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5.0043905.pdf | J. Appl. Phys. 129, 214505 (2021); https://doi.org/10.1063/5.0043905 129, 214505
© 2021 Author(s).Modeling of magnetization dynamics and
thermal magnetic moment fluctuations in
nanoparticle-enhanced magnetic resonance
detection
Cite as: J. Appl. Phys. 129, 214505 (2021); https://doi.org/10.1063/5.0043905
Submitted: 12 January 2021 . Accepted: 11 May 2021 . Published Online: 03 June 2021
Tahmid Kaisar ,
Md Mahadi Rajib ,
Hatem ElBidweihy ,
Mladen Barbic , and
Jayasimha Atulasimha
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magnetic moment fluctuations in nanoparticle-
enhanced magnetic resonance detection
Cite as: J. Appl. Phys. 129, 214505 (2021); doi: 10.1063/5.0043905
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Submitted: 12 January 2021 · Accepted: 11 May 2021 ·
Published Online: 3 June 2021
Tahmid Kaisar,1
Md Mahadi Rajib,1
Hatem ElBidweihy,2
Mladen Barbic,3
and Jayasimha Atulasimha1,a)
AFFILIATIONS
1Department of Mechanical and Nuclear Engineering, Virginia Commonwealth University, Richmond, Virginia 23284, USA
2Department of Electrical and Computer Engineering, United States Naval Academy, Annapolis, Maryland 21402, USA
3NYU Langone Health, Tech4Health Institute, New York, New York 10010, USA
a)Author to whom correspondence should be addressed: jatulasimha@vcu.edu
ABSTRACT
This study presents a systematic numerical modeling investigation of magnetization dynamics and thermal magnetic moment fluctuations of
single magnetic domain nanoparticles in a configuration applicable to enhancing inductive magnetic resonance detection signal to noise ratio
(SNR). Previous proposals for oriented anisotropic single magnetic domain nanoparticle amplification of magnetic flux in a magnetic reso-
nance imaging (MRI) coil focused only on the coil pick-up voltage signal enhancement. In this study, the numerical evaluation of the SNR hasbeen extended by modeling the inherent thermal magnetic noise introduced into the detection coil by the insertion of such anisotropic nano-particle-filled coil core. The Landau –Lifshitz –Gilbert equation under the Stoner –Wohlfarth single magnetic domain (macrospin) assumption
was utilized to simulate the magnetization dynamics due to AC drive field as well as thermal noise. These simulations are used to evaluate the
nanoparticle configurations and shape effects on enhancing SNR. Finally, we explore the effect of narrow band filtering of the broadband mag-netic moment thermal fluctuation noise on the SNR. It was observed that for a particular shape of a single nanoparticle, the SNR could beincreased up to ∼8 and the choice of an appropriate number of the nanoparticles increases the SNR by several orders of magnitude and could
consequently lead to the detectability of a very small field of ∼10 pT. These results could provide an impetus for relatively simple modifications
to existing MRI systems for achieving enhanced detection SNR in scanners with modest polarizing magnetic fields.
Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0043905
I. INTRODUCTION
Sensitivity enhancement in magnetic resonance detection con-
tinues to be an important challenge due to the importance of
nuclear magnetic resonance (NMR) and MRI in basic science,
medical diagnostics, and materials characterization.
1–5Although
many alternative methods of magnetic resonance detection havebeen developed over the years, the inductive coil detection of mag-netic resonance of precessing proton nuclear magnetic moments is
by far the most common.
6The challenge in magnetic resonance
detection stems from the low nuclear spin polarization at roomtemperature and laboratory static magnetic fields. An additionalchallenge is the fundamental requirement that the detector in amagnetic resonance experiment needs to be compatible with and
immune to the large polarizing DC magnetic field while also be
sufficiently sensitive to weak AC magnetic fields generated by theprecessing nuclear spins. The inductive coil, operating on the prin-
ciple of Faraday ’s law of induction, satisfies this requirement, and
enhancing the inductive coil detection signal to noise ratio (SNR)has been pursued through various techniques.
7–9However, an
unlimited increase of the polarizing magnetic field is cost prohibi-
tive, and technical challenges often inhibit the development of
mobile MRI units, their access, sustainability, and size. Therefore,solutions to achieving sufficient or improved SNR in NMR induc-tive coil detection in lower magnetic fields and more accessible andcompact configurations remain highly desirable.
10
A. Signal amplification by magnetic nanoparticle-filledcoil core
An idea has been put forward to increasing the magnetic field
flux from the sample through the coil by filling the coil with a coreJournal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 129, 214505 (2021); doi: 10.1063/5.0043905 129, 214505-1
Published under an exclusive license by AIP Publishingof oriented anisotropic single domain magnetic nanoparticles,11,12
as shown in Fig. 1 . The sample and the inductive coil detector are
both in the prototypical MRI environment of a large DC polariz-ing magnetic field (H
Zdc)a l o n gt h e zaxis, or a large magnetic
induction BZdc=μ0HZdc,w h e r e μ0is the permeability of free
space. This field generates a fractional nuclear spin polarization of
protons in the sample. The appl ication of RF magnetic fields
along the xaxis is subsequently used to tilt the magnetic moment
of the sample away from the zaxis and generate precession of the
sample magnetization around the zaxis at the proton NMR fre-
quency ω0=γBZdc, where γis the proton nuclear gyromagnetic
ratio. This sample moment precession around the zaxis generates
a time-varying magnetic induction BXac=μ0HXacthrough the
inductive coil detector of Nturns and sensing area Aalong the x
axis. By Faraday ’s law of induction, an AC signal voltage Vat fre-
quency ω0generated across the coil terminals is
V¼N/C1A/C1ω0/C1BXac: (1)
It is a well-known practice in electromagnet design and
ambient inductive detectors that a soft ferromagnetic core withinthe coil significantly amplifies the magnetic flux through thecoil.
13,14The challenge, however, in the configuration of NMR
detection of Fig. 1 is that the presence of the large polarizing mag-
netic field along the zaxis, HZdc, would generally saturate the detec-
tion coil core made of a soft ferromagnet along the zaxis and
render the AC magnetic field due to proton precession along the x
axis of the coil ineffective. In other words, a high polarizing mag-
netic field would saturate the coil core in its own direction andleave the core ’s magnetization unresponsive to the AC magnetic
field arising from proton precession. The solution proposed
11,12
was that the oriented anisotropic magnetic nanoparticles filling the
coil core actually have an appreciable magnetic susceptibility along
thexaxis precisely in the presence of a significant DC magneticfield along the zaxis. The pick-up coil voltage is then
V¼N/C1A/C1ω0/C1μ0/C1(HXacþMXac), (2)
where MXacis the magnetization component of the nanoparticle-
filled coil core along the x-direction (sensing direction of the coil)
due to the magnetic field HXacfrom the precessing sample nuclear
spin magnetization, MXac=χRTHXac(where χRT=ΔMXac/ΔHXacis
defined as reversible transverse susceptibility). Therefore, if the
reversible transverse susceptibility, χRT, of the magnetic
nanoparticle-filled coil core along the xaxis is significant at the
large polarizing DC magnetic field HZdcalong the zaxis, the induc-
tive coil signal voltage will be enhanced. Various theoretical15–17
and experimental investigations18–25of reversible transverse sus-
ceptibility in oriented magnetic nanostructures indeed reveal thatits magnitude can be appreciable and, therefore, might provide aviable route for magnetic resonance signal amplification, as dia-grammatically shown in Fig. 1 .
In this study, the coil signal voltage has been numerically eval-
uated by modeling individual nanoparticle magnetic momentdynamics in the Stoner –Wohlfarth (SW) uniform magnetization
approximation.
26More specifically, the AC nanoparticle moment
along the xaxis in Fig. 1 ,mXac, has been investigated in the pres-
ence of a large DC magnetic field HZdcalong the zaxis and under
the driven sample AC magnetic field HXacalong the xaxis. Though
artificially synthesized magnetic nanoparticles follow a lognormalsize distribution, for simplicity the total coil core of volume, V
c, has
been assumed to be composed of “n”number of identical oriented
single domain magnetic nanoparticles, and that for each particle,
the average x-component of the AC magnetic moment, mXac,
equally contributes to the coherent amplification of the pick-upvoltage signal of the coil detector. Therefore, the total coil AC
voltage due to the magnetic nanoparticle core contribution is
V¼N/C1A/C1ω/C1μ
0/C1n/C1mXac
Vc: (3)
B. Noise contribution by magnetic nanoparticle-filled
coil core
Essential to the SNR consideration of any NMR experimental
arrangement is the evaluation of the noise sources in the signal
chain. In this work, the focus is specifically on the magneticnanoparticle-filled inductive coil detector since the sample noisealong with the amplifier noise and the Johnson noise contributionshave been addressed in numerous works.
27–31Any magnetic mate-
rial placed inside the inductive detection coil will introduce addi-
tional pick-up voltage noise due to intrinsic magnetizationfluctuations.
32These thermal fluctuations of the coil core magneti-
zation along the xaxis, which were numerically modeled in detail
in this work, generate a total mean squared coil noise voltage,
V2/C10/C11
¼N2/C1A2/C1ω2/C1μ2
0/C1M2
X/C10/C11
: (4)
For simplicity, it is assumed that the total coil core of volume,
Vc, is composed of nnumber of identical oriented single domain
magnetic nanoparticles and that each particle magnetic moment,
FIG. 1. Schematic diagram for enhanced NMR detection with magnetic
nanoparticle-filled coil core.Journal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 129, 214505 (2021); doi: 10.1063/5.0043905 129, 214505-2
Published under an exclusive license by AIP Publishingm, undergoes random uncorrelated thermal fluctuation. Therefore,
the total mean squared coil noise voltage due to the magnetic
nanoparticle core is33,34
V2/C10/C11
¼N2/C1A2/C1ω2/C1μ2
0/C1n/C1m2
X/C10/C11
V2
C: (5)
It is to be noted that the magnetic moment fluctuation phe-
nomena have previously been investigated in various spin systems,materials, and detection modalities.
35–42However, there does not
appear to be a theoretical, numerical, or experimental study where
thermal magnetic moment fluctuations of single domain nanopar-
ticles of the configuration of Fig. 1 (where the large polarizing mag-
netic field is applied perpendicular to the nanoparticle hard axisand the coil detection axis) and their contribution to the coil noise
voltage have been carried out.
In this study, therefore, in order to assess the signal and the
noise of the configuration of the magnetic resonance coil detectorofFig. 1 , the room temperature magnetization dynamics of a single
domain nanomagnet and its thermal fluctuation in the coil core
has been simulated. Such single nanomagnet dynamics for the
macrospin Stoner –Wohlfarth (SW) model uniform magnetization
in both oblate and prolate ellipsoid geometries has been explored.The SW model assumes that the entire nanomagnet behaves like agiant classical spin. Thus, this model assumes that the spin (or
magnetization) in all regions of the nanomagnet point in the same
direction, i.e., different regions of a nanomagnet cannot have spinspointing in different directions. As a first approximation, thisassumption is valid for the nanomagnets we model as their dimen-sions are less than 100 nm.
43
This will explain the optimum nanoparticle orientation and
bias field needed to maximize SNR of the experimental arrange-ment of Fig. 1 . This analysis has been extended to scaling proper-
ties of an ensemble of nanomagnets and the effect of applying a
bandpass filter to provide an estimate on the extent to which the
insertion of magnetic particles in the sensing coil can enhance thelimits of detection of magnetic fields due to proton spin resonancesin MRI/NMR.
C. Modeling particle magnetization dynamics in the
presence of room temperature thermal noise
Modeling of the single particle magnetization dynamics was per-
formed by solving the Landau –Lifshitz –Gilbert (LLG) equation,
44–46
which was formulated for laboratory frame of reference,
d~m
dt¼/C0γ~m/C2~Heff/C0αγ[~m/C2(~m/C2~Heff)]: (6)
In Eq. (6),γis the gyromagnetic ratio (m/A s), αis the Gilbert
damping coefficient and ~mis the normalized magnetization vector,
found by normalizing the magnetization vector ( ~M) with respect to
saturation magnetization ( Ms),46
~m¼~M
Ms;m2
xþm2
yþm2
z¼1: (7)Here, mx,my,a n d mzare the normalized components of ~malong
the three Cartesian coordinates.
The effective field ( ~Heff) was obtained from the derivative of
the total energy ( E) of the system with respect to the magnetization
(~M),46,47
HQ
eff¼/C01
μ0ΩdE
dMQþHQ
thermal , (8)
where μ0is the permeability of the vacuum and Ωis the volume of
the nanomagnet.
The total potential energy in Eq. (8)is given by
E¼Eshape anisotropy þEzeeman , (9)
where Eshape anisotropy is the shape anisotropy due to the prolate or
oblate shape and can be calculated from the following equation:46
Eshape anisotropy ¼μ0
2/C16/C17
Ω[NdxxM2
xþNdyyM2
yþNdzzM2
z], (10)
where Nd_xx,Nd_yy,a n d Nd_zz represent the demagnetization
factors along the x,y,a n d zdirections, which depend on the
dimensions of the nanoparticle and follow the relation ofN
d_xx+Nd_yy+Nd_zz=148,49andMx,My,a n d Mzare the compo-
nents of magnetization vector ( ~M) along the three Cartesian
coordinates.
In Eq. (9),Ezeeman is the potential energy of nanomagnet for
an external magnetic field ( ~H), given by
Ezeeman¼/C0μ0Ω~H/C1~M: (11)
The thermal field HQ
thermalis modeled as a random field incorporated
in the manner of47,50
HQ
thermal(t)¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2kTα
μ0MsγΩΔts
/C1~G(t): (12)
In Eq. (12),~G(t) is a Gaussian distribution with zero mean
and unit variance in each Cartesian coordinate axis, kis the
Boltzman constant, Tis the temperature, Msis the saturation mag-
netization and γis the Gyromagnetic ratio. Δtis the time step used
in the numerical solution of Eq. (6)and was chosen to be 100 fs.
This was chosen to be small enough to ensure that all results areindependent of the time step.
It is to be noted that m
x,my,a n d mzare not independent and
they are related by Eq. (7)and can be represented parametrically46as
mx(t)¼sinθ(t)c o sf(t); my(t)¼sinθ(t)s i nf(t);
mz(t)¼cosθ(t):(13)
With this parametric representation, the number of variables
reduces from three ( mx,my,a n d mz)t ot w o( θ,f). When Eq. (6)is
written in the component form, three scalar equations are obtained of
which two equations are enough to solve for θandf.B ye m p l o y i n gJournal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 129, 214505 (2021); doi: 10.1063/5.0043905 129, 214505-3
Published under an exclusive license by AIP Publishingthe Euler method, the differential equations can be solved as given
in Ref. 46and temporal evolutions of θandfcan be obtained,
which provides the magnetization components along the three coordi-
nate axes.
The total angular dependence of energy can be obtained
from11,12,51
E(θ)¼[Kusin2(θ)/C0μ0HdcMssin(θ)]Ω: (14)
The first term of the equation represents the uniaxial anisot-
ropy energy and the second term stands for Zeeman energy of the
nanoparticle moment in the DC magnetic field. In the uniaxialanisotropy energy term, K
u, is the anisotropy constant that has the
value of51
Ku¼1
2μ0/C1(Na/C0Nc)/C1M2
s: (15)
NaandNcare the demagnetization factors along the hard and
easy axis of the ellipsoids, respectively, and the critical field Hcis
obtained from11
Hc¼2Ku
μ0Ms: (16)
Table I lists the values of the material properties of the
nanomagnet.II. RESULTS AND DISCUSSIONS
Consider a prolate ellipsoid of nominal volume ∼5000 nm3
(principal axes 100, 10, and 10 nm) shown in Fig. 2(a) , where it has
been assumed that the sample proton spin precession produces a
magnetic field along the easy (long) xaxis of the nanomagnet
while the DC bias field is applied along one of the hard (short)axes, viz., the zaxis. When the DC bias field is zero, there are two
deep energy wells at θ= 0°, 180° as obtained from Eq. (14). When
the magnetization is in one of these states, the magnetization
response to an AC magnetic field along the xaxis (a simplified rep-
resentation of the signal at the pick-up coil due to proton spin pre-cession of Fig. 1 ) is very small as the magnetization is in this deep
potential as seen in Figs. 3(a) and 3(b) and Table II . The corre-
sponding magnetization fluctuation due to room temperature
thermal noise [that is modeled as a random effective magnetic
field; see Eq. (8)] is also very small.
As the DC field increases along the zaxis to the point
H
dc=Hc(the DC bias field is equal to the critical field Hc), the
mean magnetization orientation is at 90°. However, the potential
well at 90° [ Fig. 2(a) ] is characterized by a flat energy profile where
the energy is nearly independent of the polar angle θ, around
θ= 90°. This leads to a large magnetization response along the x
axis to an applied AC magnetic field along the xaxis [ Figs. 3(a)
and3(b) andTable II ] in the presence of a large DC magnetic field
along the zaxis. Essentially, since the energy profile is flat in this
configuration, the particle moment fluctuation due to room tem-perature thermal noise is also high. Nevertheless, it is found thatthe signal to noise ratio (SNR) is highest at H
dc=Hc. In fact, the
magnetization response and the SNR ratio are found to increase
monotonically with the applied bias field up to Hc[see Table II
based on the selected simulations shown in Figs. 3(a) and3(b) and
all the simulations shown Fig. S1 in the supplementary material ]
and then decreases as Hdc>Hc(for example, at Hdc= 1.25 Hcin
Table II and Fig. S1 in the supplementary material ) due to an
energy well deepening at θ= 90° for Hdc>Hc[Fig. 2(a) ]. SNR isTABLE I. Material properties of CoFe.52
Parameters Material property
Saturation magnetization ( Ms) 1.6 × 106(A/m)
Gilbert damping ( α) 0.05
Gyromagnetic ratio ( γ) 2.2 × 105(m/A s)
FIG. 2. Energy profile for various DC bias magnetic fields ( Hdc) for (a) prolate: bias field along minor axis, (b) prolate: bias field along major axis, and (c) oblate: bias field
along minor axis.Journal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 129, 214505 (2021); doi: 10.1063/5.0043905 129, 214505-4
Published under an exclusive license by AIP Publishingcalculated in the following manner. First, no thermal noise is
included and the LLG simulation is performed to determine themagnetization response due to only the AC field from sampleproton spin precession. Then, another LLG simulation is per-formed with no signal and the magnetization fluctuation is studied
due to the thermal noise only. The ratio of the rms values of the
response due to the sample precessing field and that due to noise isdefined as the SNR.
It is noteworthy that the AC drive magnetic field has an
amplitude of 800 A/m (10 Oe or 1 mT) for all cases discussed in
this work, which is much larger than the typical signal due toproton spins, which may be several orders of magnitude smaller.
However, a higher drive amplitude was chosen to elicit a reasonablemagnetization response that would be easily visible in the plots andresult in reasonable SNR ratios for a single nanomagnet. In prac-tice, the number of nanomagnets placed in the detector coil could
be over n∼10
12or more resulting in sub-nT sensing capability as
discussed later. Furthermore, the proton resonance of 42.5 MHzoccurs at a DC field ∼1 T, which would change if the DC bias is
changed. However, to keep the simulations consistent, the signal
due to proton resonance is assumed as 42.5 MHz for all cases as
this would not change the qualitative findings.
FIG. 3. Magnetization dynamics with (a) 800 A/m, 42.5 MHz AC field with no thermal noise for single prolate nanomagnet with bias along minor axis and (b) only th ermal
noise at Hdc= 0 and Hdc=Hc(large magnetic response and magnetization fluctuation due to thermal noise at Hdc=Hccompared to Hdc= 0). Magnetization dynamics with
(c) 800 A/m, 42.5 MHz field with no thermal noise for single oblate nanomagnet with bias along minor axis (very large magnetic response at Hdc= 0.625 Hccompared to
Hdc= 0) and (d) only thermal noise at Hdc= 0 and Hdc= 0.625 Hc. (e) SNR vs Hdc/Hcfor single prolate and single oblate nanomagnet cases and (f) zoomed version of
SNR vs Hdc/Hcfor single prolate nanomagnet with bias along major axis.Journal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 129, 214505 (2021); doi: 10.1063/5.0043905 129, 214505-5
Published under an exclusive license by AIP PublishingNext, a prolate ellipsoid is considered as shown in Fig. 2(b) ,
where it is assumed that the proton spin precession produces a
magnetic field along one of the hard (short) axes of the nanomag-net while a DC bias field is applied along the long (easy) axis.Initially, the magnetization points downward ( θ= 180°) and at
H
dc= 0 is in a deep potential well as shown in Fig. 2(b) . As the
magnetic field applied along the + zdirection ( θ= 0°) increases, the
energy well around θ= 180° is flattened as described by Eq. (14).
Thus, for a higher field, the magnetization response increases, asdoes the magnetization fluctuation due to thermal noise as shown
inTable II (detailed simulations are shown Fig. S2 in the
supplementary material ). However, the shallow wells improve the
magnetization response due to the drive AC magnetic field morethan the increased magnetization fluctuations due to the thermalnoise (as in the prior case) and increase the overall SNR ratio
(Table II ).
However, as one approaches H
dc=Hc, the SNR drops signifi-
c a n t l y .T h i si sb e c a u s et h ee n e r g yp r o f i l ei sf l a ta t θ=1 8 0 ° b u t e v e n
small perturbations from this angle make the magnetization switchand rotate to the + zaxis ( θ= 0°) as shown in Fig. 2(b) . Once it
reaches this state, the energy well profile at H
dc=Hcandθ=0 ° i s a
deeper than the well at Hdc= 0. The reason is that the Zeeman
energy due to a field along the + zaxis makes the already deep shape
anisotropy well even deeper at θ= 0° reducing both the magnetiza-
tion response and the magnetization fluctuations due to thermal
noise, as well as the SNR ratio. Thus, the best SNR is seen atH
dc<Hc(see the high SNR at 0.875 HcinTable II )b u tc l o s et o Hc.
Finally, the case of an oblate ellipsoid of nominal volume
∼5000 nm3(principal axes 40, 40, and 6 nm) is considered, similar
to the volume of a prolate ellipsoid, shown in Fig. 2(c) . Again, it is
assumed that the proton spin precession produces a magnetic fieldalong one of the easy (long) axes of the nanomagnet while the DCbias field is applied along the hard axis. The symmetry of the
problem is such that at H
dc= 0 and the magnetization is free to
rotate in the x–yplane as there is no energy barrier to such rota-
tion. As Hdcincreases, the magnetization is still free to move in a
cone of the x–yplane at a specific angle to the zaxis that decreases
with increasing Hdc, finally coinciding with it when Hdc=Hc. Thus,
at a range of DC bias fields (for example, from Hdc= 0.25 Hcto
Hdc= 0.75 Hc) a high SNR > 1.4 is observed when a single nanomag-
net is driven by an AC magnetic field. This is due to the combina-tion of high magnetization response given the symmetry and noise
limited by the presence of the DC bias field. The simulations of
magnetic response to the AC magnetic field and magnetizationfluctuations due to random thermal noise are, respectively, showninFigs. 3(c) and 3(d) comparing H
dc= 0 and Hdc= 0.625 Hccases
with all other bias field cases shown in Fig. S3 in the supplemen-
tary material .
In summary, as far as the SNR is concerned, the prolate ellip-
soid with DC bias magnetic field along the hard axis [ Fig. 2(a) ]i s
the better choice over the prolate ellipsoid with DC bias magneticfield along the easy axis. However, the oblate geometry and config-
uration shown in Fig. 2(c) produces the highest SNR as shown in
Figs. 3(e) and3(f) andTable II [more than twice the highest SNR
for the single prolate ellipsoid configuration in Fig. 2(a) , and more
than 10 times the single prolate ellipsoid configuration of
Fig. 2(b) ]. What makes this oblate configuration even more attrac-
tive to detection coils in MRI/NMR applications is that the highSNR performance is seen over a large range of DC bias fields (e.g.,H
dc= 0.25 HctoHdc= 0.75 Hc), making it attractive for a broad
range of MRI scanner fields.
This best-case nanoparticle (oblate ellipsoid at Hdc= 0.625 Hc
with SNR = 1.71) was then taken and investigated if the SNR can
further be improved by applying a narrow band filter aroundTABLE II. Magnetization oscillations in the single nanomagnet of different geometries for different values of DC bias magnetic field (based on simulations sh own in Fig. 3 and
Figs. S1 –S3 in the supplementary material ). Boldfaced rows represent the highest SNR values for each case.
CasesValue of
bias field
(Hdc)RMS normalized magnetization ( M/Ms)
for a sinusoidal magnetic signal of
800 A/m (10 Oe) amplitudeRMS normalized magnetization
(M/Ms) due to thermal noise only
(no signal)SNR (defined here
as ratio of columns
3 and 4)
Prolate applying
bias field along
minor axis0 1.66 × 10−66.45 × 10−40.003
0.25Hc 2.82 × 10−40.0075 0.05
0.5Hc 0.002 0.0103 0.2885
0.75Hc 0.0128 0.0202 0.63
Hc 0.1035 0 .1464 0 .7042
1.25Hc 0.009 0.0484 0.193
Prolate applying
bias field along
major axis0 9.3 × 10−40.025 0.03
0.5Hc 0.0017 0.034 0.0465
0.75Hc 0.0032 0.05 0.0653
0.875 Hc 0.0063 0 .0662 0 .09
Hc 0.0018 0.025 0.075
Oblate applying
bias field along
minor axis0.25Hc 0.94 0.68 1.4
0.5Hc 0.843 0.582 1.51
0.625Hc 0.76 0 .45 1 .71
0.75Hc 0.645 0.46 1.44
Hc 0.0934 0.0921 0.95Journal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 129, 214505 (2021); doi: 10.1063/5.0043905 129, 214505-6
Published under an exclusive license by AIP Publishing42.5 MHz. The rationale is that the magnetization response driven
by the magnetic field due to proton spin precession at 1 T appliedDC field is dominant around 42.5 MHz while the magnetization
fluctuations driven by thermal noise are a broad band as evidenced
by the single-sided amplitude spectrum shown in Fig. 4(a) . When a
bandpass filter of 42 –44 MHz was applied, the SNR improved to
∼8 as shown in Fig. 4(b) .
A. Scaling of SNR with coil core nanoparticle number
While the SNR with an AC magnetic field of 10 Oe (equiva-
lent to 1 mT) shows a SNR ∼8 for optimal conditions, this was
assuming a single nanomagnet. However, for a large number ofnanomagnets nwithin the core, the magnetization response
increases as n(as more nanomagnets coherently contribute to
more magnetic moment and, therefore, greater induced voltage in
the coil) according to Eq. (3), while the magnetic noise would only
increase as √n(as the thermal noise induced fluctuations of nano-
magnets have a random phase) according to Eq. (5), leading to a
SNR increase of n/√n=√n. The scaling trend of SNR obtained
from Eq. (5)has been corroborated by performing simulations for
5, 10, 100, 150, and 300 nanoparticles under best-case conditions(oblate ellipsoid at H
dc= 0.625 Hc). The analytically calculated trend
of SNR increasing as √ntimes is compared with numerical simu-
lations using the LLG formalism incorporating noise as shown in
Fig. 5 . It should be noted that for higher than 150 nanomagnets,the simulated gain in SNR is smaller than the SNR expected due to
√nscaling. This is possibly due to numerical issues in not main-
taining random (completely uncorrelated) magnetization dynamics
between different nanoparticles as the number of particles simu-
lated increases beyond 100.
If a square detection coil of 2 cm on a side and the pitch
between nanomagnets ∼200 nm is considered, 10 billion nanomag-
nets can be accommodated in a single layer of 2 cm by 2 cm dimen-
sion. Additionally, as the single nanoparticle layer thickness is∼6 nm, the average distance between two such layers can be
∼25 nm. Thus, 400 000 such magnetic nanoparticle layers can be
accommodated in 1 cm coil thickness. Consequently, n=4 0×1 0
14
nanomagnets can be incorporated into the sensing coil. So, the inser-
tion of 40 × 1014nanomagnets in a core of 2 × 2 × 1 cm3size has
∼0.005 or 0.5% volume fraction (defined as the ratio of the volume
of the nanoparticles to the volume of coil core). Since the dipolarinteraction decreases as the nanoparticle density decreases,
53,54the
insertion of 40 × 1014nanomagnets constitutes a very low volume
fraction with ∼5 times higher pitch than the lateral dimension and
∼4 times higher separating distance in the vertical direction than the
height of the individual oblate nanomagnet. This justifies ignoringthe dipole coupling between nanoparticles. This number of nano-
magnets also leads to an increase in SNR from 8 to ∼5×1 0
8.I n
other words, with a SNR of 5 × 108, one could conceivably detect an
AC magnetic field of 1/(108) mT, i.e., an AC magnetic field of 10 pT
or better depending on the density with which nanoparticles are
inserted into the NMR detection coil. However, it should be noted
that nanoparticle pinning sites, inherent inhomogeneities, etc. can
FIG. 4. (a) Frequency spectrum of signal + noise, before filtering. (b) Frequency
spectrum of signal + noise, after filtering. Both cases for oblate nanomagnet.
FIG. 5. The scaling trend of SNR as √nfornnumber of nanoparticles
calculated analytically (red) from Eq. (5)and obtained from simulation (blue).Journal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 129, 214505 (2021); doi: 10.1063/5.0043905 129, 214505-7
Published under an exclusive license by AIP Publishingimpede magnetization dynamics55and create phase differences, thus
decreasing signal enhancement.
The key point is that merely filling the detection coil with a
soft (high permeability) core would not help as the core would besaturated under the high DC bias fields used for MRI. However, byusing anisotropic nanoparticles of appropriate geometry that are
still responsive to AC fields from proton precession in the presence
of orthogonal strong DC fields, the coil detection sensitivity can beenhanced.
III. CONCLUSION
This numerical investigation of nanoparticle magnetization
dynamics and room temperature thermal moment fluctuationsconfirm the initial zero temperature proposal for nanoparticle-based amplification of a NMR signal. Such a consideration of thethermal fluctuation allows us to predict not just idealized zero tem-
perature signal amplification values but realistic room temperature
SNR values. This analysis suggests specific orientations of aniso-tropic oblate ellipsoid particles can lead to SNR improvements overconventional air-filled MRI coils. Much will depend on the qualityof the particles used in the coil core: shape uniformity, quality of
particles orientation within the core, smoothness of the particles
and surface pinning sites (that degrade the effect of magnetizationdynamics), and uniformity of the nanoparticle aspect ratio (whichdetermines where the particle has a peak in transverse susceptibil-ity). Further consideration would have to be made of the effect of
magnetic particles on the field non-uniformity within the nuclear
spin sample that is being detected/imaged since such field distor-tions will broaden the sample spin resonance and will have to beaddressed in both the MRI scanner bore designs that incorporate
the nanoparticles within the coils, as well as in the pulse sequences
that deal with such inhomogeneous broadening. Nevertheless, theseresults provide further strong impetus for relatively simple modifi-cations to existing MRI inductive detection coils for achievingimproved SNR in scanners operating in 0.1 –2 T polarizing field
range. This promise of a higher SNR would allow for shorter MRI
scan time, more compact MRI systems, lower operating fields, andhigher accessibility.
SUPPLEMENTARY MATERIAL
In the supplementary material , figures have been provided for
normalized magnetization ( M/M
s) for (i) a sinusoidal magnetic
signal of 800 A/m (10 Oe) amplitude and (ii) thermal noise for allcases of DC bias magnetic fields for prolate nanomagnet with biasalong minor axis and major axis as well as oblate nanomagnet with
bias along minor axis.
DATA AVAILABILITY
Data sharing is not applicable to this article as no new data
were created or analyzed in this study.
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Published under an exclusive license by AIP Publishing |
1.551122.pdf | Network structure dependence of volume and glass transition temperature
Jeffry J. Fedderly, Gilbert F. Lee, John D. Lee, Bruce Hartmann, Karel Dušek, Miroslava Dušková-Smrková, and
Ján Šomvársky
Citation: Journal of Rheology (1978-present) 44, 961 (2000); doi: 10.1122/1.551122
View online: http://dx.doi.org/10.1122/1.551122
View Table of Contents: http://scitation.aip.org/content/sor/journal/jor2/44/4?ver=pdfcov
Published by the The Society of Rheology
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22:13:55Network structure dependence of volume and glass
transition temperaturea)
Jeffry J. Fedderly,b)Gilbert F. Lee, John D. Lee, and Bruce Hartmann
Naval Surface Warfare Center, West Bethesda, Maryland 20817-5700
Karel Dus ˇek, Miroslava Dus ˇkova´-Smrc ˇkova´, and Ja ´nSˇomva´rsky
Institute of Macromolecular Chemistry, Academy of Sciences of the Czech
Republic, 162 06 Prague, Czech Republic
(Received 16 February 2000; final revision received 21 April 2000)
Synopsis
A series of polyurethanes was used to determine the molar contributions of chain ends ~CE!and
branch points ~BP!to free volume and glass transition temperature Tg. The polyurethanes were
copolymers of diphenylmethane diisocyanate and poly ~propylene oxide !~PPO!with hydroxyl
functionalities of one, two, and three. The equivalent weights of all the PPOs were equal, such thatthe chemical composition of the chain segments was essentially identical. Therefore, the onlydistinctions among polymers were differences in CE and BP concentration. Theory of branchingprocesses computer simulations were used to determine the concentration of CE due to imperfectnetwork formation. Other CE contributions were from the monofunctional PPO. Polymer volumes
andT
gs were correlated to CE and BP concentrations, and the contributions of these species were
determined from least squares fits. The molar volume and Tgcontributions were then used to
determine free volume thermal expansion coefficients. These values were compared to thermal
expansion coefficients obtained from WLF parameters ( c1,c2) obtained from the measurement of
dynamic moduli as a function of temperature. © 2000 The Society of Rheology.
@S0148-6055 ~00!01204-9 #
I. INTRODUCTION
The glass transition temperature Tgand specific volume vof an amorphous polymer
network can be quantified as the summation of contributions from the chain segments,chain ends ~CEs!and branch points ~BPs!in the system. CEs increase mobility and
generate volume, BPs restrict mobility and reduce the volume in their vicinity. Tradition-ally, the polymer sets used to determine CE and BP contributions to free volume andglass transition temperature have been series of homopolymers of varying molecularweight. These polymers have CE concentrations inversely proportional to their molecularweights. If the polymer can be crosslinked without the introduction of additional species,the BP concentration can be varied without affecting the structure of the chain segments.A different approach for generating polymers with variable CE and BP concentration,while keeping chain segment properties identical, was used here. Polyurethane networkswere synthesized for which the chain segments are essentially identical and the only
a!Dedicated to Professor John D. Ferry.
b!Author to whom correspondence should be addressed; electronic mail: FedderlyJJ@nswccd.navy.mil
© 2000 by The Society of Rheology, Inc.
J. Rheol. 44 ~4!, July/August 2000 961 0148-6055/2000/44 ~4!/961/12/$20.00
Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 162.129.251.30 On: Thu, 03 Apr 2014
22:13:55differences are in network structure. The relevant network structure differences are the
CE and BP concentrations. These concentrations were calculated and their contributions
to volume and Tgwere determined.
A series of ten polyurethanes was produced from reacting diphenylmethane diisocy-
anate ~MDI!and poly ~propylene oxide !polyols with functionalities of one, two, or three.
Through the use of polyols of identical chemical composition and equal equivalentweight, the polymers were designed to have identical chain segment composition, butwidely varying network structure. The network structures exhibit significant differencesin properties such as the relative concentrations, molecular weights, and molecularweight distributions of elastically active network chains, dangling chains, and sol. How-
ever, the volumes and T
gs of amorphous copolymers such as these can be viewed as the
summation of group contributions regardless of where the groups are located within thesystem. This concept of additive group contributions to polymer properties has beendeveloped extensively by Van Krevelen ~1990!. The polymers presented here have es-
sentially the same chemical group composition and it is assumed that they have identical
T
gs except for contributions from CE and BP. The CE and BP can be viewed as addi-
tional groups in the additive properties approach. The BP concentration was determinedby the amount of trifunctional polyol in the system. The determination of the CE con-centration is more complicated as a result of the use of a monofunctional component andthe lack of complete reaction. Theory of branching processes simulations were run to aidin the prediction of the CE concentration. This work is a part of our ongoing efforts tocharacterize the properties of monofunctionally modified polymer systems @Fedderly
et al. ~1996!, Fedderly et al. ~1999!#.
Specific volumes of the polymers were measured and a least squares fit of these
volumes to a model incorporating CE and BP concentrations was made. From this, molar
volumes for CE and BP were determined. A similar treatment was performed for T
g.
Using free volume relationships, the free volume thermal expansion coefficient was de-
termined from the volume and Tgbehavior. Dynamic mechanical properties of the poly-
mer set were also measured at several temperatures using a resonance technique. Modulifrom the various temperatures were shifted in accordance to the time–temperature super-position principle. The shift factors, plotted versus temperature, were fit to the WLF
equation. The WLF parameters ( c
1,c2) were used to obtain an independent determina-
tion of the free volume thermal expansion coefficient as well as to determine a freevolume fraction.
II. EXPERIMENTAL PROCEDURES
A. Sample preparation
The polyurethanes synthesized for this study are divided into two sets. The first set is
comprised of typical polyurethane networks formed from a blend of difunctional ~2F!and
trifunctional ~3F!poly~propylene oxide !~PPO!polyols. The 2 Fand 3Fpolyols were
Poly-G 20-112 and Poly-G 30-112, respectively from Olin Chemical. Each of thesematerials has a nominal equivalent weight of 500 g/eq. The 2 Fand 3Fmaterials were
blended to have specific number average functionalities F
nranging from 2.1 to 3 ~nomi-
nal!. The second set of polyurethanes was formed from a blend of monofunctional ~1F!
and 3FPPOs. The 1 Fmaterial was a blend of UCON LB-65 ~nominally 400 g/eq !and
UCON LB-135 ~nominally 700 g/eq !from Union Carbide. The two UCON materials
were blended to achieve an equivalent weight of 500 g/eq. Polymers from this set hadpolyol functionalities ranging from 2.0 to 3.0. The polyol blends from both sets werereacted with a stoichiometric amount of MDI. The MDI used was Mondur ML from962 FEDDERLY ET AL.
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22:13:55Bayer Chemical. This material is a blend of 4,4 8-diphenylmethane diisocyanate and
2,48-diphenylmethane diisocyanate. Other data from these samples were presented in an
earlier publication and further details on the sample preparation can be found there@Fedderly et al. ~1999!#.
B. Glass transition temperature
Glass transition temperature measurements were made using a DuPont 910 differential
scanning calorimeter ~DSC!cell with a DuPont 9900 controller. Thin flat sections of
approximately 10 mg were cut from the resonance bar samples. A scanning rate of10°C/min was used. Measurements were made under a nitrogen atmosphere. Glass tran-sition temperatures were determined from the inflection points in the DSC thermograms.
C. Sol fraction
Sol fractions were measured from 10 mm 320 mm specimens cut from 2 mm thick
sheets. Samples were weighed dry in a dry room environment ~less than 0.25% relative
humidity !then immersed in sealed jars containing nominally 40 g of 2-methoxyethyl
ether as solvent. Afte r 2 d the samples were transferred to jars containing fresh solvent
and kept for an additional 6 d. The samples were then removed from the solvent andallowed to dry to constant weight. The sol fraction was determined from the difference inweight.
D. Specific volume
The specific volume ~or density !measurements were made on bar specimens follow-
ing the general procedures of ASTM D 792 ‘‘Density by Liquid Displacement’’ using
octane as the liquid. Measurements were made at 23°C and at the T
gof the polymer
being tested. The octane density was obtained at the various temperatures using a cali-brated Pyrex bob, accounting for the thermal expansion of the bob. The low temperaturemeasurements were obtained by cooling the sample and octane separately to just belowthe desired measurement temperature. The octane was in an insulated cup and the samplein a desiccator to prevent frost from forming on the surface. The sample was quicklyimmersed in the octane and placed in the balance. As the temperature reached the desiredvalue, the immersed sample mass was recorded.
E. Dynamic mechanical properties
The dynamic mechanical properties were measured using a resonance technique de-
veloped at this laboratory @Madigosky and Lee ~1983!#. This technique has been used in
a number of studies on polymer properties @Duffyet al. ~1990!, Hartmann and Lee
~1991!#. The apparatus is based on producing resonances in a bar specimen. Typical
specimen length is 10–15 cm with square lateral dimensions of 0.635 cm. Measurementsare made over 1 decade of frequency in the kHz region from 260 to 70°C at 5° intervals.
By applying the time–temperature superposition principle, the raw data are shifted togenerate a reduced frequency plot ~over as many as 20 decades of frequency !at a
constant reference temperature.
III. RESULTS AND DISCUSSION
The primary objective of this work was to determine the molar contributions of CE
and BP to the polymer glass transition temperature and specific volumes. To accomplishthis, it is necessary to have a set of polymers in which the variations in CE and BP963 NETWORK STRUCTURE DEPENDENCE
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22:13:55dominate the changes in these properties. This requires that the composition of the poly-
mers all be the same. This was achieved by using mono-, di-, and trifunctional PPOs ofequal equivalent weight. The use of equal equivalent weight polyols is a novel approachand is central to this work, so it is worth pointing out the significance of this choice. Forthe monofunctional component, the equivalent weight is 503 g/eq with a functionality of1.01, thus a molecular weight of 508 g/mol. For the difunctional component, the respec-tive values are 502 g/eq, 1.99, and 999 g/mol. For the trifunctional component, therespective values are 504 g/eq, 2.97, and 1496 g/mol. Each polyol equivalent reacts withone equivalent of the diisocyanate, 125 g/eq, giving rise to one urethane group. Thus, anystoichiometric blend of these three polyols with diisocyanate will have essentially thesame urethane concentration, the same propylene oxide concentration, the same aromaticconcentration, etc. Slight differences exist in the PPOs due to the use of different alcoholinitiators in their production. To verify that this and the slight equivalent weight differ-
ences have only a small effect on volume and T
g, the sample set was analyzed using an
additive group contribution approach similar to that described by Fedderly et al. ~1998!.
It was found that the polymer volumes are constant 60.001 cm3/g and that the Tgs are
constant 60.5°C. Given these small variations, it was felt that the contributions of CE
and BP could be determined.
The novel approach taken here can be contrasted with studies on some similar PPO
polymers from mono- and trifunctional polyols reacted with diisocyanate @Randrianan-
toandroet al. ~1997!#. There the triol used had a molecular weight of 720 g/mol or 240
g/eq while the monofunctional had a molecular weight of 136 g/mol or 136 g/eq, nearlya factor of two smaller than the trifunctional. Thus, the higher the fraction of triol in thosepolymers, the lower the urethane concentration. Also, the monofunctional componentused there was aromatic while the trifunctional component was an aliphatic poly ~propyl-
ene oxide !. In typical systems such as this, it is difficult to separate the contributions of
composition and network structure to the polymer T
gor other properties.
Some other complicating factors include the extent of reaction, which is taken into
account by aand the degree of cyclization, which has been shown by Dus ˇek~1989!to be
no more than 2%–3% for the trifunctional polymer.
The range of specific volumes and Tgs in the polymers presented here is quite small,
but as shown above, the variations should be predominantly due to differences in network
structure ~concentrations of CE and BP !. The volume and Tgdata are fit to simple models
that account for the concentration of these structural features. From the fits of the data tothe models, material constants are determined which predict the dependence of volume
andT
gon CE and BP concentration.
In other treatments, the CE and BP are specifically identified to consist of various
numbers of repeat units or atoms along the chain @Chompff ~1971!#. In this work, there
are no assumptions concerning the magnitude or the range of the CEs and BPs. Theirconcentrations simply have an overall effect on the system.
The polyurethane networks used in this study and their polyol compositions are listed
in Table I. The polymers are specified by polyol type and number average functionality
of the polyol blend. For example, a designation of 2 F13F, 2.20 indicates a polymer
made from a blend of difunctional and trifunctional PPOs having an average functionalityof 2.20.
A. Chain end and branch point concentrations
The BP concentration is determined strictly from stoichiometry. The BP concentration
is equivalent to the 3 Fpolyol concentration in the starting materials mixture ~except for
a small correction because the 3 Fmaterial does not have a functionality of exactly three !.964 FEDDERLY ET AL.
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22:13:55CE concentrations are more difficult to determine. For every monofunctional chain in the
starting formulation one CE remains in the final network. In addition to this, however,CEs remain from unreacted functional groups. Conversion of the functional groups is not100% and the resulting network is less than ideal. Dangling chains and sol are generated.Using the sol fraction, determined experimentally, the degree of conversion was calcu-lated as described below.
Computer simulations were performed on the polymer set, using the theory of branch-
ing processes. This theory is based on the generation of structures from component unitsin different reaction states @Dusˇek~1989!#. The formalism of using probability generating
functions has been used to describe and transform various distributions. For details seeDusˇek~1986!. The simulation predicts the critical gel point conversion and several prop-
erties including: molecular weight averages before the gel point, sol and gel fractions,and molecular averages of the sol as a function of conversion of functional groups. Thesimulation also offers information on the fraction of material in dangling chains and onconcentration and molecular weight averages of elastically active network chains.
Usually, all these parameters ~e.g.,X!are calculated as a function of conversion
a.
Beyond the gel point, all these parameters are also a function of the so-called extinction
probability v, which itself is a function of a
X5C~a,v~a!!. ~1!
If the determination of ais difficult experimentally, one can calculate afrom the weight
fraction of the sol ws, which is readily measured
a5F~ws,v~a!!. ~2!
The extinction probability v(a) is determined from Eq. ~3!
v5F~a,v!, ~3!
whereF(a,v) is obtained from the probability generating function for the number of
additional bonds of a unit already connected by one bond to another unit, F(a,z), by
substituting z5v@Dusˇek~1989!#.
The procedure is illustrated for the case of F3 PPO triol, component a, and diisocy-
anate, compound b!, where the OH and NCO groups, respectively, have the same reac-
tivity and the system is stoichiometric. The sol fraction is given byTABLE I. Polymer compositions, sol fractions, and degree of conversion.
Sample
FnN1F
~mole!N2F
~mole!N3F
~mole! Ws a
2F13F
2.12 0.000 0.867 0.133 0.070 0.9772.20 0.000 0.786 0.214 0.040 0.9712.40 0.000 0.582 0.418 0.017 0.9602.60 0.000 0.378 0.622 0.007 0.9582.97 0.000 0.000 1.000 0.003 0.942
1F13F
1.99 0.500 0.000 0.500 0.244 0.9392.12 0.434 0.000 0.566 0.122 0.9592.20 0.393 0.000 0.607 0.087 0.9622.40 0.291 0.000 0.709 0.041 0.9632.60 0.189 0.000 0.811 0.022 0.953965 NETWORK STRUCTURE DEPENDENCE
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22:13:55ws5wa~12a1avb!31wb~12a1ava!2, ~4!
wherewaandwbare weight fractions of components aandb, and ais the molar
conversion of isocyanate and hydroxy groups; vaandvbare extinction probabilities
determined by the equations
va5~12a1avb!2, ~5!
vb5~12a1ava!. ~6!
The extinction probabilities can be calculated explicitly by solving Eqs. ~5!and~6!and
eliminating the trivial roots va5vb21. The solution is as follows:
va5~12a2!2
a4, ~7!
vb5122a21a3
a3. ~8!
By substituting Eqs. ~7!and~8!into Eq. ~4!and solving the equation numerically with
respect to a, one gets the desired result.
The sol fraction and degree of conversion values are also shown in Table I. The
amount of unreacted OH and NCO groups are readily calculated from the degree ofconversion and the concentration of the starting materials. These values plus the concen-tration of monofunctional component are added to give the CE concentration. The CE
and BP concentrations
nEandnB, respectively, are shown in Table II.
B. Specific volume and glass transition temperature
The molar contributions of CE and BP to specific volume were determined using the
CE and BP concentrations and the measured volumes. It is assumed here that the specific
volume can be expressed as the sum of v0~the volume of an infinitely long linear
polymer !, positive chain end contributions, and negative branch point contributions, as
shown in Eq. ~9!
v5v01VEnE2VBnB, ~9!TABLE II. Chain end and branch point concentration, specific volume, and Tg.
SamplenE3104
~mol/g !nB3104
~mol/g !vg~meas!
~cm3/g!vg~pred!
~cm3/g!v23~meas!
~cm3/g)v23~pred!
~cm3/g!Tg~meas!
~°C!Tg~pred!
~°C!
2F13F
2.12 0.733 0.970 0.8969 0.8966 0.9311 0.9311 228 227.6
2.20 0.924 1.503 0.8961 0.8962 0.9305 0.9303 227 227.1
2.40 1.274 2.690 0.8953 0.8952 0.9282 0.9285 225 225.9
2.60 1.377 3.693 0.8937 0.8941 0.9267 0.9266 225 225.9
2.97 1.845 5.193 0.8929 0.8929 0.9242 0.9245 223 223.2
1F13F
1.99 5.897 3.876 0.9001 0.8998 0.9337 0.9342 229 229.7
2.12 4.529 4.120 0.8969 0.8977 0.9314 0.9313 228 227.8
2.20 4.022 4.257 0.8969 0.8969 0.9308 0.9301 228 227.0
2.40 3.086 4.558 0.8961 0.8953 0.9276 0.9279 226 225.5
2.60 2.639 4.812 0.8945 0.8944 0.9268 0.9266 224 224.6966 FEDDERLY ET AL.
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22:13:55whereVEandVBare the CE and BP molar volumes, respectively.
In the present case, Eq. ~9!leads to a set of ten simultaneous equations ~for the ten
polymers !in three unknowns ~v0,VE, andVB!which can be solved by the method of
least squares. To achieve an accurate determination of the three parameters, it is neces-sary that there be considerable variation in the CE and BP concentrations and that they bepresent in many unique ratios. This type of diversity is greatly enhanced through the use
of both conventional formulations using 2 F13Fpolyols and unique formulations using
1F13Fpolyols. This variability can be seen in Fig. 1, which shows specific volume
measured at 23°C (
v23) versus CE concentration and in Fig. 2, which shows the same
volumes versus BP concentration. In both figures, the intersection of the 1 F13Fand
2F13Flines is at the pure 3 Fsample point. Extending out from this point, the number
FIG. 1.Specific volume at 23°C vs chain end concentration.
FIG. 2.Specific volume at 23°C vs branch point concentration.967 NETWORK STRUCTURE DEPENDENCE
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22:13:55average functionality monotonically decreases. The uniqueness of the CE and BP ratios is
evident from the 1 F13Fand 2F13Fdata having dependencies of opposite slope for
CE concentration ~Fig. 1 !, and slopes of the same sign, but significantly different mag-
nitude for BP concentration ~Fig. 2 !.
Equation ~9!was evaluated using volumes measured at Tg(vg) and at 23°C ( v23).
AtTg, the values for v0,VE, andVBwere determined to be 0.8968
cm3/g, 13.0 cm3/mole, and 12.3 cm3/mole, respectively. At 23°C, the respective values
were found to be 0.9318 cm3/g, 17.4 cm3/mole, and 20.3 cm3/mole. Volumes for each
polymer were predicted from the model at both temperatures and are shown along withthe measured volumes in Table II.
In a manner similar to specific volume, T
gis assumed to be of the form shown in Eq.
~10!. It is the sum of Tg0, theTgof a linear polymer of infinite molecular weight, positive
BP contributions, and negative CE contributions.
Tg5Tg01~TBnB2TEnE!/vg, ~10!
whereTBandTEare molar Tgcontributions for BP and CE, respectively, and vgis the
specific volume at Tggiven in Table II. Again, this provides a set of ten simultaneous
equations in three unknowns ( Tg0,TB, andTE!. A least squares solution for Tg0was
determined to be 245 K, while TBandTEwere 1.21 3104and 1.05 3104Kcm3/mole,
respectively. Tgvalues were predicted for each polymer and these values along with the
measured values are given in Table II.
The parameters TEandTBare related to other material constants. Consider the simple
case of a polymer with no branch points nB50. Then the molecular weight is twice the
reciprocal of the concentration of chain ends, Mn52/nE, and Eq. ~10!reduces to the
well known result @Fox and Loshaek ~1955!#for the effect of chain ends on Tg.
Tg5Tg02K/Mn, ~11!
whereK52TE/vg. Likewise, the TBvalues can be compared to the KXparameter
which represents the contribution of branch points to Tg@Chompff ~1971!#, with the
result that KX52TB.
TheVEandVBparameters are also related to other material constants. Again consider
the case of no branch points. The change in volume from one polymer to another withdifferent molecular weight is assumed to result from a change in free volume. Thecontribution that chain ends make to free volume fraction in polymers of finite molecularweight has been given by Ninomiya et al. ~1963!in the form
f5f
01A/Mn ~12!
and it follows from Eq. ~9!thatA52VE/vg. Likewise, the VBvalues can be compared
to theAxparameter, which represents the contribution of a pair of branch points to
volume @Chompff ~1971!#, with the result that Ax52VB.
Values for the four parameters determined here are given in Table III along with the
equivalent AandKvalues. These values are typical of those found for other polymers
@Chompff ~1971!, Nielsen and Landel ~1990!#. It should be noted that the branch points
in these systems are trifunctional. Free volume relationships show that the Tgdependence
on branch points is proportional to the fractional free volume and to j22~jis the
functionality of the branch point !and inversely proportional to the free volume thermal968 FEDDERLY ET AL.
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22:13:55expansion coefficient af@Chompff ~1971!#. Therefore the TBorKxvalues shown here
would be half the value expected for a similar polymer with tetrafunctional branch points.
Chompff ~1971!has shown that a reasonable approximation of af~the free volume
thermal expansion coefficient !is given by the ratio of AXtoKX. Since these parameters
have been related to constants from Eqs. ~9!and~10!, we can write
af5AX/KX5VB/TB. ~13!
Using the values of the parameters at Tg,VB/TB510.231024°C21. Similarly, the
ratio of the total relative volume change to change in Tgwill also provide an estimate for
the expansion coefficient. This is obtained from the reciprocal of the slope of a plot of Tg
vsvg/vg0, shown in Fig. 3. A value of 10.3 31024°C21was obtained, in excellent
agreement with the value from Eq. ~13!. Note that this value is based on the ratio of two
experimentally determined values and does not depend on calculated CE or BP concen-trations.
C. Dynamic mechanical measurements
Free volume parameters can also be determined from dynamic moduli obtained as a
function of temperature. Dynamic shear moduli were measured from 260 to 70°C at 5°
intervals using the resonance apparatus described previously. Using the time–
temperature superposition principle, the G
8values were shifted in log frequency space to
obtain the shift factor log aT. The temperature at which G8versus frequency has theTABLE III. Volume and glass transition temperature parameters.
VE
~cm3/mol!VB
~cm3/mol!TE31024
(K cm3/mol)TB31024
(K cm3/mol)A
~g/mol !AX
(cm3/mol!K31024
~K g/mol)KX31024
(K cm3/mol)
Tg13.0 12.3 1.05 1.21 29.0 24.6 2.34 2.42
23°C 17.4 20.3 37.3 43.6
FIG. 3.Glass transition temperature vs relative specific volume at Tg.969 NETWORK STRUCTURE DEPENDENCE
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22:13:55steepest slope was chosen as the reference temperature T0. A plot of log aTvsTwas
produced for each polymer and least-squares fit to the WLF equation @Ferry ~1980!#
logaT52c10~T2T0!/~c201T2T0!. ~14!
Two constants c10andc20are obtained from the fit for the given reference temperature.
Thec1andc2parameters were redetermined using Tgas the reference temperature from
the following transformations:
c2g5c201Tg2T0, ~15!
c1g5c10c20/c2g. ~16!
Thec1gandc2gparameters are related to the free volume parameters afandfg~the
fractional free volume at Tg!, using the following equations @Ferry ~1980!#:
fg5B/2.303c1g, ~17!
af5B/2.303c1gc2g, ~18!
whereBis an empirical constant near unity. A typical plot of log aTvsTfit to the WLF
equation is shown in Fig. 4. The WLF and free volume parameters for the polymer set arelisted in Table IV.
Thef
gvalues appear to be fairly randomly distributed, therefore it was not possible to
correlate these value with vg. An average value of 0.043 was obtained. There was also
a fair amount of scatter in the afvalues, but the average value of 8.9 31024°C21
compares very closely with the value of 10.2 31024°C21obtained from Eq. ~13!. As-
suming a linear temperature dependence between Tgand 23°C, the rubbery thermal
expansion coefficients aLwere also determined and are also shown in Table IV. The
average value is 7.5 31024°C21. Although it would be expected that the afvalues be
FIG. 4.LogaTvsTplot for 1 F13F,Fn52.6, fit to the WLF equation.970 FEDDERLY ET AL.
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22:13:55somewhat less than those of aL, it is reasonable that the aLvalue compares very closely
with the Eq. ~13!and Eq. ~18!values.
IV. CONCLUSIONS
A novel method was used to determine volume and Tgcontributions from chain ends
and branch points in a series of polyurethane networks. The volumes and Tgso ft h e
polymers, which were identical in composition but varied in CE and BP concentration,were fit to simple models incorporating these elements. It was found that the opposing
contributions of CE and BP were nearly identical in magnitude for both volume and T
g.
AtTg, the volume contribution of a CE was found to be 13.0 cm3/mole. The volume that
is removed from the system by a BP was found to be 12.3 cm3/mole. At 23°C these
values were 17.4 and 20.3 cm3/mole, respectively. These values represent the additional
free volume associated with a chain end or lack of it by having a branch point and are not
identified with any specific atoms in the vicinity of these points. For Tg, a positive
contribution of 1.2°C for every 1024mole of BP per cm3was determined. The negative
contribution for CE was 1.1°C for every 1024mole/cm3.
The free volume thermal expansion coefficient afwas determined from the volume
andTgparameters and found to be approximately 10 31024°C21. Dynamic mechanical
moduli of the materials were also measured as a function of temperature. From WLF fits
of the shift factors, the c1andc2parameters were used to calculate free volume thermal
expansion coefficients and the free volume fractions at Tg. The average expansion co-
efficient was found to be about 9 31024°C21and compares closely with the value
obtained from the volume and Tgmeasurements. The free volume fraction fgwas also
determined from the WLF fits. A reasonable average free volume fraction of 0.043 wasdetermined.
ACKNOWLEDGMENTS
This work was supported by NATO Collaborative Research Grant No. CRG 970041,
the CDNSWC In-house Laboratory Independent Research Program sponsored by theOffice of Naval Research, and by Grant Agency of the Academy of Sciences of the CzechRepublic, Grant No. A4050808.TABLE IV. WLF parameters and thermal expansion coefficients
Sample c1gc2g
~°C! fg/Baf/B3104
~°C21)aL3104
~°C21)
2F13F
2.12 12.2 41.0 0.036 8.7 7.52.20 10.5 55.9 0.041 7.4 7.72.40 10.6 37.5 0.041 10.9 7.72.60 8.3 50.1 0.053 10.5 7.72.97 7.2 44.6 0.060 13.4 7.6
1F13F
1.99 11.6 47.9 0.037 7.8 7.12.12 11.0 62.4 0.040 6.3 7.62.20 11.0 62.5 0.040 6.3 7.42.40 10.4 44.8 0.041 9.3 7.22.60 11.0 48.3 0.040 8.2 7.7971 NETWORK STRUCTURE DEPENDENCE
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22:13:55References
Chompff, A. J., ‘‘Glass Points of Polymer Networks,’’ in Polymer Networks, Structure and Mechanical Prop-
erties, edited by A. J. Chompff and S. Newmann ~Plenum, New York, 1971 !, pp. 145–192.
Duffy, J. V., G. F. Lee, J. D. Lee, and B. Hartmann, ‘‘Dynamic mechanical properties of poly ~tetramethylene
ether!glycol polyurethanes,’’ in Sound and Vibration Damping with Polymers , ACS Symposium Series
424, edited by R. D. Corsaro and L. H. Sperling ~American Chemical Society, Washington, D.C., 1990 !, pp.
281–300.
Dusˇek, K., ‘‘Network formation in curing of epoxy resins,’’ Adv. Polym. Sci. 78, 1–59 ~1986!.
Dusˇek, K., ‘‘Formation and structure of networks from telechelic polymers: Theory and application to poly-
urethanes,’’ in Telechelic Polymers: Synthesis and Applications , edited by E. J. Goethals ~Chemical Rubber
Corp., Boca Raton, FL, 1989 !, pp. 289–315.
Fedderly, J., E. Compton, and B. Hartmann, ‘‘Additive group contributions to density and glass transition
temperature in polyurethanes,’’ Polym. Eng. Sci. 38, 2072–2076 ~1998!.
Fedderly, J. J., G. F. Lee, D. J. Ferragut, and B. Hartmann, ‘‘Effect of Monofunctional and Trifunctional
Modifiers on a Phase Mixed Polyurethane System,’’ Polym. Eng. Sci. 36, 1107–1113 ~1996!.
Fedderly, J. J., G. F. Lee, J. D. Lee, B. Hartmann, K. Dus ˇek, J. Sˇomvarsky, and M. Smrc ˇkova´, ‘‘Multifunctional
Polyurethane Network Structures,’’ Macromol. Symp. 148, 1–14 ~1999!.
Ferry, J. D., Viscoelastic Properties of Polymers , 3rd ed. ~Wiley, New York, 1980 !, pp. 264–320.
Fox, T. G. and S. Loshaek, ‘‘Influence of molecular weight and degree of crosslinking on the specific volume
and glass temperature of polymers,’’ J. Polym. Sci. 15, 371–390 ~1955!.
Hartmann, B. and G. F. Lee, ‘‘Dynamic mechanical relaxation in some polyurethanes,’’ J. Non-Cryst. Solids
131–133, 887–890 ~1991!.
Madigosky, W. M. and G. F. Lee, ‘‘Improved resonance technique for materials characterization,’’ J. Acoust.
Soc. Am. 73, 1374–1377 ~1983!.
Nielsen, L. E. and R. F. Landel, Mechanical Properties of Polymers and Composites , 2nd ed. ~Marcel Dekker,
New York, 1990 !, pp. 1–32.
Ninomiya, K., J. D. Ferry, and Y. O¯yanagi, ‘‘Viscoelastic properties of polyvinyl acetate II. Creep studies of
blends,’’ J. Phys. Chem. 67, 2297–2308 ~1963!.
Randrianantoandro, H., T. Nicolai, D. Durand, and F. Prochazka, ‘‘Viscoelastic relaxation of polyurethane at
different stages of gel formation. 1. Glass transition dynamics,’’ Macromolecules 30, 5893–5896 ~1997!.
Van Krevelen, D. W., Properties of Polymers , 3rd ed. ~Elsevier, New York, 1990 !.972 FEDDERLY ET AL.
Redistribution subject to SOR license or copyright; see http://scitation.aip.org/content/sor/journal/jor2/info/about. Downloaded to IP: 162.129.251.30 On: Thu, 03 Apr 2014
22:13:55 |
1.4995240.pdf | Low spin wave damping in the insulating chiral magnet Cu 2OSeO3
I. Stasinopoulos , S. Weichselbaumer , A. Bauer , J. Waizner , H. Berger , S. Maendl , M. Garst , C. Pfleiderer , and D.
Grundler
Citation: Appl. Phys. Lett. 111, 032408 (2017); doi: 10.1063/1.4995240
View online: http://dx.doi.org/10.1063/1.4995240
View Table of Contents: http://aip.scitation.org/toc/apl/111/3
Published by the American Institute of Physics
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Applied Physics Letters 111, 022407 (2017); 10.1063/1.4993604Low spin wave damping in the insulating chiral magnet Cu 2OSeO 3
I.Stasinopoulos,1S.Weichselbaumer,1A.Bauer,2J.Waizner,3H.Berger,4S.Maendl,1
M.Garst,3,5C.Pfleiderer,2and D. Grundler6,a)
1Physik Department E10, Technische Universit €at M €unchen, D-85748 Garching, Germany
2Physik Department E51, Technische Universit €at M €unchen, D-85748 Garching, Germany
3Institute for Theoretical Physics, Universit €at zu K €oln, D-50937 K €oln, Germany
4Institut de Physique de la Matie `re Complexe, /C19Ecole Polytechnique F /C19ed/C19erale de Lausanne, 1015 Lausanne,
Switzerland
5Institut f €ur Theoretische Physik, Technische Universit €at Dresden, D-01062 Dresden, Germany
6Institute of Materials (IMX) and Laboratory of Nanoscale Magnetic Materials and Magnonics (LMGN),
/C19Ecole Polytechnique F /C19ed/C19erale de Lausanne (EPFL), Station 17, 1015 Lausanne, Switzerland
(Received 5 May 2017; accepted 8 July 2017; published online 21 July 2017)
Chiral magnets with topologically nontrivial spin order such as Skyrmions have generated enormous
interest in both fundamental and applied sciences. We report broadband microwave spectroscopy
performed on the insulating chiral ferrimagnet Cu 2OSeO 3. For the damping of magnetization
dynamics, we find a remarkably small Gilbert damping parameter of about 1 /C210/C04at 5 K. This
value is only a factor of 4 larger than the one reported for the best insulating ferrimagnet yttrium iron
garnet at room temperature. We detect a series of sharp resonances and attribute them to confined
spin waves in the mm-sized samples. Considering the small damping, insulating chiral magnets turnout to be promising candidates when exploring non-collinear spin structures for high frequency appli-
cations. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4995240 ]
The development of future devices for microwave appli-
cations, spintronics, and magnonics
1–3requires materials with
a low spin wave (magnon) damping. Insulating compounds
are advantageous over metals for high-frequency applications
as they avoid damping via spin wave scattering at free chargecarriers and eddy currents.
4,5Indeed, the ferrimagnetic insula-
tor yttrium iron garnet (YIG) holds the benchmark with a
Gilbert damping parameter aintr¼3/C210/C05at room tempera-
ture.6,7During the last few years, chiral magnets have
attracted a lot of attention in fundamental research and stimu-
lated new concepts for information technology.8,9This mate-
rial class hosts non-collinear spin structures such as spin
helices and Skyrmions below the critical temperature Tcand
critical field Hc2.10–12Dzyaloshinskii-Moriya interaction
(DMI) is present that induces both the Skyrmion lattice phase
and nonreciprocal microwave characteristics.13Low damping
magnets offering DMI would generate new prospects by par-ticularly combining complex spin order with long-distance
magnon transport in high-frequency applications and mag-
nonics.
14,15At low temperatures, they would further enrich
the physics in magnon-photon cavities that call for materials
with small aintrto achieve high-cooperative magnon-to-pho-
ton coupling in the quantum limit.16–19
In this work, we investigate the Gilbert damping in
Cu2OSeO 3, a prototypical insulator hosting Skyrmions.20–23
This material is a local-moment ferrimagnet with Tc¼58 K
and magnetoelectric coupling24that gives rise to dichroism
for microwaves.25–27The magnetization dynamics in
Cu2OSeO 3has already been explored.13,28,29A detailed
investigation on the damping which is a key quality for mag-
nonics and spintronics has not yet been presented however.
To evaluate aintr, we explore the field polarized state (FP)where the two spin sublattices attain the ferrimagnetic
arrangement.21Using spectra obtained by two different copla-
nar waveguides (CPWs), we extract a minimum aintr¼(9.9
64.1)/C210–5at 5 K, i.e., only about four times higher than in
YIG at room temperature. We resolve numerous sharp reso-
nances in our spectra and attribute them to modes that are
confined modes across the macroscopic sample and allowed
for by the low damping. Our findings substantiate the rele-vance of insulating chiral magnets for future applications in
magnonics and spintronics.
From single crystals of Cu
2OSeO 3, we prepared two
bar-shaped samples exhibiting different crystallographic ori-
entations. The samples had lateral dimensions of 2 :3/C20:4
/C20:3m m3. They were positioned on CPWs that provided us
with a radiofrequency (rf) magnetic field hinduced by a
sinusoidal current applied to the signal (S) line surrounded
by two ground (G) lines (Fig. 1andsupplementary material ,
Table SI). We used two different CPWs with either a broad30
or narrow signal line width of ws¼1m m o r 2 0 lm, respec-
tively. The central long axis of the rectangular Cu 2OSeO 3
rods was positioned on the central axis of the CPWs. The
static magnetic field Hwas applied perpendicular to the sub-
strate with Hkh100iandHkh111ifor samples S1 and S2,
FIG. 1. Sketch of a single crystal mounted on either a broad or narrow CPW
with a signal (S) line width wsof either 1 mm or 20 lm, respectively (not to
scale). The rf field his indicated. The static field His applied perpendicular
to the CPW plane.a)Electronic mail: dirk.grundler@epfl.ch
0003-6951/2017/111(3)/032408/5/$30.00 Published by AIP Publishing. 111, 032408-1APPLIED PHYSICS LETTERS 111, 032408 (2017)
respectively. The direction of Hdefined the z-direction fol-
lowing the definition of Ref. 4. The rf field component h?H
provided the relevant torque for excitation. Components hk
Hdid not induce precessional motion in the FP state of
Cu2OSeO 3. We recorded spectra by a vector network ana-
lyzer using the magnitude of the scattering parameter S12.
We subtracted a background spectrum recorded at 1 T toenhance the signal-to-noise ratio (SNR) yielding the dis-played DjS
12j. In Ref. 7, Klingler et al. have investigated the
damping of the insulating ferrimagnet YIG and found that
Gilbert parameters aintrevaluated from both the uniform pre-
cessional mode and standing spin waves confined in the mac-roscopic sample provided the same values. We evaluateddamping parameters as follows (and further outlined in thesupplementary material ).
31When performing frequency-
swept measurements at different fields H, the obtained line-
width Dfwas considered to scale linearly with the resonance
frequency fras32
Df¼2aintr/C2frþDf0; (1)
with the inhomogeneous broadening Df0.I nF i g s . 2(a)–2(d) ,
we show spectra recorded at 5 K in the FP state of the materialusing the two different CPWs. For the same applied field H,
we observe peaks residing at higher frequency fforHkh100i
compared to Hkh111i. From the resonance frequencies,
we extract the cubic magnetocr ystalline anisotropy constant
K¼ð /C0 0:660:1Þ/C210
3J/m3for Cu 2OSeO 3[compare supple-
mentary material , Fig. S1 and Eqs. (S1)–(S3)]. The magnetic
anisotropy energy is found to be extremal for h100iandh111i
reflecting easy and hard axes, respectively. The saturation mag-netization of Cu
2OSeO 3amounted to l0Ms¼0:13 T at 5 K.22
Figure 2summarizes spectra taken with two different
CPWs on the two different Cu 2OSeO 3crystals S1 and S2,
exhibiting different crystallographic orientations in the fieldH(further spectra are depicted in supplementary material ,
Fig. S2). For the broad CPW [Figs. 2(a) and2(c)], we mea-
sured pronounced peaks whose linewidths were small. Weresolved small resonances below the large peaks [arrows in
Fig.2(b)] that shifted with Hand exhibited an almost field-
independent frequency offset dffrom the main peaks that we
will discuss later. For the narrow CPW [Figs. 2(b) and2(d)],
we observed a broad peak superimposed by a series of reso-nances that shifted to higher frequencies with increasing H.
The field dependence excluded them from being noise orartifacts of the setup. Their number and relative intensities
varied from sample to sample and also upon remounting the
same sample in the cryostat (not shown). They disappearedwith increasing temperature Tbut the broad peak remained.
It is instructive to first follow the orthodox approach
29and
analyze damping parameters from modes reflecting the exci-
tation characteristics of the broad CPW. Second, we followRef. 7and analyze confined modes.
Lorentz curves (blue) were fitted to the spectra recorded
with the broad CPW to determine resonance frequencies andlinewidths. Note that the corresponding linewidths were
larger by a factor offfiffi ffi
3p
compared to the linewidth Dfthat is
conventionally extracted from the imaginary part of the scat-tering parameters.
33The extracted linewidths Dfwere found
to follow linear fits based on Eq. (1)at different temperatures
[supplementary material , Figs. S2 and S3(a)].
In Fig. 3(a), we depict the parameter aintrobtained from
the broad CPW.34ForHkh100i[Fig. 3(a)], between 5 and
20 K, the lowest value for aintramounts to (3.7 60.4)/C210–3.
This value is three times lower compared to preliminary datapresented in Ref. 29. Beyond 20 K, the damping is found to
increase. For Hkh111i, we extract (0.6 60.6)/C210
–3as the
smallest value. Note that these values for aintrstill contain an
extrinsic contribution due to the inhomogeneity of hin the
z-direction and thus represent upper bounds for Cu 2OSeO 3.
For the inhomogeneous broadening Df0in Fig. 3(b), the data-
sets taken with Happlied along different crystal directions
are consistent and show the smallest Df0at lowest tempera-
ture. Note that a CPW wider than the sample is assumed toexcite homogeneously the ferromagnetic resonance (FMR)atf
FMR35transferring an in-plane wave vector k¼0 to the
sample. Accordingly, we ascribe the intense resonances of
Figs. 2(a) and2(c) tofFMR. Using fFMR¼6 GHz and aintr
¼3:7/C210/C03at 5 K [Fig. 3(a)], we estimate a minimum
relaxation time of s¼½2paintrfr/C138/C01¼6:6 ns.
In the following, we examine in detail the additional
sharp resonances that we observed in spectra of Fig. 2.I n
Fig. 2(a) taken with the broad CPW for Hkh100i,w e
FIG. 2. Spectra DjS12j(magnitude) obtained at T ¼5 K for different Hval-
ues using (a) broad and (b) narrow CPWs when Hjjh100ion sample S1.
