Nonlinear frequency and amplitude modulation of a nanocontact-based spin-torque oscillator
P. K. Muduli,* Ye. Pogoryelov, and S. Bonetti
Materials Physics, Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden
G. Consolo
Department of Physics, University of Ferrara, 44100 Ferrara, Italy
Fred Mancoff
Everspin Technologies, Inc., 1300 N. Alma School Road, Chandler, Arizona 85224, USA
Johan Åkerman
Materials Physics, Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden
and Physics Department, University of Gothenburg, 412 96 Gothenburg, Sweden

Received 15 October 2009; revised manuscript received 4 March 2010; published 28 April 2010
We study the current-controlled modulation of a nanocontact spin-torque oscillator. Three principally dif-
ferent cases of frequency nonlinearity
d2f /dIdc2 being zero, positive, and negative are investigated. Standard
nonlinear frequency-modulation theory is able to accurately describe the frequency shifts during modulation.
However, the power of the modulated sidebands only agrees with calculations based on a recent theory of
combined nonlinear frequency and amplitude modulation.
DOI: 10.1103/PhysRevB.81.140408 PACS number
s: 75.75.
c, 85.70.Kh, 85.75.
d, 84.30.Ng
Spin-torque oscillators
STO offer a combination of at-
tractive properties such as ultrawide band frequency
operation,1,2 extremely small footprint
without any need for
large inductors, and easy integration using well-established
magnetoresistive random access memory processes. The ba-
sic principle of a spin-torque oscillator is based on the trans-
fer of angular momentum from a spin-polarized current to
the local magnetization.3,4 The effect usually occurs
in a nanoscale device where a large current density

108 A /cm2 can drive the precession of the free layer
magnetization at GHz frequencies,5,6 thus acting as a nano-
scale oscillator. Effective modulation of the microwave sig-
nal generated from STOs is required for communication ap-
plications. However, both the STO frequency and amplitude
are typically nonlinear functions of the drive current. This
nonlinearity is related to a change in the precession angle
with the increase in the current magnitude.7–9 Experiments
have shown other sources of nonlinearities such as
temperature10 and dynamic-mode hopping.11–13 The wide
range of possible sources of nonlinear behavior is likely to
render the frequency modulation of STOs highly nontrivial.
Despite the rapidly growing literature on the many differ-
ent aspects of STOs, experimental studies of frequency
modulation are still limited to a single work by Pufall et al.14
They observed both unequal sideband amplitudes and a shift
of the carrier frequency with modulation amplitude, which
they ascribed to nonlinear frequency modulation
NFM.
While linear frequency-modulation
LFM theory assumes
that the instantaneous frequency of the modulated signal is
linearly proportional to the modulating signal,15 NFM theory
takes into account the nonlinear change in the intrinsic oper-
ating frequency during modulation. Pufall et al.14 calculated
the observed sideband amplitudes using NFM theory and
found a rather large
about 50% discrepancy between their
calculated and experimentally observed sidebands, which
they argued might be due to amplitude modulation or other
nonlinear properties of the STO.
In this work we study the frequency and amplitude modu-
lation of a nanocontact STO for various amounts of fre-
quency nonlinearity. The frequency nonlinearity is described
by the second derivative of the frequency, f , with respect to
the dc bias current, Idc, d2f /dIdc2 . Three different cases of
frequency nonlinearity
d2f /dIdc2 being zero, positive, and
negative are investigated. As expected from NFM theory,
the carrier and its associated sidebands exhibit a change in
frequency under modulation, which can be directly calcu-
lated from the experimentally determined nonlinear proper-
ties of the frequency of the free-running STO. However, the
power of the modulated sidebands is only poorly reproduced
using NFM theory and we show that it is essential to con-
sider amplitude modulation in order to reach any quantitative
agreement. Using a recently proposed theory of combined
nonlinear frequency and amplitude modulation
NFAM,16
we are able to show remarkable agreement between our ex-
perimental data and calculations, which involve no adjust-
able parameters. Despite the complex phenomena involved
in the STO nonlinearities, we show that modulation of these
devices is highly predictable.
The nanocontact metallic-based STOs studied in this work
have been described in detail in Ref. 17. Using e-beam li-
thography, a circular Al nanocontact with nominal diameter
of 130 nm is fabricated through a SiO2 insulating layer, onto
a 826 m2 pseudo-spin-valve mesa with the following
layer structure: Si /SiO2 /Cu
25 nm /Co81Fe19
20 nm /
Cu
6 nm /Ni80Fe20
4.5 nm /Cu
3 nm /Pd
2 nm. While
all data presented here has been taken on a single device,
similar behavior has been observed in several other devices
of the same size.
The low-frequency
100 MHz modulating current is in-
jected from an RF source to the STO via a circulator. The dc
bias current is fed to the device by a precision current source

