Vito Puliafito
Magnetism Research Group
Università di Messina, Italy
Modulazione non lineare
di ampiezza e frequenza
in nano-oscillatori spintronici
XXVI Riunione Annuale dei Ricercatori di Elettrotecnica, ET 2010
9-11 Giugno 2010, Napoli
Unità di Messina
ww2.unime.it/mrg
Componenti
o
o
o
o
o
o
o
Bruno Azzerboni
Alessia Bramanti
Andrea Calisto
Giancarlo Consolo
Giovanni Finocchio
Alessandro Prattella
Vito Puliafito
Principali tematiche di ricerca
Modellizzazione materiali magnetici
o Micromagnetismo
o Spintronica
o Elaborazione di segnali biomedicali
o
Principali collaborazioni:
Unità ET di Perugia, prof. Cardelli
o Università di Perugia, prof. Carlotti
o Università di Ferrara, prof. Nizzoli
o
o
o
o
o
Modulazione non lineare di ampiezza e
frequenza in nano-oscillatori spintronici
Università di Salamanca, Spagna
Università di Cornell, Ithaca, Usa
Università di Oakland, Rochester, Usa
Royal Institute of Technology, Svezia
Vito Puliafito - ET 2010, Napoli, 11/06/2010
Outline
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Introduction on analog modulation processes
Motivation of the present study
Mathematical models of modulation
Numerical analysis of spintronic nano-oscillators
Comparison between analytical, numerical and
experimental results
Conclusions
Modulazione non lineare di ampiezza e
frequenza in nano-oscillatori spintronici
Vito Puliafito - ET 2010, Napoli, 11/06/2010
Analog modulation processes
Carrier wave
Message signal (modulating)
Parameters of the carrier wave
modified by the modulating
signal:
- frequency (FM)
- amplitude (AM)
- phase (PM)
Modulazione non lineare di ampiezza e
frequenza in nano-oscillatori spintronici
Vito Puliafito - ET 2010, Napoli, 11/06/2010
LFM
carrier
c  t   Ac cos  2 f ct 
output signal
modulating
m  t   Am cos  2 f mt 
 t

s  t   Ac cos  i  m , t    Ac cos  2π  f i  m ,  d 
 0

instantaneous frequency
f i  m , t   f c  km  t 
s  t   Ac cos  2πfc t   sin  2πf m t  
S f  
Ac
J
2

n      f  f c  nf m     f 

kAm
fm
f c  nf m  
frequency
spectrum
 central frequency = fc
 symmetric sidebands
 sidebands number = ∞
Motivation of the study
[Pufall et al., APL 86, 082506, 2005]
i  t   I dc  iac  t 
 shift of the central frequency
 asymmetric sidebands
the Spin-Transfer Oscillator (STO)
works as a NFM modulator
I dc  8.5 mA
 fc
iac  Am cos  2πf mt 
10,075 GHz
f m  40 MHz
NFM
 t

s  t   Ac cos  i  m , t    Ac cos  2π  f i  m ,  d 
 0

output signal
v
f i  m , t    kh m h  t 
instantaneous frequency
k0  f c
h 0
v
 I

s  t   Ac  cos  c t    h  kh  sin  h m t  


h 1
A
S f   c
2


 p 
p 1,...,v
 v
 J
 h1

h
  h  

v
v
 



I
I
   f  f c  f m  h h     f  f c  f m  h h  
h1
h1



 
 central frequency shift
v

cI 
f 
  f c  h  kh  Am h 
2 
h 1

I
c
 asymmetric sidebands
NFM
Comparison between NFM model and
experiments [Pufall et al., APL 2005]
NFM model reproduces the central frequency shift,
but not the different amplitudes of sidebands.
Reason of the disagreement
Additive amplitude modulation effects are NOT INCLUDED.
There are theoretical, experimental and numerical evidences of
amplitude modulation.
  k  N k Bk2
frequency
amplitude
[Slavin and Kabos, IEEE Trans. Magn. 41, 2005]
[Pufall et al., APL 2005]
NFAM
output signal
 t

s  t   Ac  m , t  cos  i  m , t    Ac  m , t  cos  2π  f i  m ,  d 
 0

v
f i  m , t    kh m h  t 
instantaneous frequency
k0  f c
h 0
u
instantaneous amplitude
Ac  m , t    k m k  t 
k 0

 v
1 u
S  f     k  k     J
4 k 0
 j   i 1

i
  i  

j 1,..., v

 


 v
 
I
f

f

i


k

c
 i
 fm   
 i 1
 



 
I
  f  f c    i i  k  f m   
 i 1
 

v

 v
 
I
f

f

i


k

c
 i
 fm  
 i 1
 



  
I
 f  f c    i i  k  f m  
 i 1
  

v

quantitatively
different asymmetric
sidebands
Numerical study: framework
LLGS equation of motion:
M