Corresponding spectra taken on sample S2 for Hjjh111iare shown in (c) and
(d), respectively. Note the strong and sharp resonances in (a) and (c) when
using the broad CPW that provides a much more homogeneous excitation
field h. Arrows mark resonances that have a field-independent offset with
the corresponding main peaks and are attributed to standing spin waves. An
exemplary Lorentz fit curve is shown in blue color in (a).FIG. 3. (a) Damping parameters aintrand (b) inhomogeneous broadening Df0
forHparallel to h100i(circle) and h111i(square). aintrandDf0are obtained
from the slopes and intercepts at fr¼0, respectively, of linear fits to the
linewidth data (compare supplementary material , Figs. S2 and S3).032408-2 Stasinopoulos et al. Appl. Phys. Lett. 111, 032408 (2017)identify sharp resonances that exhibit a characteristic fre-
quency offset dfwith the main resonance at all fields (black
arrows). We illustrate this in Fig. 4(a)in that we shift spectra
of Fig. 2(a) so that the positions of their main resonances
overlap. The additional small resonances (arrows) in Fig.2(a)are well below the uniform mode. This is characteristic
for backward volume magnetostatic spin waves (BVMSWs).
Standing waves of such kind can develop if they are reflected
at least once at the bottom and top surfaces of the sample.The resulting standing waves exhibit a wave vector k¼np=d,
with order number nand sample thickness d¼0.3 mm. The
BVMSW dispersion relation f(k)o fR e f . 13(compare also
supplementary material , Fig. S4) provides a group velocity
v
g¼/C0300 km/s at k¼p=d[triangles in Fig. 4(b)]. The decay
length ld¼vgsamounts to 2 mm considering s¼6:6 ns. This
is about seven times larger than the relevant thickness d,
thereby allowing standing spin wave modes to form across thethickness of the sample. Based on the dispersion relation of
Ref. 13, we calculated the frequency splitting df¼
f
FMR/C0fðnp=dÞ[open diamonds in Fig. 4(inset)] assuming
n¼1a n d t¼0.4 mm for the sample width tdefined in Ref.
13. Experimental values (filled symbols) agree with the calcu-
lated ones (open symbols) within about 60 MHz. In the caseof the narrow CPW, which provides a broad wave vector dis-tribution,
36we observe even more sharp resonances [Figs.
2(b)and2(d)]. A set of resonances was reported previously in
the field-polarized phase of Cu 2OSeO 3.26,28,37,38Maisuradze
et al. assigned secondary peaks in thin plates of Cu 2OSeO 3to
different standing spin-wave modes38in agreement with our
analysis outlined above.
We attribute the series of sharp resonances in Figs. 2(b)
and2(d) to further standing spin waves. In Figs. 5(a) and
5(b), we highlight prominent and particularly narrow reso-
nances with #1, #2, and #3 recorded with the narrow CPWforHkh100iandHkh111i. We trace their frequencies f
ras
a function of H. They depend linearly on Hshowing that for
both crystal orientations, the selected sharp peaks reflect dis-tinct spin excitations. From the slopes, we extract a Land /C19e
factor g¼2.14 at 5 K. Consistently, this value is slightly larger
than g¼2.07 reported for 30 K in Ref. 13.F r o m g¼2.14, we
calculate a gyromagnetic ratio c¼gl
B=/C22h¼1:88/C21011rad/
sT, where lBis the Bohr magneton of the electron. Note thatFIG. 4. Spectra of Fig. 2(a) replotted as f/C0fFMRðHÞfor different Hvalues
such that all main peaks are at zero frequency and the field-independent fre-
quency splitting dfbecomes visible. The numerous oscillations seen particu-
larly on the bottom curve are artefacts from the calibration routine. The inset
depicts experimentally evaluated (filled circles) and theoretically predicted
(diamonds) values dfusing dispersion relations for a platelet. Triangles indi-
cate calculated group velocities vgatk¼p=ð0:3m m Þ. Dashed lines are
guides to the eyes.FIG. 5. Resonance frequencies as a function of field Hof selected sharp
modes labelled #1 to #3 extracted from individual spectra (insets) for (a)
Hkh100iand (b) Hkh111iat T¼5 K. (c) Lorentz fit of a sharp mode #1
forHkh100iat 0.85 T. (d) Extracted linewidth Df as a function of reso-
nance frequency fralong with the linear fit performed to determine the
intrinsic damping a0
intrfrom confined modes. Inset: Effective damping a0
effas
a function of resonance frequency fr. The red dotted lines mark the error
margins of a0
intr¼ð9:964:1Þ/C210/C05.032408-3 Stasinopoulos et al. Appl. Phys. Lett. 111, 032408 (2017)the different metallic CPWs of Fig. 1vary the boundary condi-
tions and thereby details of the spin wave dispersion relations
in Cu 2OSeO 3. However, the frequencies covered by dispersion
relations vary only over a specific regime; for, e.g., forwardvolume waves, the regime even stays the same for different
boundary conditions.
4Following Klingler et al. ,7the exact
mode nature and resonance frequency were not decisive whenextracting the Gilbert parameter.
We now concentrate on mode #1 in Fig. 5(a) forHk
h100iat 5 K that is best resolved. We fit a Lorentzian line-
shape as shown in Fig. 5(c) for 0.85 T and summarize the
corresponding linewidths Dfin Fig. 5(d). The inset of Fig.
5(d) shows the effective damping a
eff¼Df=ð2frÞevaluated
directly from the linewidth as suggested in Ref. 29. We find
thataeffapproaches a value of about 3.5 /C210/C04with increas-
ing frequency. This value is a factor of 10 smaller compared
toaintrin Fig. 3(a) extracted from FMR peaks by means of
Eq.(1). This finding is interesting as aeffmight still be
enlarged by inhomogeneous broadening. To determine the
intrinsic Gilbert-type damping from standing spin waves, we
apply a linear fit to the linewidths Dfin Fig. 5(d)atfr>10:6
GHz and obtain (9.9 64.1)/C210–5. For fr/C2010.6 GHz, the
resonance amplitudes of mode #1 were small reducing the
confidence of the fitting procedure. Furthermore, at low fre-quencies, we expect anisotropy to modify the extracted
damping, similar to the results in Ref. 39. For these reasons,
the two points at low f
rwere left out for the linear fit provid-
inga0
intr¼ð9:964.1)/C210–5.
We find Dfand the damping parameters of Fig. 3to
increase with T. It does not scale linearly for Hkh100i.A
deviation from linear scaling was reported for YIG single
crystals as well and accounted for by the confluence of a
low-kmagnon with a phonon or thermally excited magnon.5
We now comment on our spectra taken with the broad CPW
that do not show the very small linewidth attributed to the
confined spin waves. The sharp mode #1 yields Df¼15:3
MHz at fr¼16:6 GHz [Fig. 5(d)]. At 5 K, the dominant peak
measured at 0.55 T and fr¼15:9 GHz with the broad CPW
provides however Df¼129 MHz. Dfobtained by the broad
CPW is thus increased by a factor of eight. This increase is
attributed to the finite distribution of wave vectors provided
by the CPW. We confirmed this larger value on a third sam-ple with Hkh100iand obtained (3.1 60.3)/C210
–3using the
broad CPW ( supplementary material , Fig. S2). The discrep-
ancy with the damping parameter extracted from the sharpmodes of Fig. 5might be due to the remaining inhomogene-
ity of hover the thickness of the sample, leading to an uncer-
tainty in the wave vector in the z-direction. For a standing
spin wave, such an inhomogeneity does not play a role as the
boundary conditions discretize k. Accordingly, Klingler et al.
extracted the smallest damping parameter of 2 :7ð5Þ/C210
/C05
reported so far for the ferrimagnet YIG at room temperature
when analyzing confined magnetostatic modes.7The finding
of Klingler et al. is consistent with the discussion in Ref. 33.
From Ref. 33, one can extract that the evaluation of damping
from finite-wave-vector spin waves provides a damping
parameter that is either equal or somewhat larger than theparameter extracted from the uniform mode ( supplementary
material ). The evaluation of Fig. 5(d) thus overestimates the
parameter.To summarize, we investigated the spin dynamics in the
field-polarized phase of the insulating chiral magnet
Cu
2OSeO 3. We detected numerous sharp resonances that we
attribute to standing spin waves. Their effective damping
parameter is small and amounts to 3 :5/C210/C04. A quantitative
estimate of the intrinsic Gilbert damping parameter extracted
from the confined modes provides (9.9 64.1)/C210–5at 5 K.
The small damping makes an insulating ferrimagnet exhibit-ing the Dzyaloshinskii-Moriya interaction a promising candi-
date for exploitation of complex spin structures and related
nonreciprocity in magnonics and spintronics.
Seesupplementary material for further spectra, the mag-
netic anisotropy constant, and linewidth evaluation.
We thank S. Mayr for assistance with sample preparation.
Financial support through Deutsche Forschungsgemeinschaft
(DFG) TRR80 (projects E1 and F7), DFG FOR960, and ERC
Advanced Grant No. 291079 (TOPFIT) is gratefully
acknowledged.
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1.3677838.pdf | The concept and fabrication of exchange switchable trilayer of
FePt/FeRh/FeCo with reduced switching field
T. J. Zhou, K. Cher, J. F. Hu, Z. M. Yuan, and B. Liu
Citation: J. Appl. Phys. 111, 07C116 (2012); doi: 10.1063/1.3677838
View online: http://dx.doi.org/10.1063/1.3677838
View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v111/i7
Published by the American Institute of Physics.
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Downloaded 23 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsThe concept and fabrication of exchange switchable trilayer
of FePt/FeRh/FeCo with reduced switching field
T. J. Zhou,a)K. Cher, J. F . Hu, Z. M. Yuan, and B. Liu
Data Storage Institute, A*STAR (Agency for Science Technology and Research), 5 Engineering Drive 1,
Singapore 117608
(Presented 31 October 2011; received 14 October 2011; accepted 21 November 2011; published
online 8 March 2012)
We report the concept and fabrication of exchange switchable trilayer of FePt/FeRh/FeCo with
reduced switching field for heat assisted magnetic recording (HAMR). A thin layer of FeRh is
sandwiched between L10FePt and magnetically soft FeCo. At room temperature, FePt and FeCo
are magnetically isolated by the antiferromagnetic FeRh layer. After the metamagnetic transitionof FeRh layer by heating, FePt and FeCo are exchange-coupled together through ferromagnetic
FeRh layer. Therefore, the switching field of FePt can be greatly reduced via exchange-spring
effect. Simulation work was carried out to understand the exchange coupling strength and the FeCothickness effects on the switching field reduction. It is found that switching field decreases with the
increase of exchange coupling strength and FeCo thickness. The trilayer films were also
successfully fabricated. A clear change of reversal mechanism from two-step to one-step switchingupon heating was observed and a 3-time switching field reduction was demonstrated. The results
show the promise of the trilayer for HAMR applications.
VC2012 American Institute of Physics .
[doi:10.1063/1.3677838 ]
I. INTRODUCTION
Heat-assisted magnetic recording (HAMR) is believed
to have the potential to achieve multiple Tbit/in2recording
density.1The conventional HAMR technology requires to
write information at temperature close to or slight higherthan the Curie temperature,
2Tc, which is about 750 K for
FePt. Such high writing temperature puts stringent require-
ments on the overcoat and lubricants. Thiele et al. proposed
the use of FePt/FeRh bilayer as the composite HAMR media
for heat-assisted recording with reduced writing temperature
(500 K or lower).3FeRh is antiferromagnetic at room tem-
perature and it undergoes a metamagnetic transition to ferro-
magnetic state at elevated temperatures (350–400 K).4
Therefore, this FePt/FeRh structure can provide thermal sta-
bility at room temperatures while the coupling between FePt
and FeRh reduces switching field after the metamagnetic
transition by slightly heating. Zhu et al. also proposed a bi-
nary anisotropy media consisting a trilayer of a magnetic re-
cording layer with perpendicular anisotropy, a magnetic
assist layer with negative anisotropy, and a phase transitionlayer between the recording and assist layers to reduce writing
temperature.
5With such structure, simulation results showed
the switching field can be reduced to a few percentage of theanisotropy field of the recording layer.
In this work, we further develop the concept and pro-
pose the exchange switchable trilayer of FePt/FeRh/FeCowith a purpose of reducing both writing temperature and
switching field. In the trilayer, FeRh forms a very thin layer
between FePt and FeCo and works as an exchange switchinglayer to turn on/off the coupling between FePt and FeCoupon heating/cooling. As shown in Fig. 1, at room tempera-
ture, FePt and FeCo are magnetically isolated by the antifer-
romagnetic FeRh layer. After the metamagnetic transition
of FeRh by heating, FePt and FeCo are exchange-coupledtogether through ferromagnetic FeRh layer, and therefore
the switching field of FePt can be greatly reduced due to
exchange spring effects.
6,7Here the FeCo provides a higher
magnetic moment that can further reduce the switching field
compared to the FePt/FeRh bi-layer. FeCo layer also func-
tions as a soft magnetic underlayer to enhance writing. Simu-lation work was carried out to study the exchange coupling
strength and FeCo thickness effects on the switching field
reduction. The switching field decreases with both exchangecoupling strength and FeCo thickness. The trilayer films
were fabricated. About 3-time switching field reduction was
experimentally demonstrated. The results show the promiseof the trilayer for HAMR applications.
II. SIMULATION MODEL AND RESULTS
To understand the exchange coupling strength and soft
layer thickness effect on the reduction of switching field, the
FIG. 1. (Color online) Schematic representing the exchange switchable tri-
layer of FePt/FeRh/FeCo before and after metamagnetic transition of FeRh.a)Author to whom correspondence should be addressed. Electronic mail:zhou_tiejun@dsi.a-star.edu.sg.
0021-8979/2012/111(7)/07C116/3/$30.00
VC2012 American Institute of Physics 111, 07C116-1JOURNAL OF APPLIED PHYSICS 111, 07C116 (2012)
Downloaded 23 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsmagnetic switching of the proposed FePt/FeRh/FeCo trilayer
was simulated by micro-magnetic modeling8where gyro-
magnetic motion of magnetization is governed by the Lan-dau-Lifshitz-Gilbert equation given by
d^m
ds¼^m/C2~heff/C0a^m/C2ð^m/C2~heffÞ: (1)
^mis the magnetization unit vector and ais the damping
constant. ~heffis the effective field, which includes the anisot-
ropy field, exchange field, external field, thermal field, and
demagnetization field. The following parameters are used:
Anisotropy constant KFePt¼6/C2107erg/cc, KFeRh¼7/C2104
erg/cc, KFeCo¼9.5/C2104erg/cc, saturation magnetization
MsFePt¼1140 emu/cc, M sFeRh¼1400 emu/cc, M sFeCo¼1900
emu/cc, and interlayer exchange coupling constant C* is 0.4for the hysteresis loop calculation.
The simulated hysteresis loops for the trilayer at 300 K
and 473 K are shown in Fig. 2. At 300 K, a clear two-step
switching is obtained. One is around zero fields correspond-
ing to the switching of the soft layer. The other is at applied
field, H
a, of 0.75 H k, which corresponds to the switching of
hard layer. The switching field of the trilayer is defined as
that of the hard layer. At 473 K, the switching of the soft
layer is shifted to higher field and that of the hard layer ismoved to lower field—a one-step switching is observed with
a much reduced switching field of /C240.25 H
k. At 300 K, FeRh
is antiferromagnetic and the FePt and FeCo layers are mag-netically isolated. Therefore the trilayer exhibits two distinct
switching states. FeRh is ferromagnetic at elevated tempera-
tures of 473 K and the trilayer forms an exchange spring. Inexchange-spring media, magnetic reversal starts in the soft
layer by forming a Neel-type domain wall when an external
field is applied. This wall propagates toward and penetratesinto the hard layer, assisting in the switching, which facili-
tates a single state reversal at much lower switching fields.
Figure 3reveals how the switching fields change with
exchange coupling strength, C*. The initial increment of C*
from 0.25 to 0.5 reduced the switching field by a factor of 2,
which is close to one fourth the anisotropy of FePt. Subse-quent increase of exchange coupling only yields smaller
changes to the switching field. In the exchange-spring media,
the spins in the soft layer act on the magnetization of thehard layer like a (exchange) spring. The spring strength isproportional to both the exchange coupling strength and sat-
uration magnetization of the soft layer. Therefore, certainexchange coupling is needed to have high enough spring
strength in order to minimize the switching of the hard layer.
Figure 4plots the switching field as a function of FeCo
thickness at fixed exchange coupling strength of C* ¼0.5.
The switching field can be reduced to one fifth the anisotropy
field of FePt at FeCo thickness of 15 nm or thicker. Suchreduction makes it possible to use the conventional perpen-
dicular head to write information into the recording medium.
The reduction of switching field is mainly due to the springeffect plus the demagnetization effect from the bottom FeCo
layer. The demagnetization energy is proportional to the
magnetic volume of the FeCo layer, which is a function ofFeCo thickness and saturated at certain thickness. This can
explain why the switching fields decrease with FeCo thick-
ness and saturated at certain thickness.
A theoretical analysis of the magnetization reversal pro-
cess in a structure of FePt/FeRh bi-layer was conducted by
Guslienko et al. to understand the underlying physics.
9It
was concluded that the switching field was related to the
interlayer exchange-coupling strength and the saturation
magnetization of FeRh at ferromagnetic state. For the pro-posed trilayers, it is plausible to treat the bottom two layers
of FeRh and FeCo as one magnetically soft layer after
FIG. 2. (Color online) Simulated hysteresis loops before and after metamag-
netic transition of FeRh.
FIG. 3. (Color online) Switching fields vs the exchange-coupling strength.
FIG. 4. (Color online) Switching fields as a function of FeCo layerthickness.07C116-2 Zhou et al. J. Appl. Phys. 111, 07C116 (2012)
Downloaded 23 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsmetamagnetic transition of FeRh. Then, for strong interlayer
exchange coupling, we have
Hswðstrong coupling Þ
/C25Ku;FePt/C2tFePt
Ms;FePt/C2tFePtþ/C22Ms;ðFeRhþFeCoÞ/C2tFeRhþFeCo(2)
and for weak interlayer exchange coupling, the following
applies:
Hswðweak coupling Þ¼HK;FePt/C0J
tFePt/C22MFeRhþFeCo
/C21þJ
tMs;FePt
ðHK;FePt/C04p/C22Ms;FeRhþFeCo/C20/C21
;
(3)
where Jis the interlayer exchange coupling and /C22MFeRhþFeCo
is the average saturation magnetization of the bottom two
layers of FeRh and FeCo. It is clearly shown that the switch-
ing field decreases with both the exchange coupling and theFeCo thickness as observed based on simulation. Also due to
higher saturation magnetization of FeCo, the trilayer has
higher potential for the reduction of switching field com-pared with the FePt/FeRh bilayers.
III. FABRICATION OF THE TRILAYERS AND CONCEPT
DEMONSTRATION
The trilayer was fabricated. Firstly, (002) oriented FeCo
was deposited onto MgO substrate at 300/C14C. Then (001) ori-
ented FeRh was grown on FeCo at 400–500/C14C. Last, (001)
oriented FePt was deposited onto FeRh layer at 400–500/C14C.
Due to high temperature process, 0.5 nm Ta layer was
inserted between FePt and FeRh and between FeRh and FeCoto prevent the interlayer diffusion. XRD (Fig. 5) showed
good (001) orientated FePt layer on top of (001) orientated
FeRh and (001) orientated FeRh on top of (002) orientatedFeCo layers. Temperature-dependent dc demagnetization(DCD) curves of the trilayers were measured at different tem-
perature to study temperature-dependent magnetizationswitching behavior. The measured results are shown in Fig. 6.
At low temperature (250 K and 300 K), a clear two-step
switching was observed. When temperature was increased to350 K and above, a single-step switching was shown. The
switching field as reduced from 4500 Oe to about 1500 Oe,
which is about a 3-time reduction, after the metamagnetictransition of FeRh.
IV. SUMMARY
We proposed and experimentally demonstrated that the
switching field can be effectively reduced without the loss of
thermal stability in exchange switchable trilayer of FePt/FeRh/FeCo. The writing temperature of the trilayer can also
be reduced to the metamagnetic transition temperature of
FeRh, which is about 400 K. The trilayer has higher heatingefficiency because only a very thin FeRh layer is needed to
be heated above the metamagnetic transition temperature.
Although the results presented show the promise for the tri-layer structure for HAMR applications, much work needs to
be done for the improvement of magnetic properties and
microstructure in order for it to be used as practical HAMRmedia.
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FIG. 5. (Color online) XRD pattern of the exchange switchable trilayer of
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Downloaded 23 Jun 2012 to 139.184.30.132. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions |
1.5042417.pdf | An analog magnon adder for all-magnonic neurons
T. Brächer , and P. Pirro
Citation: Journal of Applied Physics 124, 152119 (2018); doi: 10.1063/1.5042417
View online: https://doi.org/10.1063/1.5042417
View Table of Contents: http://aip.scitation.org/toc/jap/124/15
Published by the American Institute of PhysicsAn analog magnon adder for all-magnonic neurons
T. Brächer and P. Pirro
Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität Kaiserslautern, 67663
Kaiserslautern, Germany
(Received 31 May 2018; accepted 2 August 2018; published online 2 October 2018)
Spin-waves are excellent data carriers with a perspective use in neuronal networks: Their lifetime
gives the spin-wave system an intrinsic memory, they feature strong nonlinearity, and they can be
guided and steered through extended magnonic networks. In this work, we present a magnon adder
that integrates over incoming spin-wave pulses in an analog fashion. Such an adder is a linearprequel to a magnonic neuron, which would integrate over the incoming pulses until a certain non-
linearity is reached. In this work, the adder is realized by a resonator in combination with a paramet-
ric ampli fier which is just compensating the resonator losses. Published by AIP Publishing.
https://doi.org/10.1063/1.5042417
I. INTRODUCTION
In certain tasks like pattern recognition, the brain outper-
forms conventional CMOS-based computing schemes by far
in terms of power consumption. Consequently, neuromorphiccomputing approaches aim to mimic the functionality of
neurons in a network to boost computing ef ficiency.
1–5In the
brain, stimuli are conveyed by short wave packets from oneneuron to another, where they lead to stimulation which adds
up and then, ultimately, triggers a nonlinear response. Thus,
it is natural to consider waves as data carriers for bio-inspiredcomputing and arti ficial neuronal networks. Certain key
components need to be accessible by the used kind of waves:
It should be possible to convey them through extended net-works as well as to store the information carried by the waves
for a certain time so that stimuli can add up. In addition, the
waves should exhibit nonlinear dynamics in order to mimicthe threshold characteristics of a neuron. Among the possible
waves that one can consider, spin waves, the collective exci-
tation of magnetic solids, are a highly attractive candidate:
6–10
The dynamics of spin-waves and their quanta, magnons, are
governed by a nonlinear equation of motion,11,12providing
easy access to nonlinearity.13–15They can be guided through
reprogrammable networks by using spintronics and nonlinear
effects and their finite lifetime provides an intrinsic memory
to the spin-wave system. In addition, their excitation energy
is very low and their nanometric wavelengths at frequencies
in the GHz and THz range promise a scalable and power-efficient platform for neuromorphic computing.
In this work, we employ micromagnetic simulations to
demonstrate an analog magnon adder, which can be regardedas a pre-step to a magnon based neuron. The adder, which is
sketched in Fig. 1(a), consists of two building blocks: a
leaky spin-wave resonator and a parametric ampli fier.
16–19
Spin waves can enter the resonator by dipolar coupling to the
input.20Within it, their amplitude is added to or subtracted
from the amplitude of the already accumulated amplitudes, asis sketched in Fig. 1(b). This process is, in principle, equiva-
lent to the arrival of excitation pulses in the axon, where the
neuron integrates over the incoming stimuli until a criticalstimulus is reached. In our scheme, the parametric ampli fieracts to counteract the spin-wave losses that arise from propa-
gation through the resonator and the leakage to the input and
output of the resonator. We show that by working at thepoint of loss compensation, the adder can add and subtract
the spin-wave amplitudes over a large range and enables to
store the sum of these calculations in the resonator.
II. LAYOUT AND WORKING PRINCIPLE
To demonstrate the magnonic adder, we perform micro-
magnetic simulations using MuMax3.21We chose dimen-
sions that are compatible with the time scales and feature
sizes accessible in state-of-the-art magnonic experiments. For
our simulations, we assume the material parameters ofYttrium Iron Garnet (YIG),
22,23a widely used material in
magnonics:6,24saturation magnetization Ms¼140 kA m/C01,
exchange constant Aex¼3:5p J m/C01, and Gilbert damping
parameter α¼0:0002, which represents the damping of the
spin-waves mainly into the phonon system. The geometry we
study consists of three w¼0:5μm wide and 40 nm thick
rectangular YIG waveguides in a row [see Fig. 1(a)], similar
to the general design proposed in Ref. 25. The length of the
central waveguide, which acts as the resonator, is L¼20μm.
The waveguide to the left of the resonator acts as input,
where spin-waves are excited by a source creating a local
magnetic field. In a magnonic network, this input could be
connected to an arbitrary number of other waveguides acting
as individual inputs. The waveguide on the right of the reso-
nator acts as output, which again could be reconnected in anetwork. In our simulations, input and output are 10 μm long
and separated from the resonator by g¼75 nm wide gaps.
Spin-waves can tunnel through this gap,
20which constitutes
the coupling channel from the resonator to the input and
output, respectively. In the present simulation, about 0 :1% of
the spin-wave amplitude is tunneling through the gap. A dif-ferent gap spacing or a different magnetization con figuration
leads to different tunneling amplitudes, resulting in different,
potentially larger losses for the resonator.
26These can be
compensated by adjusting the parameters of the ampli fica-
tion. Toward their outer edges, the damping in the input and
output is increased exponentially to mimic the transport ofJOURNAL OF APPLIED PHYSICS 124, 152119 (2018)
0021-8979/2018/124(15)/152119/5/$30.00 124, 152119-1 Published by AIP Publishing.
spin-waves out into the network that would take place in a
real extended system.
Figure 2(a)shows the simulated spin-wave dispersion27,28
of the fundamental mode in a color-coded scale. The external
field of μ0Hext¼20 mT is applied along the long axis of the
resonator. The excitation frequency f¼5:8 GHz corresponds
to the excitation of spin-waves with a wave-vector of kk¼
56 rad μm/C01(i.e., wavelength λ/C25112 nm) and the periodic
excitation source is matched to excite this wave-vector reso-
nantly. From the simulated spin-wave dispersion, a groupvelocity of v
g¼(0:88+0:05)μmn s/C01can be extracted. This
corresponds to a roundtrip time of Δt¼2/C1L=vg/C2546 ns
through the resonator. During one trip, the spin-wave amplitudendecays exponentially following A
p(t)¼Ap(0)/C1exp (/C0t=τ)
with their lifetime τ. From this, it can be inferred that during
one pass lasting Δt, the relative amplitude change is
Ap(tþΔt)
Ap(t)¼e/C0Δt
τ: (1)
As mentioned above, the dipolar coupling between the reso-
nators is very weak and only a small fraction of 0 :1%of the
spin-wave amplitude is actually coupled from the resonator
to the input and output, respectively. Consequently, thelosses of the resonator are dominated by the propagation loss.
In order to counteract these losses, we employ paramet-
ric ampli fication, also known as parallel pumping.
16–19In
this technique, the system is pumped at the frequency fp
which equals twice the resonance frequency f. One possible
driving force is Oersted fields,16,19μ0hp, where microwave
photons split into pairs of magnons as indicated in Fig. 2(a).
Here, we consider this mechanism, but also other, more
energy ef ficient realizations like the use of electric fields
have been proposed.29,30In the simplest case of adiabatic
parametric ampli fication,16the pumping at 2 fleads to theformation of wave pairs at fwith wave-vector +kkin order
to conserve momentum, as is sketched in Fig. 2(a). Parallel
pumping counteracts the damping losses with two key fea-tures:
16(1) It only couples to already existing waves and, in
the absence of nonlinear saturation, leads to an exponential
increase of the spin-wave amplitude following Ap(t)¼
Ap(0)/C1exp [( Vμ0/C1hp/C0τ/C01)t] if the energy per unit time V/C1
μ0hpinserted into the spin-wave system exceeds the losses
given by τ/C01. Here, Vconstitutes the coupling parameter of
the given spin-wave mode at ( f,kk). (2) Parallel pumping
conserves the phase of the incident spin-waves. This is
important to pro fit from the phase of the spin-wave in encod-
ing which is, for instance, vital to be able to perform subtrac-
tion in the presented magnon adder. In the simulated
structure, the local parametric ampli fier exhibits an extent of
1μm along the resonator and it is situated in the center of the
resonator. For simplicity, we only take into account the
FIG. 1. (a) Sketch of the magnon adder consisting of an input, a resonator,
and an output, as well as a parametric ampli fier to compensate the losses in
the resonator. (b) Sketch of the operation principle of the adder: Subsequent
pulses with amplitude Ap(n) (indicated by the numbers) enter the resonator,
which integrates over their amplitude. Periodically, a pulse leaves the resona-
tor at the output. The value stored in the resonator and the value in the
output are equal to the sum Sof the input amplitudes.
FIG. 2. (a) Simulated spin-wave dispersion relation at a field of μ0Hext¼
20 mT applied along the resonator long axis and illustration of the parallel
pumping process. (b) Dynamic magnetization 6 μm away from the center of
the resonator as a function of time. Red: Only excitation of a single spin-
wave pulse with amplitude “1”in the input. Green: Only periodic application
of ampli fication pulses, no stimulus at the input. Black: Periodic application
of an ampli fication pulse twice per roundtrip with an input stimulus of one
pulse with amplitude “1.”The gray shaded areas indicate the position of the
idler waves.152119-2 T. Brächer and P. Pirro J. Appl. Phys. 124, 152119 (2018)parallel component of the microwave field created by a stri-
pline.19,31Please note that a reduction of the ampli fier size
below the wavelength of the spin waves to be ampli fied
results in the nonadiabatic regime of paramteric ampli fica-
tion.16,32In this regime, two co-propagating spin waves will
be created, i.e., the idler wave runs along with the signalwave. In this case, the ampli fication is not only phase-
conserving but also phase-sensitive.
16,19This allows for alter-
native designs of the adder or a magnonic neuron. However,for the wavelength we employ here, a very small pumping
source would be required to access this regime, as the
pumping source has to provide the necessary momentum.
16
This results in a very short interaction time of the spin waves
with the pumping field which leads to a large increase of the
needed ampli fication fields.
Similar to the operation in the brain, we assume that
incident information is carried by pulses. As sketched in
Fig.1(a), these pulses can arrive at the ampli fier with differ-
ent amplitudes and at different times. They exhibit a fixed
duration of 5 ns and delayed pulses are sent to the input at
times which are integer multiples of the roundtrip time Δt.
The ampli fication is also pulsed: tp¼5 ns long pumping
pulses are applied whenever the spin-wave pulse in the reso-
nator passes the ampli fier, i.e., twice per roundtrip. When the
net increase of the spin-wave amplitude by the pumping is
equivalent to the losses, this leads to the formation of a pair
of signal and idler spin-waves running back and forth in theresonator. The general act of the parametric ampli fication is
shown in Fig. 2(b) for one single input pulse of amplitude
“1.”In our simulations, this amplitude was arbitrarily chosen
to correspond to an external excitation with a local field
amplitude of 65 μT in the input. The diagram shows the
out-of-plane dynamic magnetization component m
zas a func-
tion of time at a point 6 μm away from the resonator center.
The red curve shows the amplitude if no pumping field is
applied —the spin-waves pass the position where mzis
recorded for the first time at t¼t0/C2527 ns. They are
reflected at the end of the resonator and pass the measure-
ment position again at t/C2536 ns. Then they pass a roundtrip
through the resonator and arrive again at t¼t0þΔt/C25
73 ns, and so forth. The damping of the waves can be
clearly seen and it amounts to about 25 %per roundtrip. In
contrast, the black curve shows the time evolution of the
spin-wave amplitude if the parametric ampli fication is
switched on and is just strong enough to compensate thelosses during one roundtrip. Now, the idler pulses are
created, which give rise to additional pulses highlighted by
the gray shaded areas. After the idler is build up and aftersome initial fluctuations, the quasi-steady-state is reached and
the pulses run back and forth with constant amplitude. For
completeness, the green curve shows the dynamic magnetiza-tion if only the pumping pulses are applied, showing that for
the presented parameters, noise creation by parametric gener-
ation is negligible.
16,33
III. WORKING POINT OF THE MAGNONIC ADDER
In the following, we want to elaborate the impact of the
parametric ampli fication in more detail, since it plays acrucial role for the operation of the resonator as an adder or
as a nonlinear device. The absolute gain per roundtrip is
determined by the strength of the pumping fieldμ0hp.F o r
half a roundtrip and assuming that t¼0 is the point in time
when the pulse enters the ampli fier, we can modify Eq. (1)to
ApΔt
2/C18/C19
¼Ap(0)/C1eVμ0hp/C0τ/C01ðÞ Δtp/C1e/C0τ/C01Δt
2/C0ΔtpðÞ
¼Ap(0)/C1eVμ0hpΔtp/C0τ/C01Δt
2
¼Ap(0)/C1e0:5/C1G0(hp), (2)
with the gain G0(hp)¼2Vμ0hpΔtp/C0τ/C01Δtper roundtrip. In
the following, we will consider a normalized gain factor
G¼G0(hp)=G0(0)¼G0(hp)=(τ/C01/C1Δt), which is /C01 in the
absence of parametric ampli fication, 0 when the parallel
pumping is just compensating the losses, and which takes
positive values if more energy is inserted per roundtrip than
is lost by dissipation. Figure 3shows the gain factor Gas a
function of the applied pumping field, which has been
extracted from a linear fitt ol n [ mz(t)]/ln[Ap(t)] as is exem-
plarily shown in the insets for μ0hp¼0(G¼/C01), corre-
sponding to the intrinsic spin-wave decay with the lifetime
τ¼155 ns and for μ0hp¼24:3m T( G¼1:2). The amplitude
has hereby been integrated in time over the forward travelingsignal pulse in a time window of +4 ns, i.e., for each pulse n
from t¼(t
0/C04n sþn/C1Δt)t o t¼(t0þ4nsþn/C1Δt). As
can be seen from the linearity in the insets, the data show aclear exponential decay/growth, respectively. While the
regime G.0 is highly interesting for neuromorphic applica-
tions in general, since it provides easy access to nonlinearity,for the magnon adder, we chose the working point at G/C250.
In this case, the energy inserted is just enough to com-
pensate the losses and the current amplitude of the pulsewithin the resonator is preserved. Please note that due to the
fact that the ampli fication is proportional to the amplitude,
FIG. 3. Gain factor as a function of the applied pumping fieldμ0hp. The
insets show the amplitude of the spin-wave pulses Apas a function of time
for the case of μ0hp¼0(G¼/C01, upper inset) and μ0hp¼24:3m T
(G¼þ1:2, lower inset) on a semi-logarithmic scale. The adder is operated
at the damping compensation point G¼0, marked by the dashed lines.152119-3 T. Brächer and P. Pirro J. Appl. Phys. 124, 152119 (2018)this compensation point holds for a small and a large
amplitude spin-wave alike, as long as no nonlinearity sets
in. In the following, we will fixt h ea m p l i fication field to
μ0hp¼12:8 mT, the fie l da l s ou s e di nF i g . 2(b), to stay at
G/C250.
IV. DEMONSTRATION OF ANALOG ADDING AND
SUBTRACTING
ForG¼0, the resonator losses are compensated. In this
regime, a spin-wave pulse within it is cached as long as the
ampli fication remains switched on and the compensated reso-
nator can be used as a spin-wave adder. Since the spin-wave
dynamics in the resonator are linear, the amplitude of the
spin-wave pulse stored within the resonator corresponds tothe sum Sover all incident pulses. This sum Sis given by
S¼P
nAp(n)/C1(/C01)f(n), where Ap(n) is the amplitude of
the individual pulse nandf(n) represents its phase, being
either f(n)¼0 for a phase-shift of 0 or 2 πandf(n)¼1 for
a phase-shift of π. A phase-shifted pulse, thus, corresponds
to a negative value and allows for a subtraction. The phasecould, for instance, be given by a global reference in the
magnonic network and it could be altered by reprogramma-
ble, local phase shifters such as nanomagnets. For an inputamplitude A
p(n) ranging from “0”to“100,”individual
pulses with the respective value of Apcan be applied at the
input without signi ficant nonlinear effects, corresponding to
excitation field amplitudes ranging from 65 μTu pt o6 :5m T
in the input. Numbers /C20100 can, therefore, be injected into
the ampli fier and will be summed over in an analog fashion.
For larger excitation fields in the input, the spin-wave dynam-
ics in the input become nonlinear, which distorts the summa-
tion. Nevertheless, within the ampli fier, much larger numbers
can be handled, since only a fraction of the input spin-wave
is coupled into the resonator by the dipolar coupling.
Figure 4(a) shows the amplitude of the spin-wave pulse
within the resonator in the quasi-steady-state, which corre-
sponds to the sum S, as a function of the input amplitude on
a double-logarithmic scale. As can be seen from the figure,
the output is perfectly linearly proportional to the sum of the
input amplitudes, no matter if an individual spin-wave pulse
or a series of pulses are applied. This holds in the entiretested input range ranging from “1”up to at least “1500. ”
The latter corresponds to the sum of 15 pulses with an indi-
vidual amplitude of “100,”which are subsequently added in
the resonator. The straight line is a linear fit yielding a slope
of 1 :029+0:004, con firming the linear relationship between
input and output.
From the inputs exceeding “100,”it can already be
inferred from Fig. 4(a)that the spin-wave packets in the reso-
nator add up in a linear fashion. For instance, the output“200”corresponds to the sum of two input pulses of value
“100,”and so on. The key feature of the adder is that the
device performs the summation purely analog, and the ampli-tude of the wave running back and forth in the resonator is
directly proportional to the sum of the amplitudes of the
input pulses. To illustrate this further, Fig. 4(b) shows the
amplitude in the resonator for several combinations adding
up to 10: 1 /C2input “10,”5/C2input “2,”10/C2
input “1,”aswell as the more involved pattern “1”+“3”+“0”+“3”+“0”+
“0”+“0”+“2”+“1.”As mentioned above, the fact that spin-
waves carry amplitude and phase can also be used to do a
subtraction. This is also shown in Fig. 4(b) by the combina-
tion “20”/C0“3”/C0“0”/C0“0”/C0“3”/C0“0”/C0“0”/C0“4”/C0“0”/C0“0,”
being equal to “10.”The transitionary dynamics visible in
Fig. 2(b) at the beginning of the ampli fication process are
also visible in Fig. 4(b): For the first few roundtrips, the
stored value decreases until it reaches the steady-state. Please
note that this has no sizable impact on the adding or subtract-ing function.
It should be noted that in the demonstrated regime of
operation of the adder, the spin-wave dynamics stay linear.This allows one to add up individual spin-wave pulses. The
device already performs similar to a neuron in the brain:
Incident pulses are converted into an amplitude informationwithin the resonator and this amplitude is given by the inte-
gration over the incoming signals. In the presented adder,
small pulses carrying the amplitude of the sum are constantlyejected into the output [cf. Fig. 1(b)]. Toward a neuromor-
phic application, the resonator could be designed in a way
FIG. 4. (a) Quasi-steady-state amplitude in the resonator vs. input stimulus
on a double-logarithmic scale. The straight line is a linear fit yielding a slope
of 1 :029+0:004 con firming the linear relationship between the input ampli-
tude and the sum in the resonator. (b) Different combinations resulting in a
sum of 10 within the resonator. Since the number of applied pulses as well
as the time of their arrival is different in all cases, the sum of “10”is reached
at different times for the different combinations.152119-4 T. Brächer and P. Pirro J. Appl. Phys. 124, 152119 (2018)that its quality factor is a function of the spin-wave ampli-
tude, for instance, by a change of the dipolar coupling ef fi-
ciency associated with a nonlinear change of wave-vector.This way, a nonlinearity can open up the resonator once the
critical stimulus is overcome. In such a way, spin-wave axons
that can be conveniently integrated into extended networksbecome feasible.
V. CONCLUSION
To conclude, by means of micromagnetic simulations,
we have demonstrated a magnon adder, where the magnon
amplitude adds and subtracts in an analog fashion. The spin-
wave summation is performed in a resonator, whose lossesare compensated by a parametric ampli fier. This way, the
amplitude is stabilized and constant in time if no mathemati-
cal operation is performed. The spin-wave signal in the reso-nator is directly proportional to the time-integrated amplitude
of the incoming pulses. Hereby, the phase degree of freedom
of the spin-waves allows one to add spin-wave pulses in the
case of constructive interference between the incoming spin-
wave pulses. If a pulse is shifted by π, it will instead be sub-
tracted. The presented device can perform as a magnon cache
memory that can store an analog magnon sum on long time
scales and, thus, constitutes the first step toward an all-
magnonic neuron.
ACKNOWLEDGMENTS
The authors thank B. Hillebrands and A. Chumak for
their support and valuable scienti fic discussion. They also
gratefully acknowledge financial support by the DFG in the
framework of the Collaborative Research Center SFB/TRR-173 Spin+X (Project B01), the Nachwuchsring of the
TU Kaiserslautern, and the ERC Starting Grant No. 678309
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1.1721128.pdf | Metrization of Phase Space and Nonlinear Servo Systems
Chi Lung Kang and Gilbert H. Fett
Citation: Journal of Applied Physics 24, 38 (1953); doi: 10.1063/1.1721128
View online: http://dx.doi.org/10.1063/1.1721128
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128.114.34.22 On: Tue, 25 Nov 2014 02:25:4338 H. L. ROBINSON
eter. In both cases the experimental data are compared
with calculated curves using Young's circuital form.
Note that the difference between the experimental and
theoretical curves is greater when the diameter is one
half wavelength. The experimental data in Fig. 5 have
been normalized as follows: All values of 1/10 at the
center of an aperture one-half wavelength in diameter
were averaged, then each reading was multiplied by a
factor such that the reading at the center would have
this average value. This makes it possible to compare
the shapes of the two experimental curves. The relative
intensities over a given curve as shown by its shape are
more precise than the actual value of 1/10• The lack of
agreement near the edge of the aperture indicated that
results within a sixteenth-wavelength of the edge are not
reliable. CONCLUSION
Although Young's circuital form predicts the inten
sity in the plane of apertures a few wavelengths in diam
eter, it does not agree with experimental values for
apertures less than a wavelength in diameter. Theo
retical curves based upon it indicate neither the sharp
increase in intensity near the ends of the electric diam
eters nor the high intensity at the centers of apertures
near one-half wavelength in diameter.
ACKNOWLEDGMENT
It is a pleasure to acknowledge the assistance through
out this study of C. L. Andrews, who brought the prob
lem to the author's attention and whose guidance was
often sought. The calculations of H. S. Story, P. Pi
saniello, and R. F. Tucker, Jr., have been of great value.
JOURNAL OF APPLIED PHYSICS VOLUME 24, NUMBER 1 JANUARY, 1953
Metrization of Phase Space and Nonlinear Servo Systems*
CHI LUNG KANGt AND GILBERT H. FETTt
University of Illinois, Urbana, Illinois
(Received July 29, 1952)
By introducing a proper distance function, the phase space for a servomechanism is completely metrized.
A new approach is developed to study servo systems directly on the basis of instantaneous performance
under an arbitrary input function. A criterion for determining the effect of nonlinearity on performance is
obtained. It will serve as basis for the design of nonlinear servo systems.
INTRODUCTION
CONTROL systems that lead to the following dif
ferential equation are to be considered:
e(n)+ale(n-l)+ ... + an_le(l)+ ane
=G(e(n-ll, e(n-2), .. ·e, t), (1)
where e stands for error and superscript in parenthesis
indicate order of differentiation with respect to time.
The left side of the equation is a linear equation with
constant coefficients. It represents a basic system to
which the actual system (which may be changed from
time to time) always refers. Any nonlinearity pur
posely introduced or parasitic to the basic system is
lumped with the input function on the right side of the
equation as function G. The existence theorem of solu
tion to such a differential equation is well established.!
* This paper is part of a thesis submitted by the first named
author in partial fulfillment of requirements for the degree of
Doctor of Philosophy in Electrical Engineering at University of
Illinois. t Formerly University Fellow, University of Illinois; now with
Boonton Radio Corporation, Boonton, New Jersey. t Professor of Department of Electrical Engineering, University
of Illinois.
1 S. Lefschetz, Lectures on Differential Equations (Princeton
University Press, Princeton, 1948), p. 23. Suppose there exists an unique solution to the differ
ential equation. Then G(e(n-l), e(n-2), .. 'e, t) can be
considered as another function of time, say F(t), which
thereby becomes a forcing term to the basic linear
system.
For any system represented by an nth order differ
ential equation, its states are specified by the set
(e(n-!), e(n-2), .. 'e, t) in the phase space. Hence, the
phase space becomes the configuration space for all the
states of all the systems of nth order. By definition,
e(t) = (Jdt) -(to(t), (2)
where (Ji(t), (Jo(l) are the input and the output functions
of the servo system, respectively. Since (Jo(t) is always
continuous and (Ji(t) should be continuous almost
everywhere, the trajectory of the representing point of
the state of the system in error coordinates is continu
ous almost everywhere. Good servo performance means
that this error trajectory remains for most of the time
near to the origin. Hence, at any point in the phase
space, this state point of the system should tend to
move back to the origin quickly.
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128.114.34.22 On: Tue, 25 Nov 2014 02:25:43METRIZATION OF PHASE SPACE 39
DEFINITION OF DISTANCE FUNCTION
The notion of distance from the state point to the
origin thus comes up. Mathematically, it really does
not matter what the actual distance function is, as
long as the usual hypotheses for a distance function
are satisfied. Thus, in the three-dimensional case, a
spherical neighborhood is equivalent to an ellipsoidal
one, which incidently is what is to be adopted here.
However, a properly defined neighborhood may greatly
simplify the actual analysis. So the problem under
investigation is to choose a logical, rational, and physi
cally meaningful definition for the distance function.
Transform the given differential equation (1) with
its right side replaced by F(t) to normal coordinates.2
The following substitutions, with dot on top of the
letters indicating differentiation with respect to time,
e=el } el=e2
e~l~e7O '
lead to the vector equation,
de -=Be+f(t),
dt (3)
(4)
where e=(e1, e2, ···en),f(t)=(O,O, ···O,F(t)),andBis
a constant matrix in terms of the constant coefficients
in the left side of the given differential equation.
Let the characteristic roots of the basic linear system
be AI, A2, •• ·A2r-1, A2r, 'Y2r+l, .. ·1'70, where A2r-1, A2r are
complex conjugate pair and 'Yis are real roots. It can
be shown that, when they are distinct,
1
Al
l-'l2
P= A13
A1n-1 1
A2r
A2,2
A2r3
" n-1 1\2r 1
'Y2r+1
'Y2r+12
'Y2r+13
'Y2r+l7O-1
is a nonsingular matrix such that.
P-1BP=R, 1
'Yn
1'702
'Yn3
(5)
'Ynn-1
(6)
where R is a diagonal matrix and has the characteristic
roots as its diagonal elements.
Thus, the transformation
e=Pz (7)
gives
dz/dt=Rz+P-1f(t), (8)
2 H. Goldstein, Classical Mechanics (Addison-Wesley Press,
Cambridge, Massachusetts, 1950), p. 329. i.e.,
Z2r= A2rZ2r+q2r. 7OF(t) (9)
Z2r+l = 'Y2r+lZ2r+l+q2r+1. "F(t)
where {qi;}=P-1.
The actual trajectory of the system is the result of
the motion of the force free (i.e., F(t) = 0) trajectory
caused by the forcing function F(t). The state point
will jump from one to another force free trajectory.
These trajectories never cross each other. Thereby, it
is natural to derive the notion of distance from the force
free case. The state point can be considered as a ma
terial point, whose motion in the phase space will be
characterized by a Lagrangian function leading to the
same set of equations of motion, Eq. (9). This La
grangian function for the force free case is
i-2r B=n-2r
L=[ L: Zi2+ L: Z2r+.2]
i=1 8=1
2r ~n-2r
+![L: Alzl+ L: 'Y2r+.2Z2r+,2J. (10) i-I.-I
It is quite instructive to note that the first sum in
the above expression can be looked upon as the kinetic
energy of the system with the time derivatives of the
coordinates considered as generalized velocities; the
other sum can be considered the negative of the po
tential energy corresponding to a force field propor
tional to the. coordinates. The Hamiltonian function,
hence the total energy, is a constant and is equal to
zero. Therefore, the value of the Lagrangian function
which is twice the kinetic. energy would be a measure
of the swing of the energy content of the system away
from the equilibrium position. It can also be easily
shown that the necessary and sufficient condition that
the basic system is stable is to have the value of its
Lagrangian function on its force free trajectory tending
to zero as time tends to infinity. This property of the
Lagrangian function suggests it at once as a natural and
rational distance function. However, a distance func
tion has to be positive definite; so, in case the La
grangian function contains oscillatory terms, the en
velope to the Lagrangian function instead of the func
tion itself will be used. Thus, by making use of Eq. (9)
with F(t) = 0, the following definition of distance is
derived from the Lagrangian function:
r n-2r
D(z, 0) = 2 L: A2r-1A2rZ2r-1Z2r+ L: 'Y2r+.2Z2r+.2
r-1 .=1 (11)
and
D(z, z')=D(z-z', 0). (12)
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128.114.34.22 On: Tue, 25 Nov 2014 02:25:4340 C. L. KANG AND G. H. FETT
where
o
o
S= rA2-IA2r A2r-IA2r (13)
o
Since Zi differs from Zi by only constants, z defines
the state as well as z does. And in the space of z, which,
in general, is complex, the above distance function is
nothing but the ordinary norm in n-dimensional space
over the field of complex number. If the matrix C
= (P-I)TS(P-I) has all its characteristic roots positive
(this is always the case when the basic linear system
has only real characteristic roots), then D= constant
will be an ellipsoidal surface. This point is of prime im
portance in the present discussion and must be checked
to assure that it is satisfied. The e space is therefore
topologically Euclidian. And the above defined dis
tance function satisfies all the hypotheses for a dis
tance function and completely metrizes the phase space.
NONLINEAR SERVO SYSTEMS
In a servo system, this distance function can readily
be used as an ordering relation defining at least par
tially a preference among all the states in the phase
space. To have a smaller distance from the origin is
therefore a necessary condition for one state to be
"better" than another state in a servo system. Since
error itself, more than its time derivative, is of im
portance, some auxiliary ordering relation can be set
up to assure real improvement of the performance of
the servo system.
Now the effect of the forcing function F(t) at any
.instant is to be examined. Differentiation of D gives
r
D= 2 L: A2r-IA2r(Z2r-IZ2r+Z2r-IZ2r)
r=1
n-27
+2 L: 'Y2r+b2r+.Z2r+o. (14)
0=1
Substitution of the expressions of Eq. (9) for the
z/s gives o
r n-2r
D= 2[L: A2r-IA2r(A2r+A2r-I)Z2r-IZ2r+ L 'Y2r+.3Z2r+02J
r=l 8=1
r
+ 2F(t)[L: A2r-IA2r(q2r, nZ2r-l+q2r-I, nZ2r)
r=1
n-2r + L: 'Y2r+.2q2r+8. nZ2r+oJ (15)
0=1
= Do+ 2F(t)K,
where Do represents the rate of reduction of distance
for force free case, and the other term represents the
effect of the forcing function. It can be readily proved
that Do is always negative for a stable basic system as
can be expected. Whether the effect of the forcing func
tion is favorable (i.e., to make the D more negative)
or not depends on the sign of the term F(t)K. K is a
linear function of the coordinates; hence, K = 0 is a
plane in the phase space through the origin. The whole
space is divided into two halves by the K = 0 plane,
on one side of which a larger (algebraically) F(t) is
preferred,. and on the other side, a smaller F(t).
The functional dependence ofG(e(n-I), e(n-2), ., ·e, t)
on the error and its time derivatives depends on the
nature of the system, the magnitude of its parameters,
its nonlinearity, etc. If at any point in the phase space,
the function G(e(n-I), e(n-2), .. ·e, t), hence F(t), can be
changed favorably by some modification of the system,
whatever it may be, then the performance of the sys
tem would be improved at that point. And the effect
of any nonlinearity in a supposedly linear system can
also be studied in the light of this K plane criterion.
If the state of the system or at least the sign of its
K value can be monitored by direct measurement,
proper change can be made in the system accordingly
to improve the performance. As a special case, there
may be two linear systems, one faster in response and
the other heavily damped. They can be switched into
action alternately, as the K plane criterion permits, to
improve the servo performance. In fact, this study is
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128.114.34.22 On: Tue, 25 Nov 2014 02:25:43METRIZATION OF PHASE SPACE 41
motivated by such a heuristic attempt of switching
among systems in a composite system. And a nonlinear
system can naturally be considered as the result of
continuous switching among linear systems. Notice
that the behavior of the actual system is always ex
pressed in terms of forcing with respect to a basic sys
tem. This provides a simple and unique way to treat
the general servo system. .
It should be pointed out that the comparison of D
has been made with respect to that at the particular
point under consideration in the phase space. When the
trajectory is changed by any modification of the sys~
tem, the basis of comparison is changed too. This
makes the general study of the over-all rate of reduc
tion of distance rather difficult. An investigation for
the special case with the real parts of all the char
acteristic roots equal will help to understand the situa
tion. To insure that the rate of reduction of the abso
lute value of the error is increased, extra forcing control
should be used only when
eK>O. (16)
Thus, the general scheme of extra forcing control
for a third-order system may be
K>o>O el>O LlF<O
el<O LlF=O,
K<-o el>O LlF=O
el<O LlF>O,
IKI~o for all e LlF=O,
where LlF stands for the extra forcing control, and 0 is
introduced to give a zone about the K plane without
extra forcing. This is to avoid possible instability at
the origin due to the presence of inevitable delay in
switching. When D has been reduced to the extent that
the maximum dimension of the corresponding ellip
soidal surface of constant distance is less than 0, the
whole system will behave exactly as the basic linear
system. It is difficult to say much about the resulting
trajectory in general. While it seems hardly possible for
some trajectory to remain in the LlF=O region forever,
the question is to what extent will any trajectory come
into some region with extra control and expose itself
to it. The trajectories emerging out of the planes K = ± 0
into regions with extra forcing may be forced immedi
ately back to these planes. This situation is certainly
intolerable practically. A purposely designed hysteresis
band (not given in the above scheme) for the switching
on and off of the extra forcing around the K = ± il
planes should solve this difficulty.
The choice of the basic system is evidently an im
portant problem in the design of such composite sys
tem. To reduce the region where extra forcing is for
bidden, the normal to the K plane should make a small
angle with the error axis. This will likely give a system
more suitable for this kind of extra control.
One possible way of introducing the extra control is
suggested below. An extra control box is used to feed
an extra error signal H to the actual error, as is shown 90(:1)
FIG. 1. A possible way of introducing extra forcing control.
in Fig. 1. Thus,
e'(t)=e(t)+H. (17)
Let kN(p)/S(p) represent the forward transmission
characteristics of the servo loop where N(p) and S(p)
are polynomials in p = d/ dt. Therefore,
8o(t) = kN(p)e'(t)/ S(p) = kN(p)e(t)S(p)
+kN(p)H/S(p) (18)
= 8i(t) -e(t).
Hence,
[S(p)+kN(p)Je(t) =S(p)8 i(t)-kN(p)H. (19)
The last term in the above equation is the extra control
needed. Thus, a constant forcing term can be obtained
by making H=ct', where p is the lowest power of p
in N(p) and c is a constant. If a network with transfer
function l/N(p) is available, then the extra forcing
term of the form h(e) can be obtained by feeding this
h(e) through such a network to give the function H.
Since the extra forcing term of the form clel+c2e2
+Caea, where the c's are constants, is equivalent to a
change to another linear system, it can be achieved by
direct adjustment of the parameters of the basic system.
But either theS(p) of the system should not be changed,
or its effect on the term S(p)8i(t) should be taken into
consideration.
CONCLUSION
To facilitate the study of a general servo system
directly on the basis of performance, the phase space is
metrized by defining a distance function. The defini
tion adopted here seems to be quite natural and physi
cally meaningful. And above all, it leads to a simple
partition of the phase space and hence a simple cri
terion to determine the effect of any nonlinearity,
purposely introduced or parasitic, in the system.
Actual design of specific systems have not been at
tempted here. This work should be considered as a
new approach for the study of nonlinear systems.
Since this work is based on the differential equation,
its application should not be limited to servo systems.
It will certainly be useful in the study of nonlinear
damper for vibration.
BIBLIOGRAPHY
(1) L. A. MacCoU, Fundamental Theory of Servomechanisms
(D. Van Nostrand Company, Inc., New York, 1945).
(2) C. Lanczos, The Variational Principles of Mechanics (Uni
versity Press, Toronto, 1949).
(3) H. Goldstein, Classical Mechanics (Addison-Wesley Press,
Cambridge, Massachusetts, 1950).
(4) S. Lefschetz, Lectures on Dijferential Equations (Princeton
University Press, Princeton, 1946).
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1.3396983.pdf | Gilbert damping in perpendicularly magnetized Pt/Co/Pt films investigated by all-
optical pump-probe technique
S. Mizukami, E. P. Sajitha, D. Watanabe, F. Wu, T. Miyazaki, H. Naganuma, M. Oogane, and Y. Ando
Citation: Applied Physics Letters 96, 152502 (2010); doi: 10.1063/1.3396983
View online: http://dx.doi.org/10.1063/1.3396983
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128.255.6.125 On: Thu, 11 Dec 2014 13:23:13Gilbert damping in perpendicularly magnetized Pt/Co/Pt films investigated
by all-optical pump-probe technique
S. Mizukami,1,a/H20850E. P . Sajitha,1D. Watanabe,1F. Wu ,1T . Miyazaki,1H. Naganuma,2
M. Oogane,2and Y . Ando2
1WPI-Advanced Institute for Materials Research, Tohoku University, Katahira 2-1-1,
Sendai 980-8577, Japan
2Department of Applied Physics, Graduate School of Engineering, Tohoku University, Aoba 6-6-05,
Sendai 980-8579, Japan
/H20849Received 16 February 2010; accepted 25 March 2010; published online 13 April 2010 /H20850
To investigate the correlation between perpendicular magnetic anisotropy and intrinsic Gilbert
damping, time-resolved magneto-optical Kerr effect was measured in Pt /Co/H20849dCo/H20850/Pt films. These
films showed perpendicular magnetization at dCo=1.0 nm and a perpendicular magnetic anisotropy
energy Kueffthat was inversely proportional to dCo. With an analysis based on the Landau–Lifshitz–
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frequency and lifetime expressions to experimental data of angular variations in spin precessionfrequency and life-times. The
/H9251values increased significantly with decreasing dCobut not inversely
proportional to dCo.©2010 American Institute of Physics ./H20851doi:10.1063/1.3396983 /H20852
Spin transfer torque magnetic random access memory
/H20849STT-MRAM /H20850utilizing magnetic tunnel junctions /H20849MTJs /H20850is
one of many candidates for next-generation nonvolatilerandom access memory. Many groups are currently develop-
ing STT-MRAM, and in particular, STT-MRAM based onMTJs with perpendicularly magnetized electrodes as theseexhibit a large thermal stability factor /H9004and a very low
critical current density J
crequired for current-induced mag-
netization switching /H20849CIMS /H20850.1,2The Jcis proportional to
/H9251MsHkeffin CIMS for out-of-plane magnetization configura-
tion, where the respective /H9251,Ms, and Hkeffare the Gilbert
damping constant, saturation magnetization, and effectiveperpendicular magnetic anisotropy /H20849PMA /H20850field. On the other
hand, /H9004is also proportional to M
sHkeff. Thus, to reduce Jc
while maintaining /H9004constant, requires some intervention.
One possibility is to use perpendicularly magnetized materi-als with low
/H9251value. Gilbert damping originates intrinsically
from a quantum mechanical electron transition mediated byspin-orbit interaction.
3Roughly speaking, /H9251is proportional
to/H92642/W, where /H9264is the spin-orbit interaction energy and Wis
thed-band width.4PMA also originates from spin-orbit in-
teraction and broken symmetry and is also roughly propor-tional to
/H92642/Win the theory.5These theories imply that Gil-
bert damping tends to be stronger in materials with high-PMA and there might be a linear correlation between them.Recently, Gilbert damping in /H20851Co /Pt/H20852
Nmultilayer films with
high-PMA was investigated by the time-resolved magneto-
optical Kerr effect /H20849TRMOKE /H20850, and the /H9251value increased
with increasing stacking number Nwhile in contrast PMA
decreased.6In addition, /H9251values were deduced from domain
wall motion in Pt/Co/Pt films and were found to be indepen-dent of Co layer thickness although PMA increased withdecreasing thickness.
7The conclusion is that the relationship
between Gilbert damping and PMA is still unclear. To clarifythe Gilbert damping mechanism in materials with large-PMA, a more systematic study is required to extract the pre-cise nature of this correlation.In this paper, we report on the systematic investigation
of intrinsic Gilbert damping for Pt/Co/Pt films deduced fromangular dependence of TRMOKE and discuss its correlationwith PMA. The Pt/Co/Pt films were deposited on naturallyoxidized Si substrate at room temperature using magnetronsputtering. The base pressure was 1 /H1100310
−7Torr and Ar pres-
sure was 3 mTorr. The Pt buffer and capping layer thick-nesses were 5 nm and 2 nm, respectively, and Co layer thick-nesses d
Cowere varied from 4.0 to 0.5 nm. Structural
analysis was accomplished by x-ray diffraction /H20849XRD /H20850and
x-ray reflectivity /H20849XRR /H20850. Magnetic properties were investi-
gated using polar magneto-optical Kerr effect /H20849PMOKE /H20850and
a superconducting quantum interference device magnetome-
ter. Magnetization dynamics were investigated by x-bandferromagnetic resonance /H20849FMR /H20850and TRMOKE. Details of
FMR measurements and analyses were the same as describedin a previous report.
8In the TRMOKE measurements, a stan-
dard optical pump-probe setup was used with a Ti:sapphirelaser and a regenerative amplifier.
9Beam wavelength and
pulse width were /H11011800 nm and 100 fs, respectively, and
pump beam fluence was 3.8 mJ /cm2. The s-polarized probe
beam, for which the intensity was much less than for thepump beam, was almost normally incident on a film surfaceand the time variation in the magnetization was exhibited byPMOKE. TRMOKE measurements were obtained with anapplied magnetic field Hof 4 kOe, and the angle
/H9258Hbetween
field and direction normal to the film was varied from 0° to80° using a specially designed electromagnet.
From the XRD
/H9258-2/H9258patterns, only a Pt /H20849111/H20850diffraction
peak appeared for all films, indicating the films were /H20849111/H20850-
textured polycrystalline. The XRR analysis showed that ac-tuald
Covalues were equal to nominal values within experi-
mental error, and interface roughness and/or alloying layerthickness were /H110110.4 nm. Figure 1/H20849a/H20850shows coercivity H
Cas
a function of dCofor Pt/Co/Pt films. Perpendicular magneti-
zation appears at dCo=1.0 nm and the coercivity exhibits a
maximum value at dCo=0.8 nm, for which a PMOKE loop is
shown in Fig. 1/H20849b/H20850as an example. To obtain the Hkeffvalues,
we performed PMOKE measurements varying angle /H9258Hsub-a/H20850Electronic mail: mizukami@wpi-aimr.tohoku.ac.jp.APPLIED PHYSICS LETTERS 96, 152502 /H208492010 /H20850
0003-6951/2010/96 /H2084915/H20850/152502/3/$30.00 © 2010 American Institute of Physics 96, 152502-1
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128.255.6.125 On: Thu, 11 Dec 2014 13:23:13ject to a constant applied field of 4 kOe. Subsequently, the
data were compared to theoretical data calculated numeri-cally using the expression for magnetization angle
/H9258,
sin 2/H9258=/H208492H/Hkeff/H20850sin/H20849/H9258H−/H9258/H20850, with the adjustable Hkeffas a fit-
ting parameter. The effective anisotropy energy Kueffwas
evaluated from the relation Kueff=MsHkeff/2. The product
KueffdCowas plotted in Fig. 1/H20849c/H20850as a function of dCo. The Kueff
values were also obtained from FMR measurement for rela-
tively thicker films and are plotted with open circles; thesevalues agree with those from PMOKE. Proportionality of
K
ueffdCoagainst dCoindicates PMA is due to interface PMA,
as reported in much of the literature.10The interface PMA
energy Kswas estimated to be 0.35 erg /cm2by extrapola-
tion as indicated by the solid line in Fig. 1/H20849c/H20850. The Ksvalues
ranged from 0.3 to 1 in /H20849111/H20850-textured Co/Pt multilayer
films,10depending on interface quality and degree of texture,
and our value is relatively lower because the buffer layer isthinner than in conventional multilayers so as to increase theKerr signal intensity.
Figures 2/H20849a/H20850and2/H20849b/H20850show the representative TRMOKE
measurements for Pt/Co/Pt film with d
Co=2.0 nm and 0.8
nm, respectively, measured at /H9258H=60°. MOKE signals de-
crease suddenly in sub-ps time regime and subsequently ex-hibit damped oscillatory behavior for both films that is acommon feature observed in all-optical measurements.
11Thespin precession frequency fand life-time /H9270were evaluated
by fitting the damped harmonic function superposed with anexponential decay function, as expressed in the formAexp/H20849−Bt/H20850+Csin/H208492
/H9266ft+/H9278/H20850exp/H20849−t//H9270/H20850, using the phase of
precession /H9278and fitting parameters A,B, and C, as shown
with solid curves in Figs. 2/H20849a/H20850and2/H20849b/H20850.
Figures 3/H20849a/H20850and3/H20849b/H20850show the /H9258Hdependence of fand
1//H9270for Pt/Co/Pt film with dCo=2.0 nm and 0.8 nm, respec-
tively. With dCo=2.0 nm, the normal direction of the film is
a magnetic hard-axis, so that fincreases with increasing /H9258H.
The 1 //H9270values tend to increase with increasing fbecause
Gilbert damping acts more effectively on faster spin motions,much like viscosity, as seen in Fig. 3/H20849a/H20850. Trends in fand 1 /
/H9270
against /H9258Hbecome inverted in Fig. 3/H20849b/H20850because a magnetic
easy-axis is perpendicular to the film plane for dCo
=0.8 nm. The experimental angular-dependence data of
fand 1 //H9270were parameter-fitted with expressions derived
from the Landau–Lifshitz–Gilbert equation. Taking intoaccount PMA and arbitrary
/H9251, these expressions are f
=f0/H208811−/H208492/H9266f0/H9270/H20850−2with f0=/H20849/H9253/2/H9266/H20850/H20881H1H2//H208811+/H92512and 1 //H9270
=/H9251/H9253/H20849H1+H2/H20850//H208491+/H92512/H20850. Here, /H9253is the gyromagnetic ratio
and H1=Hcos/H20849/H9258H−/H9258/H20850+Hkeffcos2/H9258and H2=Hcos/H20849/H9258H−/H9258/H20850
+Hkeffcos 2/H9258. The /H9253and/H9251values were treated as fitting pa-
rameters, while the Hkeffvalues were fixed to those obtained
from PMOKE measurements. The magnetization angle /H9258
was calculated in the same way as those in PMOKE. Thecalculated data fitted well to the experimental data for bothfilms without invoking magnetic inhomogeneity or two-magnon scattering. This indicates that intrinsic Gilbert damp-ing is the dominant mechanism in the relaxation of magne-tization precession in these films.
Figure 4/H20849a/H20850shows the
/H9251values evaluated from
TRMOKE as a function of the reciprocal of dCo. The /H9251val-
ues increase significantly with decreasing dCoand are not
proportional to 1 /dCo. This trend is different from the linear
relationship between Kueffand 1 /dCo. The /H9251values obtained
from FMR are also shown in this figure with open circles.FMR was barely measurable at d
Co/H110211.0 nm because of sig-
nificantly large linewidths. The /H9251values from FMR show
quite good agreements with those from TRMOKE for in-plane magnetized films but these tend to deviate slightlyfrom those from TRMOKE with d
Co/H110211.0 nm. The /H9251values
in our films are of the same order of magnitude as reportedvalues,6,7and the nonlinearity of /H9251against 1 /dCois similar to
that observed in perpendicularly magnetized CoFeB filmsdespite a much different magnetic material.12To account for
this enhanced /H9251values, the relaxation frequency G, defined(a) (c)
out-of-plane
01f×dCo(erg/cm2)
00.5 1.0 1.5 2.000.10.20.30.4HC(kOe)
dCo(nm)
1nit)
in-plane (b)
0 0.5 1.0 1.5 2.0-1Kueff
dCo(nm)-4 -2 0 2 4-101MOKE (arb. u n
H(kOe)
FIG. 1. /H20849a/H20850The Co layer thickness dCodependence of coercivity HCfor
Pt/Co/Pt films. The curve is used as a visual guide. /H20849b/H20850A hysteresis loop for
a Pt/Co /H208490.8 nm /H20850/Pt film measured by PMOKE with applied field perpendicu-
lar to film plane. /H20849c/H20850The effective perpendicular magnetic anisotropy energy
Kueffmultiplied by dCoas a function of dCoas measured by PMOKE /H20849/L50098/H20850and
ferromagnetic resonance /H20849/H17034/H20850. Solid line is a fit to the experimental data.
-200nal (arb. unit )
-200
0 100 200-60-40MOKE sign
0 100 200-60-40
Pump-probe delay time (ps)/g894/g258/g895 /g894/g271/g895
FIG. 2. Signals of TRMOKE measured with applied field of 4 kOe directed
at 60 deg. from the film normal in /H20849a/H20850Pt/Co /H208492.0 nm /H20850/Pt and /H20849b/H20850Pt/
Co/H208490.8 nm /H20850/Pt films. Solid curves are the calculated damped harmonic func-
tion superposed on an exponential decay parameter-fitted to the experimen-tal data.2030
2030GHz)
Grad/s)2030
2030
Grad/s )(GHz)(a)
0 30 60 90010
010f(G
1/τ(G
θΗ(deg.)0 30 60 90010
010
θΗ(deg.)
1/τ(Gf(
(b)
FIG. 3. Magnetic field angle /H9258Hdependence of precession frequency f/H20849/H17034/H20850
and inverse precession life-time 1 //H9270/H20849/L50098/H20850for/H20849a/H20850Pt/Co /H208492.0 nm /H20850/Pt and /H20849b/H20850
Pt/Co /H208490.8 nm /H20850/Pt films. Solid and broken curves are the calculated data of f
and 1 //H9270parameter-fitted to the experimental data.152502-2 Mizukami et al. Appl. Phys. Lett. 96, 152502 /H208492010 /H20850
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128.255.6.125 On: Thu, 11 Dec 2014 13:23:13asG=/H9251/H9253Ms, is shown in Fig. 4/H20849b/H20850as a function of 1 /dCo.
The Gvalues for Pt /Ni80Fe20/H20849Py/H20850/Pt films reported previ-
ously are also shown in Fig. 4/H20849b/H20850with open triangles.8,9For
dCo/H110221.0 nm, the Gvalue for Pt/Co/Pt films seems to be
proportional to 1 /dCo, and its slope evaluated by linear fitting
was 34 /H11003108rad /s nm, which is roughly three times larger
than for Pt/Py/Pt films /H2084913/H11003108rad /sn m /H20850. The enhanced
Gilbert damping in thin Py layer in contact with a Pt layer
can be caused by a spin pumping effect. The damping fre-quency is then expressible as G=G
0+/H20849/H92532/H6036/2/H9266/H20850g↑↓/dFM, us-
ing the bulk relaxation frequency G0and mixing conduc-
tance g↑↓.13The/H9253values for Pt/Co/Pt films were almost the
same as in Pt/Py/Pt films, and the g↑↓is considered to be
almost the same for both films because it is approximatelyequal to the conductance of Pt layer in the diffusive transportregime.
13Thus, Gilbert damping in Pt/Co/Pt films could be
enhanced by an another mechanism, in addition of spinpumping. It is possible that Co 3 d–Pt 5 dhybridization ef-
fectively decreases the bandwidth Wfor the Co atomic layer
in contact with a Pt layer,
14enhancing both PMA and Gilbert
damping, as mentioned earlier. However, this hybridizationmechanism seems not to explain the significant increase in G
ford
Co/H110211.0 nm. This thickness regime is close to the inter-face roughness or alloying layer thickness, which might af-
fect Gilbert damping but the problem remains open. The an-gular dependence of TRMOKE measurement with highmagnetic field should be done in films with atomically flatinterface as further subject.
In conclusion, Gilbert damping for perpendicularly
magnetized Pt/Co/Pt films had been investigated usingTRMOKE. The effective PMA energy was shown to linearlydependent on 1 /d
Co, while /H9251andGincreased rapidly in the
regime dCo/H110211.0 nm corresponding to a switch in the mag-
netic easy-axis from in-plane to out-of-plane. No linear cor-relation between PMA and Gilbert damping was observed.TheGvalue deduced from
/H9251for Pt/Co/Pt films was much
larger than that for Pt/Py/Pt films, which was considered tobe due to the d-dhybridization effect.
This work was partially supported by Grant for Indus-
trial Technology Research /H20849NEDO /H20850and Grant-in-Aid for Sci-
entific Research.
1S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey, B. D. Terris, and E.
E. Fullerton, Nature Mater. 5, 210 /H208492006 /H20850.
2M. Nakayama, T. Kai, N. Shimomura, M. Amano, E. Kitagawa, T. Na-
gase, M. Yoshikawa, T. Kishi, S. Ikegawa, and H. Yoda, J. Appl. Phys.
103, 07A710 /H208492008 /H20850.
3V . Kambersky, Can. J. Phys. 48,2 9 0 6 /H208491970 /H20850.
4V . Kamberský, Czech. J. Phys., Sect. B 26,1 3 6 6 /H208491976 /H20850.
5P. Bruno, Physical Origins and Theoretical Models of Magnetic Aniso-
tropy /H20849Ferienkurse des Forschungszentrums Jürich, Jürich, 1993 /H20850.
6A. Barman, S. Wang, O. Hellwig, A. Berger, and E. E. Fullerton, J. Appl.
Phys. 101, 09D102 /H208492007 /H20850.
7P. J. Metaxas, J. P. Jamet, A. Mougin, M. Cormier, J. Ferre, V . Baltz, B.
Rodmacq, B. Dieny, and R. L. Stamps, Phys. Rev. Lett. 99, 217208
/H208492007 /H20850.