Keithley 6221 through a dc-40 GHz bias tee connected in
parallel with the transmission line. The signal is then ampli-
fied using a broadband 16–40 GHz, +22 dB microwave am-
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In Fig. 4, we show the detailed modulation current depen-
dence of the carrier and the first-order sideband power with
calculated results as described in the following paragraph.
While the evolution of the carrier power with modulation
current does not seem to be affected by the nonlinearity, both
the upper and lower sidebands are strongly affected by the
sign and the value of d2f /dIdc2 : the lower sideband gets mark-
edly stronger than the upper sideband for d2f /dIdc2 0
31
mA, and weaker than the upper sideband for d2f /dIdc2 0

38 mA. The position of the maximum sideband power is
also shifted up/down for the upper/lower sideband. It is note-
worthy that this shift only depends on the magnitude of
d2f /dIdc2 and does not change sign when d2f /dIdc2 goes from
positive to negative. Even for the linear case
28 mA,
d2f /dIdc2 =0, the power of the two sidebands are unequal.
The upper sideband has higher power than the lower side-
band, as expected from the positive slope of amplitude ver-
sus bias current in Fig. 2
d. This case of linear frequency
modulation provides a strong experimental evidence that am-
plitude modulation is also taking place.
In order to interpret the observed behavior and estimate
the importance of both the frequency and amplitude nonlin-
earities, we consider three qualitatively different models de-
scribing
i LFM,
ii NFM, and
iii NFAM. The latter
model is adapted from16 and specifically takes into account
nonlinearities in both output frequency and amplitude as a
function of the input bias current.
Since LFM and NFM models have already been described
in Ref. 14 and 15, we focus on the details of the NFAM
model used in our analysis. The instantaneous frequency is
assumed to depend nonlinearly on the modulating signal
f i
t = k0 + k1m
t + k2m
t2 + k3m
t3 + ¯ ,
1
where, m
t, is the modulating signal and the coefficients ki
represent the ith order frequency sensitivity coefficients.
Similarly, the output amplitude, Ac is given by
Ac
t = 0 + 1m
t + 2m
t2 + 3m
t3 + ¯ ,
2
where i is ith order amplitude sensitivity coefficient. The
coefficients ki and i are given by the nonlinear current de-
pendence of f and A of the free running STO. We use sine
wave modulation, m
t= Im sin
2fmt, where Im is the am-
plitude and fm is the frequency of modulating signal. The
resulting NFAM spectrum becomes16
S
f = 1
4h=0
3
h 
n,m,p,q=−



Jn
1Jm
2Jp
3Jq
4
f − fcI −
n + 2m + 3p + 4q + hfm
+ f − fcI −
n + 2m + 3p + 4q − hfm
+ f + fcI −
n + 2m + 3p + 4q + hfm
+ f + fcI −
n + 2m + 3p + 4q − hfm ,
3
where 1=k1Im / fm+3k3Im3 /4fm, 2=k2Im2 /4fm+k4Im4 /4fm,
3=k3Im
3 /12fm, and 4=k4Im4 /32fm are frequency-modulation
indices of different order. 0=0+2Im
2 /2, 1
=1Im+33Im
3 /4, 2=2Im
2 /2, and 3=3Im
3 /4 are amplitude-
modulation parameters. In the above we assumed that the
frequency in Eq.
1 is nonlinear up to fourth order and the
amplitude in Eq.
2 is nonlinear up to third order, which is
found sufficient to describe the experimental data. The fre-
quency spectrum S
f consists of a shifted carrier frequency
fcI = k0 + k2Im2 + 3k4Im4 /8 + ¯
4
and an infinite number of sidebands symmetrically located at
fcI
lfm, where l=n+2m+3p+4q
h is a positive integer
identifying the sideband order. The NFAM carrier shift is
identical to that obtained from an NFM model since effects
due to amplitude modulation do not enter in Eq.
4. This
shift can be readily calculated by means of the polynomial
fitting procedure shown in Fig. 2. The comparison with the
experimentally obtained values reveals a good agreement, as
shown in Fig. 3. The sideband power, on the other hand, is
strongly affected by the amplitude modulation, through the
coefficients i, and can be used to compare the NFM and
NFAM models. In a 6 mA interval around each operating
point, we expand the frequency dependence into a fourth-
order Taylor series, and the amplitude dependence into a
third-order Taylor series as shown in Fig. 2. The coefficients
along with their standard errors are summarized in Table I.
Using these coefficients we calculate the sideband power ex-
pected from NFM and NFAM, respectively,
second and
third columns in Fig. 4 and also compare with LFM theory

