Microwave Oscillators and Detectors Based …
17
6 Generation Linewidth of MTJ-Based STNOs
The understanding of the origin of linewidth in STNOs is crucial for applications. In
the following, we will discuss the linewidth of single modes and multimode STNOs
in separate subsections.
6.1 Single-Mode Excitation
The linewidth, f of the single mode in STNO devices can be described by the
single-mode theory developed by Kim et al. [14, 67, 68].
According to this theory, f of a nonlinear oscillator is given by:
f = g (1 −
I dc
I th
), for I << I th
(8)
= f L (1 + ν
2
), for I >> I th
(9)
where I dc is the dc bias current, and the nonlinear linewidth amplification is (1 +
ν
2
) = 1 +
p 0 N
p
2
, where N =
d ω
dp
is the nonlinear frequency shift, and p is the
power restoration rate (
−1
p is the correlation time of the power fluctuations); f L =
g
kT
E(p 0 )
is the intrinsic thermal linewidth—i.e., the linewidth of a linear (ν = 0)
oscillator. Here, E(p 0 ) is the total energy of the oscillator. Above threshold (I dc I th ),
the nonlinear amplification of the linewidth is controlled by the ratio of the nonlinear
frequency shift N to the power restoration rate p .
George et al. [69] successfully applied the single-mode theory to MTJ-based
STNOs. They showed that the nonlinear coefficient ν can be calculated from the
equation:
A NL = 1 +
I dc
g
df
dI dc
2
.
(10)
Here, A NL is the nonlinear amplification equal to (1 + ν
2
). Figure 10 shows the
experimental observation of the linewidth below and above the threshold current of
1 mA. Above the threshold current, the linewidth is amplified by a factor of (1 + ν
2
)
due to nonlinear oscillations maintained by the STT.
The high power of MTJ-based STNOs allows direct measurement of the signal in the time-domain, which is not possible in low-power metallic-based STNOs.
Bianchini et al. [86] have shown that it is possible to directly measure the nonlinear coefficient ν, as well as the linear linewidth, from time-domain data using the
following relation of the phase variance [66, 86]:
δφ
2
R = 2ω 0
(1 + ν
2
)t − ν
2 1 − e
−2 p
2 p
.
(11)
17
6 Generation Linewidth of MTJ-Based STNOs
The understanding of the origin of linewidth in STNOs is crucial for applications. In
the following, we will discuss the linewidth of single modes and multimode STNOs
in separate subsections.
6.1 Single-Mode Excitation
The linewidth, f of the single mode in STNO devices can be described by the
single-mode theory developed by Kim et al. [14, 67, 68].
According to this theory, f of a nonlinear oscillator is given by:
f = g (1 −
I dc
I th
), for I << I th
(8)
= f L (1 + ν
2
), for I >> I th
(9)
where I dc is the dc bias current, and the nonlinear linewidth amplification is (1 +
ν
2
) = 1 +
p 0 N
p
2
, where N =
d ω
dp
is the nonlinear frequency shift, and p is the
power restoration rate (
−1
p is the correlation time of the power fluctuations); f L =
g
kT
E(p 0 )
is the intrinsic thermal linewidth—i.e., the linewidth of a linear (ν = 0)
oscillator. Here, E(p 0 ) is the total energy of the oscillator. Above threshold (I dc I th ),
the nonlinear amplification of the linewidth is controlled by the ratio of the nonlinear
frequency shift N to the power restoration rate p .
George et al. [69] successfully applied the single-mode theory to MTJ-based
STNOs. They showed that the nonlinear coefficient ν can be calculated from the
equation:
A NL = 1 +
I dc
g
df
dI dc
2
.
(10)
Here, A NL is the nonlinear amplification equal to (1 + ν
2
). Figure 10 shows the
experimental observation of the linewidth below and above the threshold current of
1 mA. Above the threshold current, the linewidth is amplified by a factor of (1 + ν
2
)
due to nonlinear oscillations maintained by the STT.
The high power of MTJ-based STNOs allows direct measurement of the signal in the time-domain, which is not possible in low-power metallic-based STNOs.
Bianchini et al. [86] have shown that it is possible to directly measure the nonlinear coefficient ν, as well as the linear linewidth, from time-domain data using the
following relation of the phase variance [66, 86]:
δφ
2
R = 2ω 0
(1 + ν
2
)t − ν
2 1 − e
−2 p
2 p
.
(11)