8S. Mizukami, Y . Ando, and T. Miyazaki, Jpn. J. Appl. Phys., Part 1 40,
580 /H208492001 /H20850.
9S. Mizukami, H. Abe, D. Watanabe, M. Oogane, Y . Ando, and T.
Miyazaki, Appl. Phys. Express 1, 121301 /H208492008 /H20850.
10M. T. Johnson, P. J. H. Bloemen, F. J. A. den Broeder, and J. J. de Vries,
Rep. Prog. Phys. 59, 1409 /H208491996 /H20850, and references therein.
11M. van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair, L. Lagae, W. J. M.
de Jonge, and B. Koopmans, Phys. Rev. Lett. 88, 227201 /H208492002 /H20850.
12G. Malinowski, K. C. Kuiper, R. Lavrijsen, H. J. M. Swagten, and B.
Koopmans, Appl. Phys. Lett. 94, 102501 /H208492009 /H20850.
13Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88,
117601 /H208492002 /H20850.
14N. Nakajima, T. Koide, T. Shidara, F. Miyauchi, H. Fukutani, A. Fujimori,
K. Ito, T. Katayama, M. Nyvlt, and Y . Suzuki, Phys. Rev. Lett. 81, 5229
/H208491998 /H20850.0.30.40.50.61 0.5
αdCo(nm)
4060801 0.5rad/s)dFM(nm)
TRMOKE
FMR(a) (b)
FM = Co24 0.7 24 0.7
0 1 200.10.2α
1/dCo(nm-1)0 1 202040G(108
1/dFM(nm-1)FM = Ni80Fe20
Pt/FM( dFM)/Pt
FIG. 4. Inverse thickness 1 /dCodependence of /H20849a/H20850Gilbert damping constant
/H9251and /H20849b/H20850relaxation frequency Gfor Pt /Co/H20849dCo/H20850/Pt films. The values ob-
tained from the time-resolved magneto-optical Kerr effect and ferromagneticresonance are shown with the solid /H20849/L50098/H20850and open circles /H20849/H17034/H20850, respectively.
The reported values of Gfor Pt /Ni
80Fe20/Pt films are also shown with open
triangles /H20849/H17005/H20850. Solid and broken lines are fitted to the experimental data for
1/dFM/H110211.0 nm−1.152502-3 Mizukami et al. Appl. Phys. Lett. 96, 152502 /H208492010 /H20850
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128.255.6.125 On: Thu, 11 Dec 2014 13:23:13 |
1.1660764.pdf | Curvature Stabilization of the Universal Instability
Gilbert A. Emmert
Citation: Journal of Applied Physics 42, 3530 (1971); doi: 10.1063/1.1660764
View online: http://dx.doi.org/10.1063/1.1660764
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/42/9?ver=pdfcov
Published by the AIP Publishing
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131.111.185.72 On: Thu, 18 Dec 2014 14:43:45JOURNAL OF APPLIED PHYSICS VOLUME 42, NUMBER 9 AUGUST 1971
Curvature Stabilization of the Universal Instability
Gilbert A. Emmert
Department of Nuclear Engineering, University of Wisconsin, Madison, Wisconsin 53706
(Received 8 January 1971)
A graphical t~chnique for de~ermining the influence of the drift velocity in a magnetic well
on wave-particle resonance IS presented and applied to the universal instability. It is seen
that the curvature drift velocity can strongly affect the number of resonant ions and lead
to increased stabilization of the wave.
The universal instability is a low-frequency electro
static instability driven by a density gradient of the
plasma. 1-3 It is caused by electrons whose motion
parallel to B resonates with the wave. 4,5 The con
~ct~n produced by the density gradient and the
E x B drift causes the resonant electrons to give
energy to the wave and thus destabilize it. Compet
ing stabilizing effects are electron and ion Landau
damping and ion convection; these are also resonant
particle effects. Neglecting magnetic field curva
ture, the net electron contribution is destabilizing if
the frequency w is less than the diamagnetic fre
quency w* (w* = IkL v;IU~eL I, ve is the electron
thermal velocity, Oe is the electron cyclotron fre
quency, kL is the wave number perpendicular to B,
and L is the plasma-density scale length) and is
maximum when the number of resonant electrons is
large, Le., 1~/k,,1 <ve, where k" is the wave
number along B. The stabilizing ion contribution is
small when the number of resonant ions is small
L e., I wi k,,1 »v i' Thus the instability occurs '
primarily for Vi « I w Ik" I < v e'
Magnetic field curvature has usually been simulated
in slab models by a fictitious gravity g 3,6,7 which
produces a guiding center drift g /0; g is usually
chosen so that g 10 matches the drift of a thermal
particle in an actual curved field. The resonance
condition becomes w-kLg/O-k" v,,=O. In this
model the effect of curvature is to introduce a
Doppler shift of the frequency-, Wi = W -k Lg 10, and
consequently a shift of the resonant velocity v , "
= Wi Ik". In a magnetic well, the ion resonant veloc-
ity is shifted downwards (Wi < w) which increases the
ion-stabilizing contribution. For the electrons,
w' > wand their destabilizing influence is decreased.
From this it can be concluded that a magnetic well
tends to stabilize the universal instability.
In an actual curved field the resonance condition is
B
RESONANCE
ELLIPSE
f=CONSTANT A
--'------'---"IL1..'-:Iv:-e---------Ll-~VIl
FIG. 1. Locus of resonant electrons.
3530 w-{kL/OR) (v2+iv2)-k v =0 (1)
II 1 II II '
where 0 is the cyclotron frequency and R is the
radius of curvature of a field line. Unfortunately,
(1) leads to intractable integrals in the dispersion
relation. The slab model with gravity corresponds
to replacing (v~ + i v~) by its thermal average. An
alternative procedure, used by Laval et al. 8 is to
replace only the v: term by its thermal average.
The approximate resonance condition becomes
w -(kjOR) (v~ + i <v~» -k v = O. (2) ... J) II
USing (2), they found that the stabilizing influence of
a magnetic well was significantly greater than that
given by the slab model with gravity. They inter
preted this to be due to a "Landau effect in the di
rection of the drift". This paper proposes, however,
that the results of Laval can still be interpreted as
a "Doppler shift" of the resonant velocity, but of
greater magnitude than that given by the gravity
Doppler shift. We present a graphical technique for
determining the validity of various resonance approx
imations.
Recall that the universal instability occurs primarily
when the resonant ion velocity along B is much
greater than Vi' The Doppler shift is proportional to
v~ and thus the resonant ions experience a greater
Doppler shift in a curved field than a thermal ion.
Since the resonant ion contribution is proportional to
exp{ -v~ Iv~) evaluated at the resonant velocity, this
effect is Significant. The gravity model assumes that
the resonant ions experience the same Doppler shift
as the thermal ions and thus underestimates the
stabilizing effect of a magnetic well.
We can visualize the effect of the various resonance
conditions in the following way. We rewrite (1) as
V + __ II + _L_= __ + __ II
( ORk ) 2 v 2 RwO R202k2
" 2k l 2 k l 4k~' (3)
B
a
FIG. 2. Locus of resonant ions when w/k Ii Vj « (RTe/LT//~
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131.111.185.72 On: Thu, 18 Dec 2014 14:43:45CURVATURE STABILIZATION OF THE UNIVERSAL INSTABILITY 3531
o c
W/k vII
II
FIG. 3. Locus of resonant ions when w/kll vi» (R Te/L Ti) 1:2
which is the equation of an ellipse in the VII' V ~ plane
with eccentricity 1/12 and centered at v~ = 0,
VII = -nRkj2kJ.. This ellipse represents the locus
of resonant particles in the VII' V J. plane.
If we consider values typical for drift waves in fu
sion plasmas (w-w*-kJ. v;/neL, w/kll < v., and
R/L > 1), we find that for the electrons, the second
term on the right side of (3) dominates. Since ne < 0,
the ellipse for the electrons looks like that shown in
Fig. 1. A local Maxwellian distribution j is constant
on circles centered about the origin and decreases
as exp[ -(v~ + v~)/v;] in the radial direction. Reso
nant particle effects are proportional to j, oj /ov '"
and oj/ovJ. which are largest for VII' vJ. ~ve' Hence
the dominant resonant particle contribution comes
from the region on the ellipse nearest the origin,
i. e. , near point a. At a, v II ~ w /k II and hence the
curvature drift has little effect on the resonant elec
tron contribution. Laval's approximation is to re
place the ellipse by the two lines A and B as the
locus of resonant particles; line A is insignificant
and line B appears to be a reasonable approximation
to the important part of the ellipse. The gravity
approximation is to consider a single line very close
to B; this also appears reasonable.
The situation is quite different for the ions. Of the two terms on the right side of (3), either term can
dominate depending on w/kll Vi compared with (RTe/
LTy/2, (assumingw-w*). For l<w/kllv/
«(RTe/LT/)1/2, the second term dominates and the
ellipse appears as in Fig. 2. When w/kll Vi «(RTe/
LTi)1/2, we get the ellipse shown in Fig. 3. In both
cases the major part of the resonant contribution
comes from points on the ellipse near a [where VII
~ w/kll in Fig. 2 and VII ~ (Rwn/kJ.)1/2
-vi(RTe/
LTi)1/2 in Fig. 3]. Laval's approximation again con
sists of replacing the ellipse by the two lines A and
B. The gravity model uses a single line C near w/kll
which, in the case of Fig. 3, underestimates the
resonant ion contribution.
This graphical technique for determining the impor
tant regions of the resonance ellipse can be used for
other distribution functions. For example, if the ion
distribution is bi-Maxwellian with TJ. > 2 T,l' the im
portant region of the resonance ellipse in Fig. 3 is
the top. This requires a different approximation to
the resonance condition when w/kll is sufficiently
large.
This work is supported in part by the Atomic Energy
Commission and by the Wisconsin Alumni Research
Foundation.
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[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
131.111.185.72 On: Thu, 18 Dec 2014 14:43:45 |
1.3177269.pdf | Pitch bending and glissandi on the clarinet: Roles of the vocal
tract and partial tone hole closure
Jer-Ming Chen,a/H20850John Smith, and Joe Wolfe
School of Physics, The University of New South Wales, Sydney, New South Wales 2052, Australia
/H20849Received 9 March 2009; revised 15 June 2009; accepted 17 June 2009 /H20850
Clarinettists combine non-standard fingerings with particular vocal tract configurations to achieve
pitch bending, i.e., sounding pitches that can deviate substantially from those of standard fingerings.Impedance spectra were measured in the mouth of expert clarinettists while they played normallyand during pitch bending, using a measurement head incorporated within a functioning clarinetmouthpiece. These were compared with the input impedance spectra of the clarinet for the fingeringsused. Partially uncovering a tone hole by sliding a finger raises the frequency of clarinet impedancepeaks, thereby allowing smooth increases in sounding pitch over some of the range. To bend notesin the second register and higher, however, clarinettists produce vocal tract resonances whoseimpedance maxima have magnitudes comparable with those of the bore resonance, which then mayinfluence or determine the sounding frequency. It is much easier to bend notes down than up becauseof the phase relations of the bore and tract resonances, and the compliance of the reed. Expertclarinettists performed the glissando opening of Gershwin’s Rhapsody in Blue . Here, players
coordinate the two effects: They slide their fingers gradually over open tone holes, whilesimultaneously adjusting a strong vocal tract resonance to the desired pitch.©2009 Acoustical Society of America. /H20851DOI: 10.1121/1.3177269 /H20852
PACS number /H20849s/H20850: 43.75.Pq, 43.75.St /H20851NHF /H20852 Pages: 1511–1520
I. BACKGROUND
A. Pitch bending and glissandi
Pitch bending refers to adjusting the musical pitch of a
note. Usually it means a smooth variation in pitch and caninclude portamento andglissando , which refer to continuous
variation of pitch from one note to the next. The pitch ofsome instruments can be varied continuously over a widerange by adjusting the position of the hands and fingers, e.g.,the slide trombone or members of the violin family. The
pitch of some fretted string instruments, e.g., the guitar andsitar, can also be varied smoothly over a restricted range bymoving the finger that stops the string on the fingerboard,thereby changing the string tension. The pitch of lip-valveinstruments can be altered via changes in lip tension /H20849lip-
ping/H20850. In woodwind instruments, each particular configura-
tion of open and closed tone holes is called a fingering, andeach fingering is associated with one or more discrete notes.Woodwinds are, however, capable of pitch bending, either bypartially opening/closing tone holes or by playing techniquesthat involve the player’s mouth, vocal tract, breath, and inthe case of some air-jet woodwinds such as the flute andshakuhachi, adjusting the extent of baffling with the face.
On the clarinet, partially covering a tone hole can be
used to achieve pitch bending when a transition betweennotes uses a tone hole covered directly by a finger rather thanby a pad. Seven tone holes are covered directly by the fin-gers, allowing bending by this method over the range G3/H20849175 Hz /H20850to G4 /H20849349 Hz /H20850and from D5 /H20849523 Hz /H20850upwards./H20849The clarinet is a transposing instrument; clarinet written
pitch is used in this paper—one musical tone above soundingpitch. /H20850Substantial pitch bending using the vocal tract, how-
ever, is usually possible only over the upper range of theinstrument /H20849Pay, 1995 /H20850, typically above about D5 /H20849523 Hz /H20850,
although the actual range depends on the player. Further, thisbending is asymmetric: Although expert players can use theirvocal tract and embouchure to lower the pitch by as much asseveral semitones, they can only raise the pitch slightly/H20849Rehfeldt, 1977 /H20850. Similar observations apply to saxophones,
whose reed and mouthpiece are somewhat similar to those ofclarinets.
Pitch bending on the clarinet is used in several musical
styles including jazz and klezmer . In concert music, the most
famous example is in the opening bar of Gershwin’s Rhap-
sody in Blue /H20849Fig. 1/H20850, which features a clarinet playing a
musical scale over two and half octaves. At the composi-tion’s first rehearsal, the clarinettist replaced the last severalnotes in the scale with a glissando /H20849Schwartz, 1979 /H20850. This
delighted the composer and started a performance tradition.Figure 1shows the spectrogram of a performance in this
style, in which the glissando spans an octave from C5 /H20849466
Hz/H20850to C6 /H20849932 Hz /H20850at the end of the run. It is explained in
Sec. III B why the glissando usually replaces only the last
several notes, as shown in Fig. 1.
This solo in Rhapsody in Blue /H20849including the perfor-
mance tradition /H20850is such a standard part of the clarinet rep-
ertoire that it is well known to most professional clarinettists.It is therefore used as the context for part of this study intothe roles that a player’s vocal tract and partial covering oftone holes can play in pitch bending. For the rest of the
study, an artificial exercise was used: Players were simplyasked to bend down the pitch of standard notes on the clari-
a/H20850Author to whom correspondence should be addressed. Electronic mail:
jerming@phys.unsw.edu.au
J. Acoust. Soc. Am. 126 /H208493/H20850, September 2009 © 2009 Acoustical Society of America 1511 0001-4966/2009/126 /H208493/H20850/1511/10/$25.00
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 193.0.65.67 On: Tue, 09 Dec 2014 16:38:21net, using only their vocal tracts. Acoustic impedance spectra
of the clarinet bore were measured using techniques reportedpreviously /H20849Dickens et al. , 2007a ,2007b /H20850, while impedance
spectra inside the player’s mouth were measured using animpedance head built into the mouthpiece of the clarinet sothat the player could perform with very little perturbation.
B. The sounding frequency of the clarinet
The sounding frequency f0of the clarinet is determined
by several interacting effects and thus depends on a numberof parameters. Some of these effects are modest /H20849such as lip
damping, bite configuration, jaw force, and blowing pres-sure/H20850and, in this study, the aim was to hold them constant
rather than to examine them in detail. For example, f
0de-
pends on the natural frequency of the reed. For a given reed,this can be adjusted by the player during performance byapplying greater or lesser force with the lower jaw, therebyvarying its effective length and vibrating mass and thus in-fluencing the sounding pitch slightly.
The clarinet has a range of rather more than three oc-
taves and, to first order, stable reed oscillation /H20849at the sound-
ing frequency f
0/H20850occurs near one of the maxima in the
acoustic impedance Zloadthat loads the reed generator,
which, along with the pressure difference between mouth-piece and bore, determines the airflow into the instrument/H20849Fletcher and Rossing, 1998 /H20850. The acoustic pressure differ-
ence across the reed is /H9004p=p
tract−pborewhere ptractandpbore
are, respectively, the acoustic pressures in the mouth near the
reed /H20849upstream /H20850and in the clarinet mouthpiece near the reed
/H20849downstream /H20850.I fUis the acoustic flow passing through the
aperture between the reed and the mouthpiece, then Zload
=/H9004p/U.Benade /H208491985 /H20850offered a considerable simplification
of processes at the reed junction, applied continuity of vol-ume flow, and assumed that p
tractandpboreboth act on equal
areas of the reed. He then showed that the impedance loadingthe reed generator is given byZ
load=Zreed/H20849Zbore+Ztract/H20850
Zreed+Zbore+Ztract=/H20849Zbore+Ztract/H20850/H20648Zreed. /H208491/H20850
This impedance includes contributions from the bore of the
instrument /H20849Zbore/H20850and the player’s vocal tract /H20849Ztract/H20850where
both are measured near the reed. It also includes the effective
impedance of the clarinet reed /H20849Zreed/H20850itself:/H9004pdivided by
the volume flow due to reed vibration. The second expres-
sion in Eq. /H208491/H20850is included to show explicitly that, in this
rudimentary model, ZtractandZboreare in series, and their
sum is in parallel with Zreed. Consequently, under conditions
in which the vocal tract impedance is small compared to thebore impedance, Z
loaddepends only on Zboreand Zreed
alone—indeed, the maximum in measured impedance of the
clarinet bore in parallel with the reed corresponds closely tothe pitch in normal playing /H20849Benade, 1985 /H20850. On the other
hand, if the player were able to make Z
tractlarge and compa-
rable to Zbore, the player’s vocal tract could significantly in-
fluence, or even determine, the sounding frequency of theplayer-instrument system.
The interaction of bore, reed, and airflow is inherently
nonlinear and the subject of a number of analyses and ex-perimental studies /H20849e.g., Backus, 1963 ;Wilson and Beavers,
1974 ;Benade, 1985 ;Grand et al. , 1996 ;Fletcher and Ross-
ing, 1998 ;Silva et al. , 2008 /H20850. Although the nonlinear effects
must be considered to understand the threshold pressure forblowing and features of the waveform and spectrum, there isagreement that the playing frequency can be explained toreasonable precision in terms of the linear acoustics of thebore, vocal tract, and reed, using Benade’s /H208491985 /H20850model.
Specifically, the operating frequency lies close to the fre-quency at which the imaginary part of the acoustical load iszero, which in turn is very near or at a maximum in themagnitude of the impedance. The present experimental paperconsiders only the linear acoustics of the bore, the vocal tractand the compliance of the reed.
C. Influence of the player’s vocal tract
Some pedagogical studies on the role of the player’s
vocal tract in woodwind performance report musicians’ opin-ions that the tract affects the pitch /H20849Pay, 1995 ;Rehfeldt,
1977 /H20850. Scientific investigations to date include numerical
methods, modeling the tract as a one-peak resonator/H20849Johnston et al. , 1986 /H20850, electro-acoustic analog simulations
in the time domain /H20849Sommerfeldt and Strong, 1988 /H20850, and
digital waveguide modeling /H20849Scavone, 2003 /H20850.Clinch et al.
/H208491982 /H20850used x-ray fluoroscopy to study directly performing
clarinettists and saxophonists and concluded that “vocal tractresonance frequencies must match the frequency of the re-quired notes” played. Backus /H208491985 /H20850later observed that the
player’s vocal tract impedance maxima must be similar orgreater in magnitude than the instrument bore impedance inorder to influence performance, but concluded that “reso-nances in the vocal tract are so unpronounced and the im-pedances so low that their effects appear to be negligible.”On the other hand, Wilson /H208491996 /H20850while investigating pitch
bending concluded that the upstream vocal tract impedanceat the fundamental frequency in pitch bending must be largeand comparable to the downstream bore impedance, but did
FIG. 1. /H20849Color online /H20850The opening of Gershwin’s Rhapsody in Blue. The
upper figure shows the 2.5 octave run as it is written—but not as it is usuallyplayed. In traditional performance, the last several notes of the scale arereplaced with a smooth glissando . The lower figure is a spectrogram of such
a performance. The opening trill on G3 /H20849written, 174 Hz /H20850is executed from 0
t o2sa n df o llowed by a scale-like run at 2.5–3.5 s that becomes smooth
pitch rise from C5 /H20849466 Hz /H20850to C6 /H20849932 Hz /H20850at 3.5–5.7 s.
1512 J. Acoust. Soc. Am., Vol. 126, No. 3, September 2009 Chen et al. : Clarinet pitch bending and glissandi
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 193.0.65.67 On: Tue, 09 Dec 2014 16:38:21not report details on the vocal tract resonance frequency or
the magnitude of its impedance. Watkins /H208492002 /H20850summarized
several empirical studies of the use of the vocal tract and itsreported or measured geometry in saxophone performance.
The environment in the clarinettist’s mouth when play-
ing poses challenges for direct measurements of vocal tractproperties during performance. The vibrating reed generateshigh sound pressure levels in the mouth: Backus /H208491961 /H20850and
Boutillon and Gibiat /H208491996 /H20850reported sound levels of 166 dB
and exceeding 170 dB inside the mouthpiece of a clarinetand saxophone, respectively, when artificially blown. Thestatic pressure and humidity in the mouth complicate mea-surements. To avoid these difficulties, some previous mea-surements of the musician’s vocal tract /H20849Wilson, 1996 /H20850were
made under conditions that were somewhat different fromnormal performance. Fritz and Wolfe /H208492005 /H20850made acoustic
impedance measurements inside the mouth by having themusician mime with the instrument for various musical ges-tures, including pitch bending. The peaks in impedance mea-sured in the mouth were as high as a few tens of MPa s m
−3
and so comparable with those of the clarinet bore, but nosimple relation between the frequencies of the peak and thenote played was reported.
More recently, Scavone et al. /H208492008 /H20850developed a
method to provide a real-time measurement of vocal tractinfluence on saxophones while the player performs a varietyof musical effects. This method, developed from Wilson’s
/H208491996 /H20850indirect technique, uses microphones in the mouth-
piece, one on the tract side and one on the bore side. It usesthe played note and its harmonics as the measurement signal,and so measures impedance ratios of the upstream vocal tractto downstream saxophone bore at harmonics of the soundingreed. This has the advantage of simplicity, a strong signal,and is fast enough to provide real-time feedback to players.However, it does not measure vocal tract resonance fre-quency or the magnitude of impedance at the tract resonance.Using this method, they found that during pitch bending onthe alto saxophone, the pressure component at the playingfrequency was larger in the player’s mouth than in the bore,from which it may be surmised that a strong vocal tract reso-nance influences behavior of the reed.
At the same time, the authors /H20849Chen et al. , 2008 /H20850re-
ported direct impedance spectra measurements made insidethe mouth during performance and described how experttenor saxophonists can produce maxima in Z
tractand tune
them to produce notes in the very high /H20849altissimo /H20850range of
that instrument. Several amateur saxophonists, on the other
hand, were found not to exhibit tuning, and consequentlywere unable to play in the altissimo range. The saxophonehas a single reed mouthpiece, somewhat similar to that of aclarinet. A major difference, however, is that the saxophone’sbore is nearly conical, while that of the clarinet is largelycylindrical. This difference has the result that, for high notes,the maxima in Z
boreon the saxophone /H20849Chen et al. , 2009 /H20850are
rather smaller than those in the clarinet /H20849Backus, 1974 ;Dick-
enset al. , 2007b /H20850, so saxophone players can achieve the
condition Ztract/H11015Zborediscussed above with weaker reso-
nances of the vocal tract.II. MATERIALS AND METHODS
A. Measurements of bore impedance
Measurements of the acoustic impedance of the clarinet
bore for the fingering positions employed in pitch bendingwere made on a B-flat soprano clarinet, the most commonmember of the clarinet family /H20849Yamaha model CX /H20850.Z
borewas
measured using the three-microphone-two-calibration/H208493M2C /H20850method with two non-resonant loads for calibration
/H20849Dickens et al. , 2007a /H20850: one was an open circuit /H20849nearly in-
finite impedance /H20850and the other an acoustically infinite wave-
guide /H20849purely resistive impedance /H20850. This method allows the
measurement plane to be located near the reed tip “lookinginto” the clarinet bore /H20849Fig.2/H20850without the involvement of a
player. Clarinet bore impedances were then measured forstandard fingering positions, as well as for some with partialuncovering of the tone holes. The excitation signal for thesemeasurements was synthesized as the sum of sine wavesfrom 100 to 4000 Hz with a spacing of 1.35 Hz. The mea-surements of standard fingerings gave results similar to thosereported previously /H20849Dickens et al. , 2007b /H20850. A professional
clarinettist was engaged for measurements involving partialuncovering of tone holes. Fingering gestures that would betypical when executing the glissando in Rhapsody in Blue
were used, involving the gradual and consecutive sliding ofone’s fingertips off the keys in order to uncover smoothly theopen finger holes of the clarinet, starting from the lowest.Acoustic impedance spectra of the clarinet bore were mea-sured at varying stages of the finger slide while the clarinet-tist temporarily halted the slide for several seconds.
B. Measurements of effective reed compliance
Representative values for the effective compliance of the
clarinet reed during playing were measured using Benade’s
/H208491976 /H20850technique: measuring the sounding frequency pro-
duced by a clarinet mouthpiece /H20849with reed /H20850that is attached to
FIG. 2. Schematic of the 3M2C technique used to measure the acoustic
impedance of the clarinet bore. The loudspeaker provides a stimulus synthe-sized as the sum of 2900 sine waves, while three microphones at knownspacings record the response from the clarinet being measured. Not to scale.
J. Acoust. Soc. Am., Vol. 126, No. 3, September 2009 Chen et al. : Clarinet pitch bending and glissandi 1513
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 193.0.65.67 On: Tue, 09 Dec 2014 16:38:21various lengths of pipe and played mezzoforte in a “normal”
style, i.e., without using the vocal tract to adjust the pitch.Comparing the lengths of pipe for each tone with the calcu-lated lengths of simple tubes /H20849closed at one end /H20850having a
natural frequency matching the played one, the equivalentcompliance and volume of the mouthpiece under playingconditions for a wide range of frequencies can be calculatedby reducing the mouthpiece volume /H20849having the tubes as
deep as possible /H20850, then treating the calculated compliance as
the sum of the compliance of the remaining mouthpiece vol-ume and the compliance of the reed. Here, Benade’s /H208491976 /H20850
technique relies on the assumption that the impedance of theair in the gap between reed and mouthpiece does not contrib-ute to the end effect. The magnitude of the ac component ofthe flow resistance is discussed in Sec. III B.
Three Légère synthetic reeds of varying hardness /H208491
3
4,
21
2, and 3 /H20850were used in combination with cylindrical metal
pipes with internal diameter of 14.2 mm and external diam-eter of 15.9 mm, and lengths of 99, 202, 299, and 398 mm.The reed compliance thus calculated was consistent for allpipe lengths listed. A reed of hardness 3 played with a tightembouchure, typical for normal playing, yielded an averageequivalent volume of 1.1 ml, which equals the value givenbyNederveen /H208491998 /H20850. This value, treated as a pure compli-
ance, is used for Z
reedin calculations here, i.e., reed losses
are neglected. /H20849The effect of different reeds and of the lip
force applied to them was not studied in detail here. How-ever, it is worth nothing that a reed of hardness 3 played witha relaxed embouchure yielded an equivalent volume of 1.7
ml, and that the softest reed /H20849hardness 1
3
4/H20850played with a
relaxed embouchure gave 2.7 ml. Clarinettists are well awarethat one can play flat with a relaxed embouchure, and espe-cially with a soft reed. /H20850
C. Measurements of vocal tract impedance
The acoustic impedance of the player’s vocal tract was
measured directly during performance using an adaptation ofa technique reported previously /H20849Tarnopolsky et al. , 2006 ;
Chen et al. , 2008 /H20850and based on the capillary method /H20851meth-
ods reviewed by Benade and Ibisi /H208491987 /H20850andDickens et al.
/H208492007a /H20850/H20852. An acoustic current is injected into the mouth via a
narrow tube incorporated into a standard clarinet mouthpiece/H20849Yamaha 4C /H20850—see Fig. 3. The internal cross section of this
narrow tube is approximately rectangular with an area of2m m
2, giving a characteristic source impedance around
200 MPa s m−3. The sound pressure inside the mouth is
measured via an adjacent tube embedded into the mouth-piece. This cylindrical tube /H20849internal diameter of 1.2 mm /H20850is
connected to a microphone /H20849Brüel & Kjær 4944A /H20850located
just outside the mouthpiece to form a probe microphone. Thesystem thus measures the impedance looking into the vocaltract from a location inside the mouth just past the vibratingreed. It is calibrated by connecting the modified mouthpieceto the quasi-infinite tube used as a standard /H20849Smith et al. ,
1997 /H20850, which has an internal diameter of 26.2 mm /H20849compa-
rable in size with the vocal tract /H20850and length 197 m. To mini-
mize the perturbations to the players, they used their ownclarinet and a reed of their choice with the experimentalmouthpiece. They reported only moderate perturbation totheir playing—presumably because the mouthpiece geometryremains almost unchanged, except for an increase in thick-ness of about 1.5 mm at the bite point. /H20849Indeed, some players
routinely add a pad of up to 0.8 mm thickness at this point. /H20850
The acoustic impedance spectrum for each particular vo-
cal tract configuration was measured by injecting a calibratedacoustic current /H20849synthesized as the sum of 336 sine waves
from 200 to 2000 Hz with a spacing of 5.38 Hz /H20850during
playing. Each configuration was measured for 3.3 s to im-prove the signal-to-noise ratio. Because measurements aremade during playing, the signal measured by the microphonewill necessarily include the pressure spectrum of the reedsounding in the mouth. This produces clearly recognizableharmonics of the note played in the raw impedance spectrumwith amplitudes much higher than those of the vocal tractitself, and allows the sounding frequency f
0to be deter-
mined. To determine the Z/H20849f/H20850measured in the mouth, these
broadly spaced, narrow peaks are removed and replaced with
interpolation. There are also high levels of broad band noiseproduced in the mouth. For this reason, the spectrum is thensmoothed using a third order Savitsky–Golay filter typicallyover 11 points /H20849/H1100630 Hz /H20850for magnitude values and 15 points
/H20849/H1100640 Hz /H20850for phase values. An example is shown in Fig. 4.
D. Players and protocols
Five expert clarinettists /H20849four professional players and 1
advanced student /H20850were engaged for the measurements on the
players’ vocal tract. They used the modified mouthpiece ontheir own clarinet and performed the following tasks.
/H20849i/H20850 They played the opening bar of George Gershwin’s
Rhapsody in Blue /H20849Fig. 1/H20850, first normally, then later
pausing for a few seconds at various points in theglissando while their vocal tract impedance was mea-
sured.
/H20849ii/H20850They were asked to play chromatic notes in the range
G4/H20849349 Hz /H20850to G6 /H208491397 Hz /H20850using standard finger-
ings and their normal embouchure and tract configu-ration, and to hold each note while the impedance intheir mouth was measured.
/H20849iii/H20850They used standard fingerings for chromatic notes in
the range G4 /H20849349 Hz /H20850to G6 /H208491397 Hz /H20850and were
asked to bend each sounding pitch progressively
FIG. 3. Photograph of the modified mouthpiece used to measure the acous-
tic impedance of the player’s vocal tract during performance. Tube A isattached to the microphone whereas tube B is attached to the calibratedsource of acoustic current. The circular inset to the left shows a magnifiedview of the mouthpiece tip. The circular end of tube A and the rectangularcross section of tube B are visible just above the reed.
1514 J. Acoust. Soc. Am., Vol. 126, No. 3, September 2009 Chen et al. : Clarinet pitch bending and glissandi
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 193.0.65.67 On: Tue, 09 Dec 2014 16:38:21down from its standard pitch and to hold it steady
while the impedance in their mouth was measured.
III. RESULTS AND DISCUSSION
A. Effects of partial tone hole closure on the clarinet
bore impedance
Acoustic impedance measurements of the clarinet bore
/H20849Zbore/H20850for standard fingerings show the expected, well-
spaced maxima indicating the bore resonances near which
the clarinet reed operates in normal playing /H20849Fig. 5/H20850. This
same figure also shows measurements using a fingering withone tone hole partly uncovered, to different extents, by afinger slide. These fingerings with different extents of holeuncovering show impedance maxima at frequencies interme-
diate between those used for notes on the diatonic scale. Thepeaks, however, are lower than those for standard fingerings,particularly when the hole is nearly closed: The differencecan be as large as tens of percent /H20849Fig. 5/H20850. Thus the finger
slide allows sounding pitch to increase smoothly by gradu-ally raising the bore resonance, instead of moving in discretesteps as in a normal musical scale. Over part of the clarinet’srange, this technique alone, with no vocal tract or embou-chure adjustments, could contribute to the smoothly varyingsounding pitch. However, musicians report that this is onlyone of the effects used to produce a glissando .
B. Vocal tract resonance and glissandi
Figure 6shows the measured impedance of the clarinet
bore /H20849shown here with Zreedin parallel /H20850, and that of a typical
result for the impedance measured in the mouth /H20849Zmouth/H20850dur-
ing normal playing /H20849top/H20850. For comparison, it also shows a
typical measurement of the acoustic impedance in the mouthof a player performing a glissando /H20849middle and bottom /H20850. This
measured impedance in each case is simply added in serieswith the clarinet bore impedance, and then the effective reedimpedance added in parallel to obtain an estimate of the ef-fective acoustic impedance of the tract-reed-bore system ac-cording to the simple model represented by Eq. /H208491/H20850. In both
cases, the player’s fingers were fixed at the fingering used forA5.
In some cases, the impedance measured in the mouth
Z
mouth is expected to be a good approximation to that of the
vocal tract Ztract. It is complicated, however, by the presence
in the mouth of the reed and the acoustic component of vol-
FIG. 4. The magnitude of the acoustic impedance spectrum measured in the
mouth for a normal vocal tract configuration /H20849and embouchure /H20850playing the
written pitch C6 /H20849932 Hz /H20850with standard fingering. Because measurements
/H20849thin line /H20850are made during playing, harmonics of the note sounded appear
added to the impedance spectrum—at 935 and 1870 Hz in this case. Theseare removed and replaced with interpolation, and the data then smoothed/H20849thick line /H20850to produce the vocal tract impedance spectra used in this paper.
Here, the measured resonance is at 1225 /H1100625 Hz.
FIG. 5. The measured acoustic impedance at the mouthpiece of the clarinet
/H20849Zbore/H20850for different fingerings. The second maxima only shown here. Solid
lines indicate the standard fingerings for the written pitches D5 /H20849523 Hz /H20850,E 5
/H20849587 Hz /H20850,F 5 /H20849622 Hz /H20850, and G5 /H20849698 Hz /H20850, and show maxima spaced at
discrete frequencies corresponding to those notes. Dashed lines indicate asequence of fingerings that use partial covering of the hole that is opened tochange from F5 to G5.
FIG. 6. Measured input impedance of the clarinet, Zbore, shown here with the
reed compliance in parallel, Zbore/H20648Zreed/H20849pale line /H20850, and the impedance mea-
sured in the mouth, Zmouth /H20849dark line /H20850. The impedance of the reed, Zreed
/H20849dotted line /H20850, was calculated from the measured reed compliance. Zload
=/H20849Zmouth+Zbore/H20850/H20648Zreedis plotted as a dashed line. In both cases the fingering
is for the note written A5 /H20849784 Hz /H20850. Arrows indicate the frequency f0of the
sounded note. The top graph shows the impedance magnitude for the noteplayed normally. The middle graph shows that for the same fingering, asplayed in the glissando exercise. At this stage of the glissando , the sounding
frequency is 76 Hz /H20849190 cents, almost a whole tone /H20850below that of the
normal fingering. The bottom graph shows the phases of the impedanceswhose magnitudes are shown in the middle graph.
J. Acoust. Soc. Am., Vol. 126, No. 3, September 2009 Chen et al. : Clarinet pitch bending and glissandi 1515
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 193.0.65.67 On: Tue, 09 Dec 2014 16:38:21ume flow past the reed. In the simple Benade /H208491985 /H20850model,
the measured impedance would be Ztractin parallel with
/H20849Zbore+Zc/H20850, where Zcis the combined impedance of the reed
and the intermittent air gap beside it. The impedance of the
reed, assumed largely compliant, is discussed above, and hasa magnitude of a few tens of MPa s m
−3in the range of
interest. The impedance through the intermittent gap includesboth the inertance of the air in the gap and the flow resistanceacross it. For a typical blowing pressure of a few kilopascaland a cross section of about 10 mm
2, and considering only
the Bernoulli losses, this is also some tens of MPa s m−3.
Despite their geometry, the two components are not simplyin parallel, because when the reed’s motion produces a vol-ume flow into the bore, it also tends to reduce the aperture,and conversely.
Because Z
mouth is the parallel combination of Ztractand
/H20849Zc+Zbore/H20850, it may sometimes be an underestimate of Ztract,
especially when Ztractis large. An estimate of the lower
bound for Zccan be gained from the magnitude of Zmouth
measured when Zboreis small. In Fig. 6, for example, Zmouth
is 17 MPa s m−3at 942 Hz, when Zboreis a minimum at
0.4 MPa s m−3, over 40 times smaller. In the cases of pitch
bending, Zmouth is measured as high as 60 MPa s m−3at fre-
quencies when Zboreis a few MPa s m−3. Hence Zmouth is
likely to be a significant underestimate of Ztractonly when the
latter is several tens of MPa s m−3. For this reason, Zmouth is
simply shown in Fig. 6as an estimate of Ztractand is used in
estimating Benade’s /H208491985 /H20850effective impedance Zload
/H11061/H20849Zmouth+Zbore/H20850/H20648Zreed.
In normal playing /H20849top graph in Fig. 6/H20850, the magnitudes
of the peak in Zmouth /H2084920 MPa s m−3in this example /H20850is
about half that of the peaks of Zbore/H2084941 MPa s m−3here/H20850,
smaller than the effective impedance of the reed/H20849/H1101125 MPa s m
−3at these frequencies /H20850, and also smaller than
those of the peak in Zbore/H20648Zreed/H2084944 MPa s m−3here/H20850.1Con-
sequently, according to the simple Benade /H208491985 /H20850model ex-
pressed in Eq. /H208491/H20850, the combined acoustic impedance for nor-
mal playing yields a resulting maximum determined largelyby the maximum in Z
bore: The reed vibrates at a frequency
/H20849781 Hz /H20850matching the strongest peak in Zbore/H20648Zreed/H20849which is
close to the peak in Zbore/H20850.
In the glissando exercise, however, the maximum im-
pedance measured in the mouth is consistently comparable inmagnitude with that of the maximum of the bore impedanceand the effective impedance of the reed. The middle graph inFig. 6shows that, because here the peak in Z
mouth is no
longer small compared with Zbore, the peak in /H20849Zmouth
+Zbore/H20850/H20648Zreedis no longer determined solely by a peak in
Zbore. In this example, the Zmouth maximum /H2084932 MPa s m−3/H20850
centered at 705 Hz is more comparable in magnitude with
the corresponding Zbore/H20648Zreedmaximum /H2084944 MPa s m−3/H20850.
Here, the sounding frequency during the glissando /H20849indicated
by an arrow /H20850is about 76 Hz /H20849190 cents or about one whole
tone/H20850lower than that produced for normal playing, while the
peak in /H20849Zmouth+Zbore/H20850/H20648Zreedis calculated to fall at 662 Hz,
119 Hz lower than the peak in Zbore/H20648Zreed/H20849781 Hz /H20850. In most
of the glissando examples studied, sounding frequency f0did
not coincide with the peak in Zbore/H20648Zreedbut instead occurred
closer to the peak in /H20849Zmouth+Zbore/H20850/H20648Zreed. However, f0wasusually about 10–40 Hz above the peak in /H20849Zmouth
+Zbore/H20850/H20648Zreed. This difference might be due to the simplicity
of the model used to derive Eq. /H208491/H20850. Also, a smaller value of
Zreedwould give rise to a higher frequency for the peak in
/H20849Zmouth+Zbore/H20850/H20648Zreed, so the difference might also be ex-
plained if the compliance of the reed in this situation were
higher than in normal playing condition from which it wasestimated. /H20849The compliance of the reed could, in principle,
be reduced by biting harder on the reed. However, playersreport no changes in the lip force during this exercise. Un-fortunately, it would be difficult to determine independentlythe value of the reed compliance for the pitch bending play-ing condition, because Benade’s /H208491976 /H20850technique assumes a
non-negligible vocal tract impedance. /H20850
As explained above, the clarinet’s resonance dominates
in normal playing and the player’s tract has only a minoreffect, as shown in the top part of Fig. 6. However, the region
of the glissando in Rhapsody in Blue /H20849from C5 to C6,
written—i.e., 466–932 Hz /H20850lies in the clarinet’s second reg-
ister, a range where the clarinet resonances have somewhatweaker impedance peaks than those in the lower register. Inthis range, experienced players can produce a resonance inthe vocal tract whose measured impedance peak is compa-rable with or sometimes even larger in magnitude than thoseof the clarinet. Consequently, by tuning a strong resonance ofthe vocal tract and skillfully adjusting the fingering simulta-neously, expert clarinettists can perform a glissando and
smoothly control the sounding pitch continuously over alarge pitch range. In performing this glissando , the sounding
pitch here need only deviate from that of the fingered note bya semitone or so. However, greater deviations are possible.To study this, a simple pitch bending exercise was used.
C. Vocal tract resonance in normal playing and pitch
bending
Figure 7shows the resonance frequency measured in the
mouth for the five test subjects as they played. It is plottedagainst the sounding pitch for both normal playing and pitchbending in the range between G4 /H20849349 Hz /H20850and G6 /H208491397
Hz/H20850. This plot shows the extent of vocal tract tuning: If the
players tuned a resonance of the vocal tract to the noteplayed, then the data would lie close to the tuning line y=x,
which is shown as a gray line. If players maintained a con-stant vocal tract configuration with a weak resonance and thesounding pitch were determined solely by /H20849Z
bore/H20648Zreed/H20850, the
data would form a horizontal line. The magnitude of the
impedance peak is indicated on this graph by the size of thesymbol used, binned in half decades as indicated by the leg-end.
Above about 600 Hz /H20849written E5 /H20850, the data for pitch
bending /H20849black circles /H20850show clear tuning: The sounding fre-
quency f
0is always close to that of an impedance peak mea-
sured in the player’s mouth. Below this frequency, the ex-amples where the peaks in Z
mouth are large /H20849indicated by large
circles /H20850also follow the tuning line. In the range below 600
Hz, examples of /H20849intended /H20850pitch bending with relatively
small peaks /H20849small black circles /H20850sometimes deviate from the
tuning line: In these cases, the player has not succeeded in
1516 J. Acoust. Soc. Am., Vol. 126, No. 3, September 2009 Chen et al. : Clarinet pitch bending and glissandi
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 193.0.65.67 On: Tue, 09 Dec 2014 16:38:21having the instrument play at the frequency determined by a
resonance in the mouth. The legend shows, for comparison,the magnitude of the peaks in Z
borefor fingerings in the first
and second registers of the clarinet. /H20851For the purposes of this
discussion, the second register can be defined to be the notesplayed using the second peak in impedance. Loosely speak-ing, these are the notes that use the second mode of the bore.Thus the second register goes from written B4 /H20849440 Hz /H20850to
C6/H20849932 Hz /H20850./H20852Comparison with the size of the peaks in the
bore and the tract impedance gives one reason why pitchbending is easier in the second register and higher, wherepeaks in Z
boreare smaller.
The range of frequencies over which the vocal tract is
used for pitch bending in the second register of the clarinet/H20849well within the normal range of the instrument /H20850is compa-
rable with the range for which Scavone et al. /H208492008 /H20850reported
vocal tract effects for the alto saxophone /H20849520–1500 Hz /H20850.
This range is also comparable with that reported for vocaltract tuning in a study on tenor saxophones: To play in thevery high /H20849altissimo /H20850range, saxophonists tune their vocal
tract resonance /H20849Chen et al. , 2008 /H20850, but they do not do so in
the normal range. However, the tenor saxophone is a tenorinstrument and its altissimo range /H20849above written F#6, sound-
ing E5, 659 Hz /H20850corresponds approximately to the upper sec-
ond and third registers on the clarinet and to the range overwhich tract tuning is shown in Fig. 7.
The results for normal playing /H20849gray symbols in Fig. 7/H20850
are more complicated. First playing at low frequencies/H20849which were not the principal object of this study /H20850is dis-
cussed. At frequencies below about 600 Hz, the results areapproximately as expected for a configuration of the tract,which did not vary with pitch. Above about 600 Hz, how-ever, and in normal playing, the resonance measured in theplayer’s mouth occurs at frequencies about 150 Hz higher/H20849on average /H20850than the sounding pitch. The magnitudes of
these vocal tract resonances formed during normal playingare modest /H20849/H20841Z
mouth/H20841about 9 MPa s m−3on average /H20850when
compared with those of the operating clarinet impedancepeak /H20849/H1101140–90 MPa s m
−3/H20850, so they contribute little to the
series combination and thus are expected to have only a
small effect on the sounding frequency of the reed; here theclarinet bore resonance dominates as expected /H20849Nederveen,
1998 ;Dickens et al. , 2007b /H20850. Nevertheless, even though the
players are not tuning their vocal tract to the note produced,they are adjusting it as a function of the note produced. Whymight this be?
First it is noted that, in normal playing, a strong reso-
nance of the vocal tract is not needed, to first order, to deter-mine the sounding frequency f
0: Here the player can usually
allow the clarinet bore resonance to determine, at least ap-proximately, the appropriate sounding pitch. Indeed, calcula-tions show that the magnitude and frequencies of these vocaltract impedance peaks change /H20849Z
mouth+Zbore/H20850/H20648Zreedby only
several hertz at most. /H20849While even a few hertz difference is
important in accurate intonation, a raise in pitch over thewhole range can be achieved by adjusting the mouthpiece onthe barrel. /H20850One possibility is that, in this range, experienced
players learn, presumably implicitly, to keep their vocal tractresonance away from the sounding pitch to prevent it from
interfering with the bore resonance during normal playing.
Forf
0below about 450 Hz, the resonances of the bore
are stronger and unintended bending is less of a danger. Inthis range, players may keep the tract resonance at a constantfrequency /H20849near 600 Hz for these players /H20850. For the range 450
Hz to at least 1400 Hz, they raise the frequency of their tractresonance to keep it substantially above that of the bore. Aswill be discussed below, it is easier to bend a note down thanup on the clarinet. Perhaps having a tract resonance “nearby”/H20849100–200 Hz away /H20850makes for a good performance strategy:
Tuning assistance from the vocal tract can be quickly andeasily engaged by adjusting the resonance frequency andstrength appropriately, should the need arise. And perhapshaving a resonance slightly below the played note is just too
dangerous, because of the potential effects on the pitch,which will be discussed later. This strategy of keeping thetract resonance at a frequency somewhat above that of thebore resonance for normal playing may explain the results ofClinch et al. /H208491982 /H20850, who observed a gradual variation of
vocal tract shape with increasing pitch over the range ofnotes studied. These researchers used x-ray fluoroscopy tostudy the vocal tract during playing and concluded that play-ers were tuning the tract resonance to match the note played.However, as this technique can only give qualitative infor-mation about the tract resonance, it is possible that the sub-ject of their study was also keeping the tract resonance fre-quency somewhat above that of the note played.
In contrast to the results for normal playing, the mea-
surements made during pitch bending show tight tuning ofthe sounding pitch to the vocal tract resonance, the differencein frequency being typically less than 30 Hz. Here, a strongresonance measured in the mouth /H20849average /H20841Z
mouth/H20841
/H1101120 MPa s m−3/H20850is generated by the player and competes
with the clarinet bore resonance. This changes /H20849Zmouth
FIG. 7. Measured vocal tract resonance frequency plotted against clarinet
sounding frequency. Data from the five players include both normal playing/H20849pale circles /H20850and pitch bending together with the glissando exercise /H20849dark
circles /H20850in the range between written G4 /H20849349 Hz /H20850and G6 /H208491397 Hz /H20850.T h e
size of each circle represents the magnitude of the acoustic impedance forthat measurement, binned in half decade bands. For comparison two circlesat bottom right show typical magnitudes of Z
borefor fingerings in the first
and second registers of the clarinet. The gray line indicates the hypotheticalrelationship /H20849frequency of vocal tract resonance /H20850/H11005/H20849frequency of note
played /H20850.
J. Acoust. Soc. Am., Vol. 126, No. 3, September 2009 Chen et al. : Clarinet pitch bending and glissandi 1517
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 193.0.65.67 On: Tue, 09 Dec 2014 16:38:21+Zbore/H20850/H20648Zreedand, as predicted by the simple model, the reso-
nance frequency of the player’s vocal tract begins to influ-
ence the sounding frequency of the reed /H20849normally deter-
mined by the bore resonance /H20850. This can be observed for
sounding notes above 600 Hz, in agreement with Rehfeldt
/H208491977 /H20850who suggested that the lower limit to large pitch
bending on the clarinet lies about D5 /H20849587 Hz /H20850. This would
also explain why the glissando is usually only played over
the last several notes of the scale in Rhapsody in Blue.
Below written E5 /H20849/H11011600 Hz /H20850, there is less strict tuning
of vocal tract resonance. This might be because it is difficult
to produce a vocal tract resonance with a sufficiently largeimpedance peak at frequencies below this range. Scavone
et al. /H208492008 /H20850placed the lower limit for adjusting the relevant
vocal tract influence at about 520 Hz. Further, clarinet boreresonances in this lower playing range are rather stronger.However, although the extent of pitch bending using the vo-cal tract resonance is limited in this range, other strategiesare used, including partial uncovering of tone holes and tech-niques that are not studied here, such as changing the biteforce on the reed and adjusting lip damping.
Figure 8plots the same data used for Fig. 7to show the
magnitude of Z
mouth and Zbore/H20648Zreedexplicitly. Here, these
quantities are plotted as a function of the deviation of theimpedance maximum from sounding frequency f
0. For
Zbore/H20648Zreed/H20849black dots /H20850, two tight clusters of data are seen.
One cluster is those of the bore resonances involved in pro-ducing notes in the clarinet’s first register: These have mag-nitudes of about 90 MPa s m
−3, the other corresponds to the
resonances that produce the clarinet’s second register /H20849about
40–50 MPa s m−3/H20850. These bore resonances are of course
centered on the normal sounding pitch.
For the vocal tract resonances, two contrasting regimes
are seen in Fig. 8: For vocal tract resonances with impedance
peaks above about 20–25 MPa s m−3, the sounding fre-
quency is tuned closely to the resonance frequency measuredin the mouth, with typically less than 30 Hz deviation. Only
measurements made during pitch bending fall into this re-gion. For tract resonances with smaller magnitude, thesounding frequency is not necessarily tuned to the vocal tractresonance.
Normal playing /H20849light circles /H20850over this pitch range /H20849G4–
G6/H20850shows a broad scattering of weak mouth resonances that
deviate from the sounding frequency by typically 100–200Hz. The asymmetry is striking: They are nearly always abovethe sounding frequency. These weak tract resonances/H20849/H20841Z
tract/H20841/H11270/H20841Zbore/H20841/H20850do not affect the correct sounding pitch. For
pitch bending /H20849crosses /H20850, however, a strong vocal tract reso-
nance can influence the sounding frequency. When the peakinZ
mouth is large /H20849for notes in the high range of Z: dark
crosses /H20850, the tract resonance dominates and consequently
there is little deviation from the sounding pitch. The pitchbending points in the right of the graph largely correspond tothe lower range of the clarinet /H20849below about 600 Hz—pale
crosses /H20850where bore resonances are very strong /H20849dots, top
left/H20850. It may be that it is difficult to produce a vocal tract
resonance with a strong peak in this frequency range. Here,the tract resonances are weak and do not determine thesounding frequency directly.
D. Example: Pitch bending exercise on fingered C6
To elucidate the relative influence of bore, tract, and
reed on combined impedance and sounding frequency, play-ers were further asked to finger a standard note /H20849written C6,
932 Hz /H20850and to bend its sounding pitch progressively down
from the standard pitch while their vocal tract impedancewas measured. Players were able to bend the normal pitchC6/H20849932 Hz /H20850smoothly down by as much as a major third to
G#5, a deviation of 400 cents or a third of an octave. This issimilar to the average maximum downward pitch bend of330 cents found by Scavone et al. /H208492008 /H20850for the alto saxo-
phone.
Figure 9shows calculations of the combined impedance
/H20849Z
mouth+Zbore/H20850for bending a note down. A single measured
clarinet impedance spectrum /H20849for C6 fingering /H20850is added in
series to three different impedance spectra measured in aplayer’s vocal tract for normal playing and two varying de-grees of pitch bending, all using the same fingering. To in-crease pitch bending, the tract resonance moves to succes-sively lower frequencies. It also has successively increasedmagnitude. The resulting decrease in frequency of the maxi-mum in the series impedance correlates with successivelylower sounding frequencies /H20849indicated by the arrows /H20850.
E. Why is it easier to bend pitch down rather than up?
On the clarinet and saxophone, it is possible to bend
pitches down, whereas on the clarinet “only slight upwardalterations are possible” /H20849Rehfeldt, 1977 /H20850and similarly “sig-
nificant upward frequency shifts …are not possible” on the
alto saxophone /H20849Scavone et al. , 2008 /H20850. It can be shown here
that, according to the simple model of Benade /H208491985 /H20850, this is
simply because although X
reedis always compliant, Zborecan
be either inertive or compliant.
FIG. 8. The magnitudes of the maxima in acoustic impedance of Zbore/H20648Zreed,
the clarinet bore in parallel with the reed /H20849dark dots /H20850and of that measured in
the player’s mouth, Zmouth, are plotted as a function of the difference be-
tween the frequency of the relevant maximum and that of the pitch played:In other words as a function of the deviation of resonance frequency fromthe sounding pitch. Crosses indicate pitch bending together with the glis-
sando exercise; open circles indicate normal playing.
1518 J. Acoust. Soc. Am., Vol. 126, No. 3, September 2009 Chen et al. : Clarinet pitch bending and glissandi
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 193.0.65.67 On: Tue, 09 Dec 2014 16:38:21Nederveen /H208491998 /H20850and the experiments reported here
give an effective clarinet reed compliance C/H20849in a typical
clarinet embouchure on a reed of hardness 3 /H20850as about 7
/H1100310−12m3Pa−1/H20849equivalent to an air volume of 1.1 ml /H20850.A t
a frequency of 1 kHz, this gives a reactance Xreed
/H20849=−1 /2/H9266fC/H20850of about −20 MPa s m−3. Its dependence on
frequency is weak compared with that of the tract and bore
impedances near resonances. For the purposes of this simplemodel, and as argued above, the sounding frequency f
0oc-
curs near the maximum in Zload=/H20849Ztract+Zbore/H20850/H20648Xreed, where
the reactance /H20849i.e., the imaginary part /H20850is zero, i.e., when
Xtract/H20849f/H20850+Xbore/H20849f/H20850=−Xreed/H20849f/H20850. /H208492/H20850
Because Xreedis always negative and moderately large, the
net sign of /H20849Xtract+Xbore/H20850must always be positive /H20849and
equally large /H20850for resonance to occur.
In normal playing, Ztractis generally small in comparison
with the maxima in Zbore, and initially, its effect can be ne-
glected. The condition that Xbore/H20849f/H20850=−Xreed/H20849f/H20850requires that
Xborebe positive and so the sounding frequency f0must lie
on the low frequency /H20849inertive /H20850side of the resonance peak in
Zbore. A soft reed or a more relaxed embouchure will produce
a larger compliance C, a decrease in Xreed, and consequently
a decrease in sounding frequency.
Now, consider the effect of including Ztractwith maxima
of similar magnitude to that of Zboreand, initially, at the same
resonance frequency, which will be greater than f0.I ft h e
resonant frequencies of Ztractand Zboreare similar, both
Xtract/H20849f/H20850andXbore/H20849f/H20850will be positive at frequencies below
their resonances, including f0. If the magnitude of the reso-
nance in Ztractis now increased, Xtract/H20849f/H20850will also increase
and consequently the sum Xtract/H20849f/H20850+Xbore/H20849f/H20850will increase and
so the sounding frequency f0will decrease so that Xreedsat-
isfies Eq. /H208492/H20850. If the player now decreases the resonance fre-
quency of the tract, the value of Xtract/H20849f/H20850around f0will in-
crease and so f0must again decrease to satisfy Eq. /H208492/H20850.However, decreasing f0by continuing to decrease the reso-
nance frequency of the tract will become increasingly diffi-cult. This is because the contribution of X
bore/H20849f/H20850to the sum
Xtract/H20849f/H20850+Xbore/H20849f/H20850will decrease as f0moves further away
from the resonance frequency of Zboreand yet Xtract/H20849f/H20850
+Xbore/H20849f/H20850must increase to match the increase in Xreed/H20849f/H20850asf0
decreases. Eventually a player will be unable to increase
Ztract/H20849f/H20850sufficiently to match Xreed/H20849f/H20850and further downward
pitch bending will not be possible.
The situation is quite different, however, if a player
wishes to bend the pitch upwards. Again, imagine a vocaltract resonance, comparable in magnitude to that in the bore,and with initially the same resonance frequency as the bore,again above f
0. If the resonant frequency of the tract is then
increased, the sum of Xtract/H20849f/H20850+Xbore/H20849f/H20850in the frequency
range where this sum is inertive will decrease and conse-
quently f0will increase as predicted by Eq. /H208492/H20850. However,
once f0exceeds the resonance frequency of the bore, Xbore/H20849f/H20850
will suddenly change sign and the sum Xtract/H20849f/H20850+Xbore/H20849f/H20850will
decrease dramatically to a value much smaller than possible
forXreed/H20849f/H20850. Players are probably unable to increase Ztract/H20849f/H20850
sufficiently to increase f0past this point, unless they can
produce a peak in Ztractthat is significantly greater than that
inZbore, which Fig. 8shows is relatively rare. /H20849In those rare
cases where the player can produce such a peak in Ztract, then
f0is determined largely by the tract and depends less
strongly on the bore, much as is the case in the altissimoregion of the saxophone. /H20850Thus, in normal situations, the
maximum increase in sounding frequency will be of thesame order in magnitude as the decrease in resonance fre-quency of the bore due to the compliance of the reed, possi-bly not more than 50 cents.
What then are the effects of vocal tract resonances tuned
above and below that of the bore? Figure 10shows the im-
pedance of Z
borefor the note written D6, with a peak at 1070
FIG. 9. The clarinet bore impedance /H20849Zbore/H20850for the fingering C6 /H20849gray line /H20850
is shown with the series impedance /H20849Zmouth+Zbore/H20850for normal playing /H20849solid
line/H20850and increasing degrees of pitch bending /H20849dashed lines /H20850, all while main-
taining the same fingering for C6 /H20849932 Hz /H20850. Vertical arrows indicate the
sounding pitch for the three cases: The right-hand arrow shows a normalsounding pitch at C6+8 cents /H20849937 Hz /H20850, while the left-hand arrow denotes
the lowest pitch sounding at G#5+7 cents /H20849743 Hz /H20850, a deviation of 400
cents or a major third.
FIG. 10. To examine the effect of vocal tract resonances tuned above andbelow that of the bore, two hypothetical vocal tract values of equal magni-tude /H2084911 MPa s m
−3/H20850are shown /H20849dark lines /H20850, with resonance frequencies
1020 /H20849shown above /H20850and 1120 Hz /H20849shown below /H20850, along with Zreed/H20849dotted
line/H20850,Zbore/H20849peak at 1070 Hz, pale line /H20850, and Zbore/H20648Zreed/H20849peak at 1050 Hz,
pale dashes /H20850. Total impedance of the tract-reed-bore system /H20849Zbore
+Zmouth/H20850/H20648Zreedfor both cases are shown /H20849dark dashes /H20850, with respective
maxima at 980 /H20849above /H20850and 1035 Hz /H20849below /H20850.
J. Acoust. Soc. Am., Vol. 126, No. 3, September 2009 Chen et al. : Clarinet pitch bending and glissandi 1519
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 193.0.65.67 On: Tue, 09 Dec 2014 16:38:21Hz, and that of Zbore/H20648Zreed, which has a peak at 1050 Hz
/H20849near the nominal frequency, 1047 Hz, for that note /H20850.A
single measured vocal tract impedance spectrum Zmouth was
then numerically shifted in frequency so that the “same” tractimpedance peak now lay at either 50 Hz above or 50 Hzbelow the peak in 1070 Hz. Then /H20849Z
bore+Zmouth/H20850/H20648Zreedis plot-
ted for the two cases. In both cases, the frequency of the peak
/H20849Zbore+Zmouth/H20850/H20648Zreedlies below that of the peak in Zbore/H20648Zreed,
but the downward pitch bend is larger for the lower fre-
quency tract resonance. This figure also shows what happenswhen Z
mouth is much smaller than Zbore, because this is the
case for the other two bore resonances that occur in the fre-quency range shown. Finally, it is worth observing anotherpeak in Z
bore, that at about 700 Hz. At this frequency, Zmouth
is small compared with Zbore, and both are small compared
with Zreed; consequently, the peak in /H20849Zbore+Zmouth/H20850/H20648Zreed
here nearly coincides with that in Zbore.
IV. CONCLUSION
For normal clarinet playing, resonances in the clarinet
bore /H20849determined by the fingering used /H20850dominate to drive
the reed to oscillate at a frequency very close to that of thebore and reed in parallel. However, if the upstream resonancein the player’s vocal tract is adjusted to have a sufficientlyhigh impedance peak at the appropriate frequency, the vocaltract resonance competes with or dominates the clarinet reso-nance to determine the reed’s sounding frequency.
By skillfully coordinating the fingers to smoothly un-
cover the clarinet finger holes and simultaneously tuningstrong vocal tract resonances to the continuously changingpitch, expert players are able to facilitate a smoothtrombone-like glissando , of which a famous example is the
final octave of the run that opens Gershwin’s Rhapsody inBlue.
ACKNOWLEDGMENTS
The authors thank the Australian Research Council for
support of this project and Neville Fletcher for helpful dis-cussions. They thank Yamaha for the clarinets, Légère for thesynthetic reeds, and the volunteer clarinettists.
1The peak in Zbore/H20648Zreedhas a larger magnitude than that of either Zboreor
Zreedbecause it is a parallel resonance between the reed compliance and
the bore in its inertive range, i.e., at frequencies a little below the peak inZ
bore.
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net,” J. Acoust. Soc. Am. 33, 806–809.
Backus, J. /H208491963 /H20850. “Small-vibration theory of the clarinet,” J. Acoust. Soc.
Am. 35, 305–313.
Backus, J. /H208491974 /H20850. “Input impedance curves for the reed woodwind instru-
ments,” J. Acoust. Soc. Am. 56, 1266–1279.
Backus, J. /H208491985 /H20850. “The effect of the player’s vocal tract on woodwind
instrument tone,” J. Acoust. Soc. Am. 78, 17–20.
Benade, A. H. /H208491976 /H20850.Fundamentals of Musical Acoustics /H20849Oxford Univer-sity Press, New York /H20850, pp. 465–467.
Benade, A. H. /H208491985 /H20850. “Air column, reed, and player’s windway interaction
in musical instruments,” in Vocal Fold Physiology, Biomechanics, Acous-
tics, and Phonatory Control , edited by I. R. Titze and R. C. Scherer /H20849Den-
ver Center for the Performing Arts, Denver, CO /H20850, Chap. 35, pp. 425–452.
Benade, A. H., and Ibisi, M. I. /H208491987 /H20850. “Survey of impedance methods and
a new piezo-disk-driven impedance head for air columns,” J. Acoust. Soc.Am. 81, 1152–1167.
Boutillon, X., and Gibiat, V. /H208491996 /H20850. “Evaluation of the acoustical stiffness
of saxophone reeds under playing conditions by using the reactive powerapproach,” J. Acoust. Soc. Am. 100, 1178–1889.
Chen, J.-M., Smith, J., and Wolfe, J. /H208492008 /H20850. “Experienced saxophonists
learn to tune their vocal tracts,” Science 319, 726.
Chen, J.-M., Smith, J., and Wolfe, J. /H208492009 /H20850. “Saxophone acoustics: Intro-
ducing a compendium of impedance and sound spectra,” Acoust. Aust. 37,
18–23.
Clinch, P. G., Troup, G. J., and Harris, L. /H208491982 /H20850. “The importance of vocal
tract resonance in clarinet and saxophone performance, a preliminary ac-count,” Acustica 50, 280–284.
Dickens, P., Smith, J., and Wolfe, J. /H20849
2007a /H20850. “High precision measurements
of acoustic impedance spectra using resonance-free calibration loads andcontrolled error distribution,” J. Acoust. Soc. Am. 121, 1471–1481.
Dickens, P., France, R., Smith, J., and Wolfe, J. /H208492007b /H20850. “Clarinet acoustics:
Introducing a compendium of impedance and sound spectra,” Acoust.Aust. 35, 17–24.
Fletcher, N. H., and Rossing, T. D. /H208491998 /H20850.The Physics of Musical Instru-
ments /H20849Springer, New York /H20850, pp. 470–480.
Fritz, C., and Wolfe, J. /H208492005 /H20850. “How do clarinet players adjust the reso-
nances of their vocal tracts for different playing effects?,” J. Acoust. Soc.Am. 118, 3306–3315.
Grand, N., Gilbert, J., and Laloë, F. /H208491996 /H20850. “Oscillation threshold of wood-
wind instruments,” Acust. Acta Acust. 82, 137–151.
Johnston, R., Clinch, P. G., and Troup, G. J. /H208491986 /H20850. “The role of the vocal
tract resonance in clarinet playing,” Acoust. Aust. 14, 67–69.
Nederveen, C. J. /H208491998 /H20850.Acoustical Aspects of Wind Instruments , 2nd ed.
/H20849Northern Illinois University, De Kalb, IL /H20850, pp. 35–37.
Pay, A. /H208491995 /H20850. “The mechanics of playing the clarinet,” in The Cambridge
Companion to the Clarinet , edited by C. Lawson /H20849Cambridge University
Press, Cambridge /H20850, pp. 107–122.
Rehfeldt, P. /H208491977 /H20850.New Directions for Clarinet /H20849University of California
Press, Berkeley, CA /H20850, pp. 57–76.
Scavone, G., Lefebre, A., and da Silva, A. R. /H208492008 /H20850. “Measurement of
vocal-tract influence during saxophone performance,” J. Acoust. Soc. Am.123, 2391–2400.
Scavone, G. P. /H208492003 /H20850. “Modeling vocal-tract influence in reed wind instru-
ments,” in Proceedings of the 2003 Stockholm Musical Acoustics Confer-ence, Stockholm, Sweden, pp. 291–294.
Schwartz, C. /H208491979 /H20850
.Gershwin: His Life and Music /H20849Da Capo, New York,
NY/H20850, pp. 81–83.
Silva, F., Kergomard, J., Vergez, C., and Gilbert, J. /H208492008 /H20850. “Interaction of
reed and acoustic resonator in clarinetlike systems,” J. Acoust. Soc. Am.124, 3284–3295.
Smith, J. R., Henrich, N., and Wolfe, J. /H208491997 /H20850. “The acoustic impedance of
the Boehm flute: Standard and some non-standard fingerings,” Proc. Inst.Acoustics 19, 315–330.
Sommerfeldt, S. D., and Strong, W. J. /H208491988 /H20850. “Simulation of a player-
clarinet system,” J. Acoust. Soc. Am. 83, 1908–1918.
Tarnopolsky, A., Fletcher, N., Hollenberg, L., Lange, B., Smith, J., and
Wolfe, J. /H208492006 /H20850. “Vocal tract resonances and the sound of the Australian
didjeridu /H20849yidaki /H20850I: Experiment,” J. Acoust. Soc. Am. 119, 1194–1204.
Watkins, M. /H208492002 /H20850. “The saxophonist’s vocal tract,” The Saxophone Sym-
posium: J. North Am. Saxophone Alliance 27, 51–75.
Wilson, T. A., and Beavers, G. S. /H208491974 /H20850. “Operating modes of the clarinet,”
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formance and its role in sound production,” Ph.D. thesis, University ofWashington, Seattle, WA.
1520 J. Acoust. Soc. Am., Vol. 126, No. 3, September 2009 Chen et al. : Clarinet pitch bending and glissandi
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 193.0.65.67 On: Tue, 09 Dec 2014 16:38:21 |
1.3259398.pdf | Inherent spin transfer torque driven switching current fluctuations in magnetic element
with in-plane magnetization and comparison to perpendicular design
Xiaochun Zhu and Seung H. Kang
Citation: Journal of Applied Physics 106, 113906 (2009); doi: 10.1063/1.3259398
View online: http://dx.doi.org/10.1063/1.3259398
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/106/11?ver=pdfcov
Published by the AIP Publishing
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193.0.65.67 On: Mon, 15 Dec 2014 12:40:55Inherent spin transfer torque driven switching current fluctuations in
magnetic element with in-plane magnetization and comparison toperpendicular design
Xiaochun Zhua/H20850and Seung H. Kang
Advanced Technology, Qualcomm Incorporated, 5775 Morehouse Drive, San Diego, California 92121, USA
/H20849Received 24 June 2009; accepted 14 October 2009; published online 2 December 2009 /H20850
This paper presents a systematic micromagnetic modeling study on switching current fluctuations in
both in-plane and perpendicular spin-transfer-torque /H20849STT/H20850magnetoresistive random access
memory devices. For the magnetic tunnel junction /H20849MTJ /H20850with in-plane magnetization, high-order
spin wave modes are excited during a STT-driven switching, which leads to an inherently broadswitching current distribution. If the MTJ size is not sufficiently small, a stable vortex can be formedover a wide range of current amplitudes. In contrast, the excitation of such high-order spin wavesis absent in STT switching of MTJs with perpendicular magnetic anisotropy. Consequently, thefluctuation in switching current amplitude or pulse duration is significantly smaller in comparison.©2009 American Institute of Physics ./H20851doi:10.1063/1.3259398 /H20852
I. INTRODUCTION
High switching current and poor scalability pertaining to
field-driven magnetoresistive random access memory/H20849MRAM /H20850have brought only limited success in commercial
applications of the highly anticipated MRAM technology.Recently spin-transfer-torque /H20849STT/H20850
1,2MRAM has emerged
as a compelling alternative by offering solutions to suchproblems. Both in-plane and perpendicular STT-MRAMshave been designed and demonstrated experimentally.
3–9
On-going STT-MRAM development efforts are largely
on further lowering the switching current. For practicalmemory designs, however, equally important is controllingthe distribution of the switching current throughout a cyclicalswitching of the memory operation. It is often not surprisingto observe wide switching current distributions in the case ofin-plane magnetic tunnel junctions /H20849MTJs /H20850. However, the
mechanism that causes such switching current distribution isdifferent from that of the conventional field-driven MRAM.Our prior study on in-plane MTJ elements has shown thatvariations in physical parameters such as MTJ shape, aspectratio, and edge roughness do not cause a significant distribu-tion of switching current thresholds.
10
In this paper, we present a systematic micromagnetic
modeling study as an attempt to understand the switchingcurrent fluctuation in in-plane MTJ elements. We report thatthe nature of STT-driven switching in an in-plane MTJ ele-ment inherently causes a significant switching current fluc-tuation unless the element size is sufficiently small. In con-trast, such fluctuation is essentially absent in the case ofperpendicularly magnetized STT-MRAM elements.
II. MODEL
The simulated STT-MRAM elements in this work are
MTJs which consist of a storage layer, a tunnel barrier, and asynthetic antiferromagnetic /H20849SAF/H20850reference layer pinned byan antiferromagnetic layer. Assuming that the two ferromag-
netic layers in the SAF are ideally designed and that theygenerate negligible stray fields, only the storage layer ismodeled. For an in-plane STT-MRAM, the storage layer isCoFeB whose saturation magnetization /H20849M
s/H20850is
1200 emu /cm3, and has a thickness of 2.5 nm. The element
is elliptically shaped. For a perpendicular STT-MRAM, thestorage layer is assumed to be an exchange-coupled bilayerthat consists of a spin-polarization enhancing layer CoFeBand a perpendicular layer with perpendicular crystalline an-isotropy. The thicknesses are 1.5 and 3 nm for the spin-polarization enhancing layer and the perpendicular layer, re-spectively. The perpendicular layer has a saturationmagnetization of 400 emu /cm
3. The exchange constant
within the homogeneous free layer materials is assumed tobe 1.0 /H1100310
−6erg /cm. The direct interlayer exchange con-
stant A exbetween the perpendicular layer and CoFeB layer is
set to have the same exchange strength as that within thelayers. While not discussed in this paper, we found that vary-ing A
exfrom 50% to 150% of the above value did not alter
any conclusion of the work.
The storage layers are modeled as an array of mesh cells.
The size of each cell is 5 nm /H110035n m/H11003t/H20849thickness /H20850. The
intrinsic crystalline anisotropy of the perpendicular layer istuned between 4 /H1100310
6and 8/H11003106erg /cm3depending on
the MTJ size. The STT-modified Landau–Lifshitz–Gilberttheory is applied to simulate the magnetization dynamics ofthe current-induced magnetization reversal by micromag-netic modeling.
11–13A polarization factor of 0.65 is used
throughout the calculations. In addition, the current-inducedOersted field is included in the simulations. The thermal agi-tation is modeled by adding a random thermal field to theeffective field.
14The magnitude of this field follows a Gauss-
ian distribution with the variance determined by thefluctuation-dissipation theorem.