   H eff  M  
t
M0
M 
I

M


f
r
R
M   M  p  


c



t
M


0
Numerical Integration method:
- Finite-difference approach
- Fifth-order Runge-Kutta scheme
Device:
-Extended Point-Contact (800nm x 800nm x 5nm)
Parameters:
[V. Puliafito et al.,
-External field Hext=800mT directed at 80° w.r.t. the plane
IEEE Trans. Magn. 45, n.11, 2009]
-Ms (FL) = 0.7 T (FL dynamics only);
-A = 1.4×10-11 J/m
-Rc = 20nm;
Effective Field:
-Spin-torque efficiency: 0.25
-Magnetostatic, Exchange, Zeeman
-Cell size: 4nm
-NO Oersted
-a = 0.01
-NO Anisotropy
-uniform current density distribution
-No Thermal
-Abrupt Absorbing Boundary Conditions
Analisys procedure
STEP 1: CHOOSE the SETUP
In the free running condition i(t) = Idc
(NO modulation), choose a bias point and the
operating range.
STEP 2: FIT
Find the best polynomial fit of the functions
f(I) and A(I) (or P(I)) and extract the values of
amplitude (k) and frequency (kh) sensitivity
coefficients.
STEP 3: MODULATION
Apply the modulating signal:
i(t) = Idc + iac (t) = Idc + Am sin (2fmt).
STEP 4: USE NFAM MODEL
Predict the composition of the Fourier Spectrum of
the modulated signal by means of the analytical
formula.
Analysis #1: varying Am
comparing the numerical results with the analytical models:
the shift of central frequency
for both NFM and NFAM
analytical models:
v

cI 
f 
  f c  h  kh  Am h 
2 
h 1

I
c
Analysis #1: varying Am
comparing the experimental results with the analytical models:
the shift of central frequency
[Muduli et al., PRB 81, 140408(R), 2010]
Analysis #1: varying Am
comparing the numerical results with the analytical models:
the asymmetric sidebands

 l  S f  f cI  lf m


S f  f cI  lf m

l is sideband order
FULL AGREEMENT WITH THE NFAM MODEL
Analysis #1: varying Am
comparing the experimental results with the analytical models:
the asymmetric sidebands
[Muduli et al., PRB 81, 140408(R), 2010]
Analysis #2: varying fm
All the results presented so far are valid if
f m  0.9 fcI .
When the frequency of the modulating signal is increased above this value,
the modulation process vanishes (no sidebands are observed) as
frequency pulling or injection locking phenomena are observed instead.
f m  250 MHz
f m  15 GHz
A “pure” NAM modulator
We showed that it is not possible to build
a pure frequency spintronic modulator,
since there are amplitude modulation
effects that we cannot disregard.
Let’s see if it is possible to have a
pure amplitude spintronic modulator.
There is a critical angle, referred to as
“linear angle”, at which the frequency
tunability coefficient
  f I
is equal to zero.
[G.Consolo et al., PRB 78, 2008]
[G. Consolo and V. Puliafito,
IEEE Trans. Magn. 46, n.6, 2010]
Here, the frequency is kept constant with
the applied current and only the
amplitude changes.
NAM
output signal
 t

s  t   Ac  m , t  cos  i  m , t    Ac  m , t  cos  2π  f i  m ,  d 
 0

fi  m, t   fc
instantaneous frequency
u
Ac  m , t    k m k  t 
instantaneous amplitude
polynomial order u
k 0

1 u
S  f    k
4 k 0
   f  f c  kf m     f  f c  kf m 

+  f  f c  kf m     f  f c  kf m 
 central frequency = fc
 symmetric sidebands
u
 k   a j Amj  j
j 1
 number of sidebands = 2*u
Analysis #3: a pure NAM
Analysis with
no modulation
(dc current)
Analysis with
modulation
(dc+ac current)
Numeric results agree with the analytical model:
at the “linear” angle configuration, the STO works as a pure NAM!
Conclusions

We developed a general analytical model for a nonlinear
combined frequency-amplitude modulation process

It has been tested on a point-contact structure:
 it
generally works as a nonlinear modulator of both
frequency and amplitude (in the range fm<0.9fcI)
 in
the “linear angle” configuration, it works as a
nonlinear modulator of the sole amplitude
Modulazione non lineare di ampiezza e
frequenza in nano-oscillatori spintronici
Vito Puliafito - ET 2010, Napoli, 11/06/2010
ww2.unime.it/mrg
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Modulazione non lineare di ampiezza e
frequenza in nano-oscillatori spintronici
Vito Puliafito - ET 2010, Napoli, 11/06/2010