
parameters in routine programmes [34] developed for the CPP-GMR to calculate the spin 
accumulation in the spacer layer for our structure and in a standard structure Py (15 nm)/Cu (10 
nm)/Py(2 nm), respectively – 2.2 and + 2.0 in arbitrary units and check the change of sign at the 
origin of the wavy angular dependence. 
For the simulations of the magnetization dynamics, we have solved a Landau Lifschitz 
Gilbert equation including a spin transfer term of the form M1x(M1xM2) with the angular 
dependence shown in Fig.1a. The calculations are performed at zero temperature. The saturation 
magnetization µ0Ms = 0.87 T has been derived from ferromagnetic resonance experiments 
performed on a Cu(6nm)/Py(7nm)/Cu(6nm) layer at room temperature. The other parameters are 
the anisotropy field Han= 0.009 T, the gyromagnetic factor γ0 = 2.21 105(s.A/m)-1, α = 0.011. The 
area of the pillars is about 1.38 104 nm², as derived by scanning electron microscopy.  
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Correspondence and requests for materials should be adressed to V. C. The authors declare they 
have no competing financial interests 
Acknowledgements 
The authors thank M. Gmitra for the calculations of Fig 1b based on the model of Ref[16]. We 
would like also to acknowledge H. Hurdequint for FMR measurements, L. Vila for assistance in 
fabrication, O. Copie and B. Marcilhac for assistance in transport and frequency measurements 
and M.R. Pufall for discussions. This work was partly supported by the french National Agency 
of Research ANR through the PNANO program (MAGICO PNANO-05-044-02) and the EU 
through the Marie Curie Training network SPINSWITCH (MRTN-CT-2006-035327). J. B. 
acknowledges support by funds from the Polish Ministry of Science and Higher Education as a 
research project (2006-2009). 
*Present address : Instituto de Física, UFRGS, 91501-970 Porto Alegre, RS, Brazil 
Figure captions 
Figure 1 Angular dependence of the spin transfer torque for a standard and a ‘wavy’ 
angular dependence. a, Variation of the spin transfer torque on the free Py layer of a 
Au(infinite)/Cu(5 nm)/Py(8 nm)/Cu(10 nm)/Co(8 nm)/Ta(10 nm)/Cu(infinite) multilayer as a 
function of the angle ϕ between the magnetizations of the free Py and fixed Co layers for positive 
and negative currents. The solid curves are calculated in the model of Barnaś et al [17], the solid 
straight lines represent the slopes of the torque variation as the angle tends to 0 and π and have 
been derived from the small angle expression of Fert et al [16]. The parameters used in the 
calculations and mainly derived from CPP-GMR data are listed in the Section Methods. Inset : 
typical variation of the spin transfer torque as a function of the angle between the magnetizations 
of the free and fixed layers for a standard trilayer structure (case of Co/Cu/Co from Ref.[10]). b-
c, Sketches showing schematically the direction (blue arrow) of the spin transfer torque on the 
free layer for configurations close to the P and AP configurations of the free layer (m) and fixed 
layer (M) magnetizations for a standard (b) and a wavy (c) angular dependence of the torque. 
Figure 2 Transport measurements on nanopillars with standard or “wavy” angular 
dependence of the spin transfer torque. a, Differential resistance vs current measured for a 
nanopillar with a standard structure Py(15 nm)/Cu(10 nm)/Py(4 nm) at “low field” (H = 6 Oe) 
and “high field” (H = 133 Oe). In the latter case (precession), the applied field is larger than the 
coercive field equal to H = 133 Oe. Curves are offset for clarity. b-c : Transport data for a Co(8 
nm)/Cu(10 nm)/Py(8 nm). nanopillar. b, Resistance vs field at low current (I = 200 µA). c, 
Differential resistance vs current for different applied fields around zero. These fields correspond 
to the coloured symbols in b. 