15
a/H20850Electronic mail: xiaochun@qualcomm.com.JOURNAL OF APPLIED PHYSICS 106, 113906 /H208492009 /H20850
0021-8979/2009/106 /H2084911/H20850/113906/5/$25.00 © 2009 American Institute of Physics 106 , 113906-1
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193.0.65.67 On: Mon, 15 Dec 2014 12:40:55III. IN-PLANE STT-MRAM
In this section, we report the results on STT-driven mag-
netization switching in the case of in-plane MTJ elements.Figure 1shows the transient switching process for an ele-
ment of 160 /H1100380 nm
2with the averaged magnetization
components plotted as a function of time /H20849x- and y-axes are
along the long and short axes of the elliptical element, re-spectively /H20850. The corresponding transient magnetization con-
figuration at 3 ns is shown in Fig. 2/H20849a/H20850, with the color con-
trast indicating the perpendicular component of themagnetization and the arrow representing the in-plane com-ponent of magnetization. We find that STT switching alwaysstarts with the excitation of high-order spin waves along bothlong and short axes. As the switching progresses, the mag-netization precessional angles with respect to the long axis ofthe element increase, yielding nonuniform magnetizationconfigurations,
13as shown in Fig. 2/H20849a/H20850. This is attributed to
the fact that for an in-plane MTJ element, the demagnetiza-tion field varies significantly from the element edge to thecenter, causing the initial precession frequency and ampli-tude to vary across the element. Combined with the effect offerromagnetic exchange coupling within the element, thespin waves with high-order modes are generated. This exci-tation of high-order spin wave during a STT switching is
therefore inherent for an in-plane MTJ element.
The phenomenon described above leads to two impor-
tant consequences. First, the switching by the high-order spinwave excitation can result in the formation of a magnetiza-tion vortex in the middle of the element, as the one shown inFig.2/H20849b/H20850. Such vortex formation during switching is a proba-
bilistic event and it can occur over a broad range of currentamplitudes. The probability of vortex formation becomesgreater for an element with a larger width and/or a largerthickness due to the increase in magnetostatic energy. Sec-ond, the onset of irreversible magnetization switching be-comes uncertain in time domain, i.e., the switching time in-evitably fluctuates. Prior to the actual happening of anirreversible switching, neither the magnetization precessionamplitude nor the transient magnetic configuration gives asign to predict when the irreversible switching will actuallyoccur, as shown in Figs. 1and2/H20849a/H20850. If current is abruptly
turned off before reaching the critical switching time, themagnetization will return to its original state. For a fixedcurrent pulse width, this uncertainty of switching time ismanifested as current amplitude fluctuation. Figure 3plots
the switching probability as a function of current density /H208495
ns pulse width /H20850for the elements varying from 40 to 100 nm
in width. The element aspect ratio is kept as 2.5. For theelements of 80 and 100 nm in width, the switching currentdistributions are relatively broad. Furthermore, at relativelyhigh current amplitudes, the probability of forming a vortexstate is significantly high over a broad range of current den-sity magnitudes. The vortex formation can be eliminated ifthe element size can be sufficiently small. As shown in Fig.3, for the element of 40 nm in width, a stable vortex can no
longer be formed and the switching current distribution be-comes much narrower. It should be noted that if either thethickness or the saturation magnetization of the free layer isincreased, the critical element size for eliminating such vor-tex formation will decrease.
The transient STT switching process of an in-plane MTJ
element is determined by the combination of two competingeffects: the magnetostatic energy /H20849or demagnetization en-
ergy/H20850which drives the nonuniform switching and effectively
reduces shape anisotropy and the ferromagnetic exchangecoupling trying to keep magnetization uniform. If the char-acteristic exchange length in the free layer is defined as L
ex
=/H20881A/Kshape, where Kshape/H11011105erg /cm3, estimated from the
calculated shape anisotropy field, Hk,shape in this work, one
obtains an exchange length Lex/H1101530 nm. Examining Fig.
2/H20849a/H20850, this exchange length is roughly a half of the wavelength
of the excited spin waves shown in Fig. 2/H20849a/H20850. If the element
dimension is significantly larger than the exchange length,high-order spin wave modes would be excited. In contrast,for the elements with dimensions comparable to or smallerthan the exchange length, coherent or uniform magnetizationrotation mode would prevail.
It should be noted that the phenomenon described above
is not caused by the inclusion of thermal agitation.
10Our
simulation work shows that with or without thermal agita-tion, this phenomenon remains essentially the same.Norma lizedMagnetization
0 1 2 3 4 5 6-1-0.8-0.6-0.4-0.200.20.40.60.81
Time (ns)Mx
MyMz
FIG. 1. /H20849Color online /H20850Simulated volume-averaged magnetization compo-
nents as a function of time for an in-plane MTJ element of 160 /H1100380 nm2.
FIG. 2. /H20849Color online /H20850/H20849a/H20850Transient magnetization configuration during STT
switching for the in-plane MTJ with a size of 160 /H1100380 nm2. The gray scale
/H20849or the color in the online version /H20850represents perpendicular component Mz.
The arrows plot in-plane component of magnetization vectors. /H20849b/H20850The for-
mation of a magnetic vortex at the end of the STT switching process.113906-2 X. Zhu and S. H. Kang J. Appl. Phys. 106 , 113906 /H208492009 /H20850
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193.0.65.67 On: Mon, 15 Dec 2014 12:40:55IV. PERPENDICULAR STT-MRAM
The transient characteristics of STT switching of a per-
pendicularly magnetized MTJ element /H2084980 nm in diameter /H20850
are shown in Fig. 4. A circular lateral shape of the element is
assumed. The volume averaged magnetization of the storagelayer during the injection o fa5n s current pulse is illustrated
in Fig. 4/H20849a/H20850. The corresponding transient magnetization con-
figurations /H20849with the color contrast indicating the perpendicu-
lar magnetization component M
zand the arrows plotting the
in-plane components of magnetization vectors /H20850are shown in
Fig. 4/H20849b/H20850. The switching starts as the magnetization curls
around the element center, similar to the classic curlingmode.
16As the switching progresses, the middle of the mag-
netization curling structure moves toward the element edge.This is because the exchange energy for the curling center atthe edge is significantly lower than that at the center. Themagnetization angle with respect to the initial perpendiculardirection increases and the magnetization precession be-comes more and more spatially nonuniform. Consequently,the magnetization of a small area at the edge of the elementis first reversed. The reversed domain then expands through
the entire element as the switching process completes.
The above described switching process is characteristic
for the STT-driven magnetization reversal in a perpendicu-larly magnetized element with a circular shape. In contrast toin-plane MTJ elements, the onset stage of the switching pro-cess in a perpendicular element is described by a simple andwell-defined curling mode. Note that this characteristic holds
even with an inclusion of the thermal excitation. The absenceof a high-order spin wave excitation is mainly due to thecylindrical symmetry: the curling-modelike precession isnaturally suited to the circular geometry of the element sothat there is no need to excite a high-order spin wave, whichis at a higher energy state. Another important point is that fora perpendicular geometry, the magnetization precession doesnot introduce an additional demagnetization field componentas it does in the in-plane case. Accordingly, increasing theprecession amplitude only yields a reduction in the perpen-dicular demagnetization field, consequently, further reducingthe spatial nonuniformity.
For a given current pulse duration, an insufficient
switching current amplitude could leave the element in amultidomain state before the switching is complete.
17Figure
5/H20849a/H20850shows the volume averaged perpendicular magnetiza-
tion component as a function of time for a current pulse of 50 2 4 6 8 10 12
x1 0700.20.40.60.81
(a)Switc hing Pro bability Switc hing Pro bability Switc hing Pro bability
J(A/cm2)0 2 4 6 8 10 12
x1 0700.20.40.60.81
0 2 4 6 8 10 12
x1 0700.20.40.60.81
(c)(b)Unswitc hedState
Switched State
Vortex StateUnswitched StateVortex StateSwitc hedState
Switched State
Unswitched State
FIG. 3. /H20849Color online /H20850Simulated switching probability as a function of
current density for the in-plane MTJ with sizes of /H20849a/H20850250/H11003100 nm2,/H20849b/H20850
200/H1100380 nm2,a n d /H20849c/H20850100/H1100340 nm2.
0 1 2 3 4 5 6 7 8 9-1-0.8-0.6-0.4-0.200.20.40.60.81
Time (ns)(a)
(b)Norma lized MagnetizationMxMyMz
FIG. 4. /H20849Color online /H20850/H20849a/H20850Simulated volume-averaged magnetization com-
ponents as a function of time for a perpendicular MTJ element /H2084980 nm in
diameter /H20850./H20849b/H20850The corresponding transient magnetization configurations dur-
ing the STT switching. The gray scale /H20849or the color in the online version /H20850
represents perpendicular component Mz. The arrows plot in-plane compo-
nents of magnetization vectors.113906-3 X. Zhu and S. H. Kang J. Appl. Phys. 106 , 113906 /H208492009 /H20850
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193.0.65.67 On: Mon, 15 Dec 2014 12:40:55ns pulse width at a series of different amplitudes for a per-
pendicular MTJ of 80 nm in diameter. At the end of thecurrent pulse duration, the reversed domain has not yet com-pleted its expansion, leaving a two-domain configuration inthe element. In practice, the formed domain wall, such as theone shown in Figs. 4/H20849b/H20850and6, could be pinned by local
pinning sites, causing the two-domain configuration to bestable.
17If the domain wall is not pinned, the magnetization
configuration will drift slowly to a single perpendicular do-main state and the larger domain at the end of the currentpulse prevails. It is important to note that since the switchingtime is inversely proportional to the current amplitude,
6so is
the critical pulse length required for a complete switching.This is because the completion of the switching essentiallydepends on the amount of energy pumped in by STT, pro-vided that the effective damping constant of the compositestorage layer is sufficiently small. For a given current ampli-tude, an insufficiently long switching current pulse will alsoresult in a multidomain state as discussed above. As shownin Fig. 5/H20849b/H20850, which is the case for a 10 ns pulse width of
200
/H9262A switching current /H20851versus 5 ns in Fig. 5/H20849a/H20850/H20852, a longer
current pulse can yield a complete magnetization reversal atthe end of the current pulse of an intermediate amplitude.Therefore, either sufficiently high current amplitude or suffi-ciently long current pulse is required to avoid the stable mul-tidomain state in a perpendicular STT-MRAM design. Theelement-to-element variation in the pinning site characteris-tics, thus, could be the key to narrow the switching currentdistribution at a given current pulse width.
When the element size is sufficiently small, the curling
mode is no longer energetically favorable. The estimatedcritical diameter of a single domain for a composite free
layer is D=2
/H20881AK //H20849/H9266MS2/H20850/H1101540 nm. Figure 6shows the STT-
driven switching for an element of 40 nm in diameter at a 5
ns pulse width. The switching starts with essentially a coher-ent magnetization precession and progresses into a nonuni-form reversal with a reversed domain developed at one sideof the element. The reversed domain would then expandthrough the entire element. Similar to the 80 nm elementcase /H20849Fig.5/H20850, a multidomain state can be formed if either the
current pulse width or the current amplitude is insufficient tocomplete the reversal. But when the MTJ element is smalland the domain wall width becomes comparable to the ele-ment size, it becomes more difficult to form a stable multi-domain configuration. The utilization of perpendicular aniso-tropy shall enable the downsize scaling of perpendicularMTJ, if lithography is not a limiting factor, to be below 5 nmin diameter with sufficient thermal magnetic stability, consid-ering material such as FePt L1
0with anisotropy strength as
high as 7 /H11003107erg /cm3. The availability of the materials
allows the reduction in the element size and ensures the co-herent switching mode during magnetization reversal.
V. CONCLUSIONS
This paper has presented a systematic micromagnetic
study to understand STT-driven magnetization reversals inboth in-plane and perpendicular MTJ elements. The studyshows that if the in-plane MTJ elements are not sufficientlysmall, STT generates high-order spin waves within the freelayer and then evolves into nearly chaotic complex transientmagnetization configurations during a switching process.Such switching process characteristically causes both theswitching current amplitude and the switching time to fluc-tuate over a relatively wide range. If the element size is notsufficiently small, a stable vortex may also be formed in the0 2 4 6 8 10 12 14-1-0.8-0.6-0.4-0.200.20.40.60.81Normalized MagnetizationMxMy
Mz0123456789-1-0.8-0.6-0.4-0.200.20.40.60.81
I=0.20mA
I=0.25mA
I=0.30mA
I=0.35mANormalized Mz Component
Time (ns)Mz>0 Mz<0(a)
(b)
FIG. 5. /H20849Color online /H20850Simulated volume-averaged magnetization compo-
nents as a function of time for a perpendicular MTJ element /H2084980 nm in
diameter /H20850/H20849a/H20850at a pulse width of 5 ns for a series of different current am-
plitudes from 200 to 400 /H9262Aa n d /H20849b/H20850at a pulse width of 10 ns for a current
amplitude of 200 /H9262A.
-20 0 20 40 60 80 100 120 140 160-1.0-0.50.00.51.0MzComponent
Injected Current (/CID80A)
FIG. 6. /H20849Color online /H20850Simulated remanent perpendicular magnetization
components Mzas a function of injected current for a perpendicular MTJ
/H2084940 nm in diameter /H20850. The red dot represents the perpendicular layer and the
black square stands for the spin-polarization enhancing layer. The currentpulse width is 5 ns. The simulation time is 8 ns. The corresponding magne-tization configurations at the completion of four different injected pulses arealso shown /H20849the gray scale or the color in the online version indicating the
perpendicular magnetization component M
z, and the arrows plot in-plane
component of magnetization vectors /H20850.113906-4 X. Zhu and S. H. Kang J. Appl. Phys. 106 , 113906 /H208492009 /H20850
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193.0.65.67 On: Mon, 15 Dec 2014 12:40:55middle of the element due to the spatial nonuniform transient
spin waves at relatively large current amplitudes. Althoughthe phenomenon could be suppressed by reducing the size ofthe elements, the required thermal stability of such elements
may actually limit the size reduction. The critical dimen-sions, below which single domain coherent reversal woulddominate the switching, also decrease with increasing theelement thickness. In contrast, the STT-driven magnetizationreversal in perpendicular MTJ elements can be made intocircular shape and the corresponding STT-driven switching ismuch more deterministic. Although both reversed and unre-versed domains coexist during a switching process, suffi-ciently long current pulses can complete the switching deter-ministically. For a given element, the switching is repeatable.Hence, we conclude that the switching current distributionpertaining to a perpendicular STT-MRAM, if significant,would result primarily from cell-to-cell material propertyvariations.
We would also like to point out another important aspect
of the STT-driven switching for in-plane MTJ elements. Byproviding a torque to local magnetization against the damp-ing torque, STT generates a nonequilibrium state by pump-ing energy into the local spin system. In the in-plane design,the excited high-order spin wave also serves as a large en-ergy reservoir and it is only when this reservoir is filled to acertain degree that the switching would occur. In the perpen-dicular design, however, nearly all the energy injected bySTT directly contributes to the switching. Therefore, the cur-rent efficiency for STT-driven switching in the perpendicular
design is significantly greater than that in the in-plane de-sign.
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H. Yamada, M. Shoji, H. Hachino, C. Fukumoto, H. Nagao, and H. Kano,Tech. Dig. - Int. Electron Devices Meet. 2005, 459.
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Yang, H. Yu, X. Lu, W. Kula, T. Zhong, R. Xiao, A. Zhong, G. Liu, J.Kan, J. Yuan, J. Chen, R. Tong, J. Chien, T. Torng, D. Tang, P. Wang, M.Chen, S. Assefa, M. Qazi, J. DeBrosse, M. Gaidis, S. Kanakasabapathy, Y.Lu, J. Nowak, E. O’Sullivan, T. Maffitt, J. Z. Sun, and W. J. Gallagher,Tech. Dig. - Int. Electron Devices Meet. 2008,1 .
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Nishiyama, T. Daibou, M. Nagamine, M. Amano, S. Takahashi, M. Na-kayama, N. Shimomura, H. Aikawa, S. Ikegawa, S. Yuasa, K. Yakushiji,H. Kubota, A. Fukushima, M. Oogane, T. Miyazaki, and K. Ando, Tech.Dig. - Int. Electron Devices Meet. 2008, 309.
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E. Fullerton, Nature Mater. 5, 210 /H208492006 /H20850.
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5.0041027.pdf | Biomicrofluidics 15, 024105 (2021); https://doi.org/10.1063/5.0041027 15, 024105
© 2021 Author(s).A low-cost 3D printed microfluidic bioreactor
and imaging chamber for live-organoid
imaging
Cite as: Biomicrofluidics 15, 024105 (2021); https://doi.org/10.1063/5.0041027
Submitted: 18 December 2020 . Accepted: 01 March 2021 . Published Online: 06 April 2021
Ikram Khan , Anil Prabhakar , Chloe Delepine , Hayley Tsang ,
Vincent Pham , and Mriganka Sur
COLLECTIONS
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and imaging chamber for live-organoid imaging
Cite as: Biomicrofluidics 15, 024105 (2021); doi: 10.1063/5.0041027
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Submitted: 18 December 2020 · Accepted: 1 March 2021 ·
Published Online: 6 April 2021
Ikram Khan,1,a)
Anil Prabhakar,1Chloe Delepine,2Hayley Tsang,2Vincent Pham,2
and Mriganka Sur2
AFFILIATIONS
1Department of Electrical Engineering, Indian Institute of Technology, Madras 600036, India
2Picower Institute for Learning and Memory, Department of Brain and Cognitive Sciences, Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139, USA
a)Author to whom correspondence should be addressed: ikramkhan11692@gmail.com
ABSTRACT
Organoids are biological systems grown in vitro and are observed to self-organize into 3D cellular tissues of specific organs. Brain organoids
have emerged as valuable models for the study of human brain development in health and disease. Researchers are now in need of improvedculturing and imaging tools to capture the in vitro dynamics of development processes in the brain. Here, we describe the design of a micro-
fluidic chip and bioreactor, to enable in situ tracking and imaging of brain organoids on-chip. The low-cost 3D printed microfluidic bioreac-
tor supports organoid growth and provides an optimal imaging chamber for live-organoid imaging, with drug delivery support. This fullyisolated design of a live-cell imaging and culturing platform enables long-term live-imaging of the intact live brain organoids as it grows.We can thus analyze their self-organization in a controlled environment with high temporal and spatial resolution.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0041027
I. INTRODUCTION
Stem cell research has revolutionized treatment developments
for diseases such as spinal cord injury, diabetes, rheumatoid arthritis,cerebral palsy, Alzheimer ’s, Parkinson ’s, and targeted cancer
treatment.
1–9In 1907, Dr. H. V. Wilson demonstrated the potential
of dissociated siliceous sponge-like cells to self-organize and regener-
ate into a complete organism.10In vitro growth of pluripotent stem
cells allows us to grow miniaturized versions of organs called orga-noids. Pluripotent stem cells are able to self-organize and form orga-noids of complex organs such as the brain, kidney, retina, and
heart.
11–20Organoid based diagnosis also comes in handy for screen-
ing pharmaceutical compounds for many diseases.21–23Thus, orga-
noid growth must be done in a controlled incubator environment,with careful monitoring. However, such monitoring is challenging asthis involves physical handling and invasive procedures.
24
Ensuring a steady supply of nutrition to a growing organoid
continues to be one of the main issues in organoid culture. As theorganoid grows bigger, its core does not get enough nutrientsupply and gas exchange, thereby triggering cell death. Microfluidictechnologies provide a solution to grow organoids in a controlled
environment and optimized perfusion of culture media. The ability
to confine fluid in a small volume and being able to manipulate thecell to a higher degree of freedom has made bio-medical research
cost effective. This opened many new applications in the field of
lab-on-chip.
25–28Such small confinements are also advantageous
in terms of easy integration with multiple test equipment and
easier portability. Soft lithography is a common technique tradi-
tionally used for the fabrication of microfluidic devices. This isbased on transferring the micro-structures from a mold to a poly-
dimethylsiloxane (PDMS),
29,30and it also involves many other
steps that limit the possible design. Recent advancement in stereo-lithography based additive manufacturing (3D printing) has made
it possible to realize advanced microfluidic chips with simple man-
ufacturing procedures.
31–33This has dramatically brought down
the cost and the complicated steps involved in traditional manu-
facturing of microfluidic chips.
In this work, a low-cost microfluidic bioreactor was developed,
designed to support both imaging and culture on a single chip and
was fabricated using stereolithography based 3D printing. This
fully standalone compact bioreactor system provides an ideal orga-noid culture environment with controlled temperature and media
flow, avoids any chance of contamination and an imaging chamber
that allows tracking of a particular cell as it grows, which was verydifficult with other techniques. We believe that our work on theBiomicrofluidics ARTICLE scitation.org/journal/bmf
Biomicrofluidics 15,024105 (2021); doi: 10.1063/5.0041027 15,024105-1
Published under license by AIP Publishing.microfluidics based 3D printed bioreactor for live-cell imaging has
many advantages including low costs; a compact, mobile, and stand-
alone design; and an optimal confined environmental enclosure for3D live-organoid imaging. As the growth happens in a fully closedenvironment, this can be used to mimic interactions between humanorganoids host with pathogens like coronavirus and has the potential
to accelerate the development of therapeutics.
34–36
II. EXPERIMENTAL SECTION
A. Microfluidic organoid imaging platform design and
fabrication
A microfluidic chip was designed as shown in Figs. 1(a) and2
for the support of both imaging and culturing of organoids. Briefly,the design of the chip included imaging wells compatible withorganoid long-term culture and microfluidics channels for mediumflow and pre-heating. Individual organoids were placed in each well
of the chip and were embedded in Matrigel, a complex extracellular
environment matrix composed of natural polymers and growthfactors, which acts as a scaffold for long-term growth of organoidsinto a 3D spheroid form. A transparent glass disk with a thickness
of 150 μm was then placed on top of the well, acting both as the
water seal for culture medium and as an optical window forlive-organoid imaging. The chip was placed on a heating plate. The
culture medium providing all the required growth nutrients to
the organoids was then pumped into the pre-heater chamber of thechip where it is warmed to 37
/C14C and distributed to one or all wells
as in Fig. 2 . The system was automatized using micro-controllers.
The microfluidic chip was made using stereolithography based
3D printing technology. SolidWorks (Dassault Systems) 3D CAD
modeling software was used to design this chip, and a “.STL ”file
was generated for 3D printing using a desktop 3D printer (Form 2printer, Formlabs). A bio-compatible dental surgical grade resin(Dental SG Biocompatible Resin, Formslab) was chosen as the
printing material, which supports a printing resolution of about
50μm. The printed chip was then cured by exposure to UV light
for 1 h. The chip was sterilized with an autoclave after printing byexposure to pressurized steam at a temperature of 121
/C14C for
30 min. The chip was designed with four culture and imaging wells.
Dimensions of the wells were optimized for culture medium availabil-
ity and flow and to be compatible with the working distance of the16/C2=0:8 NA microscope objective (Nikon) (3 mm). Small indents of
0:5/C20:5m m
2size were added to the inner surface of the wells, as
shown in Fig. 1(b) , to restrict the detachment of the Matrigel from the
3D printed well. The top of the well was sealed with a glass disk of
150μm thickness using a bio-compatible silicone (Kwik-Sil, WPI) or
UV-curing (NOA61, Norland Products) adhesive. The chip thusallows examination through a transparent optical window, with excel-
lent isolation from environmental perturbations. Each of the wells was
designed with a thermistor port, and a drug delivery port, for continu-ous temperature logging of each well and selective drug delivery to theindividual organoids, respectively. For drug delivery, a standardcannula was inserted inside the drug delivery port and sealed using a
bio-compatible silicone (Kwik-Sil, WPI) or UV-curing (NOA61,
Norland Products) adhesive. This cannula provides a hermeticalsealing support and acts as a one-way valve for the drug delivery. Thechip was designed with microfluidics channels, organized to allowmedia flow over the heating plate (pre-heater chamber), and then in
and out the wells. One inlet for culture medium input and four outlets
for each well were connected to the tubing. The system is scalable, andthe number of wells can easily be increased.
The chip was connected to an incubator environment, as
shown in Fig. 2 , for the growth of organoids. This complete incuba-
tor and supporting devices were designed to fit into a compact
form factor, with a four-well chip size of 6 /C24c m
2for easy transfer
between different instruments.
As the organoids take nutrition from the culture media for
growth, we must periodically replace the culture media. Each well
was designed to be independently selected through the use of a sol-enoid valve for culture media replacement. The culture media wasbubbled periodically with a gas mixture of 5% CO
2, 21% O 2, and
balanced N 2. A DC motor based small peristaltic pump (#1150,
Adafruit) was used for feeding the culture media to organoids. The
fluid flow was precisely controlled using a pulse width modulation(PWM) signal from the micro-controller. The details of the quan-tity of the culture medium and Matrigel used are shown in Table I .
As the organoids require a temperature of around 37
/C14C for
growth, a bench-top incubator environment was manufactured
using aluminum blocks, where the chip is placed in close contactusing screws. These aluminum blocks are black anodized to reduce
FIG. 1. (a) Imaging and organoid setup of a well in a microfluidic chip. The
design supports both organoids culturing and imaging. (b) Organoid placementin the well.Biomicrofluidics ARTICLE scitation.org/journal/bmf
Biomicrofluidics 15,024105 (2021); doi: 10.1063/5.0041027 15,024105-2
Published under license by AIP Publishing.the back-scattered light during imaging and to improve the thermal
conductivity. A resistive heater was connected below the oven base-plate and a 10 k Ωnegative temperature coefficient thermistor was
used for feedback to regulate the temperature of the oven.The culture media entering the chip was stored at an ambient tem-
perature of around 25
/C14C, which is lower than the temperature
required for the organoid growth. Such a sudden temperaturechange can induce a thermal shock for the organoid. To avoid this
situation, an on-chip pre-heater was implemented. For uninter-rupted temperature control, a dedicated micro-controller is pro-grammed with a PID algorithm and can be controlled via USBbased virtual serial port.
Figure 3(a) shows the complete mechanical assembly of the
microfluidic bioreactor. It has three main components: an aluminumoven, a microfluidic chip, and an acrylic sheet for integration to amicroscope. The oven has two parts, the first is a base heater platethat is in direct contact with the microfluidic chip, with its tempera-
ture regulated at about 37
/C14C. The second one is an oven cap that is
used to isolate the wells from ambient temperature fluctuations.The oven cap is opened while imaging the organoids as shown inFig. 3(b) and is kept closed otherwise as shown in Fig. 3(c) .
Heat transfer through the objective lens can cause a local tempera-
ture drop at the well while imaging. Hence, the objective lens
FIG. 2. Microfluidic setup for live-cell imaging and culturing.
TABLE I. Volume of matrigel and culture medium.
Fluid Volume ( μl)
Matrigel in four wells 78 × 4
Culture medium in four wells 48 × 4
Volume of the culture medium in pre-heater 370Biomicrofluidics ARTICLE scitation.org/journal/bmf
Biomicrofluidics 15,024105 (2021); doi: 10.1063/5.0041027 15,024105-3
Published under license by AIP Publishing.temperature should be regulated close to 37/C14C using the feedback
driven thermal source, in long-term experiments.
Our design costs are significantly lower than traditional Petri
dish or spin-bioreactor based organoid culture products that can
cost tens of dollars. For 3D printing, approximately 15 ml resin was
used including the resin used for the scaffold, at a cost of onlyaround 5 USD per chip. We also note that the culture mediumused is just around 764 μl per refill.
Finally, the culture medium was pumped through each well, and
the chip was ready to be combined with a pre-heated oven, shown inFig. 4(b) ,t h a tk e p tt h et e m p e r a t u r ea ta p p r o x i m a t e l y3 7
/C14C. Before
filling the media inside the well, the chip should not be placed insidethe oven. Note that we must place the chip in the oven only afterfilling media inside the well. This ensures that the water content in the
Matrigel does not evaporate and condense on the glass surface, trap-
ping air, and affecting the quality of the optical image.
After the imaging experiments, the silicone glue used to seal
the glass coverslip on top of each well can be easily removed
and the organoids collected for post hoc assays such as IHC,
RNA/DNA/protein extraction, and quantification. After removal of
FIG. 3. Microfluidic bioreactor assembly. (a) Exploded view of bioreactor. (b) Imaging mode. (c) Incubation mode.Biomicrofluidics ARTICLE scitation.org/journal/bmf
Biomicrofluidics 15,024105 (2021); doi: 10.1063/5.0041027 15,024105-4
Published under license by AIP Publishing.the glass coverslips, the organoids, and the Matrigel, a chip can be
washed with distilled water, dried, and autoclaved; and is therefore
re-usable.
Figure 4(c) shows the complete system developed for an orga-
noid culture in an incubation environment.
B. Production of cerebral organoids
Brain organoids were produced from iPSC obtained from the
Coriell cell line repository (GM23279A) following the standard
protocol.51iPSC colonies were grown in iPSC media, consisting of
20% Knock-out Serum Replacement (KOSR) (Invitrogen), 1%penicillin/streptomycin (Invitrogen), 1% non-essential amino
acids (Invitrogen), 0.5% L-glutamine (Invitrogen), 100 μm
2-mercaptoethanol (Bio-Rad), DMEM/ F-12 (Invitrogen), supple-mented with 10 ng/mL bFGF (Stemgent). The culture media waschanged daily. iPSC colonies were passaged weekly onto six-wellplates coated with 0.1% gelatin (EMD Millipore) and pre-seeded
with a feeder layer of irradiated mouse embryonic fibroblasts
(MEFs) (GlobalStem), which were plated at a density of 200,000cells/ well. Passaging entailed lifting colonies with 2.5 mg/mlCollagenase, Type IV (ThermoFisher) for 1 h and dissociation into
smaller pieces through manual tritulation before seeding onto
feeder layer of MEFs. iPSCs were detached from irradiated MEFs
FIG. 4. Microfluidic bioreactor. (a) Organoids deployment in microfluidic chip. (b) Bioreactor integration with the control system. (c) Compact standalone bioreactor.Biomicrofluidics ARTICLE scitation.org/journal/bmf
Biomicrofluidics 15,024105 (2021); doi: 10.1063/5.0041027 15,024105-5
Published under license by AIP Publishing.and plated at 9 /C2104cells per well of an ultra-low attachment
96-well plate (Corning) in iPSC media supplemented with bFGF
(10 ng/ml) and ROCK inhibitor (50 μm; Y-27632, Tocris) (day 0).
Embryoid bodies (EBs) were subsequently transferred (day 6) to anultra-low attachment 24-well plate (Corning) with neural inductionmedia: 1% N
2supplement (Invitrogen), 1% Glutamax (Invitrogen),
1% non-essential amino acids (Invitrogen), 5 μg/ml heparin
(Sigma), DMEM/F12 (Invitrogen), supplemented with 10 μm
SB431542 (Tocris Bioscience) and 1 μm dorsomorphin (Stemgent).
After this neural induction step, EBs were embedded in Matrigel(Corning) droplets on day 11 and transferred to neural differentia-
tion media: DMEM/F12: Neurobasal (Invitrogen), 0.5% N
2supple-
ment, 1% Glutamax, 0.5% non-essential amino acids, 100 μm
2-mercaptoethanol, insulin, 1% Pen/Strep (Invitrogen)supplemented with 1% B27 without vitamin A (Gibco, LifeTechnologies). Matrigel embedded organoids can then be placed in
the chip from D11 onward. For the fluorescence imaging of radial
glial cells, organoids were infected with a pAV-CMV-GFP virusprior to embedding in the chip.
C. Organoids deployment in microfluidic chip
Initially, Matrigel of around 45 μl was filled inside each well
and let to solidify for 20 min at room temperature. Then, organoidsat day 15 of differentiation were selected and placed in the centerof each well, and another layer of Matrigel was applied on top of
the organoids, letting them solidify at room temperature. This
sandwiching process gave an extracellular scaffold for organoids togrow as a 3D spheroid. The scaffold also held organoids inside thewell against the current of culture medium. Finally, a 12 mm
microscope glass disk was kept on the top of the well and was
sealed with silicone or UV-curing adhesive. Figure 4(a) shows the
full setup of the microfluidic chip containing organoids positionedinside the wells, culture medium flow, and the thermistor probe.D. Cryosectioning and immunohistochemistry
For the analysis of viability, organoids were fixed by a 30 min
incubation in 4% paraformaldehyde solution. Fixed organoids wereincubated in 20% sucrose solution overnight at 4
/C14C, followed by incu-
bation in 30% sucrose for 3 h before embedding and freezing in
optimal cutting temperature (O.C. T.) medium. Frozen organoid tissue
was sliced into 20 μm sections using a cryostat. Permeabilization/block-
ing was performed using 3% BSA/0.1% TX100 in TBS. Incubation ofsections from cerebral organoids i n primary antibodies solution was
performed overnight at 4
/C14C and in secondary antibodies solution at
room temperature for 1 h (Alexa Fluor, Molecular Probes). The follow-ing primary antibodies were used: cleaved caspase 3 (Cell Signaling,#9661, 1:200) and Ki67 (BD Biosciences, #550609, 1:200). Coverslipswere affixed with vectashield hard set antifade mounting media with
DAPI (Vector Laboratories), and z-stacks were acquired using a Leica
TCS SP8 confocal microscope.
E. Two-photon microscopy
The imaging was performed using a Prairie Ultima IV two-
photon microscopy system with a galvo-galvo scanning module
(Bruker). A 910 nm wavelength excitation light was provided by a
tunable Ti:Sapphire laser (Mai-Tai eHP, Spectra-Physics) with disper-sion compensation (DeepSee, Spectra-Physics). For collection, weused a 16 /C2=0:8 NA microscope objective (Nikon) and a GaAsP
photomultiplier tubes (Hamamatsu) that allowed us to image a large
number of cells. Images thus acquired, using a PrairieView acquisitionsoftware, were then processed using the ImageJ software.
III. RESULTS AND DISCUSSION
A. Experimental and simulation results of the oven
For the heater, a 20 Ω, TO-126 package resistor was used with
a1 0 kΩnegative resistance coefficient (NTC) thermistor for
FIG. 5. Oven temperature controller. (a) 3D model. (b) T emperature controller block diagram.Biomicrofluidics ARTICLE scitation.org/journal/bmf
Biomicrofluidics 15,024105 (2021); doi: 10.1063/5.0041027 15,024105-6
Published under license by AIP Publishing.feedback, as shown in Fig. 5(a) . A PID control algorithm, running
on an Atmega328p microcontroller with a LM298 H-bridge driver,
was used for controlling the current through the heater. The con-troller measures the temperature of the oven and compares it withthe target temperature of 37
/C14C and computes the error signal e(t).
Based on the error response signal, u(t) is calculated with propor-
tional (P), integral (I), and derivative (D) terms as
u(t)¼Kpe(t)þKiðt
0e(t0)dt0þKdde(t)
dt: (1)The magnitude of u(t) is mapped to the duty cycle of the
PWM signal pin, as shown in Fig. 5(b) . Based on the value of u(t),
the duty cycle will be adjusted such that the error reaches zero. TheP block is a gain factor to amplify the error; I block is for calculat-ing cumulative error, integrated over time, to eliminate the residualerror; and the D block gives the rate of error change (the more
rapid the error change, the greater the damping effect). The K
p,Ki,
and Kdconstants were optimized in the program for precise tem-
perature control with fast settling, integrated over 13 min. A ther-mally conductive paste was applied between the heater and the
FIG. 6. Oven thermal simulation. (a) T emperature distribution on chip. (b) T emperature distribution less than 0.21/C14C across all the wells.
FIG. 7. Oven experimental results. (a) Real-time temperature data from oven. (b) T emperature tolerance histogram over the set temperature of 37.4/C14C.Biomicrofluidics ARTICLE scitation.org/journal/bmf
Biomicrofluidics 15,024105 (2021); doi: 10.1063/5.0041027 15,024105-7
Published under license by AIP Publishing.oven for efficient thermal control. It is important to replace this
paste after using it for a few months for ensuring good thermal
conductivity.
The thermal simulation was carried out in SolidWorks to study
the thermal gradient along the organoids in the wells. Figure 6(a)
shows the steady state result of the microfluidic chip with organoids
modeled as a spherical mass, with a mass density 1050 kg/m3,a
thermal conductivity of 0.53 W/(m K), and a specific heat of3690 J/(kg K). Their corresponding thermal gradient is shown inFig. 6(b) . We observed that the temperature was successfullymaintained around 37
/C14C, with the maximum temperature difference
across the four wells being less than 0.21/C14C at the steady state.
A PID temperature controller was connected to the oven
heater and to the feedback thermistor. The experimental tempera-ture inside the four wells was measured, as shown in Fig. 7(a) .
We notice that the oven reached a steady state after about 12 min.
The box-and-whisker plot computed for the data at steady state
temperature is shown as an inset in Fig. 7(a) . These steady state
temperature data were acquired for around 12 h, and from the boxplot, it can be observed that the median was around 37.6
/C14C and
FIG. 8. Pre-heater flow optimization.Biomicrofluidics ARTICLE scitation.org/journal/bmf
Biomicrofluidics 15,024105 (2021); doi: 10.1063/5.0041027 15,024105-8
Published under license by AIP Publishing.50% of the data was 0.2/C14C around the mean. The maxima and
minima shown by the whiskers are within 0.4/C14C. The steady state
data temperature tolerance is shown in Fig. 7(b) . We note that the
oven was regulating the temperature within +0:5% tolerance
around 37.4/C14C.B. Culture medium flow control and pre-heater
The culture medium, containing 191 μl inside the wells, was
replaced within 6 s by using a calibrated flow rate of around
33.75 μl/s. Such precise calibration was achieved by using a PWM
FIG. 9. (a) Representative images of immuno-labeling of proliferation marker
Ki67 and apoptosis marker cleaved caspase 3 (c-Cas3). Scale bar represents200μm. (b) Quantification of the percentage of positive cells.
FIG. 10. (a) Carton figure (up, left) and 3D rendering of the intrinsic originated
from the surface (856 /C2856/C2240μm) of an organoid growing in the micro-
fluidic stage-top bioincubator. Scale bar represents 200 μm. (b) Quantification of
relative growth.Biomicrofluidics ARTICLE scitation.org/journal/bmf
Biomicrofluidics 15,024105 (2021); doi: 10.1063/5.0041027 15,024105-9
Published under license by AIP Publishing.FIG. 11. Imaging of the second harmonic signal from brain organoid ventricular zones (VZs). (a) Cartoon figure of the VZ structures contained in a cortical org anoid. (b)
3D reconstruction and 2D z sections of an organoid. The field of view contained several VZ structures. The yellow square indicates the field-of-view m agnified in (c). (c)
Magnified time-lapse imaging of one VZ structure demonstrated that global cell movement can be tracked. (d) The left image is a time colored projectio n of the time-lapse
imaging. The VZ area is deliminated with yellow dashed lines. Example individual cells are indicated with colored arrows. Scale bars represent 200 μm.Biomicrofluidics ARTICLE scitation.org/journal/bmf
Biomicrofluidics 15,024105 (2021); doi: 10.1063/5.0041027 15,024105-10
Published under license by AIP Publishing.based control with a change in duty cycle translating to the speed
of the peristaltic pump. The Reynolds number for this microflui-
dic channel of 2 /C20:5m m2and 33 μl/s flow rate was around 263,
indicating a laminar flow inside the channel. The culture mediumneeds to be pre-heated to avoid any thermal shock to the organo-
ids. Figure 8 shows the simulation results of the on-chip pre-
heater. The culture media at an ambient temperature of 25/C14Ca r e
passed through the coils of the microfluidic channels, and the
FIG. 12. High resolution two-photon
microscopy imaging of GFP-labeled
radial glial cells in a brain organoid
embedded and cultured in the micro-fluidic chip and bioreactor. (a) Exampleimages of dense populations of GFP
positive radial glial cells organized radi-
ally around a ventricle-like cavity. Thefine morphology of individual cells canbe distinguished. Scale bar represents
100μm. (b) Example of high magnifi-
cation time-lapse imaging revealingchanges in cell morphology. Left imageis a time colored projection of the time-
lapse imaging. Scale bar represents
20μm.Biomicrofluidics ARTICLE scitation.org/journal/bmf
Biomicrofluidics 15,024105 (2021); doi: 10.1063/5.0041027 15,024105-11
Published under license by AIP Publishing.temperature is regulated to 37/C14C using a resistive heater. The
fluid delivered to the wells was thus optimized to be close to
37/C14C (35.49 –36.61/C14C).
C. Viability
To assess the viability of organoids grown in the stage-top
microfluidic bioreactor, we measured the percentage of cells
expressing the proliferation marker Ki67 and apoptotic marker
cleaved caspase 3 (c-Cas3). We examined the ventricular zones(VZs), structures organized around a cavity or ventricle that resem-ble the developing neocortex, and the core region of organoidsgrown for 7 days in the bioreactor or grown in a regular culture
environment. We observed no significant difference in the percent-
age of proliferative cells ( Fig. 9 ). Moreover, the percentage of apo-
ptotic cells in both the ventricular zone and the organoid core wasdecreased in organoids cultured in the microfluidic bioreactor com-pared to regular culture conditions supporting good viability of the
organoid grown in the bioreactor. One advantage offered by our
microfluidic device is that it allows continuous perfusion of theculture chamber, which more closely mimics a physiological tissueperfusion, than conventional culture, and thus reduces cell death atthe organoid core.
D. Chronic imaging of brain organoid growth
Our stage-top incubator with an automated fluidic system
containing embedded organoids was placed under a two-photonmicroscope, which allowed us to capture images despite thechallenges of high density and opacity of the organoids. We per-
formed chronic imaging and obtained high resolution imaging
of intact organoids from the surface to a depth of 100 –200μm,
capturing either intrinsic (autofluorescence, second harmonicgeneration) or fluorophore (GFP expression) signals. First, we
successfully imaged the growth of organoids embedded in the
chip during 7 days as shown in Fig. 10 . We observed an increase
in the volume of the 3D structure (1.88-fold after 7 days), sup-porting the high viability of the organoids and suggesting theorganoid can expand through normal cell division growth in the
bioreactor.
E. High resolution time-lapse imaging and cell
tracking
Brain organoids contain ventricular zones (VZs) [ Fig. 11(a) ].
These regions contain several brain cell types, including neurons,
and radial glial cells that expand their processes radially to guide
neuronal migration [ Fig. 11(a) ]. We were able to image the intrinsic
signal emitted in these regions, demonstrating the use of the micro-fluidic bioreactor for applications such as 3D and time-lapseimaging of organoid VZs, as shown in Fig. 11 . Such imaging can
be used for the tracking of single cells and the analysis of their
displacement.
Finally, we expressed a GFP construct selectively in radial glial
cells, which allow us to image with extremely high resolution the
morphology and organization of these polarized cells after organo-
ids were embedded and cultured in the microfluidic bioreactor asshown in Fig. 12(a) . We performed time-lapse imaging and
observed changes in cell morphology as shown in Fig. 12(b) .
IV. CONCLUSION
A 3D organoid in vitro culture microfluidic device was
designed and fabricated to allow chronic imaging of the dynamicsinvolved in organoid self-assembly. Traditionally, organoids were
grown in a Petri dish, which was not very efficient and required a
large volume of culture media. Improved long-term culture camefrom bioreactors like rotary cell culture system (RCCS), SpinOmega (Spin Ω), and Spin Infinity (Spin 1) that have shown signifi-
cant improvement for long-term organoid growth.
37–42However,
these bioreactors do not support a means for examination/imaging
without a physical transfer of the organoids to a separate imagingchamber.
43The transfer process is prone to organoid damage or
contamination, which can lead to inaccurate estimation and limited
reproducibility of the results. Hence, we needed a system equipped
with the capacity to permit long-term imaging and limited disturb-ance of organoids grown in an incubator-like environment.
Current practices for brain organoid live imaging typically use
commercial culture dishes such as a 96-well glass-bottom plate
(Corning #4580). These plates are costly, do not provide an opti-
mized compartment for organoid growth, and do not allow formedia change/perfusion during imaging.
44,45Some researchers
have used commercial culture dishes, combined with microfabri-cated compartments. These again involve complicated fabrication
of PDMS stamps using a metal mold.
46They also typically require
the use of an inverted confocal microscope that is associated withan environmentally controlled chamber.
44–48The use of a conven-
tional incubation system for culture, separated and different fromthe environmentally controlled chamber for imaging, induces a
variation in the culture environment and can lead to cellular stress.
In recent years, many microfluidic organoids-on-a-chip systemshave been developed, demonstrating that this culture model isuseful. These organoid-on-a-chip systems continue to improve and
are becoming useful in organ and disease modeling and drug dis-
covery studies.
49,50,26However, most organoid-on-a-chip platforms
are limited by the use of soft lithography, a technically demandingmicrofabrication process. Our system has the advantage of beingeasily manufactured, via 3D printing, and is more versatile than the
aforementioned systems published in the literature. In our design,
chip printing is cheap and easily adjustable. It is adaptable to anymicroscope, including non-standard custom made or uprightmicroscopes. Moreover, our design provides a fully enclosedsystem, allowing for a sterile and safe working condition, with con-
tinuous perfusion of a nutrient-rich culture medium. We also have
precise control of the cell environment and can keep it invariantduring both the growth and imaging steps of organoid cultures.
The 3D printed microfluidic chip with imaging chambers and
drug delivery system functions with a temperature controller and a
pump to create an isolated and compact bioreactor that can be
installed temporally into a microscope for live imaging. We demon-strated the use of this bioreactor for imaging of a brain organoidexpansion and cell morphology tracking for up to 7 days. As we
observed normal organoid growth during this long-term incuba-
tion, we expect that the organoid could grow for extended periodsBiomicrofluidics ARTICLE scitation.org/journal/bmf
Biomicrofluidics 15,024105 (2021); doi: 10.1063/5.0041027 15,024105-12
Published under license by AIP Publishing.of time with the only constraint of the dimension of the wells. Our
3D printed microfluidic bioreactor for the culture and live imaging
of 3D biological tissues can find its applications in many researchor industrial laboratories where organoids can be modeled to studydevelopment, diseases, or interactions between human host orga-noids and pathogens like coronavirus SARS-COV2. It is a low-cost
solution compared to traditional laboratory methods, with a fabri-
cation price of around 5 USD per chip, and an efficient microflui-dic delivery of the expansive culture media.
Ongoing work includes scali ng up the number of wells and
integration of additional features like electrophysiology to study
the organoid model. In future, the microfluidic valves and pumps
can also be 3D printed on-chip t o reduce the amount of culture
medium used.
AUTHORS ’CONTRIBUTIONS
I.K. and C.D. contributed equally to this work.
ACKNOWLEDGMENTS
This work was supported by National Institutes of Health
(NIH) under Grant No. R01MH085802 (M. Sur) and the Centre
for Computational Brain Research, IIT Madras. M. Sur holds theN.R. Narayanamurthy Visiting Chair Professorship inComputational Brain Research at IIT Madras. I. Khan was sup-ported by the Department of Biotechnology under Grant No.
BIRAC-BT/BIPP0946/36/15, and C. Delepine by the Picower
Fellowship Program. We also thank members of the Sur lab andthe Photonics Group, Electrical Engg., IIT Madras for theirsupport.
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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Published under license by AIP Publishing. |
1.3421029.pdf | Dissipation function of magnetic media
V. G. Bar’yakhtar and A. G. Danilevich
Citation: Low Temperature Physics 36, 303 (2010); doi: 10.1063/1.3421029
View online: http://dx.doi.org/10.1063/1.3421029
View Table of Contents: http://scitation.aip.org/content/aip/journal/ltp/36/4?ver=pdfcov
Published by the AIP Publishing
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205.208.120.10 On: Sat, 20 Dec 2014 07:52:33Dissipation function of magnetic media
V. G. Bar’yakhtara/H20850and A. G. Danilevichb/H20850
Institute of Magnetism of the National Academy of Sciences and Ministry of Education and Science
of Ukraine, pr. Vernadskogo 26-B, Kiev 03142, Ukraine
/H20849Submitted November 23, 2009 /H20850
Fiz. Nizk. Temp. 36, 385–393 /H20849April 2010 /H20850
A general method of constructing a dissipation function is developed for disordered magnetic
media and for magnetically ordered systems. As an example it is shown for a ferromagnet thatnot only the invariance with respect to uniform rotations of the body but also the law of conser-vation of magnetization must be taken into account in order to construct a dissipation function. Itis found that in ferromagnets the dissipation term in the equations of motion for the magnetiza-tion is a sum of Bloch and Landau–Lifshitz–Gilbert relaxation terms. The region of applicabilityof the relaxation term in the Landau–Lifshitz form is determined. The damping of spin waves ina ferromagnet with tetragonal symmetry is calculated. A procedure is formulated for transitioningfrom a ferromagnet with lower symmetry to a ferromagnet with a continuous degeneracy param-eter. In this case the relaxation process can be systematically described by means of the dissipa-tion function described in this article. It is shown how the relaxation term of a general form forferromagnets becomes the Bloch relaxation term for paramagnets. It is shown that the magnetiza-tion vector relaxes in two stages. First the magnetic moment relaxes in magnitude quite rapidlyas a result of exchange enhancement and then the magnetization relaxes slowly to its equilibriumdirection. The second stage qualitatively corresponds to the relaxation picture described by theLandau–Lifshitz model. © 2010 American Institute of Physics ./H20851doi:10.1063/1.3421029 /H20852
I. INTRODUCTION
One of the most promising directions for developing
new magnetic media for recording information are so-calledpatterned media where a single element is hundreds and eventens of nanometers in size.
1Brillouin light scattering2and
ferromagnetic resonance3experiments show that the high-
frequency properties of such systems differ from the proper-ties of continuous films by, first and foremost, the spatialquantization of the planar component of the wave vectorarising because the radius of a single disk is finite, whichresults in the formation of standing spin waves of a dipolenature. For such structures it is of great current interest tocalculate the relaxation of the magnitude of the magnetiza-tion and to describe the damping of spin waves.
There are two forms of the relaxation term in the current
literature. The first one was proposed by Landau andLifshitz
4and modified by Gilbert.5The second type of relax-
ation term was proposed by Bloch for paramagnets.6In Ref.
4 the relaxation term does not describe exchange relaxationand the relaxation term in the equation of motion is chosenso that the magnitude of the magnetization is conserved. Amethod of obtaining the dissipation function for a ferromag-net and determining the relaxation terms in the equation ofmotion for the magnetization according to the dissipationfunction is developed in Refs. 7 and 8. This method gives theBloch-type relaxation term.
The present work is devoted to further development of
the method for constructing the dissipation function for mag-nets with any symmetry.
II. MAGNETIZATION RELAXATION IN A FERROMAGNET
We shall construct a dissipation function for ferromag-
nets on the basis of the phenomenological principles ex-pounded by Landau and Lifshitz.4,9We recall the form of the
Landau–Lifshitz equation for the magnetic moment:
/H11509M
/H11509t=−/H9253/H20851MH /H20852+R. /H208491/H20850
In this equation Mis the magnetization, His the effective
magnetic field, /H9253is the gyromagnetic ratio, and Ris a relax-
ation term. The effective magnetic field is determined fromthe quasiequilibrium thermodynamic potential as follows:
4
H=−/H9254F
/H9254M,F=/H20885f/H20849M,/H11509M//H11509xi/H20850dV. /H208492/H20850
The quasiequilibrium thermodynamic potential Fis con-
structed on the basis of symmetry considerations and the ideathat the exchange energy is much larger than the relativisticenergies /H20849the magnetic dipole interaction energy and the
magnetic anisotropy energy /H20850:
f=−/H20849M
2−M02/H208502
8/H9273M02+1
2/H9251/H11509M
/H11509xi/H11509M
/H11509xi−1
2KMz2+h2
8/H9266−H0M
=fex+fa+fdd+fZ. /H208493/H20850
The quasiequilibrium thermodynamic potential was chosen
in the form presented in Refs. 9 and 10. The second term isthe nonuniform exchange energy, the third term is the mag-netic anisotropy energy, the fourth term is the magnetic di-pole interaction energy, and last term is the Zeeman energy.The magnetostriction energy is omitted for simplicity. Thisexpression differs from the one presented in Refs. 9 and 10by the specification of f
0/H20849M2/H20850: the function f0/H20849M2/H20850is re-
placed by the expression presented here by introducing a
longitudinal magnetic susceptibility /H9273of the ferromagnet
/H20849first term /H20850. For ferromagnets this susceptibility equals in or-LOW TEMPERATURE PHYSICS VOLUME 36, NUMBER 4 APRIL 2010
1063-777X/2010/36 /H208494/H20850/7/$32.00 © 2010 American Institute of Physics 303
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
205.208.120.10 On: Sat, 20 Dec 2014 07:52:33der of magnitude /H9273−1/H11015kTC//H9262M0/H11015J//H9262M0/H112711; here, kis
Boltzmann’s constant, TCis the Curie temperature, /H9262is the
Bohr magneton, and Jis the exchange integral.
Using Eq. /H208491/H20850and the relations for the quasiequilibrium
thermodynamic potential F/H208492/H20850and /H208493/H20850, we obtain the follow-
ing expression for the variation of Fwith time:
dF
dt=−2 Q=/H20885/H9254F
/H9254M/H11509M
/H11509tdV
=−/H20885H/H11509M
/H11509tdV=−/H20885HRdV. /H208494/H20850
Hence we have for the dissipation function Q
Q=1
2/H20885HRdv=1
2/H20885qdv. /H208495/H20850
In what follows we shall work with the density qof the
dissipation function, since it is this function that is necessaryto determine the form of the relaxation term R.
We recall that in the equilibrium state the effective mag-
netic field is zero,
9,10and the dissipation function is positive-
definite and a second-order infinitesimal with respect to thedeparture from the equilibrium state. In these cases the ap-proximation quadratic in His adequate for the dissipation
function.
7,8These considerations make it possible to repre-
sent qin the form8
q=1
2/H9261ikHiHk+1
2/H9261e/H20873/H11509H
/H11509xi/H208742
. /H208496/H20850
The first term in Eq. /H208496/H20850is of relativistic nature and the
second term is of exchange nature. For simplicity we tookthe exchange relaxation constant to be a scalar; the generali-zation is trivial. In our previous work the components of thetensor /H9261
ikwere taken in accordance with the symmetry of the
paramagnetic phase, i.e. for M=0. Actually, however, the
tensor /H9261ikdepends on the magnetization of the crystal, so that
this tensor must be expanded in powers of Min order to take
account of the magnetic structure of the medium. We shallconfine ourselves to the first term of the expansion and rep-resent qin the form
q=1
2/H9261ik/H20849M/H20850HiHk+1
2/H9261e/H20873/H11509H
/H11509xi/H208742
=1
2/H9261ik/H208490/H20850HiHk
+1
2/H9262ik,spHiHkMsMp+1
2/H9261e/H20873/H11509H
/H11509xi/H208742
. /H208497/H20850
In this relation the tensor /H9262ki,spis the expansion of the tensor
/H9261ikin powers of the magnetization.
The exchange, spin dipole, and spin-orbit interactions
are the real microscopic interactions determining the relax-ation processes in a ferromagnet. The spin-wave formalismis used in the microscopic theory of relaxation.
10The calcu-
lation of the relaxation time /H20849probabilities of the correspond-
ing relaxation processes /H20850is associated with the interactions
mentioned above. The leading orders of the perturbationtheory correspond to the terms in the dissipation functionwith higher powers of the magnetization. For example, thezeroth power of the magnetization in the relation /H208497/H20850forq
corresponds to the leading approximation in the calculationof the relaxation time. The second term in Eq. /H208497/H20850corre-sponds to the fourth order of perturbation theory. In other
words the classification of the terms with respect to the orderof smallness and the powers of the magnetization for qis
analogous to the classification of the terms in order of small-ness and the powers of the magnetization for the quasiequi-librium free energy /H208493/H20850.
9
The magnetic crystal class of the ferromagnet must be
specified in order to determine the concrete form of the re-laxation tensors /H9261
ikand/H9262ik,sp. For present analysis we shall
choose a tetragonal lattice, whose symmetry class in theparamagnetic phase is D
4h, and the time reversal operation
R.11Proceeding in the standard manner we find the invariants
of this symmetry class:
Mx2+My2,Mz2,Hx2+Hy2,Hz2, /H20849HyMx+HxMy/H208502,
HyMy+HxMx,HzMz. /H208498/H20850
This group admits the existence of a magnetization vector
directed along the tetragonal axis /H20849Zaxis /H20850. Knowing this
combination we represent the second term of the dissipationfunction /H208497/H20850as follows:
1
2/H926231/H20849Hx2+Hy2/H20850/H20849Mx2+My2/H20850+1
2/H926232/H20849Hx2+Hy2/H20850Mz2
+1
2/H926241Hz2/H20849Mx2+My2/H20850+1
2/H926242Hz2Mz2+1
2/H926255/H20849HyMx
+HxMy/H208502+1
2/H926266/H20849HxMx+HyMy/H208502+/H926267/H20849HxMx
+HyMy/H20850HzMz. /H208499/H20850
We shall rewrite this relation in a form where the relaxation
terms of the Landau–Lifshitz and Bloch types can be easilyseparated. It easy to see that the part of the dissipation func-tion that corresponds to the Landau–Lifshitz relaxation termmust have the from
q
LL=1
2/H20849/H20851M,H/H20852,/H9262/H20851M,H/H20852/H20850, /H2084910/H20850
where in accordance with the tetragonal symmetry the tensor
/H9262has the form
/H9262=/H20898/H926232 00
0/H926232 0
00 /H926231+/H926255/H20899. /H2084911/H20850
The remaining part of the dissipation function /H208497/H20850is
q3=1
2/H20849/H926232−/H926241/H20850Hz2M2+1
2/H20849/H926232−/H926241+/H926242/H20850Hz2Mz2
+1
2/H20849/H926231+/H926266/H20850/H20849Hx2Mx2+Hy2My2/H20850+/H20849/H926231+2/H926255
+/H926266/H20850HxHyMxMy+/H20849/H926232+/H926267/H20850/H20849HxMx
+HyMy/H20850HzMz. /H2084912/H20850
Using the same considerations as in the construction of the
quasiequilibrium thermodynamic potential of a ferromagnet/H208493/H20850it can be shown that the terms in the dissipation function
that contain higher powers of the magnetization vector are304 Low Temp. Phys. 36/H208494/H20850, April 2010 V. G. Bar’yakhtar and A. G. Danilevich
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205.208.120.10 On: Sat, 20 Dec 2014 07:52:33higher-order infinitesimals than the terms presented in Eq.
/H208498/H20850.
The tetragonal symmetry also determines the form of the
tensor /H9261ik:
/H9261ik=/H20898/H92611100
0/H9261110
00 /H926133/H20899. /H2084913/H20850
Thus the final form of the dissipation function of a ferromag-
net with tetragonal symmetry can be written as
q=1
2/H926111H/H110362+1
2/H926133Hz2+1
2/H20849/H20851M,H/H20852,/H9262/H20851M,H/H20852/H20850
+q3+1
2/H9261e/H20873/H11509H
/H11509xi/H208742
. /H2084914/H20850
The following notation has been used in these relations:
H/H11036/H11013Hxex+Hyey, where eiare unit vectors along the corre-
sponding axes and q3is determined by the relation /H2084912/H20850.
We shall determine the relaxation term in Eq. /H208491/H20850as a
derivative of the dissipation function with respect to the ef-fective magnetic field:
R=/H9254Q
/H9254H=/H11509q
/H11509H−/H11509
/H11509x/H11509q
/H11509/H11509H
/H11509x. /H2084915/H20850
It is easily shown that this definition corresponds not only to
the quadratic form of the dissipation function but also to anarbitrary polynomial. We obtain from the relation /H2084914/H20850a re-
laxation term of the form
R=R
1+R2+R3, /H2084916/H20850
where the following notation has been introduced:
R1=/H926111H/H11036+/H926133Hzez−/H9261e/H9004H,
R2=/H20851M,/H926232/H20851M,H/H20852/H11036/H20852+/H20851M,/H20849/H926231+/H926255/H20850/H20851M,H/H20852zez/H20852,
R3x=/H20849/H926231+/H926266/H20850HxMx2+/H20849/H926231+2/H926255+/H926266/H20850HyMxMy
+/H20849/H926232+/H926267/H20850MzMxMz,
R3y=/H20849/H926231+/H926266/H20850HyMy2+/H20849/H926231+2/H926255+/H926266/H20850HxMxMy
+/H20849/H926232+/H926267/H20850HzMyMz,
R3z=/H20849/H926232−/H926241/H20850HzM2+/H20849/H926232−/H926241+/H926242/H20850HzMz2+/H20849/H926232
+/H926267/H20850/H20849HxMx+HyMy/H20850Mz. /H2084917/H20850
It is evident from these relations and Eq. /H208491/H20850that the relax-
ation term RB=R1+R3describes relaxation processes in
which the magnitude of the magnetization changes and therelaxation of spin waves:
1
2/H11509M2
/H11509t=/H20849R1+R2/H20850M/HS110050. /H2084918/H20850
It gives Bloch-type damping.
The relaxation term R2describes relaxation processes in
which the magnitude of the magnetization is conserved:R
2·M=0, i.e. only the relaxation of spin waves—this is
Landau–Lifshitz type relaxation. It is important to note thatthe properties of the dynamical part of the Landau–Lifshitz
equation and of the relaxation term are different: the dynami-cal part
/H11509M //H11509t=−/H9253/H20851M/H11003H/H20852does not change under time re-
versal /H20849t→−t/H20850, while the relaxation term /H2084917/H20850changes sign
under such a transformation.
Landau and Lifshitz proposed that the relaxation of the
magnetization be described by means of the term4
R=/H9261LL
M2/H20851M,/H20851H,M/H20852/H20852. /H2084919/H20850
Gilbert represented this relaxation term in the form5
RG=/H9261G
M/H20851M,M˙/H20852,M˙/H11013/H11509M
/H11509t. /H2084920/H20850
The Landau–Lifshitz and Gilbert relaxation terms pre-
serve the absolute magnitude of the magnetization M2, which
corresponds to the conservation of the magnitude of theatomic spin. Both terms vanish in the equilibrium state, since
in this state H=0,M˙=0. The drawback of the relaxation
terms /H2084919/H20850and /H2084920/H20850is that a single relaxation constant is used
to describe a vector field, which the magnetization is.
Since in most fundamental works experiments were per-
formed on iron-yttrium garnet /H20849cubic ferromagnet /H20850, a single
relaxation constant is sufficient to describe them. However, itturns out that in experiments on the mobility of domain wallsin uniaxial films with a high figure of merit /H20849K/4
/H9266M02/H112711/H20850a
single relaxation constant in the Landau–Lifshitz equation is
inadequate. In addition, it has been shown in a number ofworks /H20849see, for example, Ref. 7 /H20850that for a ferromagnet with
a degenerate ground state the relaxation term /H2084919/H20850gives
qualitatively incorrect results: the damping of spin wavescalculated using Eq. /H2084919/H20850turns out to be a finite quantity,
while the frequency of the spin waves goes to zero as ak
→0. It is important to note that the dissipation function of a
ferromagnet is not discussed by Landau and Lifshitz or byGilbert or in the many works appearing after theirs. This wasfirst done in Ref. 7 and 8.
III. DISSIPATION FUNCTION IN THE UNIAXIAL CRYSTAL
MODEL
We shall now discuss the question of the formal transi-
tion from a real crystal to a model of a uniaxial crystal. Forthis we return to a ferromagnet with tetragonal symmetry andsymmetry axes /H20853Z
4;X2;Y2/H20854. Let us write out the part in the
expression for the free-energy density fthat corresponds to
the anisotropy energy:
fa=−1
2K1Mz2−1
4K2Mz4−1
2K3Mx2My2. /H2084921/H20850
For a tetragonal crystal the conservation law for the compo-
nents of the magnetization vector is not satisfied and there-fore the density of the dissipation function for a tetragonalferromagnet will have the form /H2084914/H20850.
If the anisotropy constant K
3appearing in the expression
/H2084921/H20850is set to zero, we shall arrive at the magnetic anisotropy
energy in the model of a uniaxial crystal with symmetry axisZ
/H11009. Invariants of this symmetry class will have the following
form:Low Temp. Phys. 36/H208494/H20850, April 2010 V. G. Bar’yakhtar and A. G. Danilevich 305
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205.208.120.10 On: Sat, 20 Dec 2014 07:52:33Mx2+My2,Mz2,Hx2+Hy2,Hz2,HyMy
+HxMx,HzMz. /H2084922/H20850
Comparing /H2084922/H20850with /H208498/H20850we find that the constant /H926255in the
expression for the dissipation function /H2084914/H20850must be set to
zero.
To simplify the dissipation functions further we shall
appeal to the law of conservation of the components of themagnetization along the axis of symmetry that holds for auniaxial crystal:
/H11509Mz
/H11509t+/H11509/H9016zk
/H11509xk=0 . /H2084923/H20850
The tensor /H9016ikis the flux density of the component iof the
magnetization through a unit area of surface perpendicular tothekaxis.
The effective magnetic field corresponding to the mag-
netic anisotropy energy /H2084921/H20850has the following form with
K
3=0:
H=K1Mzez+K2Mz3ez+/H9251/H115092M
/H11509xi2−/H20849M2−M02/H20850M
2/H9273M02. /H2084924/H20850
It is easy to show that the component /H9253/H20851M/H11003H/H20852zof the dy-
namical part of the Landau–Lifshitz equation reduces to a
divergence. As follows from the relations /H2084917/H20850the compo-
nent Rzof the relaxation term can be represented as a diver-
gence only if /H926133=0,/H926232=0,/H926241=0,/H926242=0, and /H926267=0.
Thus when transitioning from a tetragonal to a uniaxial
ferromagnet we must set certain relaxation constants to zero.In other words, /H9261
33,/H926232,/H926241,/H926242,/H926255, and/H926267of a tetrag-
onal crystal are functions of the constant K3that vanish as
K3→0. This gives us the basis for representing them as a
power series expansion in K3:
/H926133/H20849K3/H20850/H11015/H926133/H208490/H20850+/H20873d/H926133/H20849K3/H20850
dK3/H20874
0K3
+1
2/H20873d2/H926133/H20849K3/H20850
dK32/H20874
0K32+ ... /H110151
2/H926133/H11033K32. /H2084925/H20850
The constants /H926232,/H926241,/H926242,/H926255, and/H926267can be expanded in
a similar manner.
The first term on the right-hand side of these relations
vanishes because of the conservation law /H2084923/H20850. The second
term vanishes because the relaxation constant must be posi-tive and therefore does not depend on the sign of the aniso-tropy constant K
3. Finally, since K3/H11270K2,K4it can be con-
cluded that the relaxation constant /H926133is much smaller than
/H926111, and the relaxation constants /H926232,/H926241,/H926242,/H926255, and/H926267
are much smaller than /H926231and/H926266.
We present below the relations determining the dissipa-
tion function and the relaxation terms for the model of auniaxial crystal:
q=1
2/H926111/H20849Hx2+Hy2/H20850+1
2/H20849/H20851M,H/H20852,/H926231/H20851M,H/H20852zez/H20850
+q3+1
2/H9261e/H20873/H11509Hi
/H11509xk/H208742
,R=/H926111/H20849Hxex+Hyey/H20850+/H20851M,/H926231/H20851M,H/H20852zez/H20852
+R3−/H9261e/H115092H
/H11509xi/H11509xk, /H2084926/H20850
where
q3=1
2/H20849/H926231+/H926266/H20850/H20849Hx2Mx2+Hy2My2/H20850
+/H20849/H926231+/H926266/H20850HxHyMxMy,
R3x=/H20849/H926231+/H926266/H20850HxMx2+/H20849/H926231+/H926266/H20850HyMxMy,
R3y=/H20849/H926231+/H926266/H20850HyMy2+/H20849/H926231+/H926266/H20850HxMxMy,
R3z=0 . /H2084927/H20850
IV. DAMPING OF SPIN WAVES IN FERROMAGNETS
A. Ferromagnet with tetragonal symmetry
We shall now calculate the dispersion laws for spin
waves in a ferromagnet with tetragonal symmetry. Symmetryof this type was not chosen randomly, since quite a highpercentage of the ferromagnets used in recording elements aswell as in experimental research are tetragonal. Tetragonalsymmetry of a ferromagnet makes is possible to transition tothe higher uniaxial symmetry.
We shall present results for the ground states which are
usually studied: easy axis /H9021/H20849
/H20648/H20850, when the magnetic moment
is oriented along the chosen symmetry axis 0 z, and easy
plane /H9021/H20849/H11036/H20850, for which the magnetic moment of the ferro-
magnet lies in the basal plane x0y. For simplicity we choose
the azimuthal angle /H9278=0, since the results obtained for /H9278
=/H9266/2 are no different fundamentally.
Using the relations for the dissipation function density
/H2084914/H20850and the free-energy density /H2084921/H20850of a tetragonal ferro-
magnet as well as the equation of motion for the magnetiza-tion /H208491/H20850we find the dispersion law taking account of spin-
wave damping.
1./H9021„¸…phase: /H9278=0,/H9258=0,M2=M02„1−2/H9273¸K1…/„1+2/H9273M02K2…;
stability condition K 1+K2M02>0
The dispersion law and damping for spin waves are de-
termined by the relation
/H9275SW=−i/H20849/H926111+/H92612ek2−/H926232M02/H20850/H20851/H9251k2−/H20849K1+K2M02/H20850/H20852
/H11006/H9253M0/H20851/H9251k2−/H20849K1+K2M02/H20850/H20852. /H2084928/H20850
Spin waves will be well-defined quasiparticles, if the dissi-
pation part of the dispersion law Im /H9275SWis much smaller
than is rate component Re /H9275SW. It is easy to see that the
condition for the existence of spin waves in the ground state/H9021/H20849
/H20648/H20850is satisfied if the relaxation constants are much smaller
than the characteristic frequency /H9253M0. For simplicity we
shall write this condition in the long-wavelength approxima-tion:
/H20879Im/H9275SW
Re/H9275SW/H20879——→
k→0/H20879/H926111−/H926232M02
/H9253M0/H20879/H112701. /H2084929/H20850306 Low Temp. Phys. 36/H208494/H20850, April 2010 V. G. Bar’yakhtar and A. G. Danilevich
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205.208.120.10 On: Sat, 20 Dec 2014 07:52:33Since the relaxation term /H2084917/H20850was used, the damping
rate can be calculated not only for spin waves but also theabsolute value of the magnetic moment:
/H9275M=−i/H20853/H926133+/H208512/H20849/H926232−/H926241/H20850+/H926242/H20852M02+/H92612ek2/H20854
/H11003/H20873/H9251k2+2K2M02+1
/H9273/H20874. /H2084930/H20850
2./H9021„/c142…phase. /H9021=0,/H9258=/H9266/2,M2=M02; stability conditions:
K3<0,K1<0
The dispersion law and damping for spin waves are
given by the relation/H9275SW=−i
2/H20853/H20851/H926111−/H20849/H926231+/H926255/H20850M02+/H9261ek2/H20852/H20849/H9251k2−K3M02/H20850
+/H20849/H926133−/H926241M02+/H9261ek2/H20850/H20849/H9251k2−K1/H20850/H20854/H11006/H20881D,
where
D=1
2/H208534/H92532M02/H20849/H9251k2−K1/H20850/H20849/H9251k2−K3M02/H20850
−/H20851/H20849/H9251k2−K1/H20850/H20849/H926133−/H926241M02+/H9261ek2/H20850−/H20849/H9251k2−K3M02/H20850
/H11003/H20849/H926111−/H20849/H926231+/H926255/H20850M02+/H9261ek2/H20850/H208522/H208541/2. /H2084931/H20850
As is well known, the negative term in Drepresents the
decrease of the oscillation frequency as a result of relaxation.The condition for the second term in Eq. /H2084931/H20850to be real is
determined by the condition for the existence of spin wavesin the ground state /H9021/H20849/H11036/H20850, which holds for /H9261
ii,/H9262ii/H11270/H9253M0:
/H20879Im/H9275SW
Re/H9275SW/H20879——→
k→0/H20879/H20853/H20851/H926111−/H20849/H926231+/H926255/H20850M02/H20852K3M02/H208542+/H20851/H20849/H926133−/H926241M02/H20850K1/H208522
2/H92532M02K1K3M02 /H20879/H112701. /H2084932/H20850
The expression for the oscillation frequency of the absolute
value of the magnetization vector has the form
/H9275M=−i/H20851/H926111+/H20849/H926231+/H926266/H20850M02+/H92612ek2/H20852/H20873/H9251k2+1
/H9273/H20874. /H2084933/H20850
The time variation of the magnitude of the magnetization is
determined by the following equation:
M2/H20849t/H20850=M02+2M0m/H20849k,0/H20850exp/H20873−t
/H9270M/H20849k/H20850/H20874, /H2084934/H20850
where /H9270M/H20849k/H20850=/H20849i/H9275M/H20850−1is the relaxation time of the absolute
value of the magnetic moment. This relaxation time is short-
ened because of the small longitudinal susceptibility /H20849the fac-
tor/H9273/H20850as compared with the characteristic relaxation times of
spin waves /H9270SW/H20849k/H20850=/H20849Im/H9275SW/H20850−1, determined by the relaxation
constants /H9261ii,/H9262ik, and/H9261e. We note that relaxation of the non-
uniform and uniform deviations of the magnetization in mag-nitude occurs in the phase /H9021/H20849/H11036/H20850.
Comparing the relaxation times of the magnitude of the
magnetization and relaxation of spin waves shows that
/H9270SW/H20849k/H20850
/H9270M/H20849k/H20850/H110151
/H9273/H112711. /H2084935/H20850
This equality means that relaxation in the ferromagnet is
a two-step process. At the first, fast, stage the equilibriumdistribution of the magnetization over magnitude is estab-lished as a result of the exchange interaction. The relation/H2084934/H20850describes this process. At the second, slow, stage of
relaxation the magnetization precesses around its equilibriumvalue with the frequency of the spin waves and the amplitudeof the spin waves is damped with time
/H9270SW/H20849k/H20850. These consid-
erations concerning the two-step character of the relaxation
process in the ferromagnet are valid not only for the phase/H9021/H20849/H11036/H20850considered here but also for any other phases of theferromagnet. This is because the expression /H2084917/H20850forRcon-
tains the term /H20849M
2−M02/H20850//H208492/H9273M02/H20850with factors proportional to
the relaxation constants /H9261ii. For Eq. /H2084918/H20850governing the re-
laxation of the magnitude of the magnetization this termgives on the right-hand side of the equation the factor
2M
eq2M /H20648
2/H9273M02=m/H20648
2/H9273.
