20
P. K. Muduli et al.
There have been some recent works to extend the single-mode theory to multiple
modes [72, 96–99], which is necessary in order to explain experimental observations
of multi-mode system as well as of the generation linewidth of such systems. These
works draw on an analogy between equations describing STNO multimode generation with equations similar to those describing multimode ring lasers [100, 101] in
laser technology. The theory presented in Ref. [99] is a generalization of the singlemode theory by Slavin and Tiberkevich [14] that includes contributions to nonlinear
frequency shift as well as to nonlinear damping and pumping from mode-interactions
mediated by exchange and magnetostatic coupling. An important part of the theory
in Ref. [99] is that coupling through a bath of thermally excited magnons leads to an
effective linear mode-coupling term. This is the term that leads to mode-hopping in
the presence of a stochastic force, and it also leads to temperature-driven effects, such
as temperature-induced switching from one resonant mode to another, as observed
by Muduli, Heinonen, and Åkerman [96]. If we consider a system with an in-plane
MTJ-STNO and a total current I perpendicular to the plane of the magnetic layers and
with two dominant modes 1 and 2 between which the system exhibits mode-hopping
while the other modes are thermally populated, one can derive equations for the slow
time-evolution of amplitudes A 1 (t) and A 2 (t). Here, “slow” means slow compared
to the typical nano-second or sub-nanosecond timescales of resonant modes of such
a system. As was shown in Ref. [99], the effective equations for the amplitudes of
the coupled modes are:
dA 1 (t)
dt
= −i
η 1,1 |A 1 |
2
+ η 1,2 |A 2 |
2
A 1 − G,1
1 + P 1,1 |A 1 |
2
+ P 1,2 |A 2 |
2
A 1
+σ 0 I cos β
1 − Q 1,1 |A 1 |
2
− Q 1,2 |A 2 |
2
+ R 1,2 (T )A 2
dA 2 (t)
dt
= −i
η 2,1 |A 1 |
2
+ η 2,2 |A 2 |
2
A 2 − G,2
1 + P 2,1 |A 1 |
2
+ P 2,2 |A 2 |
2
A 2
+σ 0 I cos β
1 − Q 2,1 |A 1 |
2
− Q 2,2 |A 2 |
2
+ R 2,1 (T )A 2 .
(13)
Here, σ 0 I is the effective field that arises because of STT, β is the angle between
the reference layer magnetization and an in-plane external magnetic field, and
G,i = αω i , i = 1, 2, with ω i the linearized undamped frequency of eigenmode i.
The coefficients η i,i are the usual non-linear frequency shift due to the power of
mode i, but now there are also off-diagonal non-linear frequency shifts η i,j , i = j, of
mode i due to the power of mode j. Similarly, the non-linear damping and pumping
now have off-diagonal contributions Q i,j and P i,j , i = j, to the dynamics of mode i
because of the power in mode j, in addition to the diagonal contributions Q i,i and P i,i .
Note that while Q 1,2 = Q 2,1 it is not the case in general that P 1,2 = P 1,2 . Finally, the
complex coefficients R i,j (T ), with T the temperature, couple the amplitude of mode
j to the time-evolution of mode i. These are the couplings that arise from interactions
with the thermal bath of magnons. If the coefficients R i,j are zero, the system may
exhibit single motion according to modes 1 or 2, or may exhibit mode coexistence.
Only if R i,j are non-zero [100, 101], and in the presence of a stochastic force such
as thermal agitation, will the system exhibit mode-hopping.
P. K. Muduli et al.
There have been some recent works to extend the single-mode theory to multiple
modes [72, 96–99], which is necessary in order to explain experimental observations
of multi-mode system as well as of the generation linewidth of such systems. These
works draw on an analogy between equations describing STNO multimode generation with equations similar to those describing multimode ring lasers [100, 101] in
laser technology. The theory presented in Ref. [99] is a generalization of the singlemode theory by Slavin and Tiberkevich [14] that includes contributions to nonlinear
frequency shift as well as to nonlinear damping and pumping from mode-interactions
mediated by exchange and magnetostatic coupling. An important part of the theory
in Ref. [99] is that coupling through a bath of thermally excited magnons leads to an
effective linear mode-coupling term. This is the term that leads to mode-hopping in
the presence of a stochastic force, and it also leads to temperature-driven effects, such
as temperature-induced switching from one resonant mode to another, as observed
by Muduli, Heinonen, and Åkerman [96]. If we consider a system with an in-plane
MTJ-STNO and a total current I perpendicular to the plane of the magnetic layers and
with two dominant modes 1 and 2 between which the system exhibits mode-hopping
while the other modes are thermally populated, one can derive equations for the slow
time-evolution of amplitudes A 1 (t) and A 2 (t). Here, “slow” means slow compared
to the typical nano-second or sub-nanosecond timescales of resonant modes of such
a system. As was shown in Ref. [99], the effective equations for the amplitudes of
the coupled modes are:
dA 1 (t)
dt
= −i
η 1,1 |A 1 |
2
+ η 1,2 |A 2 |
2
A 1 − G,1
1 + P 1,1 |A 1 |
2
+ P 1,2 |A 2 |
2
A 1
+σ 0 I cos β
1 − Q 1,1 |A 1 |
2
− Q 1,2 |A 2 |
2
+ R 1,2 (T )A 2
dA 2 (t)
dt
= −i
η 2,1 |A 1 |
2
+ η 2,2 |A 2 |
2
A 2 − G,2
1 + P 2,1 |A 1 |
2
+ P 2,2 |A 2 |
2
A 2
+σ 0 I cos β
1 − Q 2,1 |A 1 |
2
− Q 2,2 |A 2 |
2
+ R 2,1 (T )A 2 .
(13)
Here, σ 0 I is the effective field that arises because of STT, β is the angle between
the reference layer magnetization and an in-plane external magnetic field, and
G,i = αω i , i = 1, 2, with ω i the linearized undamped frequency of eigenmode i.
The coefficients η i,i are the usual non-linear frequency shift due to the power of
mode i, but now there are also off-diagonal non-linear frequency shifts η i,j , i = j, of
mode i due to the power of mode j. Similarly, the non-linear damping and pumping
now have off-diagonal contributions Q i,j and P i,j , i = j, to the dynamics of mode i
because of the power in mode j, in addition to the diagonal contributions Q i,i and P i,i .
Note that while Q 1,2 = Q 2,1 it is not the case in general that P 1,2 = P 1,2 . Finally, the
complex coefficients R i,j (T ), with T the temperature, couple the amplitude of mode
j to the time-evolution of mode i. These are the couplings that arise from interactions
with the thermal bath of magnons. If the coefficients R i,j are zero, the system may
exhibit single motion according to modes 1 or 2, or may exhibit mode coexistence.
Only if R i,j are non-zero [100, 101], and in the presence of a stochastic force such
as thermal agitation, will the system exhibit mode-hopping.