first column in Fig. 4.
LFM theory completely fails to describe the strong asym-
metry between the upper and lower sidebands in all cases. In
the linear case of 28 mA Figs. 4
a–4
c both NFM and
LFM theory predict nearly the same behavior with equal
sideband power since only k1 is significant and k2 0. In
contrast, the NFAM model correctly produces both the upper
and lower sideband power, implying a much better agree-
ment, mostly captured by the amplitude modulation sensitiv-
ity coefficient 1. In fact, the mean-square error, 2 between
FIG. 4.
Color online Integrated power of the carrier
black
triangles, and the first-order upper
blue squares and lower
red
circles sidebands for the three different dc bias currents: rows
a–

c 28 mA, rows
d–
f 31 mA, and rows
g–
h 38 mA. First,
second, and third columns show the corresponding calculated inte-
grated power
solid lines as predicted by LFM, NFM, and NFAM,
respectively. The mean-square error, 2 between the experiment and
calculated results of the two sidebands improved significantly for
NFAM.
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the experiment and calculated results of the two sidebands
decreases by about 80% for NFAM theory compared to
LFM. In the two nonlinear cases, the NFM model captures
the change in sign of the sideband asymmetry, given by the
sign change in k2, but only yields a partial improvement
compared to LFM. On the contrary, when the amplitude sen-
sitivity coefficients are also taken into account the agreement
of the calculations with experiment is essentially perfect.
This agreement is only obtained when both frequency and
amplitude nonlinearities are accounted for; both k2 and 1
are significant. For 31 mA
38 mA, the mean-square error
between the experiment and calculated results of the two
sidebands decreases by about 85%
83% for NFAM theory
compared to LFM and about 10%
36% compared to NFM
theory. We emphasize that none of the presented calculations
involve any free parameters and are completely determined
by the experimentally measured nonlinear current depen-
dences of the free-running STO. The agreement with NFAM
was also found to be valid for a range of lower modulation
frequencies
down to 40 MHz over the entire range of dc
bias currents. Thus our results show that, as long as both
nonlinearities are accounted for, the proposed scheme of
combined modulation is able to accurately predict the result-
ing sideband powers and frequency shifts over a wide range
of varying operating conditions. Consequently, the STO be-
haves as an ordinary RF oscillator and should lend itself to
communication applications.
In conclusion, we have carried out a detailed modulation
study on a nanocontact STO. In particular, we have studied
the impact of different levels of frequency nonlinearity. In
the nonlinear cases, both carrier and sidebands frequencies
are shifted as a function of the modulation current. Both
frequency and amplitude nonlinearities produce a significant
asymmetry in the power of the upper and lower sidebands.
We find that a combined nonlinear frequency and amplitude-
modulation model can accurately describe all our experimen-
tal data without any adjustable parameters. The modulation
of an STO is therefore predictable and independent of the
complex mechanism behind the nonlinearity. The results are
significant for the continued development of communication
and signal processing applications of spin torque oscillators.
Support from the Swedish Foundation for Strategic Re-
search
SSF, the Swedish Research Council
VR, the Gö-
ran Gustafsson Foundation and the Knut and Alice Wallen-
berg Foundation are gratefully acknowledged. Johan
Åkerman is a Royal Swedish Academy of Sciences Research
Fellow supported by a grant from the Knut and Alice Wal-
lenberg Foundation. Giancarlo Consolo gratefully thanks
support from CNISM through “Progetto Innesco.” We thank
Randy K. Dumas for critical reading of the manuscript.
*muduli@kth.se
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TABLE I. Modulation sensitivity coefficients found from polynomial fits of frequency and amplitude of the free-running STO.
Current

mA
k0

GHz
k1

MHz/mA
k2

MHz /mA2
k3

MHz /mA3
k4

MHz /mA4
0

pW1/2
1

pW1/2 /mA
2

pW1/2 /mA2
3

pW1/2 /mA3
28 20.185 117
1 1
1 2
0.2 8
1 10.4
0.5 0.9
0.07 −0.2
0.02 −0.03
0.01
31 20.545 147
1 20
1 0.8
0.1 −1
0.1 10.9
0.6 −0.5
0.07 −0.15
0.02 0.02
0.01
38 21.779 115
1 −22.5
0.6 −3.3
0.1 1.6
0.1 10.8
1 1.3
0.07 −0.1
0.02 −0.12
0.01
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