Here m/H20648is the component of the deviation of the magnetiza-
tion along the equilibrium values Meq. As a result the struc-
ture of the relaxation equation for the magnitude of the mag-netization acquires the form
dM
2
dt=−2/H20873/H9261ii
/H9273/H20874M /H20648Meq/H11015−/H20873/H9261ii
/H9273/H20874M2/H2084936/H20850
irrespective of the phase.
B. Ferromagnet with uniaxial symmetry and degenerate
ground states
A uniaxial ferromagnet is interesting because the ground
states for which the polar angle /H9258/HS110050/H20849phase /H9021/H20849/H11036/H20850, easy
plane: /H9258=/H9266/2; canted phase /H9021/H20849/H11028/H20850: cos2/H9258=−K1//H20849K2M02/H20850/H20850are
degenerate with a continuous degeneracy parameter—the
azimuthal angle /H9278of the magnetization vector in the ground
state. As noted above, it is impossible to describe dissipationprocesses in a uniaxial ferromagnet for degenerate groundstates using relaxation terms in the Landau–Lifshitz or Gil-bert form. It follows from the dispersion laws calculated inthis manner that spin waves do not exist in degenerate states,which contradicts Goldstone’s theorem.
We shall now find the dispersion law for spin waves and
the relaxation rate of the magnitude of the magnetization toits equilibrium value for the state /H9021/H20849/H11036/H20850of a uniaxial ferro-
magnet. We shall follow Bogolyubov’s ideology concerningLow Temp. Phys. 36/H208494/H20850, April 2010 V. G. Bar’yakhtar and A. G. Danilevich 307
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205.208.120.10 On: Sat, 20 Dec 2014 07:52:33quasiaverages in systems with a continuous degeneracy
parameter.12For this we use the previously obtained expres-
sion for the damping of spin waves in a tetragonal ferromag-net for the /H9021/H20849/H11036/H20850ground state. If the constant K
3is set to
zero and the relaxation constants /H926133and/H926232,/H926241,/H926242,/H926255,
/H926267are set to zero according to Eqs. /H2084926/H20850and /H2084927/H20850, then a
transition to the model of a uniaxial crystal obtains.
1. Easy axis phase /H9021„¸…:/H9258=0,M2=M02„1−2/H9273¸K1…/
„1+2/H9273M02K2…, stability condition: „K1+K2M02…>0
We obtain from Eq. /H2084928/H20850the dispersion law and damping
for spin waves:
/H9275SW=−i/H20849/H926111+/H92612ek2/H20850/H20851/H9251k2−/H20849K1+K2M02/H20850/H20852
/H11006/H9253M0/H20851/H9251k2−/H20849K1+K2M02/H20850/H20852. /H2084937/H20850The condition for the existence of spin waves in this case
has the form
/H20879Im/H9275SW
Re/H9275SW/H20879——→
k→0/H20879/H926111
/H9253M0/H20879/H112701. /H2084938/H20850
We obtain the following expression for the oscillation
frequency of the absolute value of the magnetization vector:
/H9275M=−i/H92612ek2/H20873/H9251k2+2K2M02+1
/H9273/H20874. /H2084939/H20850
2. Easy plane phase /H9021„/c142…:/H9258=/H9266/2,M2=M02, stability
condition: K 1<0
We obtain from the expression /H2084931/H20850the dispersion law
and damping for spin waves:
/H9275SW=−i
2/H20851/H20849/H926111−/H926231M02+/H92612ek2/H20850/H9251k2+/H92612ek2/H20849/H9251k2−K1/H20850/H20852
/H110061
2/H208814/H92532M02/H20849/H9251k2−K1/H20850/H9251k2−/H20851/H20849/H9251k2−K1/H20850/H92612ek2−/H9251k2/H20849/H926111−/H926231M02+/H92612ek2/H20850/H208522. /H2084940/H20850
The condition for the existence of spin waves in this case
will be determined by the expression
/H20879Im/H9275SW
Re/H9275SW/H20879/H11015k→0, /H2084941/H20850
and the expression for the oscillation frequency of the abso-
lute value of the magnetization vector becomes
/H9275M=−i/H20849/H926111+/H92612ek2/H20850/H20873/H9251k2+1
/H9273/H20874. /H2084942/H20850
It follows from the dispersion law /H2084940/H20850and the condition
/H2084941/H20850that spin waves in the degenerate ground state /H9021/H20849/H11036/H20850of
a uniaxial ferromagnet are well-defined and weakly damped.
Thus the relaxation term in the form proposed in the presentwork permits describing correctly the damping of spin wavesin ferromagnets with degenerate ground states.
V. DISSIPATION FUNCTION OF FERROMAGNETS AND
RELAXATION TERM IN THE EQUATION OF MOTION FORTHE MAGNETIZATION
In the preceding sections we examined relaxation pro-
cesses in magnetically ordered systems—ferromagnets. Weshall show that the method presented above for describingrelaxation processes can be used for a disordered magneticcrystal.
Let us consider a uniaxial paramagnet in an external
magnetic field. We choose the quasiequilibrium thermody-namic potential Fof the paramagnet with uniaxial anisotropy
in the formF=
/H20885/H208731
2/H9273M2−1
2KMz2−MH 0/H20874dV. /H2084943/H20850
Here Mis the magnetization of the paramagnet, /H9273is the
susceptibility of the paramagnet, Kis the anisotropy con-
stant, and H0is the external magnetic field. On the basis of
the foregoing considerations the dissipation function for aparamagnet with uniaxial anisotropy can be written in theform
q=1
2/H20849/H926111H/H110362+/H926133Hz2/H20850. /H2084944/H20850
In Eq. /H2084944/H20850His the effective magnetic field
H=−/H11509F
/H11509M=H0+KM zez−1
/H9273M, /H2084945/H20850
and/H926111and/H926133are the relaxation constants for the paramag-
net.
The relaxation term in the equation of motion for the
magnetization is
R=/H926111H/H11036+/H926133Hzez. /H2084946/H20850
In writing the dissipation function for the paramagnet we
took account of the fact that the second term in the expres-sion /H208497/H20850is a second-order infinitesimal with respect to the
small paramagnetic susceptibility
/H9273. For this reason, in the
case of paramagnet the Landau–Lifshitz form of the relax-ation term is neglected. Indeed, R
2/H11015M/H11003/H20849M/H11003H/H20850, and
since M/H11015/H9273H0, we have R2/H11015/H92732H02H. If an external mag-
netic field H0is applied to the paramagnet, then the state
with nonzero magnetization Mis the ground state for it. In308 Low Temp. Phys. 36/H208494/H20850, April 2010 V. G. Bar’yakhtar and A. G. Danilevich
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
205.208.120.10 On: Sat, 20 Dec 2014 07:52:33this case the equation of motion for the magnetization con-
sists of the dynamical and relaxation parts:
/H11509M
/H11509t=−/H9253/H20851M,H/H20852+/H926111H/H11036+/H926133Hzez. /H2084947/H20850
In the ground state H=0, and we find for the equilibrium
value of the magnetization
M0/H208491−/H9273K/H20850=/H9273H0. /H2084948/H20850
Using the relations /H2084945/H20850and /H2084948/H20850we represent the equation
of motion for the magnetization /H2084947/H20850in the form
/H11509M
/H11509t=−/H9253/H20851M,H/H20852−1
T2M/H11036−1
T1/H20849Mz−M0/H20850ez, /H2084949/H20850
where T1=/H9273//H926133/H208491−/H9273K/H20850and T2=/H9273//H926111.
Let us now consider a paramagnet in the absence of an
external magnetic field. It is well known that in this case theparamagnet has no magnetization. This means that if magne-tization arises locally in the body as a result of fluctuations,then it will relax to its zero value. In this case the relaxationof the fluctuations is described by Onsager’s equation.
13The
time derivatives of the components of the magnetization willbe the generalized fluxes, for which we shall take the com-ponents of the effective magnetic field:
/H11509M
/H11509t=/H9261ikHk, /H2084950/H20850
where /H9261ikare kinetic coefficients. Evidently, the relativistic
part of the dissipation function /H208496/H20850corresponds to the relax-
ation term in this equation. Once again we arrive at Eq. /H2084949/H20850,
but now without the dynamic part and with M0=0.
It follows from the results obtained that the method pro-
posed in the present article for constructing the dissipationfunction for paramagnets leads to a relaxation term inBloch’s form.
6For a paramagnet in the absence of a mag-
netic field the proposed dissipation function leads to Onsag-er’s equation, which is generally accepted for this case. Thisshows that the approach proposed here is general and makesit possible to describe equally successfully relaxation pro-cesses in magnetically ordered crystals and in paramagnets.
In closing we express our deep appreciation to B. A.
Ivanov, Corresponding Member of the National Academy ofSciences of Ukraine, and Professor A. N. Slavin for valuablediscussions.
a/H20850Email: victor.baryakhtar@gmail.com
b/H20850Email: alek_tony@ukr.net
1C. A. Ross, Ann. Rev. Mater. Res. 31, 203 /H208492000 /H20850.
2S. Demokritov and B. Hillebrands in Spin Dynamics in Confined Magnetic
Structures I , edited by B. Hillebrands and K. Ounadjela, Springer, Berlin
/H208492001 /H20850,p .6 5 .
3G. N. Kakazei, P. E. Wigen, K. Yu. Guslienko, V . Novosad, A. N. Slavin,
V . O. Golub, N. A. Lesnik, and Y . Otani, Appl. Phys. Lett. 85, 443 /H208492004 /H20850.
4L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8,1 5 3 /H208491935 /H20850.
5T. L. Gilbert, Phys. Rev. 100, 1243 /H208491956 /H20850.
6F. Bloch, Zs. für Phys. 61, 206 /H208491930 /H20850.
7V . G. Bar’yakhtar, Zh. Eksp. Teor. Fiz. 87, 1501 /H208491984 /H20850.
8V . G. Bar’yakhtar and A. G. Danilevich, Fiz. Nizk. Temp. 32, 1010 /H208492006 /H20850
/H20851Low Temp. Phys. 32, 768 /H208492006 /H20850/H20852.
9L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media ,
Nauka, Moscow /H208491982 /H20850.
10A. I. Akhiezer, V . G. Bar’yakhtar, and S. V . Peletminski /c142,Spin Waves ,
Nauka, Moscow /H208491967 /H20850.
11L. D. Landau and E. M. Lifshitz, Fluid Mechanics , Nauka, Moscow
/H208491986 /H20850.
12N. N. Borolyubov, Quasiaverages in Problems of Statistical Mechanics ,
Dubna /H208491963 /H20850; Preprint AN SSSR OIYaI R-1451.
13L. D. Landau and E. M. Lifshitz, Statistical Physics , Nauka, Moscow
/H208491976 /H20850,P t .1 .
Translated by M. E. AlferieffLow Temp. Phys. 36/H208494/H20850, April 2010 V. G. Bar’yakhtar and A. G. Danilevich 309
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
205.208.120.10 On: Sat, 20 Dec 2014 07:52:33 |
1.4792721.pdf | Transient ultrasound propagation in porous media using Biot
theory and fractional calculus: Application to human cancellous
bone
M. Fellah
Laboratoire de Physique Th /C19eorique, Facult /C19e de Physique, USTHB, BP 32 El Alia, Bab Ezzouar 16111, Algeria
Z. E. A. Fellah
LMA, CNRS, UPR 7051, Aix-Marseille Universit /C19e, Centrale Marseille, F-13402 Marseille Cedex 20, France
F . G. Mitri
Los Alamos National Laboratory, MPA-11, Sensors Electrochemical Devices, Acoustics and Sensors
Technology Team, MS D429, Los Alamos, New Mexico 87545
E. Ogam
LMA, CNRS, UPR 7051, Aix-Marseille Universit /C19e, Centrale Marseille, F-13402 Marseille Cedex 20, France
C. Depollier
LUNAM Universite du Maine, UMR CNRS 6613 Laboratoire d’Acoustique de l’Universite du Maine UFR
STS, Avenue O. Messiaen, 72085 Le Mans CEDEX 09, France
(Received 22 July 2012; revised 18 January 2013; accepted 5 February 2013)
A temporal model based on the Biot theory is developed to describe the transient ultrasonic propaga-
tion in porous media with elastic structure, in which the viscous exchange between fluid and struc-ture are described by fractional derivatives. The fast and slow waves obey a fractional wave
equation in the time domain. The solution of Biot’s equations in time depends on the Green func-
tions of each of the waves (fast and slow), and their fractional derivatives. The reflection and trans-mission operators for a slab of porous materials are derived in the time domain, using calculations in
the Laplace domain. Their analytical expressions, depend on Green’s function of fast and slow
waves. Experimental results for slow and fast waves transmitted through human cancellous bonesamples are given and compared with theoretical predictions.
VC2013 Acoustical Society of America .
[http://dx.doi.org/10.1121/1.4792721]
PACS number(s): 43.20.Bi, 43.20.Jr, 43.20.Hq, 43.20.Gp [KML] Pages: 1867–1881
I. INTRODUCTION
More than 50 years ago, Biot1proposed a semi-
phenomenological theory which provides a rigorous descrip-tion of the propagation of acoustical waves in porous media
saturated by a compressible viscous fluid. Such diphasic
materials are supposed to be elastic and homogeneous. Biot’stheory was initially introduced for petroleum prospecting and
research. Due to its very general and rather fundamental char-
acter, it has been applied in various fields of acoustics such asgeophysics, underwater acoustics, seismology, ultrasonic
characterization of bones, etc. This theory derives the equa-
tions of motion for each phase (i.e., the solid frame and thefluid) based on energy considerations which include the iner-
tial, potential, and viscous coupling between the two phases.
For an isotropic porous medium, three different bulk modesare predicted, i.e., two compressional waves and one shear
wave. One compressional wave, the so-called wave of the
first type or fast longitudinal wave, and the transverse waveare similar to the two bulk waves observed in a linear elastic
solid. The other longitudinal wave, called a wave of the
second kind, or slow wave, is a highly damped and very dis-persive mode. It is diffusive at low frequencies and
propagative at high frequencies. An important contribution tothe understanding of Biot’s theory was made by Norris.
2We
follow Biot’s approach, owing to its simplicity.
Assessment of bone loss and osteoporosis by ultrasound
systems is based on the speed of sound and broadband ultra-
sound attenuation of a single wave.3However, the existence
of a second wave in cancellous bone has been reported and
its existence is an unequivocal signature of poroelastic
media. Cancellous bone is an inhomogeneous porous mate-rial consisting of a matrix of solid trabeculae filled with soft
bone marrow. The interaction between ultrasound and bone
is highly complex. Modeling ultrasonic propagation throughtrabecular tissue has been considered using porous media
theories, such as Biot’s theory.
4–17
Most of the work on reflection and transmission of ultra-
sonic wave in a porous medium (such as cancellous bone,
rocks, sintered glass samples) is performed in the frequency
domain.9,10,18–26This scheme is natural in connection with
wave propagation generated by time harmonic incident waves
and sources. Pulse propagation in porous media is usually
modeled by synthesizing the signal via a Fourier transform ofthe continuous wave results. On the other hand, experimental
measurements are usually carried out using pulses of finite
bandwidth. Therefore, direct modeling in the time domain ishighly desirable.
27–38The attractive feature of a time domain
J. Acoust. Soc. Am. 133(4), April 2013 VC2013 Acoustical Society of America 1867 0001-4966/2013/133(4)/1867/15/$30.00
Downloaded 30 Aug 2013 to 152.3.102.242. Redistribution subject to ASA license or copyright; see http://asadl.org/termsbased approach is that the analysis is naturally bounded by
the finite duration of ultrasonic pressures and is consequentlythe most appropriate approach for the transient signal.
However, for wave propagation generated by time harmonic
incident waves and sources (monochromatic waves), the fre-quency analysis is more appropriate.
39Previous efforts to
model the propagation in the time domain have been carried
out mainly for porous materials having rigid frame (air-satu-rated porous materials, as plastic foams) in the high and low
frequency range for solving direct and inverse scattering
problems.
30,31,33,37This time domain model is an alternative
to the classical frequency domain approach.
In the past, many authors have used fractional calculus40
as an empirical method to describe the properties of visco-
elastic materials, e.g., see Caputo,41and Bagley and
Torvik.42The observation that asymptotic expressions of
stiffness and damping in porous materials are proportional tothe fractional powers of frequency
30,32,43,44suggests that
time derivatives of a fractional order might describe the
behavior of sound waves in this kind of material, includingrelaxation and frequency dependence. Fractional derivatives
are well suited to describe wave propagation in complex
media.
45–47In our previous work,30,31,33,34the fractional
derivatives have been used only for porous materials having
a rigid structure. In this paper, fractional calculus is used to
describe viscous interaction between the fluid and the struc-ture in the time domain for a porous material (cancellous
bone), having an elastic structure, using the Biot model
modified by Johnson et al.
48At high frequencies, the viscous
losses in the porous material are expressed by the square
root of the frequency. In the time domain, these losses are
expressed by the fractional derivative, which is essential toexpress, in the time domain, the propagation equation and
the analytical expressions of the reflection and transmission
operators. Experimental results for fast and slow wavestransmitted through samples of human cancellous bone and
hydroxyapatite (substitute of bone) are given and compared
with theoretical predictions.
II. FRACTIONAL DERIVATIVE MODEL
The equations of motion of the frame and fluid are given
by the Euler equations applied to the Lagrangian density.
Here uandUare the displacements of the solid and fluid
phases. The equations of motion are given by1
q11@2u
@t2þq12@2U
@t2¼P$ð$uÞþQ$ð$UÞ
/C0N$/C217ð$/C217uÞ; (1)
q12@2u
@t2þq22@2U
@t2¼Q$ð$uÞþR$ð$UÞ; (2)
where P,Q, and Rare generalized elastic constants which
are related, via gedanken experiments, to other measurable
quantities, namely /(porosity), Kf(bulk modulus of the pore
fluid), Ks(bulk modulus of the elastic solid), and Kb(bulk
modulus of the porous skeletal frame). Nis the shear modu-
lus of the composite as well as that of the skeletal frame.The equations which explicitly relate P,Q, and Rto/,Kf,
Ks,Kb, and Nare given by
P¼ð1/C0/Þ1/C0//C0Kb
Ks/C18/C19
Ksþ/Ks
KfKb
1/C0//C0Kb
Ksþ/Ks
Kfþ4
3N;
Q¼1/C0//C0Kb
Ks/C18/C19
/Ks
1/C0//C0Kb
Ksþ/Ks
Kf;
R¼/2Ks
1/C0//C0Kb
Ksþ/Ks
Kf:
The Young modulus and the Poisson ratio of the solid Es,/C23s,
and the skeletal frame Eb,/C23bdepend on the generalized elas-
tic constant P,Q, and Rvia the relations
Ks¼Es
3ð1/C02/C23sÞ;Kb¼Eb
3ð1/C02/C23bÞ;N¼Eb
2ð1þ/C23bÞ:
(3)
qmnare the “mass coefficients” which are related to the den-
sities of solid ( qs) and fluid ( qf) phases by
q11þq12¼ð1/C0/Þqs;q12þq22¼/qf: (4)
The coefficient q12represents the mass coupling parameter
between the fluid and solid phases and is always negative
q12¼/C0/qfða1/C01Þ; (5)
a1being the tortuosity of the medium. We see in Sec. II A,
how the losses will be introduced in the porous media in the
frequency domain, and then how to express the problem in
the time domain using the concept of fractional derivative.
A. Temporal modified Biot model and fractional
derivative
To express the viscous exchanges between the fluid and
the structure which play an important role in damping the
acoustic wave in porous material, the tortuosity, a1,
becomes a function of frequency called the dynamic tortuos-ity
48a(x). The parts of the fluid affected by this exchange
can be estimated by the ratio of a microscopic characteristic
length of the medium, for example, pore size to the viscousskin depth thickness, d¼(2g/xq
f)1/2(g: fluid viscosity, x:
angular frequency). This domain corresponds to the region
of the fluid in which the velocity distribution is disturbed bythe frictional forces at the interface between the fluid and the
frame. At high frequencies, the viscous skin thickness is
very thin near the radius of the pore, r.The viscous effects
are concentrated in a small volume near the surface of the
frame d=r/C281. In this case, the expression of the dynamic
tortuosity, a(x), is given by
48
1868 J. Acoust. Soc. Am., Vol. 133, No. 4, April 2013 Fellah et al. : Fractional model in porous media
Downloaded 30 Aug 2013 to 152.3.102.242. Redistribution subject to ASA license or copyright; see http://asadl.org/termsaðxÞ¼a11þ2
Kg
jxqf !1=20
@1
A; (6)
where Kis the viscous characteristic length. Currently, the
model of Johnson et al.48is the one that best describes the
losses in the porous medium in the regime of high frequen-
cies. This model introduces the viscous characteristic length,K, which is a geometric parameter. Kis an indicator of pore
size narrow, i.e., the privileged place of viscous exchanges.
Historically, this model was developed for modeling ultra-sound propagation in porous media, for geophysical applica-
tions.
48Because bone is a porous medium, the application of
this model is quite natural.
In the time domain, the dynamic tortuosity, a(x), acts as
operator and in the asymptotic domain (high frequency
approximation), its expression is given by30,31
~aðtÞ¼a1dðtÞþ2
Kg
pqf !1=2
t/C01=20
@1
A; (7)where d(t) is the Dirac function. In this model the time con-
volution of t/C01/2with a function is interpreted as a semi-
derivative operator according to the definition of the frac-
tional derivative of order /C23given in Samko et al. ,49
D/C23½xðtÞ/C138 ¼1
Cð/C0/C23Þðt
0ðt/C0uÞ/C0/C23/C01xðuÞdu; (8)
where 0 /C20/C23<1a n dCðxÞis the gamma function. A fractional
derivative no longer represents the local variations of the func-
tion but on the contrary, it acts as a convolution integral opera-
tor. More details about the properties of fractional derivativesand about fractional calculus are given in Samko et al.
49
The tortuosity operator ~aðtÞ[Eq. (7)] was used success-
fully in modeling temporal acoustic propagation in porousmedia with rigid structure. Its introduction in Biot’s equa-
tions, to describe the inertial and viscous interactions
between fluid and structure, will express the propagationequations in the time domain. When ~aðtÞis used instead of
a
1in Eqs. (1)and(2), the equations of motion (1)and(2)
will be written as
~q11ðtÞ/C3@2uðtÞ
@t2þ~q12ðtÞ/C3@2uðtÞ
@t02dt¼P$ð$uðtÞÞ þ Q$ð$uðtÞÞ /C0 N$/C217ð$/C217uðtÞÞ;
~q12ðtÞ/C3@2uðtÞðtÞ
@t2þ~q22ðtÞ/C3@2UðtÞ
@t2dt¼Q$ð$uðtÞÞ þ R$ð$UðtÞÞ: (9)
The following notation is used for the convolution integral:
½f/C3g/C138ðx;tÞ¼ðt
0fðx;t/C0t0Þgðx;t0Þdt0; (10)
for two causal functions f(t) and g(t).
In these equations, the temporal operators ~q11ðtÞ,~q12ðtÞ,
and ~q22ðtÞrepresent the mass coupling operators between
the fluid and solid phases and are given by
~q11ðtÞ¼ð 1/C0/Þqsþ/qfð~aðtÞ/C01Þ;
~q12ðtÞ¼/C0 /qfð~aðtÞ/C01Þ;
~q22ðtÞ¼/qf~aðtÞ;where ~aðtÞis given by Eq. (7).
B. Fractional propagation equations of Biot’s waves
and Green functions
As in the case of an elastic solid, the wave equations of
dilatational and rotational waves can be obtained using sca-
lar and vector displacement potentials, respectively. Twoscalar potentials for the frame and the fluid, U
sandUf,
respectively, are defined for compressional waves giving
uðtÞ¼$~UsðtÞ;UðtÞ¼$~UfðtÞ:
Using, Eqs. (4),(5),(7), and (9), we obtain
q11q12
q12q22/C18/C19@2
@t2~UsðtÞ
~UfðtÞ/C18/C19
þA1/C01
/C011/C18/C19@3=2
@t3=2~UsðtÞ
~UfðtÞ/C18/C19
¼PQ
QR/C18/C19
D~UsðtÞ
~UfðtÞ/C18/C19
; (11)
where A¼ð2/qfa1=KÞffiffiffiffiffiffiffiffiffiffi
g=qfq
,Dis the Laplacian, and @3=2=@t3=2represents the fractional derivative following the definition
given by Eq. (8). The system of equations (11) can be written in matrix form as follows:
q11@2
@t2þA@3=2
@t3=2/C0PDq12@2
@t2/C0A@3=2
@t3=2/C0QD
q12@2
@t2/C0A@3=2
@t3=2/C0QDq22@2
@t2þA@3=2
@t3=2/C0RD0
BB@1
CCA~UsðtÞ
~UfðtÞ/C18/C19
¼0: (12)
J. Acoust. Soc. Am., Vol. 133, No. 4, April 2013 Fellah et al. : Fractional model in porous media 1869
Downloaded 30 Aug 2013 to 152.3.102.242. Redistribution subject to ASA license or copyright; see http://asadl.org/termsThe resolution of the eigenvalue problem of the matrix of Biot
(12) is that there are two distinct longitudinal modes which
are called fast and slow. The matrix (12) can be written on a
basis of fast and slow waves U1ðtÞandU2ðtÞ, respectively, by
DU1ðtÞ
U2ðtÞ/C18/C19
¼~k1ðtÞ 0
0 ~k2ðtÞ !
U1ðtÞ
U2ðtÞ/C18/C19
; (13)
where ~k1ðtÞand ~k2ðtÞare the “eigenvalue operators” of the
Biot Matrix (12). Their expressions are given by~kiðtÞ¼Ci@2
@t2þDi@3=2
@t3=2þGi@
@t;i¼1;2: (14)
Their corresponding eigenvectors are
~=iðtÞ¼AiþBiffiffiffiffiffiptp;i¼1;2; (15)
where
Ci¼1
2/C16
s1þð /C0 1Þiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s2
1/C04s3q/C17
;Di¼1
2s2þð /C0 1Þis1s2/C02s4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s2
1/C04s3p !
;
Gi¼ð /C0 1Þi/C11
4s2
2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s2
1/C04s3p /C0ðs1s2/C02s4Þ2
2ðs2
1/C04s3Þ3=2 !
;Ai¼s1/C02s5þð /C0 1Þiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s2
1/C04s3p
2s7;
Bi¼1
4s2
7s2/C02s6þð /C0 1Þis1s2/C02s4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s2
1/C04s3p !
2s7þs1/C02s5/C0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s2
1/C04s3q/C18/C19
2s6"#
;i¼1;2;
and
s1¼R0q11þP0q22/C02Q0q12;s2¼AðP0þR0þ2Q0Þ;s3¼ðP0R0/C0Q02Þðq11q22/C0q2
12Þ;
s4¼AðP0R0/C0Q02Þðq11þq22/C02q12Þ;s5¼ðR0q11/C0Q0q12Þ;s6¼AðR0þQ0Þ;s7¼ðR0q12/C0Q0q22Þ:
Coefficients R0,P0, and Q0are given by
R0¼R
PR/C0Q2;Q0¼Q
PR/C0Q2;and P0¼P
PR/C0Q2:
Equations (13) and (14) show that the fast and slow
waves, U1andU2, respectively, obey the following propaga-
tion equations along the xaxis:
@2Uiðx;tÞ
@x2/C01
v2
i@2Uiðx;tÞ
@t2/C0hi@3=2Uiðx;tÞ
@t3=2
/C0d@Uiðx;tÞ
@t¼0; (16)
where the coefficients vi,hi(i¼1,2), and dare constants,
respectively, given by
vi¼2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s2
1/C04s3p
þð /C0 Þis1q ;
hi¼1
2s2þð /C0 1Þis1s2/C02s4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s2
1/C04s3p !
;i¼1;2;
andd¼/C01
4s2
2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
s2
1/C04s3p /C0ðs1s2/C02s4Þ2
2ðs2
1/C04s3Þ3=2 !
:
These coefficients depend on the acoustic and mechanical
properties of porous media. Equations (16)are the generalizedpropagation equations in time domain of fast and slow waves,
respectively. These equations contain a fractional derivativeterm resulting from the viscous exchanges between fluid and
structure in the porous medium. Similar propagation equation
with fractional derivative term has been obtained when thestructure of the porous material is motionless,
30,31and when
the waves propagate only in the fluid (equivalent fluid model
for rigid frame approximation). In the equivalent fluid model,the coefficients of the fractional propagation equation depend
essentially on the acoustical parameters (tortuosity and vis-
cous characteristic length); the mechanical properties are notinvolved unlike this general model of Biot. This fractional
propagation equation has been solved analytically in the time
domain
50,51giving the Green function (Appendix A) of the
porous medium.
The first and second terms in the propagation equations
(16),@2Uiðx;tÞ=@x2/C0ð1=v2
iÞð@2Uiðx;tÞ=@t2Þ;i¼1;2, describe
the propagation (time translatio n) of the fast and slow wave,
respectively, via the front wave velocity vi,i¼1,2. The third
terms in the propagation equations (16),hið@3=2U1ðx;tÞ=@t3=2Þ
¼hiÐt
0ð@2p=@t2Þðx;t/C0sÞðds=ffiffiffispÞ;i¼1;2, contain a time frac-
tional derivative of order 3/2 [ see the definition of fractional
derivatives in Eq. (8)]. This term is the most important one for
describing the dispersion, memory effects due to relaxation
processes, and the acoustic attenuation in the porous materials.
These effects are due to the losses in the medium modeled bythe viscous exchanges between fluid and structure, and
described by the viscous characteristic length, K.T h i st e r m
1870 J. Acoust. Soc. Am., Vol. 133, No. 4, April 2013 Fellah et al. : Fractional model in porous media
Downloaded 30 Aug 2013 to 152.3.102.242. Redistribution subject to ASA license or copyright; see http://asadl.org/termsresults from the time convolution o f the fractional derivatives
operators of tortuosity ~aðtÞwith the second time derivative of
solid and fluid displacement (acce leration). The high frequency
components of the transient signal are the most sensitive to
this term (due to the fractional derivative).
The final term in the propagation equations (16):
dð@Uiðx;tÞ=@tÞis an attenuating term; it results on the attenu-
ation of the wave without dispersion.51This term describes
the acoustic attenuation due to the viscous interactions
between fluid and structure. The low frequency components
of the transient signal are the most sensitive to this term.
C. Solution of the Biot’s equations in time domain
The “eigenvectors operators” ~=1ðtÞand ~=2ðtÞassociated
with the “eigenvalues operators” ~k1ðtÞand ~k2ðtÞlink the fluid
and solid potentials UsðtÞandUfðtÞ, respectively, to the fast
and slow waves U1ðtÞandU2ðtÞthrough the following
relation:
~Usðx;tÞ
~Ufðx;tÞ/C18/C19
¼/C16II
~=1ðtÞ~=2ðtÞ/C17
/C3U1ðx;tÞ
U2ðx;tÞ/C18/C19
; (17)
where Iis the unit operator. Knowing the expressions of
U1ðx;tÞ¼G1ðx;tÞandU2ðx;tÞ¼G2ðx;tÞ(Green functions
given in Appendix A), solutions of the fractional propagation
of fast and slow waves (16), we deduce the following expres-
sions of fluid and solid potentials:
~Usðx;tÞ¼G1ðx;tÞþG2ðx;tÞ; (18)
~Ufðx;tÞ¼A1G1ðx;tÞþA2G2ðx;tÞ
þB1ffiffiffippðt
0G1ðx;sÞffiffiffiffiffiffiffiffiffiffit/C0sp dsþB2ffiffiffippðt
0G1ðx;sÞffiffiffiffiffiffiffiffiffiffit/C0sp ds:
(19)
Equations (18) and(19) show that the fluid and solid poten-
tials of the porous medium depend on the Green functions of
the fast and slow waves, and their fractional derivative oforder 1/2. This result is very important for the understanding
of the acoustic propagation inside the porous material in the
time domain, and to the solution of the direct problem.
In this paper, we consider the general case of acoustic
propagation in a porous medium with an elastic structure.
Biot theory used in this work takes into account the propaga-tion in the solid and fluid porous material. In our previous
work,
30,31,44,50,51we considered the case where the structure
of the porous material is rigid; we used the model of equiva-lent fluid, which is a special case of the general model of
Biot, in which the acoustic wave is propagated in the saturat-
ing fluid. The fast and slow waves predicted by the Biottheory follow an equation of propagation fractional deriva-
tive identical to that found in the context of the porous mate-
rial with a rigid structure. The complexity of this work is tocalculate analytically in temporal domain, the total wave
(reflected and transmitted by the material), which depends
on the slow and fast wave, predicted by the Biot theory.
Section IIIis devoted to the calculus of the temporal
responses (reflection and transmission scattering operators)
of the porous material using the fractional derivativepropagation equation (16), the expressions of fast and slow
waves, and the boundary conditions.
III. REFLECTION AND TRANSMISSION SCATTERING
OPERATORS
When a sound wave in the fluid impinges upon a porous
medium at normal incidence, part of it is reflected back into
the fluid, part is transmitted into the porous medium as a fastwave, and part is transmitted as a slow wave. For non-
normal angles of incidence, part of it is also transmitted as a
shear wave. In this paper, we consider only the reflectionand transmission at normal incidence. In this section some
notations are introduced. The problem geometry is shown in
Fig.1. A homogeneous porous material occupies the region
0/C20x/C20L. This medium is assumed to be isotropic and to
have an elastic frame. A short sound pulse impinges nor-
mally on the medium from the left. It generates solid andfluid displacements uandU, respectively, inside the porous
medium, which satisfies the propagation equations (9). If the
incident sound wave is launched in region x/C200, then the
expression of the pressure field in the region to the left of the
material is the sum of the incident and reflected fields
p
1ðx;tÞ¼pit/C0x
c0/C18/C19
þprtþx
c0/C18/C19
;x<0;
where p1(x,t) is the field in region x<0, and piandprdenote
the incident and reflected waves, respectively. In addition, atransmitted field is produced in the region at the right of the
boundary. This has the form
p
3ðx;tÞ¼ptt/C0ðx/C0LÞ
c0/C18/C19
;x>L
[p3(x,t) is the field in region x>L, and ptis the transmitted
field].
The incident and scattered fields are related by the scat-
tering operators (i.e., reflection and transmission operators)for the material. These are integral operators represented by
p
rðx;tÞ¼ðt
0~RðsÞpit/C0sþx
c0/C18/C19
ds
¼~RðtÞ/C3piðtÞ/C3dtþx
c0/C18/C19
; (20)
ptðx;tÞ¼ðt
0~TðsÞpit/C0s/C0L
c/C0ðx/C0LÞ
c0/C18/C19
ds
¼~TðtÞ/C3piðtÞ/C3dt/C0L
c/C0ðx/C0LÞ
c0/C18/C19
: (21)
In Eqs. (20) and(21), the functions ~Rand ~Tare the
reflection and transmission kernels, respectively, for incidence
FIG. 1. Problem geometry.
J. Acoust. Soc. Am., Vol. 133, No. 4, April 2013 Fellah et al. : Fractional model in porous media 1871
Downloaded 30 Aug 2013 to 152.3.102.242. Redistribution subject to ASA license or copyright; see http://asadl.org/termsfrom the left. Note that the lower limit of integration in Eqs.
(20) and(21) is given as 0, which is equivalent to assuming
that the incident wave front first impinges on the material at
t¼0. The scattering operators given in Eqs. (20) and(21) are
independent of the incident field used in scattering experimentand depend only on the properties of the materials. The
expressions of the reflection and transmission coefficients in
Laplace domain are given by (Appendix B)
RðzÞ¼z2½F2
4ðzÞ/C0F2
3ðzÞ/C138 þ1
½zF3ðzÞ/C01/C1382/C0z2F2
4ðzÞ; (22)
TðzÞ¼2zF4ðzÞ
z2F2
4ðzÞ/C0½zF3ðzÞ/C01/C1382; (23)
where
F3ðzÞ¼F1ðzÞcosh/C16
lffiffiffiffiffiffiffiffiffiffi
k1ðzÞp/C17
þF2ðzÞcosh/C16
lffiffiffiffiffiffiffiffiffiffi
k2ðzÞp/C17
;
F4ðzÞ¼F1ðzÞþF2ðzÞ;
FiðzÞ¼qfc0½1þ/ð=iðzÞ/C01Þ/C138
/C2ffiffiffiffiffiffiffiffiffiffi
kiðzÞp 2WiðzÞ
sinh/C16
lffiffiffiffiffiffiffiffiffiffi
kiðzÞp/C17
:WðzÞ;
=iðzÞ¼AiþBi=ffiffizp;
kiðzÞ¼Ciz2þDizffiffizpþGiz;i¼1;2; (24)
where k1(z),k2(z),=1ðzÞ, and =2ðzÞare, respectively, the
Laplace transform of the eigenvalues and the eigenvectorsoperators of the system (11). The coefficients w
1(z),w2(z),
andw3(z) are given by
W1ðzÞ¼/w2ðzÞ/C0ð 1/C0/Þw4ðzÞ;
W2ðzÞ¼ð 1/C0/ÞZ3ðzÞ/C0/w1ðzÞ;
WðzÞ¼2½w1ðzÞw4ðzÞ/C0w2ðzÞw3ðzÞ/C138; (25)
and the coefficients w1(z),w2(z),w3(z), and w4(z) are given
by
w1ðzÞ¼ð PþQ=1ðzÞÞk1ðzÞ;
w2ðzÞ¼ð PþQ=2ðzÞÞk2ðzÞ;
w3ðzÞ¼ð QþR=1ðzÞÞk1ðzÞ;
w4ðzÞ¼ð QþR=2ðzÞÞk2ðzÞ:
By putting the Laplace variable z¼jx, where j2¼/C01 and x
is the angular frequency, we find the expressions of reflec-
tion and transmission coefficients in Fourier (frequency) do-
main.9The relations (22) and(23) are complicated and it is
very hard (if not impossible) to obtain their inverse Laplace
transforms. These expressions should be simplified andtransformed for obtaining their temporal equivalent. This
work is studied in Sec. III A.
A. Temporal expressions of the reflection
and transmission operators
In this section, we will seek the temporal expression of
the reflection and transmission operators. The expressions
(22) and(23) can be decomposed as simple elements
RðzÞ¼/C0 1þ1
1/C0zðF3ðzÞ/C0F4ðzÞÞ
þ1
1/C0zðF3ðzÞþF4ðzÞÞ; (26)
TðzÞ¼1
1/C0zðF3ðzÞ/C0F4ðzÞÞ/C01
1/C0zðF3ðzÞþF4ðzÞÞ:
(27)
After some developments (see Appendix C), the expressions
ofR(z) and T(z) become
RðzÞ¼/C0 1þ2
1/C0XðzÞ/C0YðzÞþ4XðzÞZ1ðzÞ2
1/C0XðzÞ/C0YðzÞ ðÞ2
þ4YðzÞZ2ðzÞ2
1/C0XðzÞ/C0YðzÞ ðÞ2; (28)
TðzÞ¼/C04XðzÞZ1ðzÞ
ð1/C0XðzÞ/C0YðzÞÞ2/C04YðzÞZ2ðzÞ
ð1/C0XðzÞ/C0YðzÞÞ2;
(29)
where
XðzÞ¼2zqfc0½1þ/ð=1ÞðzÞ/C01Þ/C138ffiffiffiffiffiffiffiffiffiffi
k1ðzÞp W1ðzÞ
WðzÞ;
YðzÞ¼2zqfc0½1þ/ð=2ÞðzÞ/C01Þ/C138ffiffiffiffiffiffiffiffiffiffi
k2ðzÞp W2ðzÞ
WðzÞ;
Z1ðzÞ¼exp/C16
/C0lffiffiffiffiffiffiffiffiffiffi
k1ðzÞp/C17
;Z2ðzÞ¼exp/C16
/C0lffiffiffiffiffiffiffiffiffiffi
k2ðzÞp/C17
:
The obtained expressions (28)and(29)are available for large
values of zcorresponding to the condition of high frequency
range [when the viscous skin thickness d¼(2g/xq)1/2is
much smaller than the radius of the pores, r]. It is now inter-
esting to calculate the explicit relations of X(z)a n d Y(z)w i t h
respect of z. From the expressions of W1ðzÞandW2ðzÞgiven
by Eq. (25),w efi n dt h a t
ffiffiffiffiffiffiffiffiffiffi
k1ðzÞp W1ðzÞ
WðzÞ¼/ðPþQÞ/C0Qþ= 2ðzÞ½/ðRþQÞ/C0R/C138
2ðQ2/C0PRÞð= 1ðzÞ/C0= 2ðzÞÞ1ffiffiffiffiffiffiffiffiffiffi
k1ðzÞp ;
1872 J. Acoust. Soc. Am., Vol. 133, No. 4, April 2013 Fellah et al. : Fractional model in porous media
Downloaded 30 Aug 2013 to 152.3.102.242. Redistribution subject to ASA license or copyright; see http://asadl.org/termsffiffiffiffiffiffiffiffiffiffi
k2ðzÞp W2ðzÞ
WðzÞ¼/ðPþQÞ/C0Qþ= 1ðzÞ½/ðRþQÞ/C0R/C138
2ðQ2/C0PRÞð= 2ðzÞ/C0= 1ðzÞÞ1ffiffiffiffiffiffiffiffiffiffi
k2ðzÞp :
The term X(z) can be deduced as
XðzÞ
qfc0¼½1þ/ð=1ðzÞ/C01Þ/C138 ½/ðPþQÞ/C0Qþ= 2ðzÞð/ðRþQÞ/C0RÞ/C138
ðQ2/C0PRÞð= 1ðzÞ/C0= 2ðzÞÞzffiffiffiffiffiffiffiffiffiffi
k1ðzÞp : (30)
An analog expression for Y(z) is obtained by replacing the indic 1 by 2 in relation (30). By replacing =1ðzÞand=2ðzÞby their
relations, we obtain
XðzÞ
qfc0¼1þ/ðA1/C01Þþ/B1ffiffizp/C20/C21
/ðPþQÞ/C0QþA2ð/ðRþQÞ/C0RÞþB2ffiffizpð/ðRþQÞ/C0RÞ/C20/C21
ðQ2/C0PRÞA1/C0A2þB1/C0B2ffiffizp/C20/C21zffiffiffiffiffiffiffiffiffiffi
k1ðzÞp :
By taking into account the approximation of high frequency range (large values of z), we obtain at the first order in 1 =ffiffizp
zffiffiffiffiffiffiffiffiffiffi
k1ðzÞp ¼1ffiffiffiffiffiffiC1p 1/C01
2D1
C11ffiffizp/C18/C19
:
The obtained expressions of X(z) and Y(z) at the first order in 1 =ffiffizpare
X
qfc0¼1ffiffiffiffiffiffiC1pðQ2/C0PRÞðA1/C0A2Þa1c2þ1ffiffizp/c2B1þa1b2/C0a1c2B1/C0B2
A1/C0A2þ1
2D1
C1/C18/C19 /C20/C21/C20/C21
;
Y
qfc0¼1ffiffiffiffiffiffiC2pðQ2/C0PRÞðA1/C0A2Þa2c1þ1ffiffizp/c1B2þa2b1/C0a2c1B2/C0B1
A2/C0A1þ1
2D2
C2/C18/C19 /C20/C21/C20/C21
;
using the notations
a1¼1þ/ðA1/C01Þ;a2¼1þ/ðA2/C01Þ;b1¼B1ð/ðQþRÞ/C0RÞ;b2¼B2ð/ðQþRÞ/C0RÞ;
c1¼/ðPþQÞ/C0QþA1ð/ðQþRÞ/C0RÞ;c2¼/ðPþQÞ/C0QþA2ð/ðQþRÞ/C0RÞ:
Note by
X¼x1þy1ffiffizp and Y¼x2þy2ffiffizp;
with
x1¼qfc0c2a1ffiffiffiffiffiffiC1pðQ2/C0PRÞðA1/C0A2Þ;
y1¼qfc0ffiffiffiffiffiffiC1pðQ2/C0PRÞðA1/C0A2Þ/c2B1þa1b2/C0a1c2B1/C0B2
A1/C0A2þ1
2D1
C1/C18/C19 /C20/C21
;
J. Acoust. Soc. Am., Vol. 133, No. 4, April 2013 Fellah et al. : Fractional model in porous media 1873
Downloaded 30 Aug 2013 to 152.3.102.242. Redistribution subject to ASA license or copyright; see http://asadl.org/termsand an analog expression for x2andy2by interchanging the indices 1 and 2.
By putting U¼x1þx2andV¼y1þy2, we obtain
1
1/C0X/C0Y¼1
1/C0U/C0Vffiffizp¼1
1/C0Uffiffizp
ffiffizpþa¼1
1/C0U1/C0affiffizpþa/C18/C19
;
with a¼V=ðU/C01Þ, we have also
X
ð1/C0X/C0YÞ2¼1
ð1/C0UÞ2x1/C02ax1ffiffizpþaþa2x1
ðffiffizpþaÞ2þy1ffiffizp/C02ay1ffiffizpðffiffizpþaÞþa2y1ffiffizpðffiffizpþaÞ2"#
and an analog expression for Y=ð1/C0X/C0YÞ2by replacing
x1,y1byx2andy2.
It is now possible to calculate the temporal expressions of
reflection and transmission coefficients from the expressions
(28) and(29). Using the expressions of the inverse Laplace
transform given in Appendix D, we obtain the following tem-
poral expression of the reflection scattering operator:
~RðtÞ¼rðtÞþ< ð tÞ; (31)
with
rðtÞ¼1þU
1/C0UdðtÞ
þ2a
1/C0U/C01ffiffiffiffiffiptpþaexpða2tÞErfcðaffiffi
tp
Þ/C20/C21
(32)
and
<ðtÞ¼4
ð1/C0UÞ2½x1G1ðt;2LÞþx2G2ðt;2LÞ/C138
þ4
ð1/C0UÞ2½P1ðtÞ/C3G1ðt;2LÞþP2ðtÞ/C3G2ðt;2LÞ/C138:
(33)
The transmission operator is given by
~TðtÞ¼/C04
ð1/C0UÞ2½x1G1ðt;LÞþx2G2ðt;LÞ
þP1ðtÞ/C3G1ðt;LÞþx2P2ðtÞ/C3G2ðt;LÞ/C138;(34)with the notations for i¼1,2,
PiðtÞ¼yi/C02axiffiffiffipptþ2a2ðyi/C0axiÞffiffiffi
t
pr
þ½a2xið3þ2a2tÞ
/C02ayið1þa2tÞ/C138expða2tÞErfcðaffiffi
tp
Þ;i¼1;2;
Giðt;jLÞ¼L/C01exp/C16
/C0jLffiffiffiffiffiffiffiffiffiffi
kiðzÞp/C17 hi
;i;j¼1;2:
Gi(t),i¼1,2 are the Green functions of the fast and slow
waves, respectively. Their expressions are given in Appendix.
A. These functions describe the propagation inside the porousmaterial. In the expressions of the reflection and transmission
scattering operators (31) and(34), only the first reflections at
the interfaces x¼0a n d x¼Lare taken into account. The
multiple reflections effects are negligible because of the high
attenuation of sound waves in these media. In the case of a
semi-infinite medium when L!1 ;G
iðt;2LÞ! 0,i¼1;2.
This means that r(t) is equivalent to the reflection by the first
interface x¼0a n d <ðtÞto the reflection to the rear interface
(x¼L). The first term of rðtÞ:½ð1þUÞ=ð1/C0UÞ/C138dðtÞcorre-
sponds to the instantaneously response of the porous material
by the first interface x¼0. The part of the wave which is
equivalent to this term is not subjected to dispersion but sim-ply multiplied by the factor ð1þUÞ=ð1/C0UÞ. The second
term of rðtÞ:½2a=ð1/C0UÞ/C138½/C0ð 1=ffiffiffiffiffiptpÞþae
a2tErfcðaffiffitpÞ/C138
depends on the fractional derivative operator, and expressesthe attenuation and dispersion process due to the viscous and
mechanical interactions between fluid and structure. This
memory term expresses the relaxation phenomenon on thereflected wave by the first interface.
The temporal operator <ðtÞis equivalent to the reflec-
tion by the second interface x¼Lof the porous material.
This term depends on the Green functions G
1(t,2L) and
G2(t,2L), which describe the propagation and the dispersion
of the fast and slow wave making one round trip inside theporous slab.
The temporal expression of the transmission operator
(34) shows the contribution of each of the fast and slow
waves on the total transmitted wave through the Green func-
tions of each of the waves (fast and slow).
The advantage of the obtained time domain expressions
of the reflection and transmission scattering operators [Eqs.
(31)–(34)] is to show analytically the effect of the acoustical
FIG. 2. Experimental set-up for ultrasonic measurements.
1874 J. Acoust. Soc. Am., Vol. 133, No. 4, April 2013 Fellah et al. : Fractional model in porous media
Downloaded 30 Aug 2013 to 152.3.102.242. Redistribution subject to ASA license or copyright; see http://asadl.org/termsand mechanical parameters of the porous material on the
reflection contributions by the interfaces of the media.
Moreover, it is easy to see the effect of each wave (fast and
slow) on the response of porous media, and to understandthe propagation of these modes within the material, and their
reflection and transmission to outside the material.
In Sec. IV, we give an experimental validation of the
transmission operator using experimental and simulated
transmitted waves on a human cancellous bone and bone
substitute (hydroxyapatite).
IV. EXPERIMENTALVALIDATION
We will try, in this section, to experimentally validate
our temporal model by comparing theoretical and experi-mental transmitted waves. Experiments are performed in
water using two broadband Panametrics (Southend-on-Sea,
Essex, UK) A 303S piezoelectric transducers with a centralfrequency of 1 Mhz in water. 400 V pulses are provided by a
5058PR Panametrics pulser/receiver. The signals received
are amplified to 90 dB and filtered above 10 MHz to avoidhigh frequency noise (energy is totally filtered by the sample
in this upper frequency domain). Electronic interference is
removed by 1000 acquisition averages. The experimental
setup is shown in Fig. 2. The parallel-faced samples were
machined from human cancellous bone in femoral heads and
hydroxyapatite (a substitute for bone). The emitting trans-
ducer insonifies the sample at normal incidence with a short(in time domain) pulse. When the pulse hits the front surface
of the sample, a part is reflected, a part is transmitted as a
fast wave, and a part is transmitted as a slow wave. Whenany of these components, traveling at different speeds, hit the
second surface, a similar effect takes place: a part is transmit-
ted into the fluid, and a part is reflected as a fast or slowwave. It is assumed that the surface porosity is equal to the
volume porosity in the porous material. The experimental
transmitted waveforms are traveling through the cancellousbone in the same direction as the trabecular alignment ( x
direction). The fluid characteristics are bulk modulus,
K
f¼2.28 GPa, density qf¼1000 Kg m/C03, viscosity g¼10/C03
Kg m s/C01. Four samples of human bones (B1–B4) and a sam-
ple of hydroxyapatite (H) are considered in this study. Their
physical characteristics are measured (Appendix E) using
standard methods10,31,52,53and given in Table I.
The simulated signals are computed from Eq. (21),i n
which the transmission operator ~TðtÞis given by Eq. (34),
andpiðtÞis the incident signal generated by the transducer.
Figures 3(a) and3(b) show, for example, the experimental
incident signal piðtÞand its spectrum used for the bone sam-
ples B2–B4. The experimental data are deduced from the
transmitted field scattered by a slab of cancellous bone sample
(or hydroxyapatite sample) of finite depth 0 /C20x/C20L.F i g u r e s
4–8show the comparison between experimental transmitted
signals (solid line) and simulated signals (dashed line) for
bone and hydroxyapatite samples. We see from the waveformsTABLE I. Biot’s model parameters of cancellous bone and hydroxyapatite.
L(mm) /a 1 K(lm) qsKg/m3/C23s /C23b Es(GPa) Eb(GPa)
Bone (B1) 8.2 0.94 1.06 204.8 1990 0.37 0.2 13 1.57
Bone (B2) 11.2 0.64 1.06 10.44 1990 0.3 0.3 10 3.73Bone (B3) 10.2 0.73 1.1 14.97 1990 0.3 0.22 10 3.1Bone (B4) 11.2 0.8 1.1 24.54 1990 0.3 0.3 10 1.91Hydroxyapatite (H) 12.5 0.9 1.13 8.06 1990 0.35 0.31 10 2.16
FIG. 3. (a) Experimental incident signal for samples B2–B4. (b) Spectrum
of the incident signal for samples B2–B4.
FIG. 4. Comparison between experimental transmitted signal (solid line)and simulated transmitted signal (dashed line) for bone sample B1.
J. Acoust. Soc. Am., Vol. 133, No. 4, April 2013 Fellah et al. : Fractional model in porous media 1875
Downloaded 30 Aug 2013 to 152.3.102.242. Redistribution subject to ASA license or copyright; see http://asadl.org/termsof the experimental signals, the presence of two waveforms.
The first waveform corresponds to the fast wave, and the sec-ond to the slow wave. The two waves are easily distinguished.
These two waves are described by the Biot theory given in
Secs. II BandII C. In our previous work, we used a frequency
model, which required the calculation of the inverse Fourier
transform numerically, in order to trace the curves of transmit-
ted waves. A comparison between the transmitted waves cal-culated with this temporal model and the frequency model
9
gives exactly the identical result (the same in Figs. 4–8). It is
better at any time to have an analytical model for transient sig-nals, rather than a numerical model based on the calculation
of the inverse Fourier transform. This temporal model is an
interesting alternative to the frequency model developed inour previous studies. The comparison between theoretical and
experimental data shows very good agreement. This allows us
to conclude that this temporal model is well suited for thedescription of acoustic propagation in porous media obeying
the Biot theory.
Generally, the responses (reflection and transmission) of
a porous material (cancellous bone, for example) to an
acoustic excitation are calculated in the frequency
9,10do-
main or in the Laplace18–20,54domain. When simulating
waveforms propagating in such media, the inverse Fourier
transform,9,10or the inverse Laplace transform18–20,54are
used for the calculation in the time domain of the reflectedand transmitted waves. The advantage of this work is that
the analytical expressions of the responses (reflection and
transmission) of the porous material are given and expresseddirectly in the time domain. We hope, through this temporal
modeling, to solve the inverse problem directly in the time
domain, which allows us to perform ultrasonic characteriza-tion of porous materials such as cancellous bone.
V. CONCLUSION
In this paper, the problem of ultrasound propagation in a
porous medium obeying Biot’s theory has been treated. The
aim of this work was to express the responses of the porous
material (reflection and transmission) analytically in the timedomain. The concept of fractional derivative was used to
express the viscous losses in the porous material. The formu-
lation of the problem in time showed that the fast and slowwaves obey a wave equation with fractional derivative. This
equation has been studied in our previous work
30,31in the
case of a rigid porous structure. The solutions of Biot’s equa-tions in the time domain is expressed by the Green functions
of fast and slow waves. This solution will, in the future, ena-
ble us to better understand the Biot wave propagation in thetime domain. Operators of reflection and transmission show
the contribution of the first and second interface, as well as
the transmitted fast and slow responses of the material.Through this temporal modeling, solving the inverse prob-
lem directly in the time domain would be possible, allowing
ultrasonic characterization of porous materials at high fre-quencies, with a particular application to cancellous bone tis-
sue. Finally, this study illustrates the importance of using
fractional derivatives to describe, in the time domain, theacoustic propagation in porous media.
FIG. 5. Comparison between experimental transmitted signal (solid line)
and simulated transmitted signal (dashed line) for bone sample B2.
FIG. 6. Comparison between experimental transmitted signal (solid line)and simulated transmitted signal (dashed line) for bone sample B3.
FIG. 8. Comparison between experimental transmitted signal (solid line)and simulated transmitted signal (dashed line) for hydroxyapatite sample H.
FIG. 7. Comparison between experimental transmitted signal (solid line)and simulated transmitted signal (dashed line) for bone sample B4.
1876 J. Acoust. Soc. Am., Vol. 133, No. 4, April 2013 Fellah et al. : Fractional model in porous media
Downloaded 30 Aug 2013 to 152.3.102.242. Redistribution subject to ASA license or copyright; see http://asadl.org/termsAPPENDIX A: GREEN FUNCTIONS OF FAST AND SLOW WAVES
The following relations are used for the Green functions50,51Giðt;KÞof the porous material for the fast ( i¼1) and the
slow waves ( i¼2)
e/C0Kffiffiffiffiffiffiffi
kiðzÞp
¼e/C0KffiffiffiffiCipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
z2þbizffiffizpþeizp
;
bi¼Di
Ci;ei¼Gi
Ci;Di¼ðb2
i/C04eiÞ;
Giðt;KÞ¼L/C01/C16
e/C0Kffiffiffiffiffiffiffi
kiðzÞp/C17
¼0i f 0 /C20t/C20KffiffiffiffiffiCip
NiðtÞþDiÐt/C0KffiffiffiffiCip
0hiðt;nÞdnift/C21KffiffiffiffiffiCip;i¼1;2(
with
NiðtÞ¼bi
4ffiffiffippKffiffiffiffiffiCip
ðt/C0KffiffiffiffiffiCipÞ3=2exp/C0b2
iK2Ci
16ðt/C0KffiffiffiffiffiCipÞ/C18/C19
;
where hiðs;nÞhas the following form:
hiðn;sÞ¼/C01
4p3=21ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðs/C0nÞ2/C0K2Ciq1
n3=2
/C2ð1
/C01exp/C0viðl;s;nÞ
2/C18/C19
ðviðl;s;nÞ/C01Þ
/C2ldlffiffiffiffiffiffiffiffiffiffiffiffiffi
1/C0l2p a
and where
viðl;s;nÞ¼/C16
Dilffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðs/C0nÞ2/C0K2Ciq
þbiðs/C0nÞ/C172
=8n:
APPENDIX B: REFLECTION AND TRANSMISSION
COEFFICIENTS IN LAPLACE DOMAIN
To simplify the analysis, we will use the Laplace trans-
form which is appropriate for our problem. We note that
P(x,z), the Laplace transform of p(x,t), is defined by
Pðx;zÞ¼L ½ pðx;tÞ/C138 ¼ð1
0expð/C0ztÞpðx;tÞdt: (B1)
In the region x/C200, the field p1(x,t) is given by
p1ðx;tÞ¼ dt/C0x
c0/C18/C19
þ~RðtÞ/C3dtþx
c0/C18/C19 /C20/C21
/C3piðtÞ:
The Laplace transform of the field outside the materials is
given by
P1ðx;zÞ¼ exp/C0zx
c0/C18/C19
þRðzÞexp zx
c0/C18/C19 /C20/C21
uðzÞ;x/C200;
(B2)P3ðx;zÞ¼TðzÞexp/C0L
cþðx/C0LÞ
c0/C18/C19
z/C20/C21
uðzÞ;x/C21L:
(B3)
Here P1ðx;zÞand P3ðx;zÞare, respectively, the Laplace
transform of the field at the left and the right of the material,uðzÞdenotes the Laplace transform of the incident field
p
iðtÞ, and finally, R(z) and T(z) are the Laplace transform of
the reflection and transmission kernels, respectively. Let rs
ij
andrf
ijbe the frame and fluid stress tensors, respectively,
and let /C15ij¼1
2ðuijþujiÞbe the frame strain tensor. The
stress-strain equations in the porous medium are given by
rs
ij¼½ ðP/C02NÞ$uðtÞþQ$UðtÞ/C138dijþNðuijþujiÞ;
rf
ik¼/C0/pfdij¼ðR$UðtÞþ$uðtÞÞdij: (B4)
pfis the pressure of the fluid. It is assumed that the pressure
field and the normal stress in the porous medium are continu-
ous at the boundary23of the material, at x¼0 and x¼L
interfaces,
/fð0þ;tÞ¼/C0 /P1ð0/C0;tÞ;rsð0þ;tÞ¼/C0 ð 1/C0/ÞP1ð0/C0;tÞ;
rfðL/C0;tÞ¼/C0 /P3ðLþ;tÞ;rsðL/C0;tÞ¼/C0 ð 1/C0/ÞP3ðLþ;tÞ;
(B5)
where rfandrsare the solid and fluid normal stresses,
respectively, inside the porous medium.
For longitudinal waves, the expressions of rsandrfare
given by
rsðx;zÞ¼ð P/C02NÞ@2Usðx;zÞ
@x2þQ@2Ufðx;zÞ
@x2
þ2N@2Usðx;zÞ
@x2;
rfðx;zÞ¼R@2Ufðx;zÞ
@x2þQ@2Usðx;zÞ
@x2; (B6)
where UsandUfare the Laplace transforms of ~Usand ~Uf,
respectively.
J. Acoust. Soc. Am., Vol. 133, No. 4, April 2013 Fellah et al. : Fractional model in porous media 1877
Downloaded 30 Aug 2013 to 152.3.102.242. Redistribution subject to ASA license or copyright; see http://asadl.org/termsIn order to derive the reflection and transmission coeffi-
cients, the boundary conditions of flow velocity at the x¼0
andx¼Linterfaces are required. The equations of flow con-
tinuity at x¼0 and x¼Lare written as
V1ð0/C0;zÞ¼ð 1/C0/ÞVsð0þ;zÞþ/Vfð0þ;zÞ;
V3ðLþ;tÞ¼ð 1/C0/ÞVsðL/C0;tÞþ/VfðL/C0;tÞ; (B7)
where V1andV3are the acoustic velocity field in the region
(1) (x/C200) and region (3) ( x/C21L), respectively. VsandVfare
the solid and fluid acoustic fields, respectively. The expres-
sions of V1andV3are obtained using Eqs. (B2) and(B3),
and the Euler equation
qfzViðx;zÞ¼@Piðx;zÞ
@x;i¼1;3: (B8)
The expressions of Vsand Vfare obtained using the
expressions
Vsðx;zÞ¼z@UsðzÞ
@x/C18/C19
;Vfðx;zÞ¼z@UfðzÞ
@x/C18/C19
:(B9)Using the boundary conditions at the interfaces x¼0 and
x¼Lgiven by Eqs. (B5)–(B9) and using Eqs. (16) and(17)
given in Eq. (B5), we obtain the following expressions
of the reflection and transmission coefficients in Laplace
domain:
RðzÞ¼z2½F2
4ðzÞ/C0F2
3ðzÞ/C138 þ1
½zF3ðzÞ/C01/C1382/C0z2F2
4ðzÞ; (B10)
TðzÞ¼2zF4ðzÞ
z2F2
4ðzÞ/C0½zF3ðzÞ/C01/C1382: (B11)
APPENDIX C: ANALYTICAL DEVELOPMENTS OF
REFLECTION AND TRANSMISSION COEFFICIENTS
The expressions (26) and (27) of the reflection and
transmission coefficients require the calculus of F3ðzÞ
/C0F4ðzÞand F3ðzÞþF4ðzÞ, respectively. From the expres-
sions of F3ðzÞand F4ðzÞgiven by Eq. (24), we obtain
F3ðzÞ/C0F4ðzÞ
2qfc0¼½1þ/ð=1ðzÞ/C01Þ/C138ffiffiffiffiffiffiffiffiffiffi
k1ðzÞp W1ðzÞ
WðzÞtanhl
2ffiffiffiffiffiffiffiffiffiffi
k1ðzÞp/C18/C19
þ½1þ/ð=2ðzÞ/C01Þ/C138ffiffiffiffiffiffiffiffiffiffi
k2ðzÞp W2ðzÞ
WðzÞtanhl
2ffiffiffiffiffiffiffiffiffiffi
k2ðzÞp/C18/C19
;
F3ðzÞþF4ðzÞ
2qfc0¼½1þ/ð=1ðzÞ/C01Þ/C138ffiffiffiffiffiffiffiffiffiffi
k1ðzÞp W1ðzÞ
WðzÞcothl
2ffiffiffiffiffiffiffiffiffiffi
k1ðzÞp/C18/C19
þ½1þ/ð=2ðzÞ/C01Þ/C138ffiffiffiffiffiffiffiffiffiffi
k2ðzÞp W2ðzÞ
WðzÞcothl
2ffiffiffiffiffiffiffiffiffiffi
k2ðzÞp/C18/C19
:
Here k1ðzÞ,k2ðzÞ,=1ðzÞ, and=2ðzÞare, respectively, the Laplace transform of the eigenvalues and the eigenvectors operators
of the system (11). In this case, the expressions of R(z) and T(z) given in Eqs. (26) and(27) yield
RðzÞ¼/C0 1þ1
1/C0XðzÞtanhl
2ffiffiffiffiffiffiffiffiffiffi
k1ðzÞp/C18/C19
/C0YðzÞtanh1
2ffiffiffiffiffiffiffiffiffiffi
k2ðzÞp/C18/C19 þ1
1/C0XðzÞcothl
2ffiffiffiffiffiffiffiffiffiffi
k1ðzÞp/C18/C19
/C0YðzÞcoth1
2ffiffiffiffiffiffiffiffiffiffi
k2ðzÞp/C18/C19 ;
TðzÞ¼1
1/C0XðzÞtanhl
2ffiffiffiffiffiffiffiffiffiffi
k1ðzÞp/C18/C19
/C0YðzÞtanh1
2ffiffiffiffiffiffiffiffiffiffi
k2ðzÞp/C18/C19 /C01
1/C0XðzÞcothl
2ffiffiffiffiffiffiffiffiffiffi
k1ðzÞp/C18/C19
/C0YðzÞcoth1
2ffiffiffiffiffiffiffiffiffiffi
k2ðzÞp/C18/C19 ;
where
XðzÞ¼2qfc0½1þ/ð=1ðzÞ/C01Þ/C138ffiffiffiffiffiffiffiffiffiffi
k1ðzÞp W1ðzÞ
WðzÞ;
YðzÞ¼2qfc0½1þ/ð=2ðzÞ/C01Þ/C138ffiffiffiffiffiffiffiffiffiffi
k2ðzÞp W2ðzÞ
WðzÞ:
Using the following relationstanh1
2lffiffiffiffiffiffiffiffiffiffi
k1ðzÞp/C18/C19
¼1/C02exp/C16
/C0lffiffiffiffiffiffiffiffiffiffi
k1ðzÞp/C17
1þexp/C16
/C0lffiffiffiffiffik1pðzÞ/C17
and cothl
2ffiffiffiffiffiffiffiffiffiffi
k1ðzÞp/C18/C19
¼1þ2exp/C16
/C0lffiffiffiffiffiffiffiffiffiffi
k1ðzÞp/C17
1/C0exp/C16
/C0lffiffiffiffiffiffiffiffiffiffi
k1ðzÞp/C17;
the expressions of the reflection and transmission coeffi-
cients become
1878 J. Acoust. Soc. Am., Vol. 133, No. 4, April 2013 Fellah et al. : Fractional model in porous media
Downloaded 30 Aug 2013 to 152.3.102.242. Redistribution subject to ASA license or copyright; see http://asadl.org/termsRðzÞ¼/C0 1þ1
1/C0XðzÞ/C0YðzÞþ2XðzÞZ1ðzÞ
1þZ1ðzÞþ2YðzÞZ2ðzÞ
1þZ2ðzÞþ1
1/C0XðzÞ/C0YðzÞ/C02XðzÞZ1ðzÞ
1/C0Z1ðzÞ/C02YðzÞZ2ðzÞ
1/C0Z2ðzÞ;
TðzÞ¼1
1/C0XðzÞ/C0YðzÞþ2XðzÞZ1ðzÞ
1þZ1ðzÞþ2YðzÞZ2ðzÞ
1þZ2ðzÞ/C01
1/C0XðzÞ/C0YðzÞ/C02XðzÞZ1ðzÞ
1/C0Z1ðzÞ/C02YðzÞZ2ðzÞ
1/C0Z2ðzÞ;
where
Z1ðzÞ¼exp/C16
/C0lffiffiffiffiffiffiffiffiffiffi
k1ðzÞp/C17
andZ2ðzÞ¼exp/C16
/C0lffiffiffiffiffiffiffiffiffiffi
k2ðzÞp/C17
:
For large values of zcorresponding to ultrasonic wave propagation in porous media, exp/C0
/C0lffiffiffiffiffiffiffiffiffiffi
k1ðzÞp/C1
and
expð/C0lffiffiffiffiffiffiffiffiffiffi
k2ðzÞp
Þare small. We can then use the following developments:
RðzÞ¼/C0 1þ1
1/C0XðzÞ/C0YðzÞX
n/C2102n
ð1/C0XðzÞ/C0YðzÞÞnXðzÞZ1ðzÞ
1/C0Z1ðzÞþYðzÞZ2ðzÞ
1/C0Z2ðzÞ/C18/C19n
þð /C0 1ÞnXðzÞZ1ðzÞ
1þZ1ðzÞþYðzÞZ2ðzÞ
1þZ2ðzÞ/C18/C19n /C20/C21
;
TðzÞ¼1
1/C0XðzÞ/C0YðzÞX
n/C2102n
ð1/C0XðzÞ/C0YðzÞÞn/C0XðzÞZ1ðzÞ
1/C0Z1ðzÞþYðzÞZ2ðzÞ
1/C0Z2ðzÞ/C18/C19n
þð /C0 1ÞnXðzÞZ1ðzÞ
1þZ1ðzÞþYðzÞZ2ðzÞ
1þZ2ðzÞ/C18/C19n /C20/C21
:
By considering the small powers of Z1(z) and Z2(z) (since the other terms are small), we obtain the following approximations
ofR(z) and T(z):
RðzÞ¼/C0 1þ2
1/C0XðzÞ/C0YðzÞþ4XðzÞZ1ðzÞ2
ð1/C0XðzÞ/C0YðzÞÞ2ð1/C0Z2
1ðzÞÞþ4YðzÞZ2ðzÞ2
ð1/C0XðzÞ/C0YðzÞÞ2ð1/C0Z2
2ðzÞÞ;
TðzÞ¼/C04XðzÞZ1ðzÞ
ð1/C0XðzÞ/C0YðzÞÞ2ð1/C0Z2
1ðzÞÞ/C04YðzÞZ2ðzÞ
ð1/C0XðzÞ/C0YðzÞÞ2ð1/C0Z2
2ðzÞÞ: (C1)
The obtained expressions (28) and(29) are available for large values of zcorresponding to the high frequency range48
considered in this paper.
APPENDIX D: INVERSE LAPLACE TRANSFORMS
The following relations are used for the calculus of the inverse Laplace transform:
L/C01 1ffiffizpþa/C20/C21
¼1ffiffiffiffiffiptp/C0aexpða2tÞErfcðaffiffi
tp
Þ;
L/C01 1
ðffiffizpþaÞ2"#
¼/C02affiffiffi
t
pr
þð1þ2a2tÞexpða2tÞErfcðaffiffi
tp
Þ;
L/C011ffiffizp/C20/C21
¼1ffiffiffiffiffiptp; L/C01 1ffiffizpðffiffizpþaÞ/C20/C21
¼expða2tÞErfcðaffiffi
tp
Þ;
L/C01 1ffiffizpðffiffizpþaÞ2"#
¼2ffiffiffi
t
pr
/C02atexpða2tÞErfcðaffiffi
tp
Þ:
J. Acoust. Soc. Am., Vol. 133, No. 4, April 2013 Fellah et al. : Fractional model in porous media 1879
Downloaded 30 Aug 2013 to 152.3.102.242. Redistribution subject to ASA license or copyright; see http://asadl.org/termsAPPENDIX E: DETERMINATION OF THE MODIFIED
BIOT PARAMETERS
The bone and hydroxyapatite samples are drained and
their physical parameters ( a1;/;K;Eb, and /C23b) are meas-
ured by techniques31,53developed initially for air-saturated
porous materials such as plastic foams and fibrous mats.When the liquid saturating the cancellous bone is drained
from the pores and replaced by air, partial decoupling of the
Biot waves occurs
30,31due to the tremendous difference in
density between the frame and air. The fluid particles do not
have enough mass to generate motion in the heavy solid
frame, and thus the slow wave propagates in the fluid whereinit is detected (in non-contact manner) by a transducer. The
three parameters a
1;/;andKare determined by measuring
the slow wave propagating in air-saturated cancellous boneand hydroxyapatite. For example, the porosity, /, and the tor-
tuosity, a
1, are determined by measuring the wave reflected
by the first interface of the bone sample at oblique inci-dence.
53The viscous characteristic length, K, is evaluated by
measuring the transmitted wave. With contact excitation,52
the fast wave travels in the solid frame and some air particles
move along with the frame. The velocity of the fast wave
approaches the velocity in the frame as measured in vacuum
and is given by vL¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðKbþ4
3NÞ=ð1/C0/Þqsq
. By measuring
the fast wave velocity of a sample whose pores are filled with
air, one finds Kbþ4=3N. The shear modulus Ncan be eval-
uated independently by measuring the velocity of the shear
wave. The expression of the shear wave velocity is given
by43vT¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
N=ð1/C0/Þqsp
. By measuring experimentally vL
and vT, we deduce KbandN, and then the values of Eband/C23b
using the relations (3). The Young’s modulus, Es, and the
Poisson ratio, /C23s, are evaluated using experimental transmit-
ted waves on water saturated samples with the same method
as in Ref. 10.