Figure 3. Microwave power spectra for the Co(8 nm)/Cu(10 nm)/Py(8 nm) nanopillar of 
Fig.2b-c. a, Microwave power spectra for an applied field close to zero (Happl = 2 Oe) at different 
currents corresponding to the coloured symbols in the inset. Inset: dV/dI vs I for Happl = 2 Oe. b, 
Microwave spectra for different applied currents corresponding to the symbols in inset for an 
effective (applied + dipolar) field of about zero (Happ = 43 Oe). Inset in b : dV/dI vs I for Happ = 
43 Oe. c, Microwave spectra for I = 9 mA at different positive applied fields. Spectra are offset 
for clarity.  
Figure 4 Experimental and simulated spin-transfer-induced high frequency dynamics for a 
Co(8nm)/Cu10nm)/Py(8nm) nanopillar. a, Experimental integrated power between 0.1 to 8 
GHz in colour scale as a function of field and current. b, Normalized experimental resistance in 
colour scale as a function of field and current (a reference curve has been subtracted to the 
experimental R vs I curves to remove the changes in resistance due to Joule heating). c-d : 
Simulated dynamics of the magnetization in a macrospin approach c, Results of macrospin 
numerical calculations of LLG equation as a function of current and field at T = 0K. The black 
line indicates the onset of current-induced precession. Inset in c : Variation of the calculated 
frequency as a function the current for Happ = 0 Oe. d Magnetization trajectories for Happ=0 (black 
arrow in c) at several increasing applied currents.  
π/2 π
Standard angular
dependence of the spin 
transfer torque 
I > 0  
P stable       
AP unstable
(H = 0)
Wavy angular
dependence of the spin 
transfer torque 
I < 0  
P unstable       
AP stable 
(H = 0)
I < 0  
P stable       
AP stable 
(H = 0)
I > 0  
P unstable       
AP unstable
(H = 0)
Fig. 1 Boulle et al.
-10 -5 0 5 10
Current (mA)
 - 19 Oe
   - 3 Oe
     9 Oe
   22 Oe
-100 -50 0 50 100
12.95
12.97
12.99
Magnetic field (Oe)
-6 -4 -2 0 2 4 6
Current (mA)
Fig. 2 
Boulle et al.
9,5 mA
8,5 mA
7,5 mA
6,5 mA
4 6 8 10
12.25
12.39
I (mA)
I (mA)
1 2 3 4
36 Oe
24 Oe
- 4 Oe
Frequency (GHz)
-19 Oe
Happ ≈ 0  (2 Oe) a
Heff ≈ 0  
(Happ=43 Oe)
I = 9 mA
11 mA
10 mA
Fig. 3
Boulle et al.
9,5 mA
8,5 mA
7,5 mA
6,5 mA
4 6 8 10
12.25
12.39
I (mA)
I (mA)
1 2 3 4
36 Oe
24 Oe
- 4 Oe
Frequency (GHz)
-19 Oe
Happ ≈ 0  (2 Oe) a
Heff ≈ 0  
(Happ=43 Oe)
I = 9 mA
11 mA
10 mA
Fig. 3
Boulle et al.
4 6 8 10
12.25
12.39
I (mA)
I (mA)
1 2 3 4
36 Oe
24 Oe
- 4 Oe
Frequency (GHz)
-19 Oe
Happ ≈ 0  (2 Oe) a
Heff ≈ 0  
(Happ=43 Oe)
I = 9 mA
11 mA
10 mA
Fig. 3
Boulle et al.
0 50 100 150
 Magnetic field (Oe)
Power (pW)0 2.74
0 50 100 150
 Magnetic field (Oe)
  9.4 mA
  6.9 mA
  5.7 mA
  3.7 mA
0 50 100 150
4 6 8
Current (mA)
=0 Oe
Magnetic field (Oe)
plane
In plane
Parallel state
ΔRdc(mΩ)
ΔRdc(mΩ)
Fig 4.
Boulle et al.
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