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1.373163.pdf | Real time quantification of Monte Carlo steps for different time scales
R. Smirnov-Rueda, O. Chubykalo, U. Nowak, R. W. Chantrell, and J. M. Gonzalez
Citation: Journal of Applied Physics 87, 4798 (2000); doi: 10.1063/1.373163
View online: http://dx.doi.org/10.1063/1.373163
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/87/9?ver=pdfcov
Published by the AIP Publishing
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75.102.73.105 On: Fri, 21 Nov 2014 22:36:10Real time quantification of Monte Carlo steps for different time scales
R. Smirnov-Ruedaa)
Computational Magnetism Group, School of Electronic Engineering & Computer Systems,
University of Wales, Dean Street, Bangor, LL57 1UT, United Kingdom
O. Chubykalo
IBM Almaden Research Center, Magnetic Theory and Modeling, 650 Harry Road, San Jose,California, 95120
U. Nowak
Theoretische Tieftemperaturphysik, Gerhard-Mercator University of Duisburg,47048 Duisburg, Germany
R. W. Chantrell
Computational Magnetism Group, School of Electronic Engineering & Computer Systems,University of Wales, Dean Street, Bangor, LL57 1UT, United Kingdom
J. M. Gonzalez
Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, 28049 Madrid, Spain
TimequantificationofMonteCarlostepsisstudiedbytheimplementationofanewtechniquewhich
takes into account the realistic size of thermal fluctuations of magnetization along with Landau–Lifshitz–Gilbert dynamic correlations. The computational model has been specifically developedfor an ensemble of isolated single-domain particles. The numerical results have been compared withLangevin dynamics calculations and theoretically predicted Brown’s asymptotes for relaxation timeofsinglespinsystem.InadditionwedemonstratedthatrealtimequantificationofMonteCarlostepsis also possible for different time scales. Implementation of real time scales into Monte Carlocalculations for different sizes of time steps is shown to be convergent to the expected value if theMonte Carlo acceptance rate is taken into account. © 2000 American Institute of Physics.
@S0021-8979 ~00!78308-3 #
I. INTRODUCTION
The understanding of slow dynamic behavior in ex-
tended systems of many interacting magnetic moments overlarge time intervals has considerable practical implicationsfor magnetic recording and, particularly, for evaluation oflong-term stability of magnetically recorded information.The theoretical formalism for studying thermally activatedmagnetization reversal is based on the solution of theLandau–Lifshitz–Gilbert ~LLG!equation of motion with a
fluctuation term representing a random thermal force. Insome special cases, a corresponding Fokker–Planckequation
1yields characteristic relaxation time values as
asymptotic solutions in the large energy barrier limit.2This
provides an important foundation for real time quantificationof different computational techniques such as Langevin dy-namics ~LD!, based on the exact numerical solution to the
Langevin equations and Monte Carlo ~MC!method, which
uses a random generation of new spin configurations to re-produce the structure of a particular stochastic process. Withregard to the latter method, in its original form MC genera-tion of system configurations is not based on the real quan-tification of time steps. However, despite this disadvantage,the conventional MC technique is a powerful tool for simu-lation of thermally activated reversal over large energy bar-riers. In contrast, the LD approach yields explicit dynamicinformation resulting directly from the numerical solution of
the LLG equation but its application is limited to short timescales up to the order of 10
29s.
Recently, we have attempted3to provide the conven-
tional MC scheme with explicit details on the real size oftime steps along with dynamic correlations arising from theLLG equation. Comparison of the time quantified MC ap-proaches with corresponding LD calculations showed a va-lidity for the theoretically justified relationship between theMC steps and real time intervals. Another comparison of thenumerical results for the time quantified MC scheme wasmade with analytical formula for the relaxation rate in thehigh damping regime.
4,5
The work in Ref. 3 represented the first attempt to use
different time scales for quantified MC steps, testing the va-lidity of that implementation by direct comparison with LDcalculations. However, it is also important to test this proce-dure against Brown’s well-known analytical results for therelaxation time, which was predicted only for an ensemble ofisolated single-domain particles. This is thought to be a nec-essary and important preliminary step before undertakingfurther realistic calculations in micromagnetic models withmany degrees of freedom. On the other hand, it is expectedthat practical implementation of different time scales willconsiderably increase the effectiveness of MC calculations incomparison with the LD scheme in the limit of large energybarriers.
a!Electronic mail: roman@sees.bangor.ac.ukJOURNAL OF APPLIED PHYSICS VOLUME 87, NUMBER 9 1 MAY 2000
4798 0021-8979/2000/87(9)/4798/3/$17.00 © 2000 American Institute of Physics
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75.102.73.105 On: Fri, 21 Nov 2014 22:36:10II. COMPUTATIONAL METHOD
The computational scheme outlined in Ref. 3 comprises
a modification of statistical properties of the conventionalMC approach
6in order to introduce explicit and detailed
information on dynamic correlations available from the lin-earized LLG equations with the damping term and randomforce. Integrated over a time interval Dt, the LLG equation
for a system of coupled spins gives
3
mia~t1Dt!2mia~t!5(
b(
jLijabmjbDt1Fia, ~1!
wheremia5(Mia2Mia(0))/Msis a deviation of a magnetic
moment Mifrom the local equilibrium value Mi0and re-
duced to the value of saturation magnetization Ms;a,bare
Cartesian components, and Fi5fiDtrepresents the effect of
thermal fluctuations on the orientation of local magnetiza-tion.
The matrix elements L
ijof the set of linearized and
coupled LLG equations are calculated for the correspondinglocal equilibrium state. In the case of an ensemble of isolatednoninteracting particles, the correlation term L
ijis not nec-
essary, leaving in the right-hand side of Eq. ~1!only uncor-
related random displacements Fi. Nevertheless, in contrast
to the conventional MC approach,6the statistical properties
of thermal fluctuations are driven by the covariance matrix
mij,
^Fia~t1Dt!Fjb~t!&5mijabDt, ~2!
defined in the LD approach by application of the fluctuation-
dissipation theorem:7
mijab52kT~Likag@Akjgb#211Ljkag@Akigb#21. ~3!
An assumption on the equivalence of statistical proper-
ties between MC and LD random deviations gives rise to theimplementation of a realistic time scale for one MC step.According to relationship ~2!, each component of particle
magnetization generated randomly in the MC scheme has aGaussian distribution analogous to that of LD. On the otherhand, the symmetric energy matrix A
ijis evaluated from the
expansion of the micromagnetic Hamiltonian in a local equi-
librium state Mi0. Thus, fluctuations of magnetization deter-
mined by the matrix mijwill depend not only on system
parameters such as damping, volume, temperature, etc., butalso on a system configuration in the local equilibrium state.In the particular case of an ensemble of single-domain par-ticles, expression ~2!for statistical properties of fluctuating
magnetization takes the following form:
miibb5kBTaDt~12~Mib~0!!2!/kEVi~11a2!, ~4!
wherekBis Boltzmann’s constant, Tis the temperature, Viis
the volume of a single-domain particle, KEis the easy an-
isotropy constant, Dtis the time interval necessary to aver-
age the fluctuations, and ais the damping constant. In LD
calculations Dtplays the role of the time integration step for
the numerical solution of the LLG equation.
Extension of the MC calculation scheme with the infor-
mation on dynamic correlations available from LLG equa-tions gives a further extension of the above-mentioned rep-resentation of MC time units. More specifically, it isassumed that in the spirit of the LD method, each value ofMC-generated magnetization m
i(t1Dt) has to be updated
for coupled spins by including dynamic correlations fromEq.~1!. After this procedure the new orientation of magne-
tization will be accepted or rejected according to the corre-sponding MC criteria.
In this article we apply the presented computational
technique to the micromagnetic model of an isolated single-domain particle system in order to compare relaxation timeswith Brown’s theoretical predictions and analogous Lange-vin dynamics calculations.
III. MICROMAGNETIC MODEL
Here we considered an ensemble of isolated spins, i.e.,
our micromagnetic model will not include exchange andmagnetostatic interactions between particles. Every single-domain particle has an easy anisotropy axis parallel to the z
axis. In the initial state all local magnetic moments weredirected along the positive direction of the zaxis. An exter-
nal magnetic field has been measured in terms of the anisot-ropy field and was applied in the negative direction of the z
axis.
Throughout this work we have used the following simu-
lation parameters: high damping constant
a54 and the value
happ50.75 for the applied magnetic field reduced to the an-
isotropy field ( h5HAPP/HA). In order to identify an integra-
tion step Dtin the LLG equation and the size of a trial step
in MC calculations we also implemented the following spe-cific values for the easy anisotropy constant k
E54.2
3106ergs/cm3and saturation magnetization Ms51.4
3103emu/cm3. The modeled system consisted of 500 iso-
lated single-domain particles.
IV. RESULTS AND DISCUSSIONS
In our simulations using LD and the MC method we
calculated the relaxation time as a function of the corre-spondingenergybarrierwhichinthecaseofsinglespins‘‘ i’’
is defined by the normalized anisotropy and the reduced ap-plied field:
DE
i5KEVi~12happ!2, ~5!
whereViis the volume of particle ‘‘ i.’’1
A comparison was made with the characteristic time t
calculated asymptotically from the Fokker–Planck equation2
for high energy barriers:
t5t0exp~DE/kBT!, ~6!
where the prefactor t0is given by8
t05~11a2!
agHAApAT*
~12h2!~12h!. ~7!
HereT*iskBTnormalized to the maximum attainable an-
isotropy energy KEVi.
In Fig. 1 we compare MC calculations and LD simula-
tions of thermally activated reversal with the same time step.In all cases the large energy barrier dependence of relaxationtime is found to be in excellent agreement with Brown’stheoretical predictions.
8In view of getting a more effective
computational scheme based on the MC approach, we stud-ied the real time quantification of MC steps for different4799 J. Appl. Phys., Vol. 87, No. 9, 1 May 2000 Smirnov-Rueda et al.
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75.102.73.105 On: Fri, 21 Nov 2014 22:36:10values of time steps. The latter has been defined by the factor
‘‘scale’’ which represents the size of time step used in theMC approach relative to the time step in LD calculations.Large values of the scale parameter represent a significantincrease in computational speed and can be used for systemswith particularly large energy barriers.
In Fig. 2 we show the MC evaluation of relaxation times
calculated with implementation of different time scales. Ob-servable disagreement for larger scales is related to the factthat the MC acceptance rate decreases gradually as the sizeof fluctuations of magnetization is increased. For instance, inthe case of scale 510
4onlyone-tenth of the newly gener-
ated configurations pass through the MC acceptance criteriawhereas for scale 51, the probability of acceptance is very
close to1. These discrepancies can be reduced drastically if
the total relaxation time value does not include MC moveswith no modification of the system configurations.
In Fig. 3 we presented the data of relaxation time ~cor-
responding to Fig. 2 !normalized with respect to the prob-
ability of MC acceptance. As was expected, results of MCcalculations for different size of time step converged toBrown’s asymptote. A deviation from the theoretically pre-dicted value is more significant for bigger scales. This errorpresumably arises from a breakdown of the simple scalinggiven by Eq. ~2!for large fluctuations. Nevertheless, the
agreement for large scales is still reasonably good and makesthe corresponding calculation much faster. Effectiveness ofthe MC scheme provided for one time scale with respect tothe other depends on the system parameters as well as on therange of the energy barriers. However, such a variation doesnot affect very much the drastic decrease in computation
time for large scales. For instance, in the case of energybarrier equal to 2.5, the scale 510 turned out to be 9.2 times
more effective than the scale 51. The same characteristic for
thescale 510
2was 76, for the scale 51032410, and for the
scale 510421100, respectively. This fact might become an
important point for further calculations of relaxation ratesover very large energy barriers.
V. CONCLUSIONS
A new MC approach, which incorporates the informa-
tion on the size of a real time step, has been used for thecalculation of relaxation times of magnetization. Its compari-son with corresponding theoretical asymptotic solutions andLD calculations for the ensemble of single-domain particlesshowed its validity in the considered range of energy barri-ers. This computational method has also been tested on theimplementation of different time scales, which is importantfor the rising effectiveness of MC calculations. It has beendemonstrated that all calculated data are convergent to theexpected values if the probability of MC acceptance is ex-plicitly used. It is important to note that the use of large timesteps is potentially extremely important for systems of inter-acting particles where speed requirements are important. Thework presented here suggests that increases of speed up tothree orders of magnitude relative to the LD technique arepossible using the time quantified MC technique, which rep-resents an important development.
Financial support of the UK EPSRC is acknowledged.
R.S.-R. is grateful to the Spanish Ministry of Education forthe provision of postdoctoral research grant. The authorsthank D. Hinzke for helpful discussions.
1W. F. Brown, Phys. Rev. 130, 1677 ~1963!.
2W. T. Coffey, Yu. P. Kalmykov, E. S. Massawe, and J. T. Waldron, J.
Chem. Phys. 99, 401 ~1993!.
3R. Smirnov-Rueda, J. D. Hannay, O. Chubykalo, R. W. Chantrell, and J.
M. Gonzalez, IEEE Trans. Magn. 35, 3730 ~1999!.
4U. Nowak, R. W. Chantrell, and E. C. Kennedy, Phys. Rev. Lett. ~submit-
ted!.
5D. Hinzke, U. Nowak, and K. D. Usadel, Proceedings SDHS’99 Duisburg
~World Scientific, Singapore, 1999 !.
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1979!.
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Matter5, 8911 ~1993!.
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Kennedy, Phys. Rev. B 58, 3249 ~1998!.
Published without author corrections
FIG. 1. Relaxation time vs energy barrier. The LD and MC simulation data
are compared with Brown’s asymptotic formulas for the following param-
eters: a54 andhapp50.75.
FIG. 2. Comparison of MC relaxation times for different time scales. MC
acceptance probability is not taken into account.
FIG. 3. Comparison of MC relaxation times for different time scales. MCacceptance probability is taken into account.4800 J. Appl. Phys., Vol. 87, No. 9, 1 May 2000 Smirnov-Rueda
et al.
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1.2359292.pdf | Self-consistent simulation of quantum transport and magnetization dynamics in spin-
torque based devices
Sayeef Salahuddin and Supriyo Datta
Citation: Applied Physics Letters 89, 153504 (2006); doi: 10.1063/1.2359292
View online: http://dx.doi.org/10.1063/1.2359292
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/89/15?ver=pdfcov
Published by the AIP Publishing
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On: Sat, 20 Dec 2014 12:15:49Self-consistent simulation of quantum transport and magnetization
dynamics in spin-torque based devices
Sayeef Salahuddina/H20850and Supriyo Datta
School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907 and NSF
Network for Computational Nanotechnology (NCN), Purdue University, West Lafayette, Indiana 47907
/H20849Received 21 June 2006; accepted 24 August 2006; published online 11 October 2006 /H20850
This letter presents a self-consistent solution of quantum transport, using the nonequilibrium
Green’s function method, and magnetization dynamics, using the Landau-Lifshitz-Gilbertformulation. This model is applied to study “spin-torque” induced magnetic switching in a devicewhere the transport is ballistic and the free magnetic layer is sandwiched between two antiparallel/H20849AP/H20850ferromagnetic contacts. A hysteretic current-voltage characteristic is predicted at room
temperature, with a sharp transition between the bistable states that can be used as a nonvolatilememory. It is further shown that this AP pentalayer device may allow significant reduction in theswitching current, thus facilitating integration of nanomagnets with electronic devices. © 2006
American Institute of Physics ./H20851DOI: 10.1063/1.2359292 /H20852
Successful integration of nanomagnets with electronic
devices may enable the first generation of practical spin-tronic devices, which have been elusive so far due to strin-gent requirements such as low temperature and high mag-netic field. It was predicted by Slonczewski
1and Berger2that
magnetization of a nanomagnet may be flipped by a spinpolarized current through the so-called “spin-torque” effectand this was later demonstrated experimentally.
3,4However,
the early spin-torque systems were metal based that allowedonly a small change in the magnetoresistance. In addition,metallic channels are difficult to integrate with complemen-tary metal oxides semiconductor technology. Recently anumber of experiments have demonstrated current induced
magnetization switching in MgO based tunneling magnetore-sistance /H20849TMR /H20850devices at /H20849i/H20850room temperature /H20849ii/H20850with a
TMR ratio of more than 100% and /H20849iii/H20850without any external
magnetic field.
5,6Encouraged by these experimental results,
here we explore theoretically a memory device based on cur-rent induced magnetization switching in the quantum trans-port regime.
The device under consideration is shown in Fig. 1.I t
consists of five layers. The two outer layers are “hard mag-nets” which act as spin polarized contacts. There is a softmagnetic layer inside the channel whose magnetization isaffected by the current flow through the so-called spin-torqueeffect. The channel can be a semiconductor
7or a tunneling
oxide.6Note that the contacts are arranged in an antiparallel
/H20849AP/H20850configuration. We have recently showed that in this
configuration, the torque exerted by the injected electrons ona the nearby spin array /H20849in this case the soft magnet /H20850is
maximum.
8A similar prediction was also made by Berger9
based on expansion/contraction of the Fermi surface. Thepossibility of an enhanced torque and therefore a lowerswitching current is our motivation for the pentalayer con-figuration instead of the conventional trilayer geometry.
In Fig. 1, the soft magnet changes the transport through
its interaction with the channel electrons, which in turn exerta torque on the magnet and try to rotate it from its equilib-rium state. In this letter, we present a self-consistent solutionof both these processes: the transport of channel electrons
/H20851through nonequilibrium Green’s function /H20849NEGF /H20850/H20852and the
magnetization dynamics of the free layer /H20851through Landau-
Lifshitz-Gilbert /H20849LLG /H20850equations /H20852/H20851see Fig. 1/H20849b/H20850/H20852. Our calcu-
lations show clear hysteretic I-Vsuggesting possible use as a
memory. Furthermore, we show that a pentalayer device withAP contact as shown in Fig. 1/H20849a/H20850should exhibit a significant
reduction in the switching current.
Unlike the conventional metallic spin-torque systems,
where transport is predominantly diffusive, the transport insemiconductors or tunneling oxides is ballistic or quasibal-listic. This necessitates a quantum description of the trans-port. We use the NEGF method to treat the transport rigor-ously. The interaction between channel electrons and theferromagnet is mediated through exchange and it is de-scribed by H
I/H20849r/H20850=/H20858RjJ/H20849r−Rj/H20850/H9268·Sj, where randRjare the
spatial coordinates and /H9268andSjare the spin operators for the
channel electron and jth spin in the soft magnet. J/H20849r¯−R¯j/H20850is
the interaction constant between the channel electron and the
jth spin in the magnet. This interaction is taken into account
through self-energy /H20849/H9018s/H20850, which is a function of the magne-
tization /H20849m/H20850, using the so-called self-consistent Born
approximation.10In this method, the spin current flowing
into the soft magnet is given by
/H20851Ispin/H20852=/H20885dEe
/H6036i/H20851Tr/H20853G/H9018sin−/H9018sinG†−/H9018sGn+Gn/H9018s†/H20854/H20852,/H208491/H20850
where the trace is taken only over the space coordinates.
Then /H20851Ispin/H20852i sa2 /H110032 matrix in the spin space. Here, G
denotes Green’s function. The torque exerted on the magnet
is calculated from /H20851Ispin/H20852by writing Ti=Trace /H20853Si/H20851Ispin/H20852/H20854,
where i=/H20853x,y,z/H20854. The total current, which is found from a
similar expression as Eq. /H208491/H20850with the self-energy /H9018snow
replaced by the total self-energy /H9018,11is shown in Figs. 2/H20849a/H20850
and2/H20849b/H20850for two different configurations of the magnetiza-
tion. The nonlinearity in the I-Vfollows from the spin-flip
exchange interaction8which is usually ignored in a purely
barrier model.
For the calculations, the Hamiltonian was written in the
effective-mass approximation where the hopping parametera/H20850Electronic mail: ssalahud@purdue.eduAPPLIED PHYSICS LETTERS 89, 153504 /H208492006 /H20850
0003-6951/2006/89 /H2084915/H20850/153504/3/$23.00 © 2006 American Institute of Physics 89, 153504-1
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 147.143.2.5
On: Sat, 20 Dec 2014 12:15:49t=/H60362//H208492m*a2/H20850,m*=0.7 me/H20849Ref. 12/H20850denoting the effective
mass and abeing the lattice spacing. The interaction constant
Jis assumed to be 0.01 eV.13A sample set of parameters is
Ef=2 eV, barrier height=1.2 eV;12barrier width=1 nm, and
exchange splitting=1.2 eV.14However, the barrier height,
width, and exchange splitting were artificially varied to getdesired injection efficiency and TMR value. We alsomodified the perfect-contact self-energy to read /H9018=−t
/H11032exp
/H20849ika/H20850/H20849t/H11032/HS11005tandkis the momentum /H20850to simulate the reflec-
tive nature of the contact /H20849for details see Ref. 15/H20850. For
plots 2–4, an injection efficiency of 70% was assumed.
The magnetization dynamics is simulated using the LLG
equation
/H208491+/H92512/H20850/H11509m
/H11509t=/H9253/H20849m/H11003Heff/H20850−/H9253/H9251
mm/H11003m/H11003Heff
+ current torque. /H208492/H20850
Here, mis the magnetization of the soft magnet, /H9253
=17.6 MHz/Oe is the gyromagnetic ratio, and /H9251is the Gil-
bert damping parameter. The Heff=Hext+/H208492Ku2/Ms/H20850mzzˆ
−/H208492Kup/Ms/H20850mxxˆ, where Hextis the externally applied mag-
netic field, Msis the saturation magnetization, and Ku2and
Kupare the uniaxial and in-plane anisotropy constants, re-
spectively. The conventional LLG equation has to be solvedwith the current torque /H20849T
i/H20850that works as an additional
source term.
Figure 2shows the situations when the transport and
magnetization dynamics are independent of each other. Thiswill change when Eqs. /H208491/H20850and /H208492/H20850are solved self-
consistently. If we start from
/H9258=/H9266position, I-Vcurve fol-
lows the trend shown in Fig. 2/H20849a/H20850. However, once the torque
exceeds the critical field /H20849discussed later /H20850, the magnet
switches abruptly. As a result I-Vcharacteristics now follow
that shown in Fig. 2/H20849b/H20850. This results in the hysteretic I-V
shown in Fig. 3/H20849a/H20850.
Figure 3/H20849b/H20850shows current flow in the device in response
to read-write-read pulse sequence. Here, we have used readpulse of 0.5 V and write pulse of +1 V. The soft magnet isinitially in the
/H9258=/H9266position. The write pulse switches it to
/H9258=0. Note the change in the current level in response to the
read pulse before and after applying the write pulse.
A question may be raised regarding the asymmetric I-V
of Fig. 2, which is not expected if one thinks about the de-
vice in Fig. 1/H20849a/H20850as a series combination of two devices, one
antiparallel /H20849AP/H20850and one parallel /H20849P/H20850. The device, however,
is different from a mere series combination since the contactin the middle works as a mixing element for up and downspin electrons. The difference will be clear if one assumes100% injection efficiency. No current is expected to flowthrough the series combination of an AP and a P device.
However, in our device, a current can still flow because thecontact in the middle mixes the up and down spin channels.This “extra” current originating from “channel mixing” givesthe observed asymmetry in Fig. 2.
Since electronic time constants are typically in the sub-
picosecond regime which is much faster than the magnetiza-tion dynamics /H20849typcially of the order of nanoseconds /H20850,w e
have assumed that, for electronic transport, the magnetiza-tion dynamics is a quasistatic process.
16
The switching is obtained by the torque component
which is transverse to the magnetization of the soft magnet.From Eq. /H208492/H20850, considering average rate of change of energy,
it can be shown that the magnitude of the torque required toinduce switching is
/H9251/H9253/H20849Hext+Hk+Hp/2/H20850,17where Hk
=2Ku2/Msand Hp=2Kup/Ms=4/H9266Ms. This then translates
into a critical spin current magnitude of
Ispin=2e
/H6036/H9251/H20849MsV/H20850/H20849Hext+Hk+2/H9266Ms/H20850. /H208493/H20850
Here, Vis the volume of the free magnetic layer. Depending
on the magnitudes of /H9251,Ms,Hk, and thickness dof the mag-
net, the spin current density to achieve switching varies from10
5to 106A/cm2/H20849e.g., for Co, using typical values /H9251
/H110110.01, Hk/H11011100 Oe, Ms=1.5/H11003103emu/cm3, and d=2 nm,
the spin current density required is roughly 106A/cm2/H20850.
Note that this requirement on spin current is completely de-termined by the magnetic properties of the free layer. Theactual current density is typically another factor of 10–100larger due to the additional coherent component of the cur-rent which does not require any spin flip. Hence an importantmetric for critical current requirement is r=I
coherent /Ispin,
which should be as small as possible. Intuitively, with AP
FIG. 1. /H20849Color online /H20850/H20849a/H20850Schematic
showing the pentalayer device. Thefree ferromagnetic layer is embeddedinside the channel which is sand-wiched between two “hard” ferromag-netic contacts. /H20849b/H20850A schematic show-
ing the self-consistent nature of thetransport problem. The magnetizationdynamics and transport are mutuallydependent on one another.
FIG. 2. /H20849Color online /H20850Nonself-consistent /H20849with magnetization dynamics /H20850
I-Vcharacteristics of the proposed device /H20849a/H20850with the soft magnet initially
at/H9258=/H9266position. The current is larger for positive bias /H20849b/H20850with the soft
magnet initially at /H9258=0 position. The current is larger for negative bias.153504-2 S. Salahuddin and S. Datta Appl. Phys. Lett. 89, 153504 /H208492006 /H20850
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On: Sat, 20 Dec 2014 12:15:49contacts, the coherent current Icoherent /H11008t22/H9251/H9252, where tis the
hopping matrix element, /H9251is the majority /H20849minority /H20850density
of states for the injecting contact, and /H9252is the minority /H20849ma-
jority /H20850density of states of the drain contact. Similarly the
spin-flip current Isf/H11008J2/H20851/H92512/H208491−P/H9251/H20850−/H92522P/H9251/H20852, where P/H9251is the
probability of a spin in the free layer to be in state /H9251.8It
follows that
rAP=/H20879Icoherent
Isf/H20879
AP=t2
J21−Pc2
Pc+/H20851/H208491/2/H20850−P/H9251/H20852/H208491+Pc2/H20850, /H208494/H20850
where Pc=/H20849/H9251−/H9252/H20850//H20849/H9251+/H9252/H20850indicates the degree of contact po-
larization. This approximate analytical expression /H20851Eq. /H208494/H20850/H20852
agrees quite well with detailed NEGF calculations describedabove. The I
coherent andIspincan be found, respectively, from
the symmetric and asymmetric portions of the nonlinearI-Vshown in Fig. 2. Figure 3/H20849c/H20850shows the variation of g
=r
trilayer /rAP pentalayer with Pc. The plot shows that g/H112711 for
reasonable values of Pc, indicating a lower switching current
for the pentalayer device. Recent experiments on AP penta-layer devices18–20have shown similar reduction of switching
current compared to tri-layer devices. These experimentsseem to follow the general trends of Fig. 3/H20849c/H20850as the reduc-
tion factor is seen to increase with increasing TMR /H20849see Fig.
4 of Ref. 19/H20850. A detailed study of the dependence of the
reduction factor on material parameters is beyond the scopeof this letter.
The sharp transition between high and low states in Fig.
3/H20849a/H20850arises from the bistable nature of the solutions to the
LLG equation in the absence of any external field perpen-dicular to the easy axis. The intrinsic speed depends on
/H9275
=/H9253Bwhere Bcan be roughly estimated as B/H11011/H6036T//H208492/H9262B/H20850.A
higher speed will require higher current density.
In conclusion, we have shown a scheme for calculating
the “spin current” and the corresponding torque directly fromtransport parameters within the framework of NEGF formal-ism. A nonlinear I-Vis predicted for AP pentalayer devices.
Experimental observation of this nonlinearity /H20851which can
also be detected as steps or peaks in, respectively, the firstand second derivative of the I-V/H20849Ref. 8/H20850/H20852would provide
strong confirmation of our approach. We have furthercoupled the transport formalism with the phenomenological
magnetization dynamics /H20849LLG equation /H20850. Our self-consistent
simulation of NEGF-LLG equations show clear hystereticswitching behavior, which is a direct consequence of thenonlinearity described above. Finally, we have shown thatthe switching current for AP pentalayer devices can be sig-nificantly lower than that of the conventional trilayer de-vices.
This work was supported by the MARCO Focus Center
for Materials, Structure and Devices.
1J. C. Slonczewski, J. Magn. Magn. Mater. 159,L 1 /H208491996 /H20850.
2L. Berger, Phys. Rev. B 54, 9353 /H208491996 /H20850.
3S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoe-
lkopf, R. A. Buhrman, and D. C. Ralph, Nature /H20849London /H20850425,3 8 0
/H208492003 /H20850.
4J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph,
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5H. Kubota, A. Fukushima, Y . Ootani, S. Yuasa, K. Ando, H. Maehara, K.
Tsunekawa, D. D. Djayaprawira, N. Watanabe, and Y . Suzuki, Jpn. J.Appl. Phys., Part 2 44, L1237 /H208492005 /H20850.
6S. S. P. Parkin, C. Kaiser, A. Panchula, P. M. Rice, B. Hughes, M. Samant,
and S. H. Yang, Nat. Mater. 3,8 6 2 /H208492004 /H20850.
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Harris, and S. S. P. Parkin, Phys. Rev. Lett. 94, 056601 /H208492005 /H20850.
8S. Salahuddin and S. Datta, Phys. Rev. B 73, 081301R /H208492006 /H20850.
9L. Berger, J. Appl. Phys. 93, 7693 /H208492003 /H20850.
10S. Datta, Proceedings of the International School of Physics Enrico Fermi,
Italiana di Fisica, 2005, p. 1.
11S. Datta, Electronic Transport in Mesoscopic Systems /H20849Cambridge Univer-
sity Press, Cambridge, 1995 /H20850.
12W. H. Rippard, A. C. Perrella, F. J. Albert, and R. A. Buhrman, Phys. Rev.
Lett. 88, 046805 /H208492002 /H20850.
13A. H. Mitchell, Phys. Rev. 105, 1439 /H208491957 /H20850.
14F. J. Himpsel, Phys. Rev. Lett. 67, 2363 /H208491991 /H20850.
15S. Datta, Quantum Transport: Atom to Transistor /H20849Cambridge University
Press, Cambridge, 2005 /H20850.
16S. Salahuddin and S. Datta, http://www.arxiv.org/cond-mat/0606648.
17J. Z. Sun, Phys. Rev. B 62, 570 /H208492000 /H20850.
18G. D. Fuchs, I. N. Krivorotov, P. M. Braganca, N. C. Emley, A. G. F.
Garcia, D. C. Ralph, and R. A. Buhrman, Appl. Phys. Lett. 86, 152509
/H208492005 /H20850.
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222510 /H208492005 /H20850.
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FIG. 3. /H20849Color online /H20850/H20849a/H20850The hysteretic I-Voriginating from a self-consistent solution of transport and LLG. At a certain bias, the current torque produced
by the conduction electrons is strong enough to flip the magnet. These transition points are indicated in the figure. /H20849b/H20850Response to a Read-Write-Read pulse.
The write pulse switches the magnet from /H9258=/H9266to/H9258=0. The corresponding change in the current can be clearly seen during the write pulse. /H20849c/H20850Variation of
the ratio of rtrilayer /rAP pentalayer /H20849r=Icoherent /Ispin/H20850, showing the possible reduction of switching current for the AP penta layer device compared to the 3-layer
device.153504-3 S. Salahuddin and S. Datta Appl. Phys. Lett. 89, 153504 /H208492006 /H20850
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On: Sat, 20 Dec 2014 12:15:49 |
1.4946123.pdf | Structural and magnetic properties of ion beam sputtered Co 2FeAl full Heusler alloy
thin films
Sajid Husain , Ankit Kumar , Sujeet Chaudhary, , and Peter Svedlindh
Citation: AIP Conference Proceedings 1728 , 020072 (2016); doi: 10.1063/1.4946123
View online: http://dx.doi.org/10.1063/1.4946123
View Table of Contents: http://aip.scitation.org/toc/apc/1728/1
Published by the American Institute of Physics
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Co2FeAl Heusler thin films grown on Si and MgO substrates: Annealing temperature effect
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Perpendicular magnetization of full-Heusler alloy films induced by MgO interface
Applied Physics Letters 98, 242507 (2011); 10.1063/1.3600645Structural and Magnetic Properties of Ion Beam Sputtered
Co 2FeAl Full Heusler Alloy Thin Films
Sajid Husain 1, Ankit Kumar 2, Sujeet Chaudhary 1,a) and Peter Svedlindh 2
1Thin Film Laboratory (Spintronics), Indi an Institute of Technology Delhi, India
2Department of Engineering Sciences, Uppsal a University Sweden, 751 21 Uppsala, Sweden
aE-mail: sujeetc@physics.iitd.ac.in
Abstract. Co 2FeAl full Heusler alloy thin films grown at different temperatu res on Si(100) substrates using ion beam sputtering
system have been investigated. X-ray diffraction (XRD) patterns revealed the A2 disordered phase in these films. The deduced
lattice parameter slightly increases with increase in the growt h temperature. The saturation magnetization it is found to incr ease
with increase in growth temperature. The magnetic anisotropy ha s been studied using angle dependent magneto-optical Kerr effec t.
In the room temperature deposited film, the combination of cubi c and uniaxial anisotropy have been observed with weak in-plane
uniaxial anisotropy which increases with growth temperature. Th e uniaxial anisotropy is attributed to the anisotropic interfac ial
bonding in these Co 2FeAl /Si(100) heterostructures.
INTRODUCTION
The performance of spintronic devices critically depends on the spin polarization of the electronic current. Due to
this fact the highly spin polar ized materials are being widely searched for realizing the working for spintronic devices.
To fulfil the requirement of high spin polarization, a half met allic ferrimagnets (HMFs) are the typical need for this
purpose. These HMFs are defined by the characteristic density o f states profile at Fermi level ( EF) such that one of the
spin channel shows metallic nature and the other spin channel e xhibits either semiconducting (energy gap at EF) or
insulating characteristics , leading to 100% spin polarization 1. Recent theoretical reports show that the Co-based full
Heusler alloy behaves like half-metallic nature 2, 3.
At present Co 2FeAl (CFA) is emerging as the most promising full Heusler alloy . It has a very low Gilbert damping
constant, high Curie temperature (~1000K) and magnetic tunnel j unctions (MTJs) fabricated using CFA have
exhibited giant tunnel magnetor esistance (G TMR) ratio (360%) a t room temperature 4. However, the use of CFA as
a ferromagnetic (FM) electrode in devices needs precise knowled ge and control of its magnetic properties. In this
sense, one of the key parameters is the magnetic anisotropy, wh ich should be large for magnetic storage and low for
magnetic switching applications. Fundamentally, the band hybrid ization and the spin orbit interaction (SOI) at
surface/interfaces leads to the a nisotropic phenomena. In this respect, the substrate material, its orientation, choice of
substrate temperature 11 and the film thickness 12 all play important role in influencing the magnetic anisotropy of the
magnetic thin films. Therefore, modi fications of the electronic structure in bulk , surface, or interfaces are expected to
lead to important changes in the magnetic anisotropy 5.
In this work, we have grown CFA thin films on Si(100) substrate s by dual ion beam sputtering at different substrate
temperatures and carried out magneto-optical Kerr effect (MOKE) measurements on them. We obtained the substrate-
temperature dependent uniaxial anisotropy constant which would be very useful for designing the spintronics devices
for magnetic random access memory (MRAM) applications.
SAMPLR PREPARATION AND EXPERIMENTAL METHODS
The Co 2FeAl (CFA) thin films of 57nm thicknesses were deposited at dif ferent temperatures Ts (RT, 300, and 400 Ԩ)
on Si(100) substrate using NORDIKO-3450 ion-beam sputtering sys tem. All films were capped with Ta(2nm) to
protect them from the oxidation. Prior to the deposition, the S i(100) wafer was cleaned with acetone and propanol
using ultrasonic cleaning bath. The Si substrates were then loa ded for deposition by load-lock system after removing
the native oxide (SiO 2) layer. A turbo molecular pump was used to obtain an ultimate pressure of ~8 u10-7 Torr, and
working pressure kept during deposition of ~2.4 u10-4 by maintaining an Ar gas flow rate of 3.8sccm. The CFA alloy
International Conference on Condensed Matter and Applied Physics (ICC 2015)
AIP Conf. Proc. 1728, 020072-1–020072-4; doi: 10.1063/1.4946123
Published by AIP Publishing. 978-0-7354-1375-7/$30.00020072-1target of 6” dia. was sputtered at 45 ι angle which was fixed on a water cooled turret. The 100W rf po wer was used to
generate plasma in the 10 cm diameter rf-ion source from Nordik o. The hysteresis loops have been recorded using
home-made longitudinal magneto-optic Kerr effect (L-MOKE) set-u p and the saturation magnetization was measured
using vibrating sample magnetometer ( Quantum Design make Ev ercool-II PPMS model-6000 ). In-plane magnetic
anisotropy has been analyzed by recording the hysteresis loops at various azimuthal angles ( )) between applied field
H and the long axis of the rectangular size (4mm× 5mm) film (see Fig.1). The later coincided with the [110] of the
underlying Si(100) substrate. The structural characterization w as performed using glancing angle (1 ι) X-ray
diffraction (GAXRD) employing Philips X’pert PRO (Model PW 3040) diffractometer. The use of glancing angle
technique is crucial in obtaining acceptable signal to noise ra tios in these thin films studied. The thickness was
accurately estimated by X-ray reflectivity (XRR) measurements ( not shown here for brevity). It may be noted that no
magnetic field is applied during the deposition .
RESULTS AND DISCUSSIONS
Structural properties
Figure.2 shows glancing angle X- ray diffraction patterns for CFA thin films grown at Ts - RT, 300, and 400oC.
Analysis of the XRD patterns (Fig.2) illustrates that in additi on to the peak corresponding to the Si(100) substrate
(indicated by ‘*’), all the samples exhibit only the (220) peak of the CFA Heusler alloy. Theoretically, in terms of the
chemical order, the CFA crystal may be in a perfectly chemicall y ordered L2 1 phase, a B2 phase is characterized by
anti-site disorder between Fe and Al while Co atoms occupy regu lar sites, and the A2 phase, w hich is totally disordered
with respect to Fe, Al, and Co atoms . In our samples, the presence of the (220) CFA re flection indicates that the films
contain the A2 disordered phase. Calculated lattice parameters (a), shown in Fig. 2 (inset), significantly increases with
the increasing Ts, similarly to samples grown on MgO substrates, where a direct correlation exists with the
enhancement of the chemical order 6. However, the lattice parameters remain smaller than the repor ted one in the bulk
compound with the L2 1 structure (0.574 nm).
20 30 40 50 60 70 80 90 1000 100 200 300 400 500.5660.5680.5700.5720.574
Lattice constant (nm)
Ts (qC)Bulk ' a' value
(422) RT
300qC
400qCIntensity (a.u)
T degs. (220)
*
Figure. 1. Schematic of the directions of applied field, laser
incidence and the easy and hard axes of the CFA
film, respectively oriented along the [11 0] and
[ͳത10] directions of Si (100) substrate. The film is
rotated about [001] direction for obtaining MH
loops along different ). Figure. 2. GAXRD profiles of Si/Co 2FeAl/ Ta films grown at
different Ts. Inset shows the variation of calculated
lattice constants with Ts.
Magnetic Properties
The MOKE M-H loops were recorded on CFA film grown at room temperature at various azimuthal field orientations
(0 to 360º) between [110] and magnetic field. Figure. 3 (a-f) show the MO KE M-H loops recorded for different
azimuthal angles. (for conciseness only 0-180º loops are shown) In the hysteresis loop recorded for )=0º, an abrupt
magnetization switching, identifying an easy axis (EA) of magne tization, is clearly visible. The EA coincides with the
020072-2long axis of the rectangular film corresponding to [110] crystallograp hic direction of Si substrate shown (as shown in
Fig.1). It can be seen that the shape of magnetization reversal MOKE M-H loops changes with the magnetic field
orientation at each angle due to the magnetic anisotropy. From Figs. 3(a-f), it is seen that the film possess uniaxial
magnetic anisotropy. Fig.3 (e) further suggests the coexistence of cubic and uniaxial anisot ropy. Similar results were
reported in Fe/GaAs 7, Co/GaAs 8 a n d C o 2FeAl/GaAs 9 a n d C o 2FeAl/MgO10. The anisotropy in cubic crystal is
generated mainly due to the cubic symmetry of fcc crystal lattice. In Fig. 4(a) and (b) show MOKE M-H loops for
)=0 and 90º directions for CFA films sputtered at 300ºC and 400º C. One can clearly see that the competition between
cubic and uniaxial becomes weak at 300ºC and cubic anisotropy a lmost vanish at 400ºC. It is apparent that in-plane
uniaxial anisotropy exists at higher Ts range. We suggest that the observed anisotropy could originate due to interfacial
bonding between substrate and film. In addition, a significant lattice mismatch (5.3%) between Si and CFA films
could also lead to the strain induced in-plane anisotropy.
-30 -20 -10 0 10 20 30-1.0-0.50.00.51.0Norma lized Kerr Signal (a.u)
Magnetic Field (Oe)) 0o
(a) -30 -20 -10 0 10 20 30-1.0-0.50.00.51.0)= 20oNormalized Kerr Signal (a.u)
Magnetic Field (Oe)
(b) -30 -20 -10 0 10 20 30-1.0-0.50.00.51.0)= 40oNormalized Kerr Signal (a.u)
Magnetic Field (Oe)
(c)
-30 -20 -10 0 10 20 30-1.0-0.50.00.51.0)= 60oNormalized Kerr Signal (a.u)
Magnetic Field (Oe)
(d) -30 -20 -10 0 10 20 30-1.0-0.50.00.51.0)= 90oNormalized Kerr Signal (a.u)
Magnetic Field (Oe)
(e) -30 -20 -10 0 10 20 30-1.0-0.50.00.51.0)=180oNormalized Kerr Signal (a.u)
Magnetic Field (Oe)
(f)
Figure 3. (a-f) MOKE M-H loops recorded on sample deposited at RT at different azimuthal angles ) in the range of
0-180͑
The anisotropy parameter can be evaluated from the empirical re lation of t he “effective” magnetic anisotropy constant
(Keff) which is generally described in terms of volume ( KV) and interface/surface ( KS) contributions and can be written
as
ܭൌܭʹܭ௦Ȁݐ
Where t is the thickness of the FM film and factor 2 comes due to the presence of two adjacent interfaces shared by
FM film. It is to be noted that Ks is very sensitive to thickness of the FM layer, and it is gene rally prominent in lower
thickness regime (~few nm). In present case, thickness is highe r so that the surface anisotropic contribution is not
expected to be significant and maximum contribution comes from the volumetric term. Hence, the effective anisotropy
constant can be calculated using
ܭൌͳ
ʹܯௌൈܪ
020072-3Here, ܯௌ is the saturation magnetization and ܪ (difference between saturation fields Hs along the EA and HA) is the
anisotropy field. The abrupt switching of magnetization observe d in Figs. 3(a &f) and 4(a&b) suggests the coherent
rotation of magnetization along EA direction, i.e., [110]. Ther efore, we can directly calculate the effective anisotropy
constant using Hs (along HA) and Ms values for different Ts. Figure.4(c) shows the calcu lated parameters. It seen that
the Keff , Ms and Hs increases with Ts which may be due to improvement in the crystalline quality of CFA at higher Ts.
Here the coercivity is found to decrease with increase in Ts which may results from the lesser pinning of domain
boundaries owing to reduction of the grain boundaries expected at higher growth temperature. \
Figure 4. MOKE M-H loop recorded on sample deposited at (a) 300 and (b) 400oC respectively at different azimuthal
angles at 0 and 90ι (c) Variation of anisotropy energy Ku, saturation magnetization Ms, saturation field along
HA and coercivity Hc variation with Ts.
CONCLUSION
In summary, a combined presence of cubic and uniaxial anisotrop ies has been evidenced in CFA Heusler alloy films
grown by Ion-beam sputtering on Si substrates. Cubic anisotropi c component is found to vanish at high growth
temperatures. We calculated the anisotropy component using simp le empirical relation and it has been found that Keff
increases with growth temperature, which is the key material re quirement for use in the MRAM applications.
ACKNOWLEDGEMENT
SH thankfully acknowledges the Department of Science and Techno logy India for providing the INSPIRE fellowship.
REFERENCES
1. R.A. de Groot, F.M. Mueller, P.G. van Engen, K.H.J. Buschow, Ph ys. Rev. Lett. 50 2024 (1983).
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S.M. Chérif, P. Moch, Phys. Rev. B 87 (2013) 184431.
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10. M.S. Gabor, T. Petrisor Jr, C. Tiusan, M. Hehn, T. Petrisor, Phys. Rev. B 84 (2011) 134413.
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Appl. Phys. Lett. 115 (2014) 043918.
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(2014). -30 -20 -10 0 10 20 30-1.0-0.50.00.51.0
)=0R
)=90R
Magnetic Field (Oe)Normalized Kerr Signal (a.u)
(a) -30 -20 -10 0 10 20 30-1.0-0.50.00.51.0 )=0q
)=90qNormalized Kerr Signal (a.u)
Magnetic Field (Oe)
(b) 0 50 100 150 200 250 300 350 400 450755760765770775780785
0.40.60.81.01.21.41.61.82.0
1520253035404550
7891011121314Ms(emu/cc)Ku(u104 erg/Oe)
Hs (Oe)
Hc(Oe)
Ts (in degree Celcius ) Ms Ku
Hs Hc
(c)
020072-4 |
1.3554208.pdf | Theory of dipole-exchange spin waves in metallic ferromagnetic nanotubes
of large aspect ratio
Tushar K. Das and Michael G. Cottam
Citation: J. Appl. Phys. 109, 07D323 (2011); doi: 10.1063/1.3554208
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Downloaded 16 Apr 2013 to 128.143.22.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsTheory of dipole-exchange spin waves in metallic ferromagnetic nanotubes
of large aspect ratio
Tushar K. Dasa)and Michael G. Cottam
Department of Physics and Astronomy, University of Western Ontario, London, Ontario N6A 3K7 Canada
(Presented 15 November 2010; received 14 September 2010; accepted 21 November 2010;
published online 28 March 2011)
A macroscopic continuum theory is presented for the dipole-exchange spin waves in nanometer-
sized cylindrical tubes with a large length-to-diameter aspect ratio. The magnetization and applied
magnetic field are taken parallel to the cylinder axis, and the properties of the hybridized surface
and bulk magnetic excitations are studied in relation to the inner and outer interfaces of thenanotubes. The calculations describe the radial and angular quantization of the different modes for
both unpinned and effective pinned cases. The results for tube geometries are found to be in
contrast with the limiting (single-interface) special cases of wires and antiwires. Numericalexamples are presented mainly for Ni materials.
VC2011 American Institute of Physics .
[doi:10.1063/1.3554208 ]
Ferromagnetic nanowires and nanotubes, as well as
arrays of these structures, have attracted much attention for
their spin dynamics. The spin-wave properties are important,for example, in the development of magnonic analogs to
photonic crystals
1and in various device applications,2,3as
well as being of fundamental interest. In particular, Brillouinlight scattering (BLS) has proved to be a useful technique for
probing spin waves (SW) in these low-dimensional struc-
tures.
1,3–5Most studies have been applied to nanowires hav-
ing a rectangular cross section (i.e., a ferromagnetic stripe
geometry), as in Refs. 3and4, and there has been relatively
less attention given to long wires (and tubes) with a cylindri-cal geometry. Some exceptions are the BLS studies of quan-
tized SW in wires
5and tubes.6On the theoretical side, for
cylindrical geometries, SW calculations in nanowires for themagnetostatic limit (at small wave numbers where the dy-
namical effects of exchange are negligible compared to
dipole–dipole interactions) were made by Sharon and Mara-dudin
7and later generalized by us to tubes8and multi-inter-
face structures.9Macroscopic dipole-exchange calculations
for cylindrical wires have been reported10and then used by
the authors of Ref. 5to interpret their BLS data. The motiva-
tion for our present work is to use the macroscopic (or con-
tinuum) method to study the dipole-exchange SW in tubes,where additional quantization effects arise due to the two
interfaces.
Following Ref. 8, we consider a ferromagnetic nanotube
modeled as a long circular hollow cylinder of inner and outer
radii, R
1andR2, respectively, with its symmetry axis parallel
to the zaxis, which is also the direction of the applied mag-
netic field B0and the static saturation magnetization M0.A
large length-to-diameter aspect ratio is assumed allowing
end-effects to be ignored. In cylindrical polar coordinates(r,h,z), a nonmagnetic medium fills the regions r<R
1and
r>R2, while a magnetic medium is in the region
R1<r<R2. Two limiting cases of our geometry correspondto the wire (taking R1!0,R2=0) and the antiwire (taking
R1=0,R2!1 ). To study the SW dynamics with both
dipolar and exchange effects included, we start from theLandau–Lifschitz torque equation with a phenomenological
Gilbert damping term:
11
dM
dt¼/C0cM/C2Beffþa
M0M/C2dM
dt: (1)
Here the total magnetization is M¼M0^zþmðr;h;zÞ
expð/C0ixtÞ, with xdenoting the angular frequency of the SW
andm/C28M0in the linear SW regime. The gyromagnetic ratio
isc, and the total effective magnetic field is
Beff¼B0^zþfbdðr;h;zÞþbexðr;h;zÞgexpð/C0ixtÞ:(2)
By analogy with Ref. 10, the dipolar field bdis deduced from
Maxwell’s equations, and the exchange field bexcan be
expressed in the form /C0kM/C0Dr2Mwhere kis an
exchange factor and Dis the exchange stiffness constant.
Finally, we remark that the inclusion here of a damping
term, proportional to the dimensionless constant ain Eq. (1),
is important in the tube geometry (with its two interfaces) inorder to get a more realistic description of the coupled SW
modes and to describe the wire and antiwire limiting cases.
In brief and by analogy with dipole-exchange SW theo-
ries
10for other geometric samples, the calculation proceeds
by substituting the total MandBeffterms into Eq. (1)and
linearizing the resulting expression in terms of componentsofm. There are two types of boundary conditions to be
applied at each of the interfaces r¼R
1andr¼R2, specifi-
cally the usual electromagnetic boundary condition8and the
so-called “exchange” boundary condition. The latter allows
for the inclusion or absence of an effective pinning or it may
take a mixed form.12The expression for minside the mag-
netic material has the form of fn(r) exp( inh) exp( ikz), where
the integer nis the azimuthal quantum number, kis the wave
number along the zaxis of translational symmetry, and the
radial function fhas the form of a linear combination ofa)Electronic mail: tdas@uwo.ca.
0021-8979/2011/109(7)/07D323/3/$30.00 VC2011 American Institute of Physics 109, 07D323-1JOURNAL OF APPLIED PHYSICS 109, 07D323 (2011)
Downloaded 16 Apr 2013 to 128.143.22.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsBessel functions InðjirÞandKnðjirÞ. Here the ji(with i¼1,
2, 3) are found from the roots of a sixth order indicial equa-tion, which depends on x,k,B
0,M0, and the damping a.
They play the role of wave numbers in the radial direction,
and so there is an admixture of three terms with weighting
factors that depend on the boundary conditions. In general,
thejiare complex and thus contain information about the
spatial localization of the SW modes. When the damping is
small, some of the jiare approximately real for the localized
surface (or interface) modes and approximately pure imagi-nary for the radial bulk modes, but they become complex in
the regions of hybridization (mode mixing). Taking account
of the dipolar fields in the core ( r<R
1) and external ( r>R2)
regions and the two sets of boundary conditions, we deduce
that the vanishing of an 8 /C28 determinant for the amplitude
coefficients yields the implicit dispersion relation of the SWmodes. There is a reduction to 4 /C24 determinant conditions
in the limiting cases of the wire and antiwire geometries.
Our results generalize the previous wire calculations
10to
include damping.
Next, we illustrate the analytic theory by some specific
numerical calculations of dispersion relations for the hybri-dized dipole-exchange SW modes. For the main applica-
tions, we consider Ni nanotubes since this magnetic material
was employed in BLS experiments,
5,6and the relevant pa-
rameters are well known: we take l0M0¼0.603 T, D¼3.13
T/C1nm2, and c¼30.9 GHz/T. To allow comparison with a
case where the exchange effects are less pronounced, wealso consider EuS nanotubes, taking l
0M0¼1.53 T,
D¼0.20 T /C1nm2, and c¼28.0 GHz/T. In all the examples
shown here, we assume that the damping is small, corre-sponding to a¼0.001, and that the effective pinning is small
at both interfaces.
In Fig. 1, we show numerical results to compare the SW
frequencies versus longitudinal wave number kfor the
single-interface limits of a Ni wire surrounded by a nonmag-netic material and its corresponding antiwire (a nonmagnetic
wire surrounded by Ni). A comparison with the surface mag-
netostatic modes (which occur between upper and lowerbounds
8indicated by the horizontal lines) is also included,
so that the effects of exchange on the dispersion relations for
the two structures are seen. The dispersion curves in themagnetostatic limit are known
7,8to have negative slope due
to the dipole–dipole interactions. By contrast, the analogous
dipole-exchange SWs are shifted upward and eventuallyhave a positive slope due to the exchange. This effect is seen
to be more pronounced in the wire geometry compared to the
antiwire case as a consequence of the higher mode localiza-tion in the former case. In addition, there are exchange-type
bulk modes (with positive slope) in other frequency regions.
In Figs. 2and3, we show nanotube calculations for a
fixed geometry with R
2/R1¼2 taking EuS and Ni as the fer-
romagnetic material, respectively. For a tube, there are gen-
erally two surface-mode branches in the magnetostatic limit(see the full curves), with one mode being mainly associated
with the inner interface and the other mainly with the outer
interface. The exchange effects are relatively weaker for EuS(see Fig. 2), and we see in the dipole-exchange case that
there are several “exchange-dominated” radially quantized
modes that become hybridized with the modified surfacemagnetic modes. For the case of Ni (see Fig. 3), the stronger
exchange means that the radially quantized exchange modes
are more separated by frequency. The results shown here forn¼1 are qualitatively similar to those for larger n, but the
SW frequencies are generally higher.
In conclusion, we have developed a general theory for
the dipole-exchange SW in nanotubes, as well as in the limit-
ing cases of wires and antiwires, and we have shown some
numerical examples of dispersion relations for Ni and EuS atwave numbers typical of BLS experiments. It is shown that
the magnetostatic results are considerably modified by the
role of exchange. The numerical examples were for the caseof weak pinning, which is consistent with the analysis given
FIG. 1. Frequencies of the dipole-exchange SW with azimuthal quantum
number | n|¼1 vs longitudinal wave number kfor the two different single-
interface cases of a wire (black circles) and an antiwire (open circles). For
comparison the dispersion curves for the surface magnetostatic modes are
shown by the solid and dashed lines for the wire and antiwire, respectively.
The applied field is B0¼0.1 T, and the magnetic/nonmagnetic interface is at
radius 15 nm.
FIG. 2. The hybridized SW frequencies vs wave number kfor a nanotube
with R1¼15 nm and R2¼30 nm in the case of relatively weak exchange,
taking parameters appropriate to EuS. The dipole-exchange frequencies cor-
respond to the open circles and, for comparison, the surface magnetostatic
modes are represented by the solid lines. The applied field is B0¼0.3 T, and
only the modes for | n|¼1 are shown.07D323-2 T. K. Das and M. G. Cottam J. Appl. Phys. 109, 07D323 (2011)
Downloaded 16 Apr 2013 to 128.143.22.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsfor the BLS data5of Ni nanowires. We have also studied the
effects of large and intermediate pinning, finding that the
mode frequencies are typically shifted downward (in somecases by several gigahertz) due to large pinning, thereby
affecting the hybridization schemes. Numerical calculations
have also been carried for larger damping by takinga¼0.05. It is found that this produces smaller changes than
those due to varying the pinning conditions, but the fre-
quency shifts may still be appreciable. The previous BLSmeasurements emphasized the size dependence (in the case
of Ni wires
5) and the field dependence (in the case of Ni
tubes6). The macroscopic theory is broadly consistent with
these findings, as already noted in both of the cited papers,
but a more complete comparison with our theory would be
possible if further BLS experiments studied the effect ofvarying the longitudinal wave number kby varying the scat-tering geometry. We note that the effects of core removal as
identified here in long tubes is quite distinct from the effect
in flat disks,
13where there is vortex formation as a conse-
quence of the different (mainly in-plane) magnetization ori-
entation. Magnetostatic mode calculations in tubes with an
elliptical (rather than circular) cross section have recentlybeen reported,
14and it would be of interest to generalize
these to include explicitly the exchange effects, and the con-
sequent mode mixing, by following the approach used here.
We gratefully acknowledge partial support of this work
by the Natural Sciences and Engineering Research Council(NSERC) of Canada.
1Z. K. Wang, V. L. Zhang, H. S. Lim, S. C. Ng, M. H. Kuok, S. Jain, and
A. O. Adeyeye, Appl. Phys. Lett. 94, 083112 (2009).
2M. L. Plumer, J. van Ek, and D. Weller, The Physics of Ultra-High-Den-
sity Magnetic Recording (Springer, Berlin, 2001).
3B. Hillebrands and K. Ounadjela, Spin Dynamics in Confined Magnetic
Structures (Springer, Berlin, 2001).
4G. Gubbiotti, S. Tacchi, G. Carlotti, P. Vavassori, N. Singh, S. Goolaup,
A. O. Adeyeye, A. Stashkevich, and M. Kostylev, Phys. Rev. B 72,
224413 (2005).
5Z. K. Wang, M. H. Kuok, S. C. Ng, L. L. Tay, D. J. Lockwood, M. G. Cot-tam, K. Nielsch, R. B. Wehrspohn, and U. Gosele, Phys. Rev. Lett. 89,
027201 (2002).
6Z. K. Wang, H. S. Lim, H. Y. Liu, S. C. Ng, M. H. Kuok, L. L. Tay, D. J.
Lockwood, M. G. Cottam, K. L. Hobbs, P. R. Larson, J. C. Keay, G. D.
Lian, and M. B. Johnson, Phys. Rev. Lett. 94, 137208 (2005).
7T. M. Sharon and A. A. Maradudin, J. Phys. Chem. Solids. 38, 977 (1977).
8T. K. Das and M. G. Cottam, Surf. Rev. Lett. 14, 471 (2007).
9T. K. Das and M. G. Cottam, IEEE Trans. Magn. 46, 1544 (2010).
10R. Arias and D. L. Mills, Phys. Rev. B 63, 034439 (2001).
11D. D. Stancil and A. Prabhakar, Spin Waves: Theory and Applications
(Springer, New York, 2009).
12K. Yu. Guslienko and A. N. Slavin, Phys. Rev. B 72, 014463 (2005).
13F. Hoffmann, G. Woltersdorf, K. Perzlm aier, A. N. Slavin, V. S. Tiberkevich,
A .B i s c h o f ,D .W e i s s ,a n dC .H .B a c k , P h y s .R e v .B 76, 014416 (2007).
14M. A. Popov and I. V. Zavislyak, Ukr. J. Phys. 53, 702 (2008) .
FIG. 3. The same as in Fig. 2, but showing now the case of relatively stron-
ger exchange effects, taking parameters appropriate to Ni.07D323-3 T. K. Das and M. G. Cottam J. Appl. Phys. 109, 07D323 (2011)
Downloaded 16 Apr 2013 to 128.143.22.132. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissions |
1.3048466.pdf | The two Ernests—I
Mark L. Oliphant
Citation: Physics Today 19, 9, 35 (1966); doi: 10.1063/1.3048466
View online: http://dx.doi.org/10.1063/1.3048466
View Table of Contents: http://physicstoday.scitation.org/toc/pto/19/9
Published by the American Institute of PhysicsThe Two Ernests—I
Some personal recollections of Ernest Rutherford and Ernest
Lawrence in the period 1927-1939. Rutherford, who dominated
the Cavendish Laboratory, gave his physicists a minimum of
equipment hut a maximum of personal interest in their re-
search. Lawrence developed the Radiation Laboratory into a
prototype facility for research with large, expensive equipment.
Both inspired others to produce and interpret nuclear reactions.
by Mark L. Oliphant
ON 11 JANUARY 1939 after a visit to
Berkeley, I wrote a letter to Ernest
Lawrence that contained the following
paragraph:
"I find it very difficult to thank
you for the magnificent and in-
structive time which I had in Berke-
ley. It was truly fine of you to be
so liberal of time and of thought on
my behalf. I know of no laboratory
in the world at the present time
which has so fine a spirit or so
grand a tradition of hard work.
While there I seemed to feel again
the spirit of the old Cavendish, and
to find in you those qualities of a
combined camaraderie and leader-
ship which endeared Rutherford to
all who worked with him. The es-
sence of the Cavendish is now in
Berkeley. I am sincere in this, and
for these reasons I shall return again
some day, and I hope very soon."
^'ow, in 1965, after many subse-
quent visits to the Radiation Labora-
tory, which Lawrence created and
which » now named after him, I re-
gain intrigued by both the many
similarities, and the differences, be-
Uveen Rutherford and Lawrence.
John Cockcroft and Ernest Walton
tot observed nuclear transformations
produced by artificially accelerated par-
tlcIes, and James Chadwick discovered
^neutron, in the Cavendish Labora-tory, Cambridge, in 1932. Lawrence
conceived the cyclotron principle in
1929, in the University of California,
Berkeley. By 1932, with his colleagues
Niels Edlefsen and M. Stanley Liv-
ingston, he h:id made the cyclotron a
successful instrument with which he
was able to confirm the results of
Cockcroft and Walton, and carry
them to much higher bombarding ener-
gies. The period between these great
discoveries and that of the fission proc-
ess by Otto Hahn and Fritz Strassmann
in 1938, was of the greatest importance
in the development of modern phys-
ics. In this article, I endeavor to set
down some recollections of that pe-
riod and of two individuals who gave
it such momentum that it changed
the whole course of physics and led,
inexorably, to the development of nu-
clear weapons and nuclear energy. No
pretense is made that this account is
complete, or that the facts presented
are in accordance with the recollections
of others who lived through those stir-
ring days. The study of the effects pro-
duced in the atomic nucleus by bom-
barding it with nuclear projectiles had
transformed knowledge of matter and
its properties. The parts played by
Rutherford and Lawrence, directly
and indirectly, will remain outstanding
contributions to that work.
Ernest Rutherford and Ernest Law-rence, in two succeeding generations,
built around them great schools of in-
vestigation that laid the foundations
of physics as it is practiced today.
These two men, so much alike, and
yet so strangely different, were parts
of totally different worlds. Together,
their lives spanned the period of the
greatest revolution in knowledge of
the physical universe since Newton's
time. Each was a pioneer, and each
was the descendant of pioneering
parents who chose to build a new
life in a land far removed from the
home of their ancestors. It is revealing:
to review the early life of each.
Rutherford, early years
Rutherford's grandfather, George
Rutherford, migrated from Scotland to
New Zealand in 1842. His son James,
then three years of age. grew up in
Sir Mark Oliphant,
K. B. E., F. R. S.,
worked with Ernest
Rutherford in the
Cavendish until
1937, when he went
to the Univ. of
Birmingham. In 1950
he became professor
of particle physics
and director of the
Research School of Physical Sciences at
the Australian National University.
PHYSICS TODAY • SEPTEMBER 1966 • 35HOUSE, in South Island, New Zealand
where Rutherford lived as a child.
the colony and followed his father's
trade as a wheelwright. James met
and married a widow, Caroline
Thompson, who had left England for
New Zealand with her parents, in
1855. They settled near Nelson, in
the South Island, where James Ruther-
ford had a small farm and worked as
a contractor building the railways.
Ernest Rutherford was born on 30
Aug. 1871, the second son in a large
family of twelve children. When Er-
nest was eleven years of age, the fam-
ily moved a short distance to Have-
lock, where his father established a
mill to treat the native flax of the
area, and a small sawmill. At the
primary school there, Ernest was in-
fluenced by his teacher, J. H. Rey-
nolds, who taught so well that Ernest
won a scholarship to Nelson College,
with almost full marks in the examina-
tion. He entered the College at 15
years of age, and was much helped
by one of the masters, W. S. Little-
john, a classicist who taught also
mathematics and science. Ernest had a
broad education, excelling in mathe-
matics, but winning distinctions in
Latin, French, English literature, his-
tory, physics and chemistry, and be-
coming head of the school. He was a
scholar of distinction, but played
games reasonably well and entered ful-
ly into the life of the school. A. S. Eve,
in his biography of Rutherford, quotes
a fellow student as saying, "Ruther-
ford was a boyish, frank, simple and
very likable youth, with no pre-
cocious genius, but when once he
saw his goal, he went straight to the
central point." He took photographs
with a home-made camera, dismem-
bered clocks, made model water wheels
such as his father used to obtain pow-
er for his mills. Under the influenceof his mother and his fine teachers,
Rutherford developed a wide taste for
literature and read avidly all his life.
He became especially interested in bi-
ographies.
In 1889 he won a scholarship to
Canterbury College, Christchurch, a
component college of the University of
New Zealand. There, as one of 150
pupils in the small institution, he en-
joyed five very full years, obtaining
successively his B.A. and M.A. de-
grees, the first in Latin, English,
French, mathematics, mechanics and
physical science, and the second, at
the end of his fourth year, with a
double first in mathematics and physi-
cal science. During his fifth year,
Rutherford concentrated on his sci-
ence, carrying out many experiments
on the electromagnetic waves discov-
ered by Heinrich Hertz, and investi-
gating the effects of the damped oscil-
lations of the Hertzian oscillator upon
the magnetization of steel needles and
iron wires. He showed that the mag-
netization was confined to a thin,
outer layer of the metal, by dissolving
away the surface in acid.
Rutherford was able to use these
magnetic effects to detect the wireless
waves from his oscillator, and demon-
strated that these waves travelled for
considerable distances, passing through
walls on the way. He reproduced
Nikola Tesla's experiments on the
high voltages that could be produced
with a resonant transformer, and de-
veloped techniques for measuring in-
tervals of time as small as 10 microsec.
He spoke to meetings of the Science
Society on his work and on the evo-
lution of the chemical elements, and
he published two papers in the Trans-
actions of the New Zealand Institute.
He found it necessary to supplement
his scholarship by coaching students,
and went to live with a widow, Mrs.
de Renzy Newton, whose daughter
Mary he later married.
In 1895 Rutherford applied for an
1851 Scholarship, which was awarded
to a New Zealand student in alternate
years. The examiners of the 1851 Royal
Commission, in London, awarded this
to a chemist, J. C. Maclaurin, but
were impressed enough by Rutherford
to urge the award of a second scholar-
ship, which was not given. However,
Maclaurin gave up the scholarship toaccept an appointment in the civil
service; so Rutherford was offered the
award. He elected to go to the Caven-
dish Laboratory, in Cambridge, to
work under J. J. Thomson, and had
to borrow the money to pay for his
passage to England. He and John S.
Townsend, of gas-discharge fame, ar-
rived at the Cavendish Laboratory al-
most simultaneously, to become the
first of the new category of research
student recently established in the
University of Cambridge. There he
joined Trinity College and began
fresh experiments on the detection
of electromagnetic waves by use of the
effects of high-frequency currents upon
the magnetization of iron wires. He
soon established himself as a research
worker of great promise, of whom
Andrew Balfour wrote, "We've got
a rabbit here from the antipodes and
he's burrowing mighty deep." Ruther-
ford was ambitious and anxious to
qualify for a post that would enable
him to marry Mary Newton. He
thought that the detector using very
fine magnetized steel wires surrounded
by a solenoid in which high-frequency
currents reduced the magnetization
might make his fortune. Before Gu-
glielmo Marconi, he was able to de-
tect radio waves at a distance of half
a mile.
Rutherford developed early an ex-
traordinary ability to recognize, and
concentrate upon, the puzzling prob-
lems of frontier knowledge in phys-
cis. He was never content to follow
pedestrian paths of measurement or
rounding off of investigations initiated
by others. George P. Thomson, in his
Rutherford Memorial Lecture, pointed
out that Rutherford was working in
the Cavendish Laboratory when two
completely new physical phenomena
were discovered. These were the dis-
coveries of x rays, by Wilhelm Roent-
gen, and of radioactivity, by Henri
Becquerel and each opened up hither-
to unsuspected areas of investigation
destined to change the course of phys-
ics. It is not surprising, therefore, that
when J. J. Thomson invited Ruther-
ford to join him in the investigation
of the ionization produced in gases by
x rays, Rutherford seized the oppor-
tunity to move into more exciting
fundamental studies.
Rutherford showed that the ioniz-
36 SEPTEMBER 1966 ° PHYSICS TODAYing effect of x rays was due to the
production of positive and negative
ions in equal numbers and devised
ingenious methods for measuring the
velocity of drift of these ions in an
electric field. Then in 1898 he in-
vestigated the ions produced when
ultraviolet light fell on a metal plate,
showing that they were all negative
ions and that their properties were
identical with the ions produced in the
gas by x rays. Upon hearing that
the radiations discovered by Bec-
querel to be spontaneously emitted
by uranium and thorium were able
to ionize gases, Rutherford made ob-
servations of the properties of the
ions produced, and found them identi-
cal with those that he had investi-
gated previously. He showed that two
kinds of radiation were present, an
easily absorbable and strongly ionizing
component which he called "alpha
rays," and a much more penetrating
radiation to which he gave the name
"beta rays." He had found the field
of physics in which he was to spend
his life.
In August 1898 Rutherford was ap-
pointed to a professorship of physics
at McGill University. He had applied
for the post reluctantly, after assessing
his prospects in Cambridge, mostly be-
cause of his desire to get married,
but, having made the decision he ac-
cepted enthusiastically. Upon arrival
in Montreal he rapidly established
himself, and was soon at work on the
further studies of radioactivity that
were to establish him as the greatest
experimental physicist of his day. In
the summer of 1900, he went to New
Zealand to collect his bride, returning
to McGill in the autumn. In 1901 their
only child, a daughter, was born.
Rutherford's subsequent work in
Montreal, Manchester, and Cambridge,
K part of the history of science, in
every textbook.
Lawrence, early years
Lawrence's grandfather, Ole Lawrence,
left his home in Norway to settle in
Madison, Wisconsin, in 1840. There
he became a school teacher in a primi-
tlve, pioneering community. He sent
his son, Carl, to the University of
Wisconsin, from which he graduated
in 1894. Carl followed his father's pro-
fession as a teacher and showed thathe inherited the pioneering spirit, for
he moved farther west to South Dako-
ta as a Latin and history master.
He became superintendent of public
schools in the small community of Can-
ton, and while there, married Gunda
Jacobsen, the good-looking daughter
of Norwegian immigrants, in 1900. Er-
nest Lawrence was born to them on
8 Aug. 1901.
Ernest's parents were good people,
in the old-fashioned sense of these
words. Although his father had a de-
gree in arts, and had taught the hu-
manities, he was not a scholar. The
mother, a teacher of mathematics be-
fore her marriage, became an excellent
wrife and mother. She was a strict
Lutheran, mingling high principles and
loving care in the upbringing of her
two sons, Ernest and John. From his
parents Ernest acquired a strict moral
code and a belief in the inherent
decency of most human beings. Carl's
ability as an administrator, combined
with his integrity, led to his becoming
in turn head of the Southern State
Teachers' College in Springfield, and
then of Northern State Teachers' Col-
lege in Aberdeen, South Dakota. So,
the family enjoyed modest means, but
not sufficient to enable the boys to
indulge in extravagances without earn-
ing money for themselves.
Ernest grew to be a tall, gangling
youth. Unlike Rutherford, he did not
enjoy the rough and tumble of team
games like football but enjoyed ten-
nis, which he played well, if not bril-
liantly, throughout his life. His career
at high school was not outstanding,
and though he showed promise in sci-
ence, he performed indifferently in
English. He read very little, and in
later life was sarcastic about and im-
patient of his humanist colleagues, see-
ing little practical good in their work.
He was never a cultured man and
had few of the social graces so that he
made few friends among girls and did
not shine in extracurricular activities
of the school. However, he was by no
means antisocial, these traits arising
from indifference towards any activity
that did not fire his interest. He was
ambitious and worked hard and con-
sistently, so that he graduated from
high school at 16 years of age after
three, instead of the usual four years.
During the Ions* summer vacations,RUTHERFORD AT 21, while a student
at Canterbury College, University of
New Zealand. Photo from A. S. Eve,
Rutherford, Cambridge University Press.
LECTURING AT McGILL University,
1907, after Rutherford left Cambridge.
PHYSICS TODAY SEPTEMBER 1966 37JESSE BEAMS shares a laboratory with Lawrence at Yale University, 1927, where
they developed a technique to observe the lifetimes of excited atomic states.
Lawrence worked on farms in the dis-
trict, as a salesman for aluminum
ware and in other ways earned the
money required to buy the necessities
of an American boy with a mechani-
cal turn of mind—motor cars of various
vintages, radio receiving equipment,
tools and electrical gadgets, and so on.
No doubt under the influence of the
concern for others of his parents,
he decided upon a career in medicine,
and he was sent to a small private
college, St. Olaf's in Minnesota, to
begin his preliminary studies. He was
too young and unsettled to do well
there. After a year he moved to the
University of South Dakota. He soon
applied to the dean, Lewis E. Akely,
for permission to build and operate a
radio transmitting equipment. Akely
was much impressed with the knowl-
edge and ambition of the youth, and
persuaded him to turn to physics,
providing him with individual tuition
in the subject in order to give him a
start. After graduation in chemistry-
he had not abandoned his ambition
to do medicine—Lawrence was persuad-
ed by his close friend, Merle Tuve,
and by the offer of a fellowship, to
move to Minneapolis. There he
worked with W. F. G. Swann, an Eng-
lish immigrant who had been working
in geophysics in Washington, but who
had joined the University of Minne-
sota in order to work in more basic
physics. Leonard Loeb recalls that
Swann was not popular with his col-
leagues but that he got on extremely
well with young graduate students,inspiring them to do research of qual-
ity and encouraging them with help
and discussion. Under his influence,
Lawrence abandoned his desire for a
medical career. Swann introduced him
to the exciting field of experiment
arising from development of the quan-
tum theory. His early interest in elec-
tromagnetism was stimulated and de-
veloped. He took his master's degree
early in 1923, and later that year
moved with Swann to Chicago.
In Chicago Lawrence found himself
in a very different environment where
research was vigorously pursued by
an outstanding group of physicists.
He was stimulated greatly by contact
with Arthur Compton, at the time
completing his work on the Compton
effect. But he found himself also in a
department run on strictly European
lines, where the professor was all-
powerful and status determined the re-
lationships among members of the lab-
oratory. Neither Swann nor Lawrence
was at ease in this atmosphere, and
when Swann accepted a post at Yale,
a year later, Lawrence went with him.
In Chicago Lawrence had learned the
real meaning of research, and he threw
himself into it with complete devo-
tion. But it was at Yale that his gifts
as an experimenter, aided by his ener-
gy and enthusiasm, really flowered.
For his PhD he worked on the pho-
toelectric effect in potassium vapor,
carrying out beautiful experiments
that demonstrated clearly that he was
a physicist of high quality. Under a
National Research Council Scholar-ship, and after appointment to an as-
sistant professorship, Lawrence con-
tinued with his researches. He made
precise observations of the ionization
potential of mercury vapor, of im-
portance in the determination of the
value of Planck's constant h and de-
vised an elegant method of measuring
the ratio of charge to mass of the
electron. With Jesse Beams, who be-
came his firm friend, he developed a
beautiful technique for measuring very
short time intervals, which was ap-
plied to observations of the lifetimes
of excited states of atoms.
In 1928 Lawrence was offered an
associate professorship at the Univer-
sity of California, in Berkeley, having
turned down an earlier offer of an
assistant professorship. A lengthy cor-
respondence with Elmer Hall, the
chairman of the physics department,
and with Raymond Birge, who had
called on Lawrence in Yale and was
much attracted by him, has been faith-
fully recorded by Birge in the history
of the department that he is writing.
It seems that Lawrence was attracted
to California by the opportunity to
teach an advanced course and to di-
rect the work of research students, ac-
tivities reserved in Yale for more senior
members of staff. Birge pointed out
the good opportunities for rapid ad-
vancement of a good man in Berkeley,
contrasting this with the policies at
Yale, Harvard and Princeton, where it
was almost impossible to "get any-
where, after one was there, except
under very special circumstances. . . ."
Lawrence wrote to Birge saying that
some men in Yale were very "sore"
that he should even consider a posi-
tion in California to be comparable
with one in Yale. "The Yale ego is
really amusing. The idea is too pre-
valent that Yale brings honor to a
man and that a man cannot bring
honor to Yale."
Lawrence accepted the offer from
Berkeley, and arrived there in August
1928. He set to work at once to con-
tinue his work on the photoionization
of cesium vapor, used the techniques
which he had developed with Beams
for the measurement of short time in-
tervals in observations of the early
stages of the spark discharge, and one
of his research students, Frank Dun-
nington, developed his method for
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PHYSICS TODAY • SEPTEMBER 1966 • 39measuring the charge-to-mass ratio of
the electron. He was not committed
to this type of investigation, however.
He felt that the current challenge in
physics was the investigation of the
atomic nucleus, rather than of the
atom as a whole. He was impressed
by the limitations of the methods of
investigation developed by Rutherford,
who bombarded nuclei with alpha par-
ticles emitted by naturally occurring
radioactive substances. Like Cockroft,
he appreciated Rutherford's desire to
be provided with much more intense
beams of even more energetic particles
with which to probe the internal struc-
ture of nuclei.
Lawrence has recorded how, early
in 1929, he read a paper by Rolf
Wideroe on the use of high-frequency
voltages for accelerating charged par-
ticles. He recognized that it should be
possible to use a magnetic field to curl
the paths of such particles into a spiral,
and that because the Larmor time-
of-revolution in the field was inde-
pendent of the energy, they could re-
main in resonance with the voltage
across an accelerating gap. Robert
Brode has told me of a visit to him
by Lawrence the day after seeing the
article, enquiring whether the mean
free paths of ions could be made long
enough for them to suffer negligible
scattering by residual gas in their very
long spiral paths. Lawrence's colleagues
agreed that his calculations were cor-
rect, but they were dubious whether
the method could be applied in prac-
tice.
In 1930, Edlefsen, who had com-
pleted his PhD thesis, constructed
crude models of the system and ob-
served some resonance effects. Living-
ston joined LawTence, after Edlefsen
left that summer, and built an im-
proved model that showed resonances
corresponding with the rotation times
of molecular and atomic ions of hy-
drogen. By Christmas 1930, a 6-in mod-
el surprisingly like a modern cyclotron,
was in operation, producing hydrogen
ions with energies of 80 000 eV.
The "magnetic-resonance accelera-
tor," as the cyclotron was first named,
had become a reality. Lawrence had
found his life's work.
In 1932 Lawrence married Molly
Blumer, daughter of a distinguished
medical man, whom he had met whileat Yale and whom he had courted
for some years. They had six children,
two boys and four girls. He was happy
with his family, and the children en-
riched the life of both. Lawrence ap-
pears to have been a normal scientist-
father, much preoccupied with his
work, alternatively indulgent and too
strict, with his serene and capable wife
holding the balance and creating the
home.
The two compared
The similarity between the early ca-
reers of the two men is apparent. The
earliest interest of each was in radio.
However, while Rutherford abandoned
that field completely when he turned
to the study of radioactivity, the radio-
frequency problems of the cyclotron
kept alive the interest of Lawrence.
With David Sloan and Livingston he
built his own oscillators, and after the
war he developed a picture tube for
color television that is now manu-
factured by the Japanese firm, Sony.
Each moved from radio into atomic
physics, and then to the study of the
atomic nucleus. Each wras single-mind-
ed, working indefatigably towards a
goal once it was chosen. Each showed
tremendous enthusiasm, which he was
able to convey to others.
In his early work, Lawrence showed
an insight into physics very like that
of Rutherford. Whereas Rutherford
continued throughout his life to ex-
plore in the frontiers of knowledge,
however, Lawrence chose to contrib-
ute to physics less directly. After the
discovery and successful development
of the cyclotron, Lawrence's flair for
organization and his business ability
enabled him to build the first of the
very large laboratories in which mas-
sive and expensive equipment was de-
signed, built and used by the able
teams of men he attracted to work
with him for investigations into basic
problems in physics in which he played
little part, personally. This pattern of
research has become the modern ap-
proach all over the world. Rutherford,
on the other hand disliked large and
expensive equipment. He preferred to
remain involved, personally, in almost
all the work going on in his laboratory.
His interest and ability in administra-
tion and finance were rudimentary. He
dominated the laboratory by his sheergreatness as a physicist and provided
for his colleagues and students only
the very minimum of equipment re-
quired for an investigation. Ruther-
ford, with his roots in the soil and the
hard, practical life of New Zealand,
bucolic in appearance, became the deep
thinker and the originator of new
physical concepts. Lawrence, brought
up in an academic atmosphere, im-
pressive and scholarly in appearance,
became the originator of new tech-
niques and of the large-scale engi-
neering and team-work approach to
discovery.
Both men were extroverts and good
"mixers" in company. Donald Cook-
sey recalls that when Lawrence entered
a room filled with great industrialists
or successful politicians, his presence
was at once noticed, and his impact
upon them was profound. Rutherford,
however, could be taken for a farmer
or shopkeeper, and it was not till he
spoke that he was noticed by those
who did not know him. Neither was
a good speaker or lecturer; yet each
influenced and inspired more col-
leagues and students than any other
of his generation. Both built great
schools of physics that became peopled
with other great men, and Nobel
prizes went naturally to members of
their laboratories. Each was most gen-
erous in giving credit to his junior
colleagues, creating thereby extraor-
dinary loyalties.
Rutherford and Lawrence were self-
confident, assertive, and at times over-
bearing, but their stature was such
that they could behave in this way
with justice, and each was quick to
express contrition if he was shown to
be wrong.
Neither Rutherford nor Lawrence
could tolerate laziness or indifference
in those who worked with them.
Rutherford said to a research student
from one of the dominions, at tea be-
fore a meeting of the Cavendish Physi-
cal Society, "You know, X, I do not
believe that you are in and at work
because your hat is hanging behind
your door!" Such a remark was far
more effective than any reprimand.
During the hectic days of the Man-
hattan Project in the war years, Law-
rence spoke to me several times of
individuals whom he felt did not share
his sense of urgency and complete
40 SEPTEMBER 1966 PHYSICS TODAYdedication to the task in hand. "I
don't know what has gone wrong
with Y. He's lazy and his attitude is
affecting those round him. I think
we'd better get rid of him."
Rutherford had a great and affec-
tionate regard for Niels Bohr, who
had worked with him in Manchester.
Lawrence could not understand the
attitude of the gentle theoretician, who
had been smuggled out of Denmark
by the British and brought to Los
Alamos, where it was thought that his
genius could aid the design of a nu-
clear weapon. While the task was not
completed, Lawrence could see no
sense in Bohr's worries about how it
should be used, or his concern about
the part the devastating new weapon
could play in the creation of a world
without war. Great as was his admira-
tion for the man who had made a liv-
ing reality of Rutherford's nuclear
atom, he felt that Bohr was actually
holding back progress and would be
better away from the project. On his
part, Bohr found it difficult to under-
stand the complete objectivity of Law-
rence over an undertaking which cre-
ated a crisis in human affairs to which
men of science could not be indiffer-
ent.
Although wholly dedicated to the
pursuit of scientific knowledge, both
Rutherford and Lawrence delighted in
the company of men who had achieved
greatness in other spheres. Because of
their positions and reputations, they
made many contacts and a multitude
of friends among industrialists, poli-
ticians, lawyers, medical men and the
higher echelons of the civil service.
They were at home in such company
and enjoyed the good living which
many such men accepted as part of
their existence. But there was one-
great difference. Rutherford enjoyed
what has been called smoking-room
humor. Although his own memory
for such stories was not good, his
great roar of booming laughter was
to be heard after dinner as he savored
the subtlety of some lewd tale. I never
heard Lawrence swear, under any cir-
cumstances, and his reaction to off-
color humor was not encouraging.
Both Lawrence and Rutherford could
be devastatingly blunt and uncom-
promising when faced with evidence
°f lack of integrity, or of gullibility,RUTHERFORD, IX 1926, visits New Zealand as Cawthron Lecturer.
LAWRENCE AT CONROLS of the 37-in. Berkeley cyclotron, about 1938.
in scientific work. I recollect an oc-
casion when Rutherford was asked to
advise whether die inventor of a diag-
nostic machine, which had been report-
ed upon favorably by one of the Royal
physicians, should be paid a large sum
of money for rights to use his equip-
ment. Diseases were alleged to be diag-
nosed by connecting electrodes to the
patient and observing the deflections
of meters indicating excess or defect
of various elements in the patient's
body. The inventor explained that the
"black box" contained radioactive va-
rieties of each of the elements, where-upon Rutherford became very angry,
pouring scorn on both the fraudulent
inventor and the gullible physicians
who believed in the efficacy of his
machine. 1 am told that Lawrence
was invited to examine the claims of
a chemist in Berkeley who maintained
that isotopes of the chemical elements
could be detected, and their propor-
tions measured, in incredibly small
concentrations, by observation of
certain optical resonances in polarized
light, which were characteristic for each
individual isotopic mass. Looking
through the eyepiece, he could find
PHYSICS TODAY SEPTEMBER 1966 • 41INSTRUMENTATION SPECS
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1no evidence whatever of the maxima
and minima which were said to exist.
He burst into laughter, in a cruelly
embarrassing manner, at the self-de-
lusion of the young observer, who had
been persuaded by the senior perpe-
trator of the hoax that there was
something to observe.
Politics
In politics, Rutherford was what
would be called nowadays, a woolly
liberal. My wife and I spent many
periods with the Rutherfords at their
country cottage, "Celyn", in the beau-
tiful Gwynant Valley of North Wales,
and later at "Chantry Cottage" in
Wiltshire, where the walking was less
arduous. He and I often had political
arguments, which were particularly hot
at the time of the abdication of Ed-
ward VIII. I thought that no harm
would come if Edward were allowed
to marry Mrs Simpson, whereas Ruth-
erford argued that it would do irrepar-
able harm to the monarchy. His main
concern was that science should be
used properly in the development of
the economy, and on one of his rare
appearances in the House of Lords, he
advocated the establishment of a
ministry of prevision to keep the gov-
ernment informed about the advance
of science and technology and the prob-
able impact upon industrial develop-
ment. He was most generous and open-
hearted, and did all that he could to
aid the victims of Nazi persecution.
He was as suspicious of communism as
he was of extreme conservatism, but
he liked Stanley Baldwin, one of the
most conservative prime ministers Brit-
ain ever had. At heart, he was apo-
litical, but when pressed, declared
that he was a liberal.
Ernest Lawrence was both an idealist,
who cared intensely about the future
of his children and all mankind, and
a pragmatist, who saw little good in
the obsession of some of his colleagues
with the examination of social and po-
litical schemes for alleviating the lot
of humanity. Sometimes during the
war, he and I walked up or down the
hill between the Radiation Laboratory
and the campus of the university. The
downward trip usually began by his
drinking a carton of cold milk, which
I loathed, the liquid portion of which
often fertilized one of the stately euca-lyptus trees planted on the hillside.
We would pause on the way to gaze
down over the unforgettable beauty of
San Francisco Bay. Then, and while
walking, he would tell me of his deep
concern that science be used fully to
aid the development of the human
race, and of his admiration for the
practical steps that Franklin Roosevelt
was taking to enable this to happen
in the United States. He would out-
line what he could see ahead in the
application of physical knowledge in
communications, and the productivity
of industry and agriculture. He would
express his conviction that knowledge
of matter and radiation would trans-
form the biological sciences and pro-
Aide tools for medicine that would
alleviate, cure and prevent disease. He
felt that this was a task for mankind,
and not only for America, and he was
anxious to help create a world situa-
tion in which all knowledge could be
shared by all men. In a practical way
he did this whole-heartedly, helping
us all, wherever we were, to build
cyclotrons, by providing freely draw-
ings, full details, and even his thoughts
about improvements upon what had
been built in Berkeley. Of course he
could not escape entirely the atmo-
sphere of the times, and after the end
of the war, he veered somewhat to-
wards a more restricted and less gen-
erous view of the part that his great
country should play in maintaining
the peace and assisting other nations.
But this was true only of his politics,
and his deep commitment to the de-
fense of America. In his science, he
remained the same open-hearted be-
liever in openness and in the value
of exchange of knowledge and of in-
formation in the removal of interna-
tional misunderstandings.
However, Lawrence was genuinely
apolitical. He had inherited liberal
democratic leanings from his parents,
but he could not become excited about
political issues. For instance, he was
quite unaffected by the "loyalty oath,"
which the university imposed upon
members of its staff, and which caused
great dissension among some of them.
Although unable to appreciate the
strong objections of many of his col-
leagues to what he regarded as a trivial
obligation imposed by those who gen-
erously supported his laboratory, never-theless, he fought hard for them as
individuals.
Advice on cyclotrons
It is interesting here to recall that
the first inquiry Lawrence received
from anyone about the possibility of
construction of a cyclotron elsewhere,
was from Frederic Joliot, of Paris. On
14 June 1932, he wrote from the Lab-
oratoire Curie, saying that he had read
with great interest Lawrence's publica-
tion on the production of ions with
high velocity. "Votre travail me parait
remarquable, et les etudes que Ton
peut faire avec de tels rayons sont
dun grand interet." [Your work seems
remarkable to me, and the studies that
can be made with such rays are very
interesting.] He would like to build
an apparatus of a similar type, and
to do it rapidly. To this end, he re-
quested two reprints of the article,
and any details of construction of the
"points les plus delicats" [the most
delicate points]. On 20 Aug. Law-
rence replied, apologizing for the de-
lay, and told Joliot that he might be
able to obtain a magnet made for a
Poulson arc radio transmitter, similar
to one that Lawrence had obtained
in the United States, which he under-
stood was being dismantled at Bor-
deaux.
The generous attitude of Lawrence
towards others desiring to build cyclo-
trons of their own is well illustrated
by the following extract from a
letter to Kenneth Bain bridge, dated
6 Feb. 1935:
"I have just received a letter from
Professor [George] Pegram at Co-
lumbia, saying that they want to
embark upon the construction of a
cyclotron provided that I have no
objections. I am writing him that,
rather than having objections I am
more than delighted that they are
planning to build a cyclotron. The
cyclotron to my mind is by far the
best ion accelerator for nearly all
nuclear work, and it would give me
a great deal of pleasure if many
laboratories would build them."
On 27 Nov. 1935 Lawrence wrote to
Chadwick, congratulating him on the
award of a Nobel Prize, and offering
to give him every help in building a
magnetic-resonance accelerator in Liv-
erpool. He said that the Cavendish
PHYSICS TODAY • SEPTEMBER 1966 • 43must miss Chadwick greatly, but that
this was compensated by the fact that
he would build in Liverpool another
great center of nuclear physics. Chad-
wick replied that he felt rather lucky
to get a Nobel Prize and thanked Law-
rence for his offer to help to build
"your magnetic-resonance accelerator,
which ranks with the expansion cham-
ber as the most beautiful piece of ap-
paratus I know." In letters about the
construction of cyclotrons by others,
Lawrence always emphasized that, con-
trary to the ideas of many, the cyclo-
tron was not a difficult piece of equip-
ment to get into operation.
The wrord "cyclotron" did not ap-
pear in any publication from the Ra-
diation Laboratory till 1935, in a paper
by Lawrence, Edwin M. McMillan and
Robert Thornton,1 where the follow-
ing footnote is inserted:
"Since we shall have many occa-
sions in the future to refer to this
apparatus, we feel that it should
have a name. The term 'magnetic-
resonance accelerator' is suggest-
ed. . . . The word 'cyclotron,' of
obvious derivation, has come to be
used as a sort of laboratory slang
for the magnetic device/'
Running their laboratories
The Cavendish Laboratory, under
Rutherford and his predecessors, was
always short of money. Rutherford had
no flair and no inclination for raising
funds. Only under extreme pressure,
first from the ebullient Peter Kapitza,
and later from Cockcroft and me, was
he prepared to fight hard for money
for large or complex equipment. He
never sought riches and died a com-
paratively poor man. Lawrence, on the
other hand, had shrewd business sense
and was adept at raising funds for
the work of his laboratory. Apart
from his early interest in medicine,
he realized early the medical possi-
bilities of the radiations produced by
the cyclotron, and did not hesitate
to use these in his search for funds.
In 1935 he wrote to Bohr:
"In addition to the nuclear in-
vestigations, we are carrying on in-
vestigations of the biological effects
of the neutrons and various radio-
active substances and are finding
interesting things in this direction.
I must confess that one reason wehave undertaken this biological work
is that we thereby have been able to
get financial support for all of the
work in the laboratory. As you well
know, it is so much easier to get
funds for medical research."
Similarly, after the war, he made full
use of the wartime achievements of
the Radiation Laboratory in raising the
support required for the very large ex-
pansion of its activities. However, it
was his concern for the defense of his
country and his belief that it was un-
wise to confine the development of
nuclear weapons to Los Alamos, which
led him to establish a branch of the
laboratory devoted to this work at
Livermore.
Lawrence's phenomenal success in
raising money for his laboratory was
undoubtedly due to his able handling
of executives in both industry and gov-
ernment instrumentalities. His direct
approach, his self-confidence, the qual-
ity and high achievement of his col-
leagues, and the great momentum of
the researchers under his direction bred
confidence in those from whom the
money came. His judgment was good,
both of men and of the projects they
wished to undertake, and he showed a
rare ability to utilize to the full the di-
verse skills and experience of the vari-
ous members of his staff. He became the
prototype of the director of the large
modern laboratory, the costs of which
rose to undreamt of magnitude, his
managerial skill resulting in dividends
of important scientific knowledge fully
justifying the expenditure. But in
achieving this, he had to give up per-
sonal participation in research. His
influence on die laboratory programs
remained profound, and his enthusi-
asm radiated into every corner of the
institution. William Brobeck, who
joined the Radiation Laboratory in
1936 as an engineer, recalls that Law-
rence took an animated part in all dis-
cussions of technique and showed an
extraordinary ability to see a piece of
equipment as a whole, avoiding be-
coming bogged down in detail. Law-
rence was a regular visitor to each
section of the laboratory until illness
caused him to appear very seldom out-
side his office.
Rutherford's method of running a
laboratory was in striking contrast to
that of Lawrence. He was not muchinterested in the apparatus for its own
sake, believing that techniques grew
from the demands of the experiment.
Like Lawrence, he advocated a simple,
preliminary approach, a sort of skir-
mish into the territory to be explored,
followed by refinement if the recon-
noiter showed promise. He would roam
round the laboratory, discussing results
and the physical knowledge they re-
vealed, rather than apparatus. His
stimulus was enormous, and his in-
fluence direct. A glance at any list of
publications from the Cavendish Lab-
oratory, or from the laboratories in
McGill or Manchester in his periods
there, reveals how deep was his influ-
ence on the researches carried out.
Lawrence worked to give others the
opportunity to achieve important re-
sults; Rutherford was so great a physi-
cist that almost every member of his
laboratory found himself working upon
some problem that Rutherford had sug-
gested, or that arose directly from
Rutherford's own work. This domi-
nance was not imposed upon his col-
leagues and students. They often be-
gan work along lines of their own
choosing, but rapidly found that die
instinct of Rutherford's genius was a
surer guide to interesting and im-
portant results.
Both Rutherford and Lawrence gave
coherence to laboratories inhabited
by workers of differing temperaments
and varying abilities. LInder their in-
fluence, each gave of his best; all re-
joiced in the outstanding achievement
of one of their number, and each felt
himself to be part of the whole, shar-
ing its triumphs and its vicissitudes.
Seventh Solvay Congress
Although Lawrence had made a very
rapid tour of Europe with his friend
Beams in the summer of 1927, he and
Rutherford did not meet till 1933. In
that year, the Seventh Solvay Con-
ference, held in Brussels from 22 to
29 Oct., was devoted to nuclear phys-
ics, and, naturally, Lawrence was in-
vited to attend. He was eager to go,
since this would give him the oppor-
tunity to meet the principal workers
in his field. Those taking part
included:
From Cavendish Laboratory:
Ernest Rutherford
James Chadwick
44 • SEPTEMBER 1966 • PHYSICS TODAYJohn Cockcroft
Patrick Blackett
Paul Dirac
Cecil Ellis
Rudolf Peierls
Ernest Walton ^
From Institut du Radium. Paris:
Marie Curie
Irene Joliot-Curie
Frederic Joliot
M. S. Rosenblum
From the Physical Institute, Leipzig:
Werner Heisenberg
Peter Debye
From elsewhere:
Neils Bohr (Institute of Theoreti-
cal Physics, Copenhagen)
Albert Einstein (then living in Bel-
gium)
Erwin Schrodinger (Physical Insti-
tute, University of Berlin)
Wolfgang Pauli (Physical Institute,
Zurich)
Louis de Broglie (France)
Marcel de Broglie (France)
Enrico Fermi (Physical Institute,
University of Rome)
George Gamow (Institute of Mathe-
matical Physics, Leningrad)
Abraham Joffe (University of Phys-
ics and Mechanics, Leningrad)
Walther Bothe (Physical Institute,
University of Heidelberg)
Lise Meitner (Kaiser Wilhelm In-
stitute, Berlin)
Francis Perrin (Institute of Chem-
istry and Physics, Paris)
Leon Rosenfeld (Institute of Phys-
ics, University of Liege)
H. A. Kramers (Institute of Phys-
ics, University of Utrecht)
Nevill Mott (University of Bristol)
Ernest Lawrence, the only American
invited, naturally was greatly pleased
to find himself among this group of
eminent physicists who, together, rep-
resented almost all that was then
known, from experimental and theo-
retical investigation, of the atomic nu-
cleus. His invitation from the Presi-
dent, Paul Langevin, asked him to
participate in 'Texamen de questions
relatives a la constitution de la ma-
tiere" [the examination of questions
relative to the constitution of matter],
and reports were to be read by Ruther-
ford, Chadwick, Bohr, Heisenberg, Ga-
mow, Cockcroft, and M and Mme
Joliot. It was clearly to be an exciting
meeting, as it was only a year earlierthat the neutron had been discovered,
and transmutation of nuclei by arti-
fically accelerated beams of charged
particles had been achieved.
In a letter to Langevin, dated 4 Oct.
1933, written after he had read the
papers that had been circulated to
those invited, Lawrence stated that he
wanted particularly to make some rath-
er extensive observations on Cock-
croft's report, and that he might wish
to comment on papers by Chadwick,
Joliot, and possibly Gamow. He was
able to obtain funds to meet the costs
of his trip, but owing to his commit-
ments in Berkeley, he could stay in
Europe for only a very limited period.
At this time, Lawrence and his co-
workers had used the cyclotron to con-
firm the results of Cockcroft and Wal-
ton on the disintegration of lithium by
proton bombardment, and had extend-
ed their observations on this and other
transformations to higher energies.
Lawrence had eagerly availed himself
of the opportunity offered by the suc-
cess of Gilbert N. Lewis, at Berkeley, in
producing almost pure samples of
heavy water, and had accelerated the
nuclei of the new hydrogen isotope in
the cyclotron. His team observed an
enormous emission of protons and
neutrons from every target that was
bombarded, and this similarity of re-
sults, irrespective of target material,
had led Lawrence to put forward the
hypothesis that the nucleus of heavy
hydrogen, called the "demon" by
Lewis, was unstable, breaking up in
nuclear collisions into a proton and
neutron. Meanwhile, Lewis had pre-
sented samples of heavy water to many
investigators, including Rutherford,
and we had been making observations
in the Cavendish Laboratory that were
not in accord with Lawrence's view
that the deuton was unstable.
Lawrence went to the Solvay Con-
ference prepared to defend his hy-
pothesis and to back the cyclotron
as the type of accelerator most versatile
for experimental work in nuclear
physics. The marginal notes made by
him on the copies of the reports pre-
sented, give interesting information
about his attitudes. Some of these
are vigorous, as the large cross over
Cockcroft's assertions that "only small
currents are possible" from the cyclo-
tron, and when Cockcroft restated thisCYCLOTRON MODEL is held by Law-
rence in 1930, year after conception.
later, he wrote, "Not true," boldly in
the margin. In several places he com-
plained that the deuton-breakup hy-
pothesis received no mention, and it
becomes clear that he did not appreci-
ate fully the calculations of neutron
mass given by Chadwick, or the observa-
tions of Cockcroft, and of Rutherford
and me, which were not in accord with
his idea. He showed particular interest
in those observations reported by the
Joliots on gamma rays produced from
atoms bombarded by alpha particles,
both those collisions that result in cap-
ture of the alpha particle, and those
in which a nucleus is excited, without
actual capture.
Lawrence's meticulous care to give
credit to his colleagures for their part
in the work in his laboratory is evi-
dent from his insistence upon the addi-
tion of their names—Malcolm Hender-
son, Milton White, Sloan, Lewis and
Livingston—wherever Cockcroft's paper
mentioned only Lawrence.
Chadwick recalls, in a letter to me,
that Rutherford was much impressed
by the vigorous young Lawrence, and
remarked to Chadwick, "He is just like
I was at his age."
Lawrence paid a brief visit to the
PHYSICS TODAY • SEPTEMBER 1966 • 45Cavendish Laboratory after the Solvay
Conference, and it was then that I
met him. We had a vigorous discus-
sion, with Lawrence sticking firmly to
his concept of an unstable deuton.
When he had gone, Rutherford, said,
"He's a brash young man, but he'll
learn!"
Cooksey tells me that he met Law-
rence at the boat in New York on his
return to America. Lawrence was bub-
bling over with enthusiasm for all that
he had seen and learned. He was par-
ticularly enthusiastic about the great
power of the neutron as an agent for
disintegrating nuclei, and expressed
the view that, before long, these would
make possible a self-propagating reac-
tion, and hence the practical release
of energy from nuclei. A truly pro-
phetic remark.
Deuton instability
After his return from the Solvay Con-
ference, Lawrence wrote to Cockcroft
informing him that, with Livingston
and Henderson, he would concentrate
upon the origin of the protons, with a
range in air of about 18 cm, which
were emitted from all targets bom-
barded with deutons. Firstly, they
would try to clear up the uncertainty
about contamination of the targets,
and if this did not turn out to be the
source of the particles, they would
"continue the experiments to shed
further light on the origin of the 18
cm protons." He reported also that,
on his way back, he had visited Wash-
ington, where Tuve had a beam of
protons with an energy of 1.5 MeV
from his Van de Graaff accelerator.
"I persuaded Tuve to investigate
the origin of the 18 cm protons
and the hypothesis of the disinte-
gration of the deuton right away.
I want to get the matter cleared up
as soon as possible and it will be a
great help if Tuve, with his inde-
pendent set-up, will investigate the
problem."
He wrote also to Gamow on 4 Dec.
1933, saying that he had been paying
particular attention to the hypothesis
of the disintegration of the deuton,
using clean targets and carefully puri-
fied materials. "However, we find that
the yield of protons and neutrons pro-
duced by the bombarding deutons is
quite independent of our endeavorsto clean the targets." They found that
2.8-MeV deutons produced disintegra-
tion protons in the same proportions
as observed at 1.2 MeV. On 28 Dec.
1933 he wrote again to Gamow:
"The experimental evidence that
the deuton disintegrates is growing.
Lately, we have observed the emis-
sion of long range protons (up to
about 20 cms) resulting from the
bombardment by protons of targets
containing heavy hydrogen. Though
perhaps the matter cannot be re-
garded as entirely settled yet . . .
certainly it must be admitted that
the evidence is preponderantly in
favor of the hypothesis of the ener-
getic instability of the deuton."
Cockcroft, in a letter to Lawrence
of 21 Dec. 1933, reported further work
on the long range protons produced
by bombardment with deutons from
lithium, carbon and boron, and noted
that while iron gave a small yield of
protons, none were observed from cop-
per, gold or copper oxide.
"We have so far not worked be-
yond 600 kV, and it may well be
that some groups appear at higher
voltages. I feel myself, however, that
the evidence so far is against your
interpretation of the break up of
H2."
Lawrence replied on 12 Jan. 1934:
"It seems to me that you are hardly
justified in feeling that the evidence
obtained by you so far is against
the interpretation of the break-up of
the deuton, since you have not
worked at voltages above 600 kV
... it seemed pretty evident from
our first preliminary observations
that the yield of the group of pro-
tons which we ascribe to deuton dis-
integration is in all cases very small
below eight or nine hundred thou-
sand volts. Despite your greater
intensities, on the basis of our ob-
servations we would hardly expect
that you would observe the disinte-
gration of the deuton at the voltage
you have been using. ... I hope that
you will soon raise your voltage to
eight or nine hundred thousand.
Meanwhile I have written Tuve your
results and asked him to look into
the matter, as I understand he is
able to work now above a million
volts. I am anxious that the hypothe-
sis of demon disintegration will besettled to everyone's satisfaction, and
to that end it seems essential that
independent experiments be carried
out in another laboratory."
Cockcroft wrote again on 28 Feb.
1934:
"We have been working steadily
on the question of disintegrations
by heavy hydrogen. In addition to
the results on lithium I reported to
you in my last letter, we find three
groups of protons from boron. . . .
We have been investigating copper,
copper oxide, iron, iron oxide, tung-
sten and silver, with stronger heavy
hydrogen, and we find from all of
these we get three groups of par-
ticles of identically the same range.
The first is an alpha particle group
having a maximum range of 3.5 cm,
the second is a proton group of
about 7 cm, and the third is a pro-
ton group of about 13 cm. This
latter group is the one which you
ascribe to the break up of the deu-
ton. It seems in the first place clear
that these three groups cannot all
be due to this break up, and we
therefore feel strongly that the alpha
particle group and the 7 cm proton
group are at any rate due to an
impurity which is probably oxygen.
We are not yet certain about the
13 cm group, but are carrying out
experiments with white hot tung-
sten targets which I hope may finally
dispose of this possibility. We can
observe all these groups at voltages
as low as 200,000, and the voltage
variation shows the standard Gam-
ow tail to the curve. . . .
"I feel, however, that we have
still very good justification for re-
fusing to commit ourselves to your
hypothesis of the deuton break up
until further experimental work has
been carried out."
To this typewritten letter, Cockcroft
added the following handwritten post-
script:
"We have now found that on boil-
ing in caustic and cleaning thorough-
ly the 13 cm group is reduced by a
factor 10; on heating to 2,600 by a
further factor. The 2.5 and 7 cm
groups disappear on heating and re-
appear on oxidation and seem due
to oxygen. . . . Oliphant is getting
queer results with H2 + H2."
Lawrence replied on 14 March 1934,
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PHYSICS TODAY SEPTEMBER 1966 • 47agreeing that Cockcroft's observation
that boiling tungsten in caustic re-
duces the 13-cm group by a factor 10
showed clearly that diis is due to a
contamination.
"I think it is quite possible that
the effects we observed when bom-
barding targets of heavy hydrogen
with hydrogen molecular beams were
due, as [C. C] Lauritsen sug-
gested, to an increase in demon con-
tamination resulting from partial de-
composition of the targets. I cannot
understand my stupidity in not rec-
ognizing this possibility when the
experiments were in progress. Need-
less to say, I feel there is now little
evidence in support of the hypothesis
of deuton instability. ...
"Rather than continuing with
preliminary and exploratory experi-
ments at higher voltages, we have de-
cided to embark on careful investiga-
tions of the nuclear effects brought to
light and we shall make as precise
and trustworthy measurements as we
can. These recent experiences have
impressed upon us forcibly the fact
that much of our work has been
of too preliminary character to be
of value. I regret very much that
the question of deuton instability
involved you in so much work, and
I want to thank you very much for
stepping in and clearing the matter
up so effectively and so promptly."
Lawrence and his colleagues were
relatively new to nuclear physics, and
it is not at all surprising that they made
mistakes in interpretation of a com-
plex phenomenon. It was characteris-
tic of the young Lawrence that he held
tenaciously to his concept of deuton
instability, but that when presented
with definite evidence that it was
wrong, he immediately set to work
to change the approach of his team
to its experiments in such a way as
to avoid similar pitfalls in the future.
Deuton stable after all
Meanwhile, the explanation of the ori-
gin of the proton group that had led
Lawrence astray had been found in the
Cavendish Laboratory. On 13 March
1934, Rutherford wrote to Lawrence:
"I have to thank you for the Aery
interesting letter you sent me some
time ago giving an account of your
work. The whole subject is certainlyin an interesting stage of develop-
ment and reminds me very much
of my early 'radioactivity' days be-
fore the theory of transformations
cleared things up.
"I think you have heard from
Cockcroft about some of our-obser-
vations the last few months. Oli-
phant and I have been particularly
interested in the bombardment of
D with D ions, and I am enclosing
a note from Oliphant giving an ac-
count of our results. I personally be-
lieve that there can be little doubt
of the reaction in which the hydro-
gen isotope of mass 3 is produced,
for the evidence from all sides is in
accord with it. The evidence for the
helium isotope of mass 3 is of course
at present somewhat uncertain but
it looks to me not unlikely.
"You will see that Oliphant like
myself is inclined to believe that the
proton group which you observe for
so many elements arises from the
reaction I have mentioned. We
have made a large number of obser-
vations with beryllium and other ele-
ments but the results are not easy
of interpretation. We think the in-
formation we have found about the
D-D reaction will be helpful in dis-
entangling the data. As you no doubt
appreciate, it takes a lot of work
to make a reasonably complete analy-
sis of the groups of particles from
any element and then it has to be
done all over again with the other
compounds to try and fix the origin
of the groups. There is an enormous
amount of work that will have to be
done with the lighter elements to
be sure we are on firm ground.
"You will have seen about Cock-
croft's results due to the bombard-
ment of carbon by protons. This no
doubt produces the radio-nitrogen
of the Joliots but we can obtain
quite strong sources of positrons by
this method. I heard that Lauritsen
or yourself had observed similar ef-
lects with D bombardment. The
whole subject is opening up in fine
style. You will also have seen that
Oliphant and Co have separated the
lithium isotopes and confirmed the
tentative conclusions we put forward
before." My note went as follows:
"You may have heard of the ex-
periments which we have carried outduring the last week or two on the
effects observed when heavy hydro-
gen is used to bombard heavy hy-
drogen. As I believe these are in-
timately related to your own work,
I should like to tell you what we
have found."
The letter went on to give details of
the results, and of their interpretation
as due to two competing reactions,
the first leading to the production of
hydrogen of mass 3 and a proton, with
ranges of 1.6 cm and 14.3 cm respec-
tively, and the second to helium of
mass 3 and a neutron.
"We suggest, very tentatively, that
your results may be explained as
due to the bombardment of films
of D and of D compounds. Our re-
sults with C, Be, etc., could all be
accounted for by the presence of
less than one monomolecular layer
of D. .. ."
On 4 June 1934 Lawrence replied
to my note, saying that the late answer
was due to his desire to be able to
send some news of interest.
"Your experiments on diplons, to-
gether with Cockcroft and Walton's
recent work, have certainly cleared
things up in beautiful fashion. There
can no longer be any doubt that
our observations which we ascribed
to diplon break-up, are in fact the
results of reactions of diplons with
each other."
He ended his letter with a reference
to Cockcroft's contention, in his Sol-
vay Conference paper, that the cyclo-
tron gave only small currents:
"Dr. Cockcroft might be interested
to know also that we are gradually
increasing our currents of high ve-
locity ions, and that now we are
working regularly with more than a
microampere of either 3 MV diplons
or 1.6 MV protons and several mi-
croamperes of 3 MV hydrogen mole-
cule ions."
Lawrence had already replied to
Rutherford's letter on 10 May 1934,
saying:
"I want to thank you for your
very much appreciated letter. Every-
one here was delighted to learn of
the extraordinarily interesting exper-
iments you have been doing on the
reactions of D-ions with each other
(perhaps I should say diplons. I
do appreciate the force of your argu-
48 • SEPTEMBER 1966 • PHYSICS TODAYmerits in support of diplon,* but
all of us here have become quite
accustomed to deuton and it would
be some effort to change).
"It is difficult for me to under-
stand how we could have failed to de-
tect the effect of diplons on each
other. We did notice about twice as
many long range protons from the
heavy hydrogen target under bom-
bardment by diplons, but the differ-
ence between the targets was much
greater under proton bombardment.
The fact that the calcium hydroxide
targets decompose readily may in
some way account for our observa-
tions. Professor Lewis has prepared
some ammonium chloride targets and
we shall investigate the matter soon.
"The manuscript of Cockcroft and
Walton's admirable paper has just
arrived. There can hardly be any
doubt any longer that most of the
effects which we ascribe to disintegra-
tion of diplons are in fact due
largely to a general contamination
of heavy hydrogen in our apparatus.
I certainly appreciate the manner
in which this complexity of nuclear
phenomena already brought to
light makes it clear that it is easy
to fall into error, and that a good
deal of cautious work must be done
for trustworthy conclusions.
"Fermi's observation of radio-ac-
tivity induced by neutron bombard-
ment is a case in point. When we
bombard various targets with three
million volt deutons, large num-
bers of neutrons are always pro-
duced, which among other things
produce the types of radio-activity
discovered by Fermi. On receiving
Fermi's reprint announcing the ef-
fect, we looked for it and found
that it was no small effect at all. For
' The evident confusion in nomenclature
arose in this way. G. N. Lewis had proposed
the name "deuton" for the nucleus of the
atom of heavy hydrogen. Rutherford ob-
jected strongly to this, feeling that it would
inevitably lead to confusion with neutron,
especially in the spoken word. After discus-
sion with his classical colleagues, he pro-
posed the name "diplon," for the nucleus,
and 'diplogen' for the atom, terms derived
from Greek, and analogous to proton and
hydrogen. The dual nomenclature was given
UP eventually, and the compromise "deu-
teron" and "deuterium" was accepted. It
was said by one cynic that Ernest Ruther-
ford was happy when his initials were in-
serted into deuton!example, we found that a piece of
silver placed outside of the vacuum
chamber about three centimeters
from a beryllium target bombarded
by a half micro-ampere of three mil-
lion volt deutons became in the
course of several minutes radio-ac-
tive enough to give more than a
thousand counts per minute when
the silver piece was placed near a
Geiger counter. We are now study-
ing this type of radio-activity in-
duced in various substances and will
not return to the effects produced
by diplon and proton bombardment
until we understand pretty well the
neutron effects.
"Dr. [Franz] Kurie has been pho-
tographing with the WiLon cham-
ber the recoil nuclei and disintegra-
tions in oxygen produced by neu-
trons from beryllium bombarded by
deutons. Although the Wilson
chamber is about twenty inches
from the neutron source and there-
fore subtends a rather small solid
angle, the neutron intensity is suffi-
ciently great to give him something
like five or ten recoil oxygen nuclei
in each picture and about one dis-
integration fork per ten pictures.
Most of the disintegrations appear to
result in C13 and an alpha-particle,
but Kurie has a dozen or so which
seem to involve the emission of a
proton and therefore the formation
of Nir>. But these conclusions are
highly tentative. At the moment
Kurie is busy making measurements
on his photographs.
"We have sent off for publica-
tion a manuscript on the transmuta-
tion of fluorine by proton bom-
bardment and I am enclosing the
essential curves of the experimental
results. As far as we can determine,
the alpha-particles from fluorine
have a range of between six and
seven centimeters, depending on the
energy of the bombarding proton.
These results support the possibility
suggested in your paper that the
4.1 cm alpha-particles observed by
you are due to boron.
"Dr. McMillan has been studying
gamma radiation from various sub-
stances and finds among other things
that fluorine emits under proton
bombardment, a five million volt
monochromatic gamma radiation ofconsiderable intensity. Some day
perhaps a short range group of
alpha particles from fluorine will be
found to account for this gamma ra-
diation.
"But possibly the most interest-
ing result that McMillan has found
about this radiation is its absorp-
tion coefficient. He finds that the
absorption per electron of the five
million volt gamma radiation varies
approximately linearly with atomic
number, reaching a value for lead
double that for oxygen. In other
words, nuclear absorption (pair pro-
duction presumably) is so great that
in going from two and a half to
five million volts the absorption co-
efficient in lead does not decrease
a great deal.
"I am glad to hear that you are
very well. You need not have told
me that you are kept very busy in
the laboratory, but I was very glad
to hear that the government has giv-
en you a substantial grant of money
for research and that you are re-
sponsible for its disbursement. Also
your comparison of your early radio-
activity days with the present is very
much appreciated. I remember in
the course of my graduate studies
what a 'kick' I got out of reading of
the early work on radio-activity, but
I did not even hope at that time
that I would have the opportunity
to work in a similarly interesting
new field of investigation. . . .
"Please tell Dr. Oliphant that I
appreciated his letter very much and
that I will be writing him directly
before long."
Rutherford's brash young man
learned very quickly, as Rutherford
predicted he would. From that time
onward, the contributions made to
nuclear physics in the Radiation Lab-
oratory were above reproach and of
rapidly increasing importance, as the
energy and intensity of beams avail-
able from the cyclotron increased. •
(This is the first of tico articles on
Ernest Rutherford and Ernest Law-
rence. The second will appear in the
next issue.)
Reference
1. E. L. Lawrence, E. M. McMillan, R. L.
Thornton, Phys. Rev. 48, 493 (1935).
PHYSICS TODAY • SEPTEMBER 1966 • 49 |
1.4868495.pdf | 1 Vector Spin Modeling for Magnetic Tunnel Junctions
with Voltage Dependent Effects
Sasikanth Manipatruni, Dmitri E. Nikonov, and Ian A. Young
Supplementary Online material
A. 4X4 spin conduction matrix for a magnetic tunnel junction
The expressions for the elements of the 4x4 spin conduction matrix
SL FLFL SLFN
G GG GG GG G
G
0 00 00 00 0
(A1)
has been derived in [1] for a single interface between a ferromagnetic and a non -magnetic metal.
[It is understood that all element in this matrix may be dependent on the applied voltage.] The
above expression is valid in a coordinate system where the x -axis is aligned to the direction of
magnetization. For an arbitrary direction
mˆ of magnetization, the spin conduction matrix is
)ˆ()ˆ( )ˆ( )ˆ(1mRx GmR m GFN FN (A2)
where the rotation matrix
22 23 24
32 33 33
42 43 441 0 0 0
0ˆ()0
0r r rRmr r r
r r r
(A3)
The magnetic tunnel junction consists of two interfaces of a tunneling oxide with each of the two
ferromagnets . In general, they may have two directions of magnetization
1ˆm and
2ˆm . Thus the
junction needs to be represented by two conductances in series, as shown in Fig. 1b .
2 Spin conduction matrix elements expressed in terms of spin reflection and transmission
coefficients
The elements of the conduction matrix are expressed, as per [1], via the conductances of majority
)(
and minority
)( spins as follows
G GG GG G G GG G G G
G GG GG GG G
G
SL FLFL SLFM
Re2 Im2 0 0Im2 Re2 0 00 00 0
0 00 00 00 0
(A4)
These conductances can be related [1] to the ab -initio calculated reflection and transmission
coefficients of the interface
nmnmtheG
,22
(A5)
nmnmtheG
,22
(A6)
n mnm nmrrheG G 12
(A7)
where ⁄ is the conductance per spin of a ballistic channel with ideal contacts [ 1];
are the transmission coefficients for majority and minority spin electrons; are the
reflection coefficients of the majority and minority spin electrons; n is the index of modes in the
non-magnetic material , and m is the index of modes in the ferromagnet . From the above
expressions we note relations
nmnm nm
SL r rheG G
,2 2
(A8) 3 from which it follows that
G GSL and
nmnm nm nm nm
FL rr rrhei G
,*2
(A9)
If the complex phase
of the reflection coefficients in the above equations is zero, then and
0FLG
as it happens at zero applied voltage. Assuming that the phase grows linearly at small
voltages
2/ " )0( ) 2cos1( )0(2
,2
VG G rrheG GSL SL
nmmn nm nm
SL SL
(A8)
VG rrheGFL
nmmn nm nm
FL ' 2sin2
,2
(A9)
In other words, the Slonczewski conductance is even vs. voltage and the field -like conductance is
odd. These are examples of the dependences in Eq. (4).
Phenomenological expressions for the spin conduction matrix
The conduction matrix for a magnetic tunnel junction does not easily lend itself to the above
treatment. Transmission and reflection coefficients make sense for the entire tunneling barrier
rather than two interfaces separately. For this reason it is advantageous to exp ress its values via
experimentally measured quantities – resistances for parallel
PR and anti -parallel
APR
magnetizations of the two ferromagnets, or equivalently, magnetoresistance
P APR R MR /
(A10)
In these cases, the two top rows of the 4x4 matrices separate from the bottom ones, and simple
algebra yields that 4
PR G /2
211
MR (A11)
In general, the generalized Ohm’s law (1) equations are solved for a circuit in Fig. 1b to find the
current, spin current through the conductances s well as the voltage and spin voltages at the node
N1. If we neglect spin flip processes in the tunneling oxide, the conductance Gsf can be neglected,
and the spin current is conserved in this node.
Simple analytical expressions can be obtained for a case of magnetization
2ˆm rotated in the x -y
plane relative to
x m ˆˆ1 by an angle
. We also set for simplicity
0FLG and
G GSL . Then
the spin voltages at N 1 are
cos14VVx
sin4VVy (A12)
and the spin currents are
2cos22 VGIx
sin4VGIy (A13)
The perpendicular spin current
yI is responsible for spin torque in Eq.(5). The charge current is
2sin 122 2VGI
(A14)
which is equivalent to Eq.(3).
5 B. Comparison of the AC and DC transfer characteristics with experimental work:
The proposed model can be used to describe DC magneto -resistance as a function of applied
voltage. The proposed model can accurately describe the loss of TMR and current difference
between RP and RAP states.
Figure A ppendix 1. Comparison of DC Magneto -resistance (A1)& (A2) Comparison to [ 2], (B1)
& (B2) comparison to [3], (C1) & (C2 ) comparison of AC magneto -resistance to [4]
6 Comparison of the voltage dependence of torque :
Figure 2. Comparison of in -plane and field like spin transfer torques ; (A1) In-plane spin torque
experimental estimation [2], (A2) Compact model for in -plane spin torque , (B1) Field like torque
experimental estimation [2] , (B2) Compact model for field like spin torque
C. Thermal noise of nanomagnets
The dynamics of nanomagnets are strongly affected by the thermal noise . The thermal noise
can be considered as a result of the microscopic degrees of freedom of the conduction electrons
and the lattice of the ferromagnetic element. At room temperature T, the thermal noise is
7 described by a Gaussian white no ise (with a time domain Dirac -delta auto -correlation). The noise
field acts isotropically on the magnet. The internal field in equation -4 is described as:
(C1)
0)(tHl (C2)
lk
sB
k l ttVMTktHtH ) (2)()(2
0 (C3)
The initial condit ions of the magnets should also be randomized to be consistent with the
distribution of initial angles of magnet moments in a large collection of magnets. At temperature
T, the initial angle of the magnets follows [52]:
(C4)
Figure 3. Comparison of thermal noise induced variation in the free layer with thermodynamic
model .
)ˆ ˆ ˆ( )( zHyHxH H THk j i eff eff
ani sHVMkT
02
8
Figure 4. Modeling examples for non -ideal behavior in MTJs. A) Effect of residual angle of the
free layer, B) Back -hopping due to voltage induced barrier change
The proposed model also captures failure modes of the MTJ arising from thermally induced state
change, thermal variability and non -ideal features of the magneto -resistance and spin torque
dynamics. As an example, we s how the effect of a residual angle in the free layer of an MTJ in
Fig 4A. Another, commonly observed failure mode for MTJs is back -hopping descried by a
voltage dependent energy barrier i.e. where the thermal barrier of the MTJ E b is a function of the
9 applied voltage across the MTJ . The proposed model shows a back -hopping occurrence in figure
4B.
[1] A. Brataas et al. , Physics Reports 427 (2006) 157 – 255
[2] Kubota, H. et al. Quantitative measurement of voltage dependence of spin -transfer torque in
MgO -based m agnetic tunnel junctions. Nature Phys. 4, 37 –41 (2008).
[3] T. Kawahara et. al., Proceedings of ISSCC (2007 ).
[4] Sankey, J. C., Cui, Y. T., Sun, J. Z., Slonczewski, J. C., Buhrman, R. A., & Ralph, D. C.
(2007). Measurement of the spin -transfer -torque vect or in magnetic tunnel junctions. Nature
Physics, 4(1), 67 -71.
[5] Brown, W. F. Thermal fluctuations of a single -domain particle. Phys.Rev.130,1677 -1686
(1963).
[6] Sun, J. Z.; "Spin angular momentum transfer in current -perpendicular nanomagnetic
junctions," IBM Journal of Research and Development , vol.50, no.1, pp.81 -100, Jan. 2006
[7] M. d’Aquino, C. Serpico, G. Coppola, I. D. Mayergoyz, and G. Bertotti. (2006). Midpoint
numerical technique for stochastic Landau -Lifshitz -Gilbert dynamics. J. App l. Phys. 99(8), pp.
08B905 -1–08B905 -3.
[8] Spedalieri, F.M.; Jacob, A.P.; Nikonov, D.E.; Roychowdhury, V.P.; "Performance of
Magnetic Quantum Cellular Automata and Limitations Due to Thermal Noise," Nanotechnology,
IEEE Transactions on, vol.10, no.3, pp.53 7-546, May 2011 .
|
1.3070967.pdf | New Books
Citation: Physics Today 25, 8, 65 (1972); doi: 10.1063/1.3070967
View online: http://dx.doi.org/10.1063/1.3070967
View Table of Contents: http://physicstoday.scitation.org/toc/pto/25/8
Published by the American Institute of Physicsbonding properties, of the problems of
interpreting hindered internal rotation,
and of the analysis of absorption band
shapes through correlation functions.
The presentation of these topics,
which occupy about the last third of
the book, emphasizes that, after all,
the goal of spectroscopy, like other
branches of science, is to discover
something about the structure and be-
havior of matter. It gives a beginning
student a good idea of what research in
vibrational spectroscopy is really all
about.
The beginning chapters are devoted
to a development of the basic, nuts-
and-bolts theory of vibrational spec-
troscopy. These contain such neces-
sary topics as the theory of small vi-
brations, normal coordinates, matrix
algebra, symmetry and point-group
representation theory, sum and prod-
uct rules, and so forth.
In many important areas, the discus-
sions are quite brief and in my opinion,
fall short of giving an uninitiated read-
er enough to work out simple examples
for himself. These weaknesses could
have been removed by giving a little
more detailed discussion of a few fun-
damental points. For example, the
admittedly complicated topic of sepa-
rating the internal and external degrees
of freedom in a molecular system, is
glossed over when it first appears in an
early chapter. This creates a problem
that haunts the latter parts of thebook. When associated problems are
encountered, such as in describing the
use of vectors to compute the kinetic
energy matrix, or in explaining where
the various terms in the Teller-Redlich
product rule come from, a little more
discussion of the basic problem of sep-
arating the degrees of freedom is given.
This could have been handled in a
more efficient and connected manner
by an adequate discussion in the be-
ginning.
In another example, Rayleigh's in-
equality rule is misstated as "... in-
creasing any mass in a periodically vi-
brating system without changing the
force field must [emphasis added] de-
crease all frequencies." As might have
been anticipated, this statement is
shortly followed by many contradictory
examples. Steele's momentary logical
lapse would not have been too serious
had he given just a little more detail
on the origin of Rayleigh's rule (only a
few words and one equation would
have sufficed) so that a beginner could
have discovered what he should have
written, and thereby derived some sat-
isfaction from outsmarting the author.
In spite of these flaws, this book
would be a good text for a lecture
course, or for an independent student
who is willing to look at one of the
older books on vibrational spectroscopy
for more explanatory detail.
WILLIAM T. KING
Brown University
new books
CONFERENCE PROCEEDINGS
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PHYSICS TODAY/AUGUST 1972 67 |
1.4885354.pdf | Testing epitaxial Co1.5Fe1.5Ge(001) electrodes in MgO-based magnetic tunnel
junctions
A. Neggache, T. Hauet, F. Bertran, P. Le Fèvre, S. Petit-Watelot, T. Devolder, P. Ohresser, P. Boulet, C. Mewes
, S. Maat, J. R. Childress, and S. Andrieu
Citation: Applied Physics Letters 104, 252412 (2014); doi: 10.1063/1.4885354
View online: http://dx.doi.org/10.1063/1.4885354
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/104/25?ver=pdfcov
Published by the AIP Publishing
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132.203.227.62 On: Thu, 02 Oct 2014 12:55:26Testing epitaxial Co 1.5Fe1.5Ge(001) electrodes in MgO-based magnetic
tunnel junctions
A. Neggache,1,2T. Hauet,1F . Bertran,2P. L e F e `vre,2S. Petit-Watelot,1T. Devolder,3
P . Ohresser,2P . Boulet,1C. Mewes,4S. Maat,5J. R. Childress,5and S. Andrieu1,a)
1Institut Jean Lamour, UMR CNRS 7198, Universit /C19e de Lorraine, 54506 Vandoeuvre le `s Nancy, France
2Synchrotron SOLEIL-CNRS, L’Orme des Merisiers, Saint-Aubin BP48, 91192 Gif-sur-Yvette, France
3Institut d’Electronique Fondamentale, CNRS, UMR 8622, 91405 Orsay, France
4Department of Physics and Astronomy/Center for Materials for Information Technology,
University of Alabama, Tuscaloosa, Alabama 35487, USA
5San Jose Research Center, HGST, a Western Digital company, San Jose, California 95135, USA
(Received 24 May 2014; accepted 13 June 2014; published online 26 June 2014)
The ability of the full Heusler alloy Co 1.5Fe1.5Ge(001) (CFG) to be a Half-Metallic Magnetic
(HMM) material is investigated. Epitaxial CFG(001) layers were grown by molecular beam
epitaxy. The results obtained using electron diff raction, X-ray diffraction, and X-ray magnetic
circular dichroism are consiste nt with the full Heusler structur e. The pseudo-gap in the minority
spin density of state typical in HMM is exam ined using spin-resolved photoemission.
Interestingly, the spin polarization found to be negative at E Fin equimolar CoFe(001) is
observed to shift to positive values when ins erting Ge in CoFe. However, no pseudo-gap is
found at the Fermi level, even if moderate magne tization and low Gilbert damping are observed
as expected in HMM mater ials. Magneto-transport proper ties in MgO-based magnetic tunnel
junctions using CFG electrodes are investigated via spin and symmetry re solved photoemission.
VC2014 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4885354 ]
The material science community is putting a large effort
into developing original magnetic materials for the imple-
mentation in low-energy spintronic devices. An important
part of this work is devoted to growing electrode materialsfor Magnetic Tunnel Junctions (MTJs) for Magnetic
Random Access Memory (MRAM), oscillators, and sensors.
Thin magnetic electrodes having high spin-polarization, lowdamping, moderate magnetization, and high perpendicular
magnetic anisotropy are investigated. The family of
Half-Metallic Magnetic (HMM) compounds fulfill theserequirements as ab initio calculations performed on many
Heusler or Half-Heusler compounds (NiMnSb, Co
2FeSi,
Co2MnSi…) have predicted a 100% spin polarization at the
Fermi energy (leading to infinite tunnel magneto-resistance
in theory), extremely low Gilbert damping, high Curie
Temperature, and moderate magnetization.1Moreover, per-
pendicular anisotropy has been experimentally achieved for
some Heusler alloys.2
Recent work performing FerroMagnetic Resonance
(FMR) and Giant Magneto-Resistance (GMR) measurements
tend to show that Co 1.5Fe1.5Ge (CFG) compound could be
very promising since low damping3and high GMR4were
reported. Ab initio calculations of the related full Heusler
alloy Co 2FeGe Density Of States (DOS) show that a pseudo-
gap exists in the integrated minority spin DOS. Nevertheless,the Fermi level sits in the higher edge of this band gap.
5,6
Takahashi et al. substituted Ge by Ga atoms to change the
number of valence electrons and, thus, to shift the Fermilevel towards the middle of the band gap.
7Assuming a L2 1
Co2FeGa xGe1/C0x(CFGG) structure, they reported up to 70%spin-polarization measured with point contact Andreev
reflection as well as more than 40% GMR at room tempera-
ture in CFGG/Ag/CFGG metallic spin-valves. In parallel, ab
initio calculations3show that substituting Co by Fe atoms—
thus maintaining a B2 ordered phase—will shift the Fermi
level into the minority band gap. In addition, increasing the
Fe relative to the Co concentration favors optimized chemi-cal ordering since Co atoms in Fe or Ge sites destroy the
pseudo-gap, although Fe atoms in Ge sites keep it unaf-
fected. Calculations seem to be confirmed by the spin-waveDoppler technique which was used to measure the spin-drift
velocity of the magnetization in current-carrying CFG and
yielded an up to 95% current polarization for CFG.
8Finally,
the Curie temperature of Co 2FeGe has been measured close
to 1000 K,6and perpendicular anisotropy for iron rich
Co20Fe50Ge30has been demonstrated when sandwiched
between MgO layers,2due to the perpendicular interface ani-
sotropy induced by MgO.9Therefore, all these features make
CFG a promising candidate for magnetic-RAM electrode.However, CFG has not been successfully integrated in
MTJ’s so far and its spin-polarization at and below the Fermi
level has not yet been measured.
In this Letter, we report on experimental studies of sin-
gle crystal Co
1.5Fe1.5Ge(001) layers as single films or as
electrodes in MgO-based MTJ’s. First, the growth and struc-tural characterization of CFG on MgO(001) substrates is
discussed. Gilbert damping and magnetic features are
extracted from FMR and magnetometry measurements. Thespin-polarization of occupied electronic states is determined
using spin-resolved photoemission spectroscopy at the
SOLEIL synchrotron source. Finally, transport measure-ments performed on CFG/MgO/CFG(001) and Co
25Fe75/
MgO/CFG(001) MTJs are presented.a)Author to whom correspondence should be addressed. Electronic mail:
stephane.andrieu@univ-lorraine.fr
0003-6951/2014/104(25)/252412/4/$30.00 VC2014 AIP Publishing LLC 104, 252412-1APPLIED PHYSICS LETTERS 104, 252412 (2014)
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132.203.227.62 On: Thu, 02 Oct 2014 12:55:26Single CFG films were grown by molecular beam
epitaxy (base pressure 10/C010Torr) on MgO(001) single-
crystal substrates. The CFG single layers were grown by
co-evaporating Co, Fe, and Ge using 3 Knudsen cells whosefluxes were calibrated by using a quartz microbalance
located at the sample position. The uncertainty on the con-
centrations is estimated to be 65%. The films were deposited
on MgO at room temperature, and annealing temperatures
were tested in the range of 350
/C14C–700/C14C. Three types of
MTJs were grown on MgO(001) substrates. Here, we will
mostly discuss MTJ’s with structure: Fe (20 nm)/CFG
(4 nm)/MgO (2.4 nm)/CFG (8 nm)/Co (20 nm)/Au (10 nm)and Co
25Fe75(50 nm)/MgO (2.4 nm)/CFG (30 nm)/Co
(20 nm)/Au (10 nm). The CFG electrodes were deposited at
room temperature and then annealed during 30 min at tem-peratures between 350
/C14C and 500/C14C. The whole stacking
was subsequently patterned by UV-photolithography and a
conventional physical etching processes to get MTJ cellswith a junction’s size of 10 /C210 to 50 /C250lm
2.
The structural properties of single CFG films were first
analysed during growth using Reflection High EnergyElectron Diffraction (RHEED). At room temperature, the
RHEED patterns are the same as observed when growing
pure Fe films (Fig. 1). The unit cell is, thus, a square of halfthe size of the Heusler (001) face, meaning that no chemical
ordering occurs. Annealing at 250
/C14C is sufficient to observe
new1=2streaks appearing along the (11) azimuth of the initial
square lattice (Fig. 1), meaning that the lattice cell has
doubled in size as expected for the Heusler L2 1structure. It
should be noted that this superstructure cannot be observed
in the case of the full Heusler X2YZstructure with chemical
disorder between Y and Z (B2 phase). In the case of the
X1.5Y1.5Zphase, however, a more complex chemical arrange-
ment cannot be ruled out. This simple observation is a clear
indication that some chemical ordering takes place even at
low annealing temperatures <300/C14C.
To go further, X-ray diffraction experiments were per-
formed using a Cu K a1anode ( k¼0.154 nm) along the (00 l)
direction. Fig. 1shows the typical X-ray h–2hdiffraction
pattern for an annealed CFG film. In addition to the MgO
(002) peak (not shown), the pattern shows only the (002) and
(004) peaks typical of the Heusler structure. No additionalphase (e.g., a parasitic hexagonal phase) has been detected.
The lattice spacing is equal to 0.573 nm in agreement with
values already reported.
4,6,10The full determination of the
species distribution within the lattice is not possible here
because the Co and Fe scattering factors are very close using
Ka1X-rays. However, it is possible to get some information
on the location of Ge atoms by fitting the (002) and (004)
peaks. A Rietveld refinement was used and an excellent
agreement is obtained (simulated curves in Fig. 1) assuming
the full Heusler L2 1structure. To go further simulations
were performed assuming the substitution of some Fe or Co
atoms by Ge. We thus were not able to fit the experimentaldata. However, this simulation does not allow us to eliminate
the possibility of some chemical disorder between Fe and Co
atoms. Such Co and Fe permutation is predicted to destroythe pseudogap.
3
The magnetization is also very sensitive to chemical dis-
order. We performed magnetometry experiments using acommercial VSM with automatic sample rotation. The mag-
netization at saturation on average is 1000 6100 kA/m
(emu/cm
3) equivalent to 5 60.5lBper unit cell, which is
similar to calculated values for Co 2FeGe6and experimental
values in Ref. 4. A cubic magnetic anisotropy is deduced
from the hysteresis loop shape, especially from the value ofremanent magnetization as a function of the applied mag-
netic field angle. X-ray magnetic circular dichroism
(XMCD) was also performed on the DEIMOS beamline atSOLEIL (Fig. 2(a)) at the Fe, Co, and Ge 2p edges. A
dichroic signal is observed for the 3 elements. The atomic
magnetic moments of Fe and Co were found parallel, andantiparallel to the Ge one. The application of the sum rules
(using the Fe and Co number of holes in the bulk) gives
2.060.1l
B/at and 1.5 60.1lB/at for Fe and Co, respec-
tively. The Ge magnetic moment is difficult to determine but
is very small (small dichroic signal). Theoretical calculations
based on first principles within density-functional theoryusing the plane-wave ultrasoft pseudo-potential method
within the generalized gradient approximation for the
exchange-correlation functional
11,12have been performed to
determine the magnetic moments. To study the effect of
permutations we have used a 16 atom super cell. The
fully relaxed structure has a total magnetic moment of
FIG. 1. RHEED patterns obtained on a CFG film as-deposited at RT (top
left) and annealed (top right). New1=2streaks appear after annealing typical
of chemical ordering. The (00 L) diffraction peak intensities are also well
fitted assuming the full Heusler L2 1structure (bottom).252412-2 Neggache et al. Appl. Phys. Lett. 104, 252412 (2014)
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132.203.227.62 On: Thu, 02 Oct 2014 12:55:26approximately 5.3 lBper unit cell with a Ge magnetic
moment equal to /C00.1lB/at. Using this value of the Ge
moment, we found a magnetic moment for the unit cell
of (1.5 þ2)/C21.5/C00.1¼5.15lBper unit cell in very
good agreement with calculations. Another important theo-retical result is that the total magnetic moment decreases to
4.8l
B/cell when Co atoms take the place of Fe atoms in the
FeGe lattice as shown in Fig. 1. Finally, Vector Network
Analyzer FerroMagnetic Resonance (VNAFMR see Ref. 13)
has been performed at room temperature on a series of CFG
samples (Figs. 2(b) and2(c)). In saturating conditions, the
FMR frequency is linear with the perpendicular field and
gives an effective magnetization equal to 1.25 60.05 T
(¼995640 kA/m in agreement with SQUID measurements)
with a Land /C19e factor equal to 2.07 þ//C00.02. The evolution of
the linewidth with the resonance frequency indicates slight
contributions of the inhomogeneous broadening, whose influ-ence is gradually suppressed for frequencies up to 17 GHz,
above which the subsequent linewidth evolution is consistent
with a Gilbert damping aof 0.007 60.001. For the 30 nm
thick layer, a faint perpendicular standing spin wave mode is
detected at 8 GHz above the uniform resonance mode.
This splitting is indicative of an exchange stiffness ofA¼1361 pJ/m.
14The low damping value is consistent
with typical HMM behavior for which the damping coeffi-
cient is expected to be very small due to the lack of minorityspin DOS at the Fermi level. All these results are strong indi-
cations that chemical disorder is very limited in our films.
The CFG spin polarization was investigated using Spin-
Resolved PhotoEmission Spectroscopy (SR-PES) performedon the CASSIOPEE beamline at the SOLEIL synchrotron.
Single CFG films were epitaxialy grown in a molecular
beam epitaxy (MBE) chamber connected to the beamline
15
so that surface contamination is prevented. First SR-PES
experiments were conducted with 37 eV p-polarized light at
a4 5/C14incident angle. The photoelectron detection was per-
formed along the (001) normal axis of the films using the
largest aperture ( þ//C08/C14) leading to investigating 80% of the
Brillouin Zone (BZ). The spin resolution was achieved by
using a Mott detector. The sample temperature was main-
tained below 100 K during the experiments. Figures 3(a)and
3(b) show the majority N "and minority N #photoemission
spectra (PES) obtained on single CoFe (as reference layer)
and single CFG thin films annealed at around 550/C14C. The
corresponding spin polarizations P ¼(N"/C0N#)/(N"þN#)
are plotted in Figs. 3(a) and3(b). In the reference CoFe
layer, the expected negative spin polarization is found at E F
(Fig. 3(a)). The situation is clearly different for CFG films.
The main difference between CoFe and CFG consists in a
strong majority spin contribution around /C00.2 eV leading
to a positive polarization peak at the same energy value
(Fig. 3(b)). This new majority spin band pushes the spin
polarization to become positive close to E F. The occurrence
of this strong majority spin contribution may be at the origin
of the greatly enhanced magnetoresistive properties obtained
in CFG spin-valves compared to CoFe.4However as might
be expected our experiment on CFG spin-valves films shows
that the actual spin polarization is <100%.
A final test to evaluate HMM behavior is to use CFG as
electrodes in MgO-based MTJs. A first experiment was to
grow CFG on Fe/MgO(001) underlayers since the resulting
MgO barrier quality is excellent and well understood in thiscase. The upper electrode was a 10 nm thick CFG layer cov-
ered by a thick Co layer resulting in a magnetic coercive
FIG. 2. Magnetic properties of (CoFe) 3Ge film using X-Ray absorption
spectroscopy (XAS and XMCD) and FMR. (a) The XMCD signal on Ge is
opposite to the Fe and Co signals, suggesting that the Ge magnetic moment
is coupled antiparallel to Fe and Co. (b) FMR frequency versus perpendicu-
larly applied field. The red line is a linear fit, yielding a Land /C19e factor of 2.07
and a magnetization of 1.25 T. Inset: Polar magneto-optical Kerr effect loop
(10 nm film). (c) FMR linewidth versus frequency in perpendicular applied
field conditions (30 nm film). Inset: Real and imaginary parts of the perme-
ability for a field of 2 Tesla.
FIG. 3. PES spectra (top) and spin polarization (bottom) for annealed (a)equimolar CoFe(001) and (b) (CoFe)
3Ge films.252412-3 Neggache et al. Appl. Phys. Lett. 104, 252412 (2014)
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132.203.227.62 On: Thu, 02 Oct 2014 12:55:26field for the upper electrode substantially higher than that of
the lower Fe electrode. The CFG layer was grown at RT and
annealed up to 550/C14C. The Co layer was grown on top of
CFG at RT to avoid intermixing. The Tunnel Magneto-Resistance (TMR) vs. H curves are shown in Fig. 4. The
TMR reaches 125% at RT and more than 220% at 15 K con-
sistent with a good epitaxial process and high spin polariza-tion in the tunneling process. However, these values are
smaller than state of the art TMR values obtained on Fe/
MgO/Fe(001) (around 200% at RT and 450% at 10 K, seeRef. 16). The TMR is even lower using two CFG electrodes
(Fig. 4), where the TMR is only around 15% at RT and 60%
at 15 K. To better understand these magneto-transportresults, we carefully studied the symmetry of the electronic
states near E
F. The high TMR values in Fe/MgO/Fe system
are actually due to the lack of D1minority spin DOS at E F.
We thus performed the SRPES measurement with s-polar-
ized light (Fig. 4) in order to identify the symmetry of the
observed transitions ( D1orD5see Refs. 15and17). We
found that the majority spin peak around /C00.2 eV has a D1
symmetry since its contribution using p-polarized light is
strongly reduced when measuring with s-polarized light. But
more interestingly, we also observe some D1contribution in
the minority spin DOS. Such minority spin channel with a
D1symmetry lowers the overall spin polarization and is det-
rimental to TMR, which explains the low TMR values
obtained using CFG in our MgO-based MTJs.To summarize, single crystalline Co 1.5Fe1.5Ge (001)
films were grown by molecular beam epitaxy. The structure
of the films is cubic with the expected lattice constant. All
the structural and magnetic characterizations clearly indicatechemical ordering consistent with the full Heusler structure.
In particular, very low Gilbert damping coefficients are
obtained. However, some chemical disorder involving Coatoms occupying in Y sites instead of Fe cannot be ruled out
here, which would lead in accordance to our theoretical
investigations to a suppression or reduction of the pseudo
gap at the Fermi level. The spin-polarization of CFG(001) at
E
Fis observed to be positive opposed to the negative spin-
polarization of FeCo(001), and, furthermore it is not 100%.
The pseudo-gap for minority spin at E Fis not observed.
Some minority spin DOS with D1symmetry was observed at
the Fermi energy explaining the modest TMR values
observed in MgO-based MTJs using CFG electrodes.
The authors thank G. Lengaigne (IJL) for patterning the
magnetic tunnel junctions.
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FIG. 4. Top—TMR as a function of magnetic field amplitude measured on a
Fe/MgO/CFG(001) MTJ (left) and on a CFG/MgO/CFG(001) MTJ (right).
Bottom—SR-PES using p and s polarization of the photon. D1 symmetry
states are excited using p-light but not with s-light. D5 symmetry states are
excited using both light polarizations.15,17The detection of D1#states at E F
explains the limited TMR values using CFG electrodes.252412-4 Neggache et al. Appl. Phys. Lett. 104, 252412 (2014)
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1.1851731.pdf | Direct correlation of reversal rate dynamics to domain configurations in micron-sized
permalloy elements
J. W. Lau, M. Beleggia, M. A. Schofield, G. F. Neumark, and Y. Zhu
Citation: Journal of Applied Physics 97, 10E702 (2005); doi: 10.1063/1.1851731
View online: http://dx.doi.org/10.1063/1.1851731
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/97/10?ver=pdfcov
Published by the AIP Publishing
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129.120.242.61 On: Fri, 28 Nov 2014 20:38:13Direct correlation of reversal rate dynamics to domain configurations
in micron-sized permalloy elements
J. W. Laua!
Department of Applied Physics and Applied Mathematics, Columbia University, New York, New York 10027
and Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York 11973
M. Beleggia and M. A. Schofield
Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York 11973
G. F. Neumark
Department of Applied Physics and Applied Mathematics, Columbia University, New York,New York 10027
Y. Zhu
Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York 11973
sPresented on 9 November 2004; published online 10 May 2005 d
The distribution of states upon the removal of applied magnetic field in an array of 7.5
37.5mm2permalloy square elements, as observed by transmission electron microscopy in Lorentz
mode, shows a predominance of two states: the vortex state and the seven-domain state. Thedistributional dependence of these two states on the rate of change of the reversal field isestablished. Micromagnetic simulations suggest that vortex nucleation and the subsequentdomain-wall propagation are the two primary mechanisms for magnetization reversal. The kineticsof the two pathways is examined in a manner that conforms to the observed distribution of states. ©2005 American Institute of Physics .fDOI: 10.1063/1.1851731 g
INTRODUCTION
Data processing advancements have placed increasing
challenging demands on reading and writing speeds in mag-netic recording media, yet the transient mechanics that resultfrom the rapid switching of magnetic fields remain largelyunexplored. It is well documented that the reversal field rate
binfluences domain-wall mobility1,2and coercivity3of a fer-
romagnet. We use in situtransmission electron microscopy
sTEM dtocorrelatethemagneticreversaldynamicstosteady-
state domain configurations in micron-sized permalloy ele-ments. Combining Lorentz microscopy with micromagneticsimulations, we are not only able to determine the reversalprocedure in these elements, but also the manner in which
b
affects a fundamental reversal mechanism, nucleation of themagnetic vortex.
Our observations show that if we bring an array of ,10
4
square permalloy elements to magnetic saturation and then
remove the external field, a different ground state occursstatistically depending on the field removal rate. Two stablestates are seen in these experiments: the single vortex state,and the seven-domain state,
4also known as the diamond and
the Landau state.5These two states are very common in mi-
cron and submicron ferromagnetic films with square or rect-angular geometries.
5–7It was not clear6why the seven-
domain state emerges, since the vortex state is lower inenergy in the absence of an applied field.
Experimentally, we found that the number of occur-
rences of these two states is strongly correlated to the rate offield removal. Figure 1 shows twoTEM images documentingthis phenomenon. On the left, the elements relax to the vor-
tex state after the saturation field is removed. On the right,the same elements relax primarily into the seven-domainsstate after the same saturation field is removed at a slowerrate. By examining the statistical correlation in the Lorentzmicrographs, combined with micromagnetic simulations, wehave determined the exact reversal mechanism, thereby un-derstanding the nature of this population inversion.
We verified that for the array with this geometry, inter-
action between elements can be safely neglected. This is duemainly to the small aspect ratio s1/375 dand to the separation
between the elements, which is of the same order as their
width. We have estimated by theoretical analysis and micro-magnetic calculations, that interaction energy between theelements is 4% of the self-energy when the elements aresaturated. Furthermore, the differences in domain structuredo not occur until the Sstate is well established. The mag-
netization at this point, being ,0.6 of its saturated value,
further reduces the significance of a neighbor–neighbor in-teraction.
adElectronic mail: lau@bnl.gov
FIG. 1. Permalloy elements after removal of the applied field. The vortex
state sleftdis the ground state if the external field is removed rapidly. The
seven-domain state sright dis favored when the field is removed slowly. The
arrows indicate the direction of saturation.JOURNAL OF APPLIED PHYSICS 97, 10E702 s2005 d
0021-8979/2005/97 ~10!/10E702/3/$22.50 © 2005 American Institute of Physics 97, 10E702-1
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129.120.242.61 On: Fri, 28 Nov 2014 20:38:13EXPERIMENTAL DETAILS
The specimens are prepared by e-beam evaporation of a
permalloy target sNi81Fe19dunder UHV conditions. A
shadow mask overlay during deposition provides the square
lattice pattern. Each square measures 7.5 mm on the sides
and 20 nm in thickness.The substrate is a 50-nm-thick amor-phous silicon nitride membrane. A 1-nm carbon cap layerwas deposited afterwards to prevent specimen oxidation. Thepermalloy elements are polycrystalline, with a fairly uniformgrain size of ,10 nm. The diffraction patterns revealed no
obvious evidence of a textured film.
The entire experiment is conducted in situwith a JEOL
JEM3000F transmission electron microscope. Elements arebrought to saturation either with the strong vertical field ofthe microscope objective lens, or with a specimen holderoutfitted with homemade Helmholtz coils to provide an in-plane field. We previously calibrated the magnetic field as afunction of lens potential
8and found that a 3-T field is ap-
plied to the specimen when the objective is fully excited, andit is a linear function of lens voltage below 1 T. The Helm-holtz coils are calibrated by diffraction techniques
9and can
provide a steady in-plane field of 15 mT. Images are col-lected throughout various stages of reversal. Lorentz contrastprovides domain boundary information within the film plane.
We begin the experiment by bringing the permalloy ele-
ments to saturation with either the objective lens or with theHelmholtz coils. By stepping down either the lens voltage orthe coil current, we alter the external field.
bis defined as the
number of steps taken between saturation and removal of thisfield. For the objective lens and Helmholtz coils setup, theduration of each
bstep is 400 and 17 mS, respectively. Large
bcorresponds to a large number of stops along the way,
whereas b=1 corresponds to an abrupt switch off of the
applied field. In the Helmholtz coil setup, a short burst ofcurrent, producing 60 mT is initially applied to ensure fullsaturation. When the objective lens is used, the specimen isfirst exposed to the 3-T field. The controlled stepping downof lens voltages begins at 1 T, taking advantage of the linearregion. A large area-of-view image is taken when the exter-nally applied field is removed. Each captured image shows,between 300 and 400 elements, a number which we define asn
0.We also define nvas the population of the vortex state and
n7=n0−nvas the population of the seven-domain state. The
number of vortex states snvdis counted up in each image, and
the fractional population of the vortex state snv/n0d,a sa
function of b, is tallied. Care is taken to ensure that the
identical area is imaged for the entire experimental range of
bvalues.
MECHANISM OF MAGNETIZATION REVERSAL
A succession of Lorentz images during an in-plane re-
versal experiment offers clues to the mechanism involved inmagnetic reversal. Figure 2 is a flowchart that illustrates theevolution of a reversal half-cycle, from saturation to the stateand that results when the external field is removed. Experi-mental images are in grayscale; simulations are in color. Fo-cusing only on the grayscale images for now, we have inframe 1, a saturated element. Relaxing the field, the elementgoes into the Sstate
5in frames 2 through 3. Further reduc-
tion of the applied field causes the abrupt collapse of the S
state either into the vortex or into the seven-domain state, as
seen, respectively, in frames 5a sred arrow dand 6 sblue ar-
rowd. No experimental images are available between the col-
lapse of the Sstate and either of the two final states. This
collapse and subsequent domain rearrangement occur ontime scales not resolvable by the eye and therefore we turn tomicromagnetics for an explanation.
We use Landau–Lifshitz–Gilbert sLLG dmicromagnetics
simulator v.2.55 sRef. 10 dto study this problem. Due to com-
putational limitations, we simulated a permalloy solid withdimensions of 4
mm34mm320 nm2with N x:Ny:Nz
=512:512:1 pixels. We used saturation magnetization sMs
=813 erg/cm3d,11exchange coupling constant sA
=1.05 merg/cm d,11and a tiny uniaxial anisotropy constant
sKu=1.0 erg/cm3dsRef. 12 das input parameters. As we are
interested only in the final state, we used a damping factor
a=1.13
Watching the relaxation on the picosecond time scale
provided a likely scenario from which both the vortex andseven-domain states are created. There is a very good agree-ment between the simulation and experiment results, as weexamine again the flowchart sFig. 2 d.As in experiment, upon
release from saturation, sframe 1 d, the simulated element
manages the Sstate sframes 2 and 3 d. Note here in frame 3,
two small squares labeling regions along the upper and loweredges of the element, magnetic vortices will most likelynucleate here. Lorentz images are in precise agreement withsimulation to this point and corroborate the dynamics sug-gested by the simulation. The details of the transformation tothe vortex or seven-domain state, invisible in experiment, arerevealed beginning at frame 4. A vortex nucleates along thetop edge and proceeds to travel towards the center in the nextframe. In an effort to minimize total energy, domains growand shrink while boundaries are rearranged. Both the vortexand the seven-domain states have this common ancestry untilframe 5. If the domains continue to expand until the extra-neous wall is expelled, the result is the vortex state s5ad. If,
however, a second vortex nucleates prior to the completeexpulsion of the extraneous wall, as in frame 5b, the subse-quent events will continue on this path until the seven-domain state is reached sframe 6 d. A white vortex state with
a counterclockwise circulation would occur, instead of the
FIG. 2. sColor dEvolution of the vortex and seven-domain states. Single
nucleation event leads to the vortex state; two nucleations produce theseven-domain state. The color corresponds to the magnetization direction ofthe local domain. Red, green, blue, and yellow correspond to right, left, up,and down, respectively.10E702-2 Lau
et al. J. Appl. Phys. 97, 10E702 ~2005 !
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129.120.242.61 On: Fri, 28 Nov 2014 20:38:13one in frame 5a, if the vortex had nucleated instead from the
bottom edge in frame 3. Note that, contrary to the vortexstates, the seven-domain state lack randomness in direction-ality, as we always observe a white wall on the left and blackwall on the right. This is due to the fact that the elements areall saturated along the same axis. This in turn predeterminesa single available pathway for an element during relaxation;should both branches of the Sstate collapse during reversal,
the course for the seven-domain state is set.
DOMAIN STRUCTURE CORRELATION
Figure 3 is the fractional population of the vortex state
snv/n0das a function of bfor both in- and out-of-plane mag-
netization reversal experiments. The trend is very similar for
both directions of field application.Apopulation inversion isclearly established between the two
bextremes. The seven-
domain state is favored at large bsslow switching dwhereas
the vortex state is the favorite at small bsfast switching d.
From micromagnetic simulations, we found that the differ-ence between the two states lies in the number of nucleationevents. Our data show that slow switching processes gener-ally favor the nucleation of vortices. Such probability of vor-tex nucleation may be further explored in terms of activatedbarriers and Arrhenius rate processes. The details of thiswork will be made available in a separate publication.CONCLUSION
We identified a single reversal mechanism that leads to
vastly different minimum energy states in micron-sized per-malloy square elements. Reversal occurs by vortex nucle-ation during the collapse of the Sstate. A single nucleation
leads to the vortex state whereas two nucleations produce theseven-domain state. This mechanism is likely to be valid insquare elements up to ten times smaller.
14Furthermore, our
data show that the reversal pathways are rate dependent. Inthe slow switching regime, the system stops frequently dur-ing the process, and a number of attempts at nucleating avortex are taken during each of these pauses.Therefore, eachelement is given more chances to overcome nucleation bar-riers. In the fastest regimes, on the other hand, the system isgiven far fewer chances for nucleation. It is known that
b
influences coercivity and domain configurations;15we are
now in the position to model the attempt mechanics of mag-netic vortex nucleation as a function of applied field and rateof field change.
ACKNOWLEDGMENTS
The authors gratefully acknowledge M. R. Scheinfein
for his valuable input. This work was supported by the U.S.Department of Energy, Basic Energy Sciences, under Con-tract No. DE-AC02-98CH10886.
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FIG. 3. Fractional vortex population as a function of reversal rate b.T h e
solid squares are the data from out-of-plane magnetic reversal, while opensquares are the in-plane data. Fast reversals ssmall
bdoverwhelmingly favor
the vortex state whereas the opposite is true for slow reversals slarge bd.The
single error bar associated with the first data point in the out-of-plane seriesrepresents the standard deviation obtained from fast toggling of the objec-tive lens 30 times, imaging, and counting between toggles.10E702-3 Lau
et al. J. Appl. Phys. 97, 10E702 ~2005 !
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129.120.242.61 On: Fri, 28 Nov 2014 20:38:13 |
1.4943758.pdf | Spin Hall effects in metallic antiferromagnets – perspectives for future spin-
orbitronics
Joseph Sklenar, , Wei Zhang, , Matthias B. Jungfleisch , Wanjun Jiang , Hilal Saglam , John E. Pearson , John
B. Ketterson , and Axel Hoffmann
Citation: AIP Advances 6, 055603 (2016); doi: 10.1063/1.4943758
View online: http://dx.doi.org/10.1063/1.4943758
View Table of Contents: http://aip.scitation.org/toc/adv/6/5
Published by the American Institute of Physics
Articles you may be interested in
Perspective: Interface generation of spin-orbit torques
AIP Advances 120, 180901 (2016); 10.1063/1.4967391AIP ADV ANCES 6, 055603 (2016)
Spin Hall effects in metallic antiferromagnets –
perspectives for future spin-orbitronics
Joseph Sklenar,1,2,aWei Zhang,1,bMatthias B. Jungfleisch,1Wanjun Jiang,1
Hilal Saglam,1,3John E. Pearson,1John B. Ketterson,2and Axel Hoffmann1
1Materials Science Division, Argonne National Laboratory, Lemont IL 60439, USA
2Department of Physics and Astronomy, Northwestern University, Evanston IL 60208, USA
3Department of Physics, Illinois Institute of Technology, Chicago IL 60616, USA
(Presented 14 January 2016; received 6 November 2015; accepted 2 December 2015;
published online 7 March 2016)
We investigate angular dependent spin-orbit torques from the spin Hall e ffect in a
metallic antiferromagnet using the spin-torque ferromagnetic resonance technique.
The large spin Hall e ffect exists in PtMn, a prototypical CuAu-I-type metallic anti-
ferromagnet. By applying epitaxial growth, we previously reported an appreciable
difference in spin-orbit torques for c- and a-axis orientated samples, implying aniso-
tropic e ffects in magnetically ordered materials. In this work we demonstrate through
bipolar-magnetic-field experiments a small but noticeable asymmetric behavior in the
spin-transfer-torque that appears as a hysteresis e ffect. We also suggest that metallic
antiferromagnets may be good candidates for the investigation of various unidirec-
tional e ffects related to novel spin-orbitronics phenomena. C2016 Author(s). All
article content, except where otherwise noted, is licensed under a Creative Commons
Attribution 3.0 Unported License. [http: //dx.doi.org /10.1063 /1.4943758]
I. SPINTORQUEFERROMAGNETICRESONANCE:METROLOGYFORSPIN
HALLEFFECTS
The spin Hall e ffect (SHE) in metallic systems has become the leading way for inter-converting
charge currents to spin currents and vice versa through the inverse e ffect.1–3Within the research
field that focuses on the SHE, an ever growing subset of experiments are interested in using the SHE
to create spin currents that can be injected into a neighboring ferromagnet to induce ferromagnetic
resonance (FMR).4–6If a spin current injected into a ferromagnet (FM) is polarized non-collinear
to the magnetization of the FM a torque is exerted on the FM; this is called a spin-transfer-torque
(STT).7–11A STT does not act on the FM as a field-like torque does, rather a STT acts as a
(anti)damping-like torque. A (anti)damping-like torque arises because if a spin current is injected
into a FM, it is the component of the spin that is not collinear to the magnetization that is ab-
sorbed.12The experiments that utilize the SHE as an agent of STT follow a useful experimental
approach that was established by Liu et al ., where the SHE in Pt was used to exert a STT on
an adjacent permalloy (Py) layer.4More specifically, the experiments tend to use bilayer samples
where a spin Hall metal and ferromagnetic metal are fabricated directly on top of one another. A
microwave charge current is then passed through the device to generate FMR,13and the resulting
lineshape of the FMR is detected and used to obtain information about the STT. In these spin-torque
FMR (ST-FMR) experiments the FMR is detected through a rectification process where the aniso-
tropic magnetoresistance14(AMR) of the ferromagnetic metal mixes with the charge current when
FMR is occurring. This rectification process is well understood and has been observed in other
experimental configurations that include nanopillars, nanowires, and thin films.15–20Variants on the
ST-FMR experiments described above are being explored. One example is where the spin Hall
magnetoresistance in a Pt metal deposited on the ferrimagnetic insulator Y 3Fe5O12(YIG) lead to
aPresent address: Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
bAuthor to whom correspondence should be addressed. Electronic mail: zwei@anl.gov.
2158-3226/2016/6(5)/055603/10 6, 055603-1 ©Author(s) 2016
055603-2 Sklenar et al. AIP Advances 6, 055603 (2016)
FIG. 1. A schematic of the shorted coplanar waveguide for ST-FMR measurement is shown in (a) together with the bias-tee
configuration. This enables the simultaneous dc voltage measurement across the trilayer as microwaves pass through the
structure. The inset in (a) defines the polar angle coordinate scheme that is discussed later. An optical image of the actual
device is shown in (b). The large contact pads are Au electrical lines and the center line is bridged to the ground line by
a Py/Cu/PtMn trilayer. (c) An example of the ST-FMR detected lineshape is shown for the a-axis PtMn sample that was
measured atφ=−45◦andφ=135◦. The black data points are fit to the red curve which is given by Eq. (2).
rectification.21–23A second example of note is STT generated by the Rashba e ffect in Py /Ag/Bi as
opposed to spin Hall e ffects.24,25For this work, the experimental set-up used to detect the AMR
rectification is shown in Fig. 1(a) and 1(b) and a ST-FMR dc detected signal is shown in (c).
II. ANTIFERROMAGNETS:MATERIALSOFINTEREST
The e fficiency of the SHE can be characterized by the spin Hall angle ( θSH),2,3which is
typically determined by the intrinsic spin-orbit coupling of the materials involved, and therefore
cannot readily be varied by additional external parameters. Thus, an ongoing goal is to seek new
materials with SHEs that can generate large STT. On the other hand, it was recently found that
the magnetic-proximity-induced magnetization states of heavy metals (Pt and Pd) also a ffects their
intrinsic SHE;26this result suggests that magnetically ordered materials may o ffer additional oppor-
tunities to tune the intrinsic SHE via their atomic spin magnetic moments. Such a mechanism is
different from the macroscopic “magnetization e ffect” in ferromagnetic materials. In the spirit of
searching for new materials along these lines, we demonstrated that the inverse spin Hall e ffect
exists in metallic antiferromagnets (AFs) through the direct dc detection of a spin pumping process
from an adjacent Py layer.27Our work strongly corroborated a previous measurement of the inverse
spin Hall e ffect in IrMn28by extending it to a wide variety of CuAu-I-type antiferromagnetic alloys,055603-3 Sklenar et al. AIP Advances 6, 055603 (2016)
FIG. 2. Schematic illustration showing (a) the chemical and magnetic structures of PtMn, and (b) a spin current generated
transverse to a charge current that will result in STT to an adjacent ferromagnet.
such as PtMn, which consists of both heavy-metal elements (Pt) and an atomic-level, staggered
magnetization (Mn).
The observation of inverse spin Hall e ffects in AFs quickly triggered a series of studies in
the context of AF-spintronics.29–36Given the growing interest in metallic AFs,37another natural
question is whether they are capable of generating e fficient spin currents through the direct spin
Hall e ffects, as predicted by Onsager reciprocal relations. We recently confirmed large spin current
generation from such metallic AFs, in particular PtMn, through ST-FMR experiments.38Further-
more, we showed that the strength of the STT in ST-FMR experiments could be controlled by
utilizing di fferent crystal structures of the same AF materials.38A schematic illustration of the
chemical and magnetic structures of PtMn and its spin Hall e ffect induced spin current generation
are Fig. 2(a) and 2(b).
Here, we will build on this work by presenting a systematic study of the ST-FMR lineshapes
as a function of the in-plane angular direction. We will also present so-called “hysteresis e ffects”
in the AF systems that are present in the STT. Angular dependent studies are of particular interest
because measurement at one in-plane angle may not reveal all the physics within a given system. To
wit, the presence of new phenomenological torques acting on the system can be overlooked,39,40and
hysteresis e ffects which may be impacting interface transparency41,42could be missed. An example
of a system that has recently benefited from a complete in-plane study is the magnetic insulator
system YIG and the well understood spin Hall metal Pt configured into a ST-FMR experiment.22,23
Here, the complete in-plane angular dependence uncovered an additional dc voltage contribution
that was not included in the theoretical model;43the extra dc signal is thought to arise from the
on-resonant microwave heating of the sample which in turn provided incoherent spin pumping.
The incoherent spin pumping was detected as a dc signal via the inverse spin Hall e ffect. In that
work it was shown that quantitative studies on the parameters describing the YIG /Pt interface can
dramatically change depending on how the unexpected signal was accounted for Ref. 22.
III. THECLASSIC“TWO-TORQUE”MODEL
Our work on ST-FMR for AF metals used the popular “lineshape analysis” for ST-FMR exper-
iments.4As will be described below, to successfully rely on such an analysis experimental design is
critical, and a proper device geometry usually consists of a shorted coplanar waveguide (CPW) as seen
in Fig. 1(a) and 1(b), where a center Ti /Au conduction line is bridged to two return ground lines by a
narrow bar. The narrow bar is a trilayer following PtMn /Cu/Py. As an example, we show PtMn grown
on MgO single-crystal substrates in an a-axis configuration.38,44The thickness of the AF layer is 10 nm,
the interfacial dusting Cu layer is 1 nm, and the top Py layer is 5 nm. The dimensions of the trilayer
completing the shorted CPW configuration are 90 ×20µm2. As indicated by the circuit schematic
in Fig. 1(a), a bias-tee configuration is used to simultaneously allow microwaves to pass through the
device, as well as for the longitudinal dc voltage that develops across the bar to be measured. It is
generally accepted that the rectification mechanism leading to this dc voltage is due to the oscillating055603-4 Sklenar et al. AIP Advances 6, 055603 (2016)
AMR mixing with the microwave charge current passing through the device. If the device is well
designed there should only be two torques acting on the ferromagnetic layer as seen here:
dˆm
dt=−|γ|ˆm×Hef f+αˆm×dˆm
dt+|γ|τFˆm׈y+|γ|τSTTˆm×(ˆy׈m), (1)
where Hef fincludes both the experimentally applied magnetic field and thin film demagnetization
fields,τFis the magnitude of the field-like torque that is coming from the Oersted field in the
y-direction, and τSTTis the spin-transfer-torque magnitude that arises from the spin polarization
being injected into the FM. The direction of the spin polarization for this type of device is the
y-direction. Additionally, the coe fficientαis the Gilbert damping factor and γis the gyromagnetic
ratio. More quantitative studies are often interested in breaking both τFandτSTTinto functions of
the film thicknesses, the charge current through the device, and important material specific parame-
ters like the spin Hall angle and spin mixing conductance. For the materials considered here this has
been done elsewhere.38
In this work the phenomenological torque model combined with an angular dependent study
strengthens previous results38and points to new directions of future work. Because the two torques
in Eq. (1) have di fferent mathematical forms, they will not tend to drive the magnetization preces-
sion at the same phase. Thus under idealized conditions the STT is responsible for the symmetric
part of the rectified lineshape and the field-like torque is responsible for the antisymmetric part
due to mixing with the charge current. By varying the in-plane angle φin Fig. 1, the amplitude
of the lineshape will change for the symmetric and antisymmetric components. Assuming this
two-torque model and a rectification through AMR, both the symmetric and antisymmetric ampli-
tude will follow a sin 2 φcosφangular dependence.17,45The sin 2φcontribution is related to the
amplitude of the oscillating AMR at a given angle φ, while the cos φcontribution is related to the
cross product( m׈y) in the equation of motion. By fitting the rectified dc voltage to the following
function40at many values of φone can test the phenomenological torque model:
Vdc=S∗∆
(H2−H2
F M R)2+∆2+A∗(H2−H2
F M R)
(H2−H2
F M R)2+∆2. (2)
Here the symmetric and antisymmetric amplitude coe fficients Sand Aare proportional to the
magnitude of τSTTandτFrespectively. It is these two coe fficients that should follow the prescribed
sin 2φcosφangular dependence. Additionally, the parameter ∆is proportional to the square of the
linewidth and is a fitting parameter, while HF M R is the field where ferromagnetic resonance occurs.
In the experiments to be presented below we will fix the microwave charge current frequency at
5.5 GHz and sweep the applied magnetic field H. An example of ST-FMR curves that are typically
measured and fit to Eq. (2) are shown in Fig. 1(c).
The antisymmetric lineshape component is well described by a sin 2 φcosφangular depen-
dence. These results are shown in Fig. 3(a), in which the red curves are fits that are only the
expected sin 2 φcosφfunction. Note that the amplitude of the antisymmetric amplitude has been
normalized to unity. If a poorly designed device created an e ffective field in the x-direction there
can be an additional sin 2 φsinφterm that describes the antisymmetric angular dependence.45We
did not see any contribution from a sin 2 φsinφin the antisymmetric amplitude above the 1% level
relative to the sin 2 φcosφamplitude. The angular dependence of the symmetric amplitude was
mostly described by sin 2 φcosφ, however this was not the only term of significance. As shown in
Fig. 3(b), the angular dependence has an additional sin 2 φdependence that has an amplitude 5% of
the sin 2φcosφterm. In Fig. 3(b) the blue curve is the contribution from the sin 2 φcosφamplitude
while the green curve is the sin 2 φamplitude. The red curve that best fits the data is the sum of
both functions. From a phenomenological point of view such an angular dependence can arise from
an out-of-plane, field-like torque that would be ∝ˆm׈z.45We also attempted to extract any sin φ
angular dependence from the data but nothing significant was found. As mentioned above, in the
case of magnetic insulators, an additional sin φangular dependence has been shown to exist in the
symmetric component of the ST-FMR lineshape.22
In the above analysis we searched for additional torques in the AF system. For completeness,
here we summarize the in-plane (IP) angular dependence that di fferent field-like and damping-like055603-5 Sklenar et al. AIP Advances 6, 055603 (2016)
FIG. 3. (a) and (b): The antisymmetric and symmetric amplitude factors for the a-axis PtMn sample are each plotted as a
function ofφrespectively. In (a) the red curve is the fit and corresponds only to the function sin2 φcosφ. In (b) the red curve
is the total fit but it is the addition of both the blue and green curves shown which are sin2 φcosφand sin2φrespectively.
torques should have on symmetric and antisymmetric components of ST-FMR lineshapes. Skinner
has previously stated the angular dependence the three x, y, z field-like torques45which make up the
first three entries of Table I. Here we add the additional x, y, z damping-like torques which make up
the final three entries of Table I. The method of deriving the angular dependences we employed is
outlined elsewhere.46
To conclude, the full in-plane angular dependence confirms that by using a shorted CPW
design, for the Py /Cu/PtMn samples, a two torque phenomenological model that utilizes AMR as
a means of rectification seems highly appropriate to describe most of the physics. The presence of
small field-like torques originating from STT are known to exist in insulating ferromagnets; these
torques are best described by an imaginary spin mixing conductance parameter.22,47Our in-plane
angular dependence may provide early evidence that a similar e ffect is occurring within the AF
system; there is a di fference however as the field we extract here is pointing in the z-direction. This
raises the possibility that the field-like torque is intrinsic to the design of the sample; an imperfect
shorted co-planar waveguide device may naturally create a field-like torque in the z-direction.
IV. POSSIBLEEXCHANGEBIASEFFECTSFORSPINRECTIFICATION
Unidirectional anisotropy is universally seen in many magnetic systems. The exchange bias
is one good example which is still stimulating new scientific findings nowadays [Fig. 4(a)]. There
are often subtle e ffects that break the symmetry and give rise to various unidirectional asymme-
try, such as the Dzyaloshinskii-Moriya interaction breaking the momentum inversion symmetry
TABLE I. The angular dependence of the symmetric and antisymmetric components of the ST-FMR lineshape are given for
differing driving torques. The first three entries are the x, y, z field-like torques, while the final three entries are the x, y, z
damping-like torques.
Torque Form “S” In-Plane “A” In-Plane
(ˆm׈x) 0 sin φsin2φ
(ˆm׈y) 0 cos φsin2φ
(ˆm׈z) sin2φ 0
[ˆm×(ˆx׈m)] sinφsin2φ 0
[ˆm×(ˆy׈m)] cosφsin2φ 0
[ˆm×(ˆz׈m)] cosφsin2φ sin2φ055603-6 Sklenar et al. AIP Advances 6, 055603 (2016)
FIG. 4. Summary of typical unidirectional e ffects: (a) Exchange bias, (b) spin wave dispersion shift, (c) unidirectional spin
Hall magnetoresistance, and (d) unidirectional spin rectification e ffect.40
for the spin-wave dispersion [Fig. 4(b)]48and the spin-accumulation giving rise to the unidirec-
tional spin-Hall magnetoresistance [Fig. 4(c)].49Antiferromagnets are well-known for their strong,
intrinsic anisotropies. Along with the full angular dependence, we are paying extra attention to the
bi-polar dependence of the lineshapes, which is so far an under-explored ingredient of the ST-FMR
experimental paradigm. Spin-transfer-torque originating from the SHE in AFs may be an ideal
system to study such behaviors involving possibly unidirectional asymmetry.
Before showing in-plane results with the PtMn sample we will summarize a recent realiza-
tion of unidirectional asymmetry in a more conventional ST-FMR experiment. This asymmetry is
achieved in a Py /Pt bilayer by tipping the magnetization out of the plane of the sample [Fig. 4(d)].40
First consider a general description of the magnetization where there is an out-of-plane (OOP)
component. In this case the magnetization can be described by two angles, ˆm(φ,ψ ), whereψis a
polar angle (see the inset of Fig. 1(a)). For the in-plane case ψ=90◦. ST-FMR can be generated
in an arbitrarily magnetized slab just like for the in-plane case.21,22,40When AMR is the main
rectification mechanism there are two notable di fferences compared to the pure in-plane (IP) case.
The first di fference is that the oscillating AMR that leads to rectification now has a contribution
from the polar dependence of the resistance:
TABLE II. For an arbitrarily magnetized magnetic film undergoing ST-FMR a single torque will contribute both a symmetric
and antisymmetric component to the lineshape. When the magnetic field is reversed from a given configuration the polarity
of each component from a given torque may or may not switch. The columns labeled S (+H→−H)and A (+H→−H)
correspond to the symmetric and antisymmetric components respectively. A value of 1 indicates that the polarity is not
switched while a value of -1 indicates that the polarity is switched upon magnetic field reversal.
Torque Form S (+H→−H) A(+H→−H)
(ˆm׈x) 1 -1
(ˆm׈y) 1 -1
[ˆm×(ˆx׈m)] -1 1
[ˆm×(ˆy׈m)] -1 1
[ˆm×(ˆz׈m)] 1 -1055603-7 Sklenar et al. AIP Advances 6, 055603 (2016)
Vdc∝dR
dφδφ+dR
dψδψ. (3)
The first term in Eq. (3) leads to the angular dependences summarized in Table I when ψ=90◦
anddR
dψ=0. The second term is the additional voltage that is gained from the OOP AMR recti-
fication. The change in resistance with respect to the magnetization orientation is assumed to be:
dR/dφ∝sin 2φsinψanddR/dψ∝sin 2ψ. The quantities δφandδψare the amplitudes of the
angular coordinates of the magnetization about the equilibrium position when FMR is occurring.
One can relate δφandδψto the actual amplitude of the oscillating magnetization which will in turn
be described by the actual torques driving the system.40,46This brings forward the second notable
difference between IP and OOP ST-FMR experiments. For a general magnetization orientation a
given torque can contribute to both the symmetric and antisymmetric component of the lineshape.
Furthermore, for the configuration where φ=45◦is fixed and ψvaries, it has been shown that
FIG. 5. In (a) and (b) the bipolar ST-FMR traces are shown for the a-axis and c-axis sample respectively. The data is
plotted with black dots and the total fit which is the red curve is made from the antisymmetric function in blue and the
symmetric function in green. The sweep direction for the field is indicated by the black arrows above or below each trace.
Additionally, each trace has a label I-IV . Note that traces II and IV both have suppressed symmetric amplitudes. In (c) the
angular dependence of the c-axis symmetric amplitude is shown. ST-FMR traces that were swept from 0 kOe to ±1 kOe are
plotted in red while the traces that were swept from ±1 kOe towards 0 kOe are plotted in black.055603-8 Sklenar et al. AIP Advances 6, 055603 (2016)
reversing the magnetic field (letting φ→180◦+φandψ→180◦−ψ) does not necessarily reverse
the sense of a given symmetric or antisymmetric component. As an example, a field-like torque of
the form given in Eq. (1) will contribute to a symmetric component of the lineshape for an arbitrary
ψvalue, but this component will not change signs under field reversal. This is in contrast to the
antisymmetric component that the field-like torque contributes to which will change signs under
field reversal. In Table II, shown below, a summary of how the lineshape from each individual type
of torque changes shape upon magnetic field reversal summarizes the spin-torque unidirectional
rectification e ffect.
In the ST-FMR traces shown in Fig. 1(c), and the angular analysis in Fig. 3, the field was swept
from a high field magnitude (-1 kOe) to a low field magnitude. Specifically, all data shown and
analyzed in the preceding figures started with the field at ±1 kOe sweeping towards 0 kOe. While
performing our experiments we also recorded bi-polar data where the field, after coming from a
saturated state, is swept from 0 kOe to the opposite field direction (stopping at a field of magnitude
1 kOe). An example of two bi-polar traces where the field is swept from - 1 kOe to +1 kOe and
then back to - 1 kOe yielding four ST-FMR traces is shown in Fig. 5(a) and 5(b) for a-axis and
c-axis samples respectively. For each plot four traces are labeled (I-IV) where traces I and III are
swept through starting with a saturated state. On the other hand traces II and IV are the new traces
that are swept through starting at a field of 0 Oe. One can readily see by eye that the magnitude
of the antisymmetric amplitude of the lineshape is generally the same for all traces (I - IV). This
is not the case for the symmetric amplitude where there is a noticeable di fference in the magnitude
amongst these traces. In fact, the magnitude of the symmetric amplitude is suppressed for II and IV
by approximately a factor of three. We have termed this a hysteresis e ffect and it is best illustrated in
Fig. 5(c). In Fig. 5(c) black circles are the non-suppressed data points where the field is swept from
a high magnitude and the red circles are the suppressed amplitude traces at the same angle when
the field is swept from 0 Oe. The angular dependence is mostly described again by sin 2 φcosφbut
it is far more interesting to note how the amplitude is suppressed. It would be strange to observe a
hysteresis e ffect in the antisymmetric component of the lineshape; in principle the Oersted field-like
torque generated by the coplanar waveguide geometry should be insensitive to the magnetic history
of the system. It is perhaps less strange that we observe hysteresis in the symmetric component of
the lineshape. It is easier to imagine that the spin current leading to a STT may encounter interface
transparency41,42issues related to magnetic history and /or interfacial unidirectional anisotropy. This
raises an interesting question: is the apparent reduction of a STT due to saturation e ffects within the
ferromagnet itself or is it related to the AFs due to unidirectional exchange bias e ffect. A thorough
experiment correlating interfacial exchange bias and the spin torque (transparency) e fficiency would
be necessary to answer such a question.
V. SUMMARYANDOUTLOOK
The ST-FMR experimental technique has become a powerful tool when combined with a line-
shape analysis. Using a prototypical AF material, i.e., PtMn, as our spin current source for ST-FMR
experiments, we point out that there may be additional detail requiring further analysis. Careful
angular dependent studies in combination with magnetic history experiments are straightforward
approaches to gain a deeper insight into underlying physical mechanisms, especially when studying
new materials.
A well-suited material system is often the key to accessing the many intriguing physics ef-
fects. Antiferromagnetic alloys such as CuAu-I-type AFs with both heavy elements (possessing
large atomic spin-orbit coupling) and magnetic atoms (possessing spin magnetic moment) serve
as an ideal model for fundamental investigation for spin-orbitronics. Moreover, their chemical
and magnetic robustness, high critical temperature and controllable growth may further lead to
practical applications in future magnetic devices. Furthermore, epitaxial samples with well-defined
interface properties and enhanced structural coherency o ffer new platforms for the study of novel
spin-orbitronics phenomena. The anisotropic spin Hall e ffect is only one example of new tunability
and functionality enabled by optimized material synthesis. Other new functionalities may be also055603-9 Sklenar et al. AIP Advances 6, 055603 (2016)
enabled including convenient control of spin currents and other interface properties by novel optical
and electrical means.
ACKNOWLEDGEMENTS
W.Z. is very grateful for many past and ongoing insightful discussions with his collaborators
and colleagues, Frank Freimuth and Yuriy Mokrousov (Jülich). This work was supported by the
U.S. Department of Energy, O ffice of Science, Materials Science and Engineering Division. Lithog-
raphy was carried out at the Center for Nanoscale Materials, which is supported by DOE, O ffice of
Science, Basic Energy Science under Contract No. DE-AC02-06CH11357. Work at Northwestern
was supported by the US Department of Energy, O ffice of Basic Energy Sciences, Materials Science
and Engineering Division under grant number DE-SC0014424.
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1.3673855.pdf | Reversal and excitations of a nanoscale magnetic domain by sustained pure spin
currents
Han Zou, Shuhan Chen, and Yi Ji
Citation: Applied Physics Letters 100, 012404 (2012); doi: 10.1063/1.3673855
View online: http://dx.doi.org/10.1063/1.3673855
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142.157.129.8 On: Sat, 13 Dec 2014 21:13:26Reversal and excitations of a nanoscale magnetic domain by sustained pure
spin currents
Han Zou, Shuhan Chen, and Yi Jia)
Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, USA
(Received 28 August 2011; accepted 10 December 2011; published online 5 January 2012)
Spin-transfer effects induced by pure spin currents are explored in nonlocal spin valves by using
sustained injection currents. Compared to pulsed injection currents used in previous experiments,
this approach provides persistent spin-transfer torques and preserves the history of the reversalprocess. A nanoscale domain in a magnetic wire can be switched reversibly by the sustained pure
spin currents. In addition, dips in nonlocal spin signal curves are observed at high magnetic fields
for only one polarity of the injection currents. This indicates stable-state magnetization precessionaround the external field driven by the sustained pure spin currents.
VC2012 American Institute of
Physics . [doi: 10.1063/1.3673855 ]
Spin-transfer effects1have attracted attention because it
is an effective and scalable approach to switch magnetiza-tions in nanoscale spintronic devices. Most experiments and
theories focus on the nanopillar structures,
2,3where a spin-
polarized charge current is utilized for spin-transfer. It ischallenging but intriguing to explore spin-transfer in nonlo-
cal spin valves (NLSVs),
4–10which are nanoscale multi-
terminal lateral structures and generate pure spin currents.Without a net charge current, a pure spin current offers a
spin angular momentum flow but induces little Joule heating
or electromigration, which is detrimental to high-density andultra-small devices.
There have been several experiments dedicated to spin-
transfer switching by a pure spin current. Most are based onthe lateral nonlocal structures,
11–14but one is based on a ver-
tical nonlocal structure.15The baseline of nonlocal resistance
varies dramatically under a bias current16such that the varia-
tion is usually much larger than the nonlocal resistance
change (spin signal) associated with magnetization reversal.
This makes it challenging to detect spin-transfer switching.Consequently, most experiments to date have been done by
applying a direct current (d.c.) for a period up to 2s to induce
switching, and after the withdrawal of the current, the nonlo-cal resistance is measured to probe the remanent magnetic
state of the nanomagnet.
It is essential to explore spin-transfer with a sustained
d.c. injection current, which gives rise to a sustained pure
spin current in NLSVs. Current pulses do not provide detailsof the reversal process such as how abrupt the reversal is.
The reversal by a pulse cannot be preserved if the nanomag-
net is not bistable. In addition, a sustained spin current gener-ates stable-state magnetization precession,
3which is an
important aspect of spin-transfer effects but has yet to be
demonstrated by using pure spin currents. Furthermore, theabrupt change of current associated with a pulse may induce
damage to the NLSV structures. A sustained injection cur-
rent, on the other hand, is varied gradually and avoids a sud-den change. In this work, we demonstrate spin-transfer
switching and stable-state magnetization procession inNLSV structures using sustained d.c. currents. The magnetic
entity excited by spin-transfer is a nanoscale domain(<80/C2100 nm
2) in a thin, narrow, but extended magnetic
nanowire, in contrast to an isolated nanomagnet used by
others.12
Fig.1(a) is a scanning electron microscope (SEM) pic-
ture of a typical NLSV device. The spin injector (F 1) and
spin detector (F 2), both made of permalloy (Py), are bridged
by a nonmagnetic Cu channel. The dimensions of the device
used in this work are the followings: The widths of F 1,F2,
and Cu are 200 nm, 80 nm, and 100 nm, respectively; thecenter-to-center distance between F
1and F 2is 400 nm. The
device is fabricated by evaporating different materials from
different angles through a suspended shadow mask createdby electron beam lithography.
17
The injection current is directed through terminals I þ
and I/C0, and the current carries spin accumulation into the Cu
channel. The diffusion of spin accumulation drives spin cur-
rents toward both ends of the Cu channel. A pure spin cur-
rent flows in the Cu channel from F 1toward F 2. Nonlocal
FIG. 1. (a) A scanning electron microscope picture of a NLSV structure. (b)
The cross sectional side view along dashed line in (a). (c) Nonlocal resist-
ance versus magnetic field ( Rsversus H) curve at 4.5 K. (d) Spin transfer
curve ( Rsversus Idc) with current pulses at 4.5 K.a)Electronic mail: yji@physics.udel.edu.
0003-6951/2012/100(1)/012404/3/$30.00 VC2012 American Institute of Physics 100, 012404-1APPLIED PHYSICS LETTERS 100, 012404 (2012)
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142.157.129.8 On: Sat, 13 Dec 2014 21:13:26voltage is measured between terminals V þand V/C0. The
cross sectional view of the device along the dashed line in
Fig. 1(a) is illustrated in Fig. 1(b).F1consists of a 20 nm
thick Py layer, and F 2consists of a 4 nm thick Py layer and a
6 nm thick Cu underlayer. The thickness of the Cu channel is
70 nm. A layer of AlO x(2 nm thick) is placed between Py
and Cu to enhance the interfacial spin polarization.18
The active region of F 2electrode for spin-transfer exci-
tation is nominally the overlapped area between F 2and Cu,
/C2480/C2100 nm2. This domain can be switched or excited by
the pure spin current. The resistances of the F 1/Cu and F 2/Cu
interfaces are measured using two terminal methods. For thedevice used in this paper, two-terminal resistances for F
1/Cu
and F 2/Cu are 90 Xand 460 Xat 4.5 K, respectively.This
NLSV structure requires only a single step of electron beamlithography, a distinct advantage compared to other
structures.
11,12
For spin signal measurement, an alternating current
(a.c.) Iacof 0.1 mA at 346 Hz is used as the injection current
and the a.c. nonlocal voltage Vacis measured using lock-in
technique. The nonlocal resistance Rs,(Rs¼Vac/Iac), as a
function of magnetic field ( H) at 4.5 K is shown in Fig. 1(c).
The magnetic field is applied parallel to the F 1and F 2elec-
trodes and swept between positive and negative values. Thereversal fields of F
1and F 2are different due to different
dimensions. At high fields, the magnetizations of F 1and F 2
are parallel (P). As the field is swept towards the opposite
polarity, an antiparallel (AP) state of F 1and F 2is established
after the reversal of F 1but before the reversal of F 2. The Rs
value is high in the P state and low in the AP state, and the
difference DRsis known as the spin signal. In Fig. 1(c),
DRs¼9mXat 4.5 K.
In Fig. 1(d), we show reversible spin-transfer switching
achieved with d.c. current pulses at 4.5 K for the same de-
vice. An a.c. current ( Iac) of 0.1 mA at 346 Hz is applied for
generating a.c. spin signal Rs, and a d.c. current ( Idc)i s
injected for inducing spin-transfer. The magnitude of the
pulse is progressively increased and the duration of the pulse
is 2 s. The Rsvalue is measured after the d.c. current pulse is
withdrawn and is plotted as a function of Idcin Fig. 1(d).
From the standard understanding of spin-transfer, a positive
Idcfavors F 2being aligned antiparallel with F 1and a nega-
tiveIdcfavors F 2being aligned parallel with F 1. This is con-
sistent with what is observed in Fig. 1(d). The transition
from P to AP state (high Rsto low Rs) occurs where
Idc¼Icþ¼þ3.5 mA, and the transition from AP to P state
(low Rsto high Rs) occurs where Idc¼Ic/C0¼/C03.7 mA. The
values of Icþand Ic/C0are known as critical switching
currents.
The difference between Rsvalues of P and AP states in
Fig.1(d) is 3.5 m X, which is less than the DRsof 9 m Xin
Fig. 1(c). This difference has been consistently observed in
several samples.14There are mainly two reasons: first, the
active domain can be smaller than the physically overlappedarea between Cu and F
2; second, there could be a surface an-
isotropy of the thin F 2layer, and the magnetization of F 2is
tilted out of plane in the zero-field resulting in compromisedP and AP states in the R
sversus Idccurve.
Another Rsversus Idccurve with current pulses meas-
ured for the same device at 100 K is shown in Fig. 2(a), andIcþ¼2.7 mA and Ic/C0¼/C0 2.0 mA. Both values are reduced
from those at 4.5 K due to the elevated temperature. Fig. 2(b)
is the Rsversus Idccurve achieved with sustained d.c. cur-
rents. The current Idcis scanned between /C04.0 mA and 4.0
mA with small increments and the Rsis measured, using
lock-in with a.c. current Iac, while the Idcis sustained in the
circuit. A dramatic shift of Rsas a function of Idcis seen in
Fig.2(b). This phenomenon is commonly observed in metal-
lic NLSV devices and can be explained by redistribution of
the injection current.16A clear hysteresis of the Rsversus Idc
curve is obtained in Fig. 2(b). Two distinct branches of the
curve represent the P state (higher Rs) and the AP state
(lower Rs). The Rsvalues merge onto the lower branch (AP
state) where Idc>Icþ¼2.5 mA and onto the higher branch
(P state) where Idc<Ic/C0¼/C01.9 mA. This is consistent with
measurements in Figs. 1(d) and2(a) as well as the common
understanding of spin-transfer. The critical currents for
switching in Fig. 2(b) are lower than those in Fig. 2(a)
because the heating effect of a sustained current is more pro-nounced than a current pulse. In Fig. 2(b), the difference
between the R
svalues on two branches at zero bias is 4.0 m
X, the same as the DRsin Fig. 2(a), reaffirming that the hys-
teretic curve is due to the spin-transfer effects. The Cu chan-
nel has a cross sectional area of 70 /C2100 nm2. At 100 K, the
injection current densities for switching are 3.6 /C2107A/cm2
from P to AP and 2.7 /C2107A/cm2from AP to P, averaging
3.1/C2107A/cm2.
Evidence for high frequency (GHz) magnetization pre-
cession induced by spin-transfer is also observed in NLSV
devices with sustained d.c. currents. Fig. 3shows Rsversus
Hcurves at 4.5 K with sustained currents applied. Again Rs
FIG. 2. (a) Spin transfer curve ( Rsversus Idc) with current pulses at 100 K.
(b) Spin transfer curve ( Rsversus Idc) with sustained injection currents at
100 K.012404-2 Zou, Chen, and Ji Appl. Phys. Lett. 100, 012404 (2012)
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142.157.129.8 On: Sat, 13 Dec 2014 21:13:26is measured using lock-in with a.c. current Iac. The field is
scanned back and forth between /C00.4 T and þ0.4 T, but the
branch with decreasing field (from þ0.4 T to /C00.4 T) is not
shown to avoid overcrowding. Injection currents of þ2.0
mA,/C02.0 mA, þ3.0 mA, and /C03.0 mA are applied for Figs.
3(a)–3(d), respectively. The dips in the low field range
(/C00.1 T<H<0.1 T) correspond to the nonlocal spin signal
as shown in Fig. 1(c). Additional dips in the Rsversus H
curve appear at high fields but only with the positive cur-
rents. In Fig. 3(a), broad dips centered at /C2460.2 T are
observed with þ2.0 mA. In Fig. 3(c), sharper dips are
observed at /C2460.3 T with þ3.0 mA. The dips are absent in
curves with negative currents ( /C02.0 mA in Fig. 3(b) and
/C03.0 mA in Fig. 3(d)). These polarity-dependent dips at high
fields indicate spin-transfer dynamics, as explained in the
following.
There are three torques exerted on the magnetization
vector of the active domain in F 2. The torque by the external
magnetic field induces precession of magnetization aroundthe external field. The Gilbert damping torque drives the
magnetization toward equilibrium and dissipates the preces-
sion. The spin-transfer torque is parallel to the damping tor-que for the negative injection current, but anti-parallel to the
damping torque for positive injection current.
Under a negative current, the spin-transfer torque merely
enhances the damping and drives the F
2magnetization into
being parallel with external field and the F 1. The Rsvalue
remains high in fields higher than coercive fields of F 1and
F2, as seen in Figs. 3(b) and3(d). Under a positive current,
the spin-transfer torque balance the damping torque and pre-
cession of F 2magnetization around external field is sus-
tained. The precession of F 2magnetization results in a
decrease of time-averaged nonlocal resistance Rsfrom the P
states because the F 1and F 2magnetizations are no longer
parallel. Therefore, dips in Rsversus Hcurves emerge in
high field and only for positive bias. The dip moves to higher
fields as the positive bias current is increased because ahigher spin current can excite the precession in a higher
field.
These features are similar to the differential resistance
peaks observed in nanopillars2,3and point-contacts,19,20where
the peaks appear in only one polarity of the current and moves
to higher field as the current is increased. In the case with
NLSVs, the features are dips instead of peaks because the Pstate has a high R
svalue and AP state has a low Rsvalue, op-
posite to the case of giant magnetoresistance in nanopillars.
The frequencies of spin-transfer driven precession in nanopil-
lars and point-contacts are in the GHz range.3A similar fre-
quency range is expected for the precession induced by purespin currents. The results in Fig. 3demonstrate the feasibility
of inducing spin-transfer dynamics by a pure spin current.
Direct measurement of nonlocal voltage oscillation in the fre-quency domain is of interest as future work.
In conclusion, we have explored spin-transfer effects in
nonlocal spin valves with sustained injection currents and,thereby, sustained pure spin currents. Both reversible switch-
ing and magnetization precession are achieved by spin-
transfer. The injection current density for switching is3.1/C210
7A/cm2at 100 K, compared favorably to previous
works. Polarity-dependent high field features in nonlocal re-
sistance provide evidence for dynamic precession. The activeregion switched or excited by the spin current is a nanoscale
domain smaller than 80 /C2100 nm
2in the extended permal-
loy spin detector.
We acknowledge the use of NanoCenter facilities at
University of Maryland. We acknowledge the funding sup-
port from DOE Grant No. DE-FG02-07ER46374.
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FIG. 3. (a) Nonlocal resistance versus magnetic field ( Rsversus H) meas-
ured at 4.5 K with a bias current of (a) 2.0 mA, (b) /C02.0 mA, (c) 3.0 mA,
and (d) /C03.0 mA. The magnetic field is scanned from negative values to
positive values.012404-3 Zou, Chen, and Ji Appl. Phys. Lett. 100, 012404 (2012)
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142.157.129.8 On: Sat, 13 Dec 2014 21:13:26 |
1.4824304.pdf | Localized parametric generation of spin waves in a longitudinally magnetized Ni81Fe19
waveguide
T. Brächer, P. Pirro, A. A. Serga, and B. Hillebrands
Citation: Applied Physics Letters 103, 142415 (2013); doi: 10.1063/1.4824304
View online: http://dx.doi.org/10.1063/1.4824304
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93.180.53.211 On: Wed, 06 Nov 2013 09:27:23Localized parametric generation of spin waves in a longitudinally
magnetized Ni 81Fe19waveguide
T. Br €acher,1,2P . Pirro,1A. A. Serga,1and B. Hillebrands1
1Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universit €at Kaiserslautern,
D-67663 Kaiserslautern, Germany
2Graduate School Materials Science in Mainz, Gottlieb-Daimler-Strasse 47, D-67663 Kaiserslautern,
Germany
(Received 29 August 2013; accepted 19 September 2013; published online 3 October 2013)
We demonstrate that in a longitudinally magnetized Ni 81Fe19waveguide spin waves can be
generated via parallel parametric generation by a microstrip antenna. By employing microfocusBrillouin light scattering spectroscopy, we show that this method provides an efficient excitation
source for backward volume spin waves. We analyze the spatial distribution of the generated spin
waves, proving that odd and even waveguide modes can be excited. Furthermore, we study thespin-wave propagation along the Ni
81Fe19waveguide, revealing that the generation process takes
place underneath the antenna due to its threshold nature. VC2013 AIP Publishing LLC .
[http://dx.doi.org/10.1063/1.4824304 ]
The excitation of spin waves, or magnons, in ferromag-
netic microstructures has been extensively studied in the
past years.1–7This is due to the fact that, for instance, the
utilization of magnonic currents, i.e., the angular momen-
tum which is transported by propagating spin waves, offers
a viable alternative to electric currents.8–10These currents
are predicted to be able to interact with topological objects
such as domain walls.2,11This offers promising potential for
the realization of logic devices based on domain wallmotion.
12,13For this purpose, the efficient excitation of spin
waves in a longitudinally magnetized magnonic waveguide
is of great importance as the large fields that are needed totransversally magnetize a waveguide in general lead to the
absence of domain walls. However, in a longitudinally mag-
netized microstructured magnonic waveguide, the excitationof spin waves needs to be further explored, as the excitation
by the direct torque of a microwave current flowing through
a microstrip antenna is inefficient and non-local in thisgeometry.
1,7,14
In this Letter, we demonstrate localized parallel paramet-
ric generation of backward volume spin waves, i.e., the paral-lel parametric amplification of thermal spin waves,
4,15–17in a
longitudinally magnetized Ni 81Fe19waveguide (see geometry
in Fig. 1(a)). The in-plane component of the Oersted field cre-
ated by a microwave current flowing through the microstrip
antenna placed across the waveguide exerts no torque on the
static magnetization. Still, it can be used to parametricallygenerate spin waves at half the frequency of the applied
microwave current.
15By employing microfocus Brillouin
light scattering (BLS) spectroscopy,18we analyze the spatial
intensity distribution of the created spin waves, showing that
odd as well as even waveguide modes can be generated. To
prove that the excitation takes place underneath the microstripantenna, we analyze the propagation characteristics along the
waveguide. The observed exponential spin-wave decay length
ofd¼0:63lm is in good agreement with the value expected
from the theory of spin waves in thin ferromagnetic films.
19
The investigated magnetic structure is a ws¼2:2lm
wide and ts¼20 nm thick magnonic waveguide madefrom Ni 81Fe19(Permalloy). On top of this waveguide, a
wa¼1:2lm wide and ta¼400 nm thick Cu microstrip has
been patterned. A schematic view of the investigated sampleand the Oersted field components along the long axis of the
waveguide created by the microwave current are shown in
Fig.1. The latter have been calculated assuming a rectangular
current distribution inside the Cu microstrip.
The microstrip can be considered as an antenna that
emits spin waves by the direct torque of the created Oerstedfield on the magnetization.
1,5This concept has proven to be
very successful in transversely magnetized magnonic wave-
guides, where the in-plane component, which is stronglylocalized underneath the microstrip antenna (cf. Fig. 1(b)), is
oriented perpendicular to the static magnetization and, thus,
can excite spin waves efficiently. In a longitudinally magne-tized waveguide, the in-plane field component is oriented
parallel to the static magnetization. Hence, only the out-
of-plane field component can exert a direct torque. However,the spin-wave excitation by the latter is inefficient, as it only
couples to the out-of-plane component of the dynamic mag-
netization which is small in thin films due to the strong shapeanisotropy.
19
In a longitudinally magnetized waveguide, an additional
problem arises, as the group velocity is small and, thus, thepropagation distance of spin waves in this geometry is rather
short. In thin films, this can lead to the peculiar case that spin
waves created in one point decay much faster in space thanthe out-of-plane component which excites them. As a conse-
quence, this excitation can mask the propagational character
of the excited spin waves.
The shape anisotropy mentioned above leads to the fact
that the out-of-plane component of the dynamic magnetiza-
tion in thin films is smaller than its in-plane component,resulting in a strongly elliptic precession. This gives rise to a
longitudinal component of the dynamic magnetization,
which oscillates along the direction of the static magnetiza-tion with twice the frequency f
SWof the transverse magnet-
ization precession.20If a dynamic Oersted field is applied
parallel to the static magnetization at twice the precession
0003-6951/2013/103(14)/142415/5/$30.00 VC2013 AIP Publishing LLC 103, 142415-1APPLIED PHYSICS LETTERS 103, 142415 (2013)
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93.180.53.211 On: Wed, 06 Nov 2013 09:27:23frequency fp¼2fSW, it couples to this longitudinal compo-
nent of the dynamic magnetization and, by the annihilation
of microwave photons, pairs of magnons are created.15The
in-plane component of the dynamic Oersted field hzcreated
by the current flowing through the microstrip antenna fulfills
this geometric condition and, thus, can excite spin waves bymeans of parallel parametric generation. Still, this generation
can only be achieved if such a longitudinal component of the
dynamic magnetization already exists and the threshold forparametric amplification is exceeded.
15,16In this Letter, the
initial precessional motion is given by the thermally excited
waveguide modes. The threshold is in general determined bythe damping of the waveguide modes at f
SW¼1=2fpand by
their coupling to the microwave field hz.4,16,21The combina-
tion of this threshold with the fast decrease of hzalong the
waveguide leads to a strong localization of the parametric
excitation.
In Fig. 2, the color coded spin-wave intensity (logarith-
mic scale) is shown for an applied microwave frequency of
fp¼12 GHz as a function of the applied microwave power P
and the external bias magnetic field l0Hextat a spin-wave fre-
quency of fSW¼fp=2¼6 GHz. The microwave current is
applied in T¼10 ns pulses and a s¼30 ns repetition rate to
avoid sample heating. In a field range from l0Hext¼0m Tt ol0Hext¼42 mT, the parametric generation of spin waves can
be observed with threshold powers of around Pth/C259 dBm
while for fields larger than l0Hext¼42 mT the threshold
drastically increases. For applied microwave powers below
the threshold, the spin-wave intensity is given by the thermalexcitations of the system. In particular, these thermal spin
waves are clearly visible in the field range from l
0Hext
/C2542 mT to l0Hext/C2532 mT where their detection efficiency
is the highest. To make sure that the observed intensity at
fSW¼6 GHz above the threshold powers Pthis indeed due to
parallel parametric generation and not due to a perpendicularparametric instability
22resulting from a strong direct excita-
tion of spin waves at fSW¼12 GHz, the spin-wave intensity
atf¼12 GHz has also been recorded (not shown). In general,
this direct excitation is weak in the field range where the exci-
tation at fSW¼6 GHz can be observed and shows no signs of
a premature saturation.22Hence, a transverse parametric
amplification by secondary processes from decaying magnons
atfSW¼12 GHz can be excluded, and we conclude that the
spin waves at fSW¼6 GHz are indeed generated parametri-
cally by the in-plane component of the antenna field.
To investigate which particular waveguide mode is
amplified, the spin-wave intensity is recorded across thewaveguide right in front of the microstrip antenna as a func-
tion of the magnetic field. The result of this measurement is
shown in Fig. 2by the mode numbers n—which represent
the number of anti-nodes of the observed waveguide mode—
and dashed lines which mark the fields where a transition
between the modes occurs. In addition, Fig. 3(a) shows ex-
emplary mode profiles, recorded at the points of maximal in-
tensity in Fig. 2.
In order to understand these transitions between the
amplified waveguide modes and the strong increase of the
power needed to excite spin waves parametrically above
l
0Hext¼42 mT, one should consider the spin-wave disper-
sion relation in the Ni 81Fe19waveguide. Since the waveguide
is only 20 nm thick, standing spin-wave modes across the
thickness of the film can be neglected in our discussion as theirfrequencies lie well above 12 GHz. Still, one has to consider a
set of transversal waveguide modes due to the finite width of
the waveguide.
23Figure 3(b) shows the spin-wave wave
FIG. 1. (a) Schematic view of the investigated sample and setup: A Cu
microstrip (width wa¼1:2lm, thickness ta¼400 nm) has been patterned
across a Ni 81Fe19waveguide (width ws¼2:2lm, thickness ts¼20 nm) and
is connected to a microwave generator. A magnetic bias field is appliedalong the long axis of the Ni
81Fe19waveguide. (b) Spatial distribution of the
square of the in-plane ( hz) and out-of-plane ( hx) Oersted field along the
waveguide which is created by a microwave current flowing through the Cu
microstrip. The shaded area represents the extent of the microstrip antenna.FIG. 2. Color coded spin-wave intensity (logarithmic scale) as a function of
the applied magnetic bias field l0Hextand the applied microwave power P
for a spin-wave frequency of fSW¼6 GHz. The dashed lines indicate the
fields at which transitions between different waveguide modes noccur.142415-2 Br €acher et al. Appl. Phys. Lett. 103, 142415 (2013)
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93.180.53.211 On: Wed, 06 Nov 2013 09:27:23vectors kzas a function of the applied magnetic bias field for a
fixed frequency of fSW¼6 GHz following Ref. 24,u t i l i z i n ga
calculated effective width25weff¼2:3lm, a thickness
ts¼20 nm, a saturation magnetization Ms¼800 kA m/C01,a n d
an exchange constant A¼1:6/C110/C011Jm/C01, for the first
four waveguide modes nfrom 1 to 4. The lowest available
waveguide mode n¼1 has the highest cut-off field of
l0Hext/C2542 mT, above which no real wave vector kzexists
anymore. This explains the strong decrease of the spin-wave
intensity above this field value. For smaller fields, more thanone spin-wave wave vector for a given field can exist. Among
these possible spin-wave modes, the one with the smallest
wave vector will have the lowest threshold. This is because ofthe decrease of the coupling efficiency of the waveguide
modes to the in-plane microwave field with increasing wave
vector as the ellipticity of the precession decreases.
4,21
Following this argument, the mode n¼1 is preferably gener-
ated at fields above l0Hext/C2531 mT as the mode n¼2f e a -
tures a larger kzandky. One would expect the most efficient
excitation of the mode n¼1f o r l0Hext/C2531 mT due to the
highest possible coupling. However, spin waves with
kz/C250r a d lm/C01are confined under the antenna due totheir very small group velocities. Hence, the position of the in-
tensity maximum for the mode n¼1 in Fig. 2, which occurs
at a field slightly larger than the fields where kz/C250r a d lm/C01,
are determined by the competition between most efficient ex-citation and propagation out of the antenna region.
At fields lower than l
0Hext/C2531 mT, the mode n¼1c a n
only exist with very large wave vector components kz. Since
the difference between the possible wave vector components
kzof the modes n¼1a n d n¼2 is much larger than the differ-
enceDky¼p=weff/C251:4 rad lm/C01between their wave vec-
tor components ky, the total wave vector kof the mode n¼2
is much smaller. Therefore, it exhibits a higher coupling tothe microwave field and, thus, at l
0Hext/C2531 mT, a transition
from mode n¼1 to mode n¼2 occurs. For the same reasons,
between l0Hext/C2531 mT and l0Hext/C2519 mT and, respec-
tively, between l0Hext/C2519 mT and l0Hext/C257 mT the
modes n¼2a n d n¼3 are preferably amplified while from
l0Hext/C207m Tt o l0Hext¼0 mT the mode n¼4 takes over.
It should be noted that, as the width of the waveguide
determines the separation of the waveguide modes n, the ac-
cessible wave vector regime for a given mode can be chosenby the dimensions of the waveguide.
25By changing the
working frequency this allows for a selection of the ampli-
fied mode even at l0Hext¼0 mT.
In order to analyze whether the parametrically generated
spin waves propagate along the long axis of the waveguide
with an exponential decay, as is expected by the theoreticalpredictions for spin waves in a thin film,
19we study how these
spin waves evolve in space. To do so, the spin-wave intensity
along the waveguide is measured in the center of the wave-guide. This propagation characteristic is shown in Fig. 4(a)
for a power of P¼10 dBm, which is slightly above the para-
metric generation threshold, and, for reference, at a muchhigher power of P¼24 dBm. An exponential fitting of the
decay at P¼10 dBm, i.e., in the vicinity of the threshold
power, yields a decay length of the spin-wave intensity ofd
exp¼0:6260:05lm. The dashed line in the figure repre-
sents the theoretical curve which is obtained following
Ref. 19assuming the material parameters stated above and a
standard Gilbert damping for Ni 81Fe19ofa¼0.01. The calcu-
lation yields a theoretical decay length of d¼0:63lma n di s
in good agreement with the experimental findings. In addition,the components h
2
zandh2
xof the antenna field are shown. It is
clearly visible that the fields do not follow an exponential
decay and that their spatial distribution is not connected to thedecay of the spin-wave intensity. Especially the strong local-
ization of the in-plane component of the antenna field in com-
parison to the spin-wave intensity is clearly visible.
If the same analysis is applied to the decay at P¼24
dBm, one obtains a significantly reduced decay length
d
exp¼0:4560:01lm. Hence, the propagation distance of
the generated spin waves is reduced by the application of
large microwave powers. If such large powers are applied, a
strong direct excitation of spin waves at fSW¼12 GHz by
the out-of-plane component of the antenna field might lead
to a transverse parametric generation of spin waves at
fSW¼6G H z .22To rule out the occurrence of such spin
waves, Fig. 4(b) shows the distribution of the spin waves at
P¼24 dBm and fSW¼12 GHz in comparison to the decay
of the parametrically excited spin waves at fSW¼6G H zFIG. 3. (a) Experimentally determined profiles of the generated waveguide
modes at the bias fields where the maxima in Fig. 2occur. The fact that the
spin-wave intensity between the anti-nodes does not drop to zero is due to
the convolution of the finite shape of the laser spot with the real profiles of
the waveguide modes. (b) Wave vectors kzfor the modes n¼1t on¼4a s
a function of the applied bias magnetic field for a fixed frequency of
fSW¼6 GHz. The full lines highlight the mode with the highest coupling.142415-3 Br €acher et al. Appl. Phys. Lett. 103, 142415 (2013)
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93.180.53.211 On: Wed, 06 Nov 2013 09:27:23and the antenna field. It is obvious that the decay of the
directly excited spin waves is following the square of theout-of-plane component of the antenna fields, which indi-
cates that these spin waves are really created by the direct
torque of this component, as assumed above. On the otherhand, the exponential decay of the parametrically generated
spin waves does not scale with the direct excitation. Hence,
we deduce that the observed reduction of the decay length isnot a result of the influence of the directly excited spin
waves at f
SW¼12 GHz. We assume that it is a consequence
of a reduction of the static magnetization of the waveguideas the number of magnons is increased, since, at the consid-
ered point of the spin-wave dispersion relation, a decrease
in the static magnetization by only 1% already causes a dropin the group velocity v
gby 30%. This can well explain the
observed reduced decay length. Independently of the
applied microwave power, the decay of the spin waves fol-lows an exponential law and is very different from the
antenna field distribution. These are strong indications that
the spin waves are generated underneath the antenna and areindeed propagating out of the antenna region along the
waveguide not interacting with the out-of-plane field of the
antenna.In conclusion, we have shown parallel parametric gen-
eration of traveling spin waves in a longitudinally magne-
tized waveguide by the in-plane component of the Oersted
field of a microstrip antenna. This method shows severaladvantages over the conventional excitation by the direct
torque of the out-of-plane component of the antenna field.
For instance, it allows for the excitation of even and oddwaveguide modes that propagate along the waveguide. We
have demonstrated that this generation is localized under-
neath the antenna and that the spin-wave propagation can be
described by the theoretical predictions for spin waves in
thin ferromagnetic films. We emphasize that this efficientlocal generation also works without the application of an
external bias magnetic field, which opens up the way for
experiments on the interaction of spin waves with domainwalls and the creation of bias-free magnonic devices.
Furthermore, we have shown that the application of large
applied microwave powers leads to a reduction of theobserved spin-wave decay length.
The authors thank the Nano Structuring Center of the
Technische Universit €at Kaiserslautern for their assistance in
sample preparation. T. Br €acher is supported by a fellowship
of the Graduate School Materials Science in Mainz
(MAINZ) through DFG-funding of the Excellence Initiative(GSC 266). Financial support by the DFG (TRR49) is greatly
acknowledged.
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(Springer, New York, 2009).FIG. 4. (a) Spin-wave intensity as a function of the distance from the centerof the antenna for f
SW¼6 GHz at two different applied microwave powers
(P¼10 dBm, red dots and P¼24 dBm, black squares) for an applied bias
field l0Hext¼32 mT in comparison to the square of the field created by the
microstrip antenna. The dashed line represents the theoretical decay
expected from thin film theory. (b) Comparison of the spatial decay of the
excited spin waves at fSW¼6 GHz and fSW¼12 GHz for an applied power
ofP¼24 dBm.142415-4 Br €acher et al. Appl. Phys. Lett. 103, 142415 (2013)
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93.180.53.211 On: Wed, 06 Nov 2013 09:27:2320A. G. Gurevich and G. A. Melkov, Magnetization Oscillations and Waves
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93.180.53.211 On: Wed, 06 Nov 2013 09:27:23 |
1.1355355.pdf | Stiffness analysis for the micromagnetic standard problem No. 4
Vassilios D. Tsiantos, Dieter Suess, Thomas Schrefl, and Josef Fidler
Citation: Journal of Applied Physics 89, 7600 (2001); doi: 10.1063/1.1355355
View online: http://dx.doi.org/10.1063/1.1355355
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/89/11?ver=pdfcov
Published by the AIP Publishing
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129.174.21.5 On: Thu, 18 Dec 2014 13:32:37Stiffness analysis for the micromagnetic standard problem No. 4
Vassilios D. Tsiantos,a)Dieter Suess, Thomas Schrefl, and Josef Fidler
Vienna University of Technology, Institute of Applied and Technical Physics,
Wiedner Hauptstrasse 8-10/137, A-1040, Vienna, Austria
In this article solutions to micromagnetic standard problem No. 4, a 500-nm 3125-nm-wide NiFe
film, are presented. A three-dimensional-finite element simulation based on the solution of theGilbert equation has been used. The simulations show that two different reversal mechanisms occurforthetwodifferentappliedfields.Forafieldat170°counterclockwisefromthesaturationdirectionthere is a nonuniform rotation of magnetization towards the direction of the applied field, with themagnetization at the ends rotating faster than the magnetization in the center. For a field at 190°counterclockwise from the saturation direction the magnetization at the ends and in the center rotatein opposite directions leading to the formation of a 360° wall after 0.22 ns associated with a peakin the exchange energy. Moreover, the time for the magnetization component parallel to the longaxis to cross the zero is 0.136 and 0.135 ns for field 1 and field 2, respectively. The stiffness of theproblem has been investigated solving the system of ordinary differential equations with a nonstiffmethod ~Adams !and a stiff one ~backward differentiation formula, BDF !. For the measure of
stiffnesstheratioofthetotalnumberoftimesteps ~nst!takenbythetwosolvers,thatisnst ~Adams !/
nst~BDF!, has been used. This ratio is 0.784 for field 1 and 0.593 for field 2, which means that the
nonstiff method ~Adams !uses larger time steps than the stiff method ~BDF!and consequently the
systems are not stiff. The average time step for the Adams method was 0.2 ps for both fields.©2001 American Institute of Physics. @DOI: 10.1063/1.1355355 #
I. INTRODUCTION
The magnetic material defined by micromagnetic
~mMAG !standard problem No. 4 is a rectangle NiFe film,
with thickness t53 nm, width d5500 nm, and length L
5125 nm. The initial state is an equilibrium sstate. The s
state is obtained after applying and slowly reducing a satu-rating field along the @1,1,1#direction to zero. Standard prob-
lem No. 4 is focused on the dynamic aspects of micromag-netic computations.
1The problem has been studied using a
three-dimensional-finite element simulation based on the so-lution of the Gilbert equation. The problem runs for twodifferent applied fields, one at 170° ~field 1 !and the other at
190° ~field 2 !counterclockwise from the positive xaxis. Fig-
ure 1 shows the magnetization distribution of the sstate, the
coordinate system and also the field directions.
The required output for the comparison is twofold: the
(x,y,z)components of the spatially averaged magnetization of
the sample as a function of time from t50 until the sample
reaches equilibrium in the new field, and also an image ofthe magnetization at the time when the xcomponent of the
spatially averaged magnetization first crosses zero. The mag-netization images will be used to check for any differences inthe reversal mechanisms if the time data between solutionsare different.
II. MODEL AND SIMULATION METHOD
In micromagnetics the magnetic polarization is assumed
to be a continuous function of space. The time evolution ofthe magnetization follows the Gilbert equation of motiondJ
dt52ug0uJ3Heff1a
JsJ3]J
]t, ~1!
which describes the physical path of the magnetic polariza-
tionJtowards equilibrium. The effective field Heffis the
negative functional derivative of the total magnetic Gibb’sfree energy, which can be expressed as the sum of the ex-change energy, the magneto-crystalline anisotropy energy,the magnetostatic energy, and the Zeeman energy.
2The term
g0is the gyromagnetic ratio of the free electron spin and ais
the damping constant. To solve the Gilbert equation numeri-cally the magnetic particle is divided into finite elements. Ahybrid finite element boundary element method
3is used to
calculate the scalar potential uon every node point of the
a!Electronic mail: v.tsiantos@computer.org
FIG. 1. Plot of the magnetization arrows for the sstate, the coordinate
system used and the direction of the two applied fields.JOURNAL OF APPLIED PHYSICS VOLUME 89, NUMBER 11 1 JUNE 2001
7600 0021-8979/2001/89(11)/7600/3/$18.00 © 2001 American Institute of Physics
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129.174.21.5 On: Thu, 18 Dec 2014 13:32:37finite element mesh. The demagnetizing field, which contrib-
utes to the effective field, is the negative derivative of thescalar potential u. The effective field H
i
effat the node point i
of an irregular finite element mesh can be approximated us-ing the box scheme
H
eff,i52SdEt
dJD
i521
Vi]Et
]Ji, forVi!0, ~2!
whereViis the volume of the surrounding node i, such that
(
iVi5V, andViøVj50 foriÞj. ~3!
The discretization of the Gilbert equation leads to an
ordinary differential equation for every node for each com-ponent. In the case of a nonstiff problem it is advisable to usean appropriate method, such as Adams,
4whereas in stiff
problems a backward differentiation formula ~BDF!method
could be an option for the time integration. BDF method isimplicit, so at each time step a nonlinear algebraic systemmust be solved. For the solution of the nonlinear system amethod, such as Newton, has to be used which leads usuallyto a very large system of linear equations. In this article thelatter is solved using the scaled preconditioned incompletegeneralized minimum residual method ~SPIGMR !,
5based on
the generalized minimum residual method proposed bySaad.
6SPIGMR belongs to the family of Krylov subspace
methods, which are iterative methods for solving systems oflinear equations. SPIGMR has been explored in micromag-netics by Tsiantos, Miles, and Middleton,
7,8and also used by
Yang and Fredkin.9
The stiffness of the problem has been investigated solv-
ing the system of ordinary differential equations with twodifferent solvers. A nonstiff method ~Adams !and a stiff one
~backward differentiation formula, BDF !have been used to
measure the stiffness of the problem. For the latter the ratioof the total number of time steps ~nst!taken by the two
FIG. 2. Magnetization for the three components for field 1 during the simu-
lation for ;2.2 ns ~Adams method !.
FIG. 3. Magnetization for the three components for field 2 during the simu-
lation for ;3n s~Adams method !.
FIG.4. Thestrayfield,Zeeman,exchangeandtotalenergyforfield1during
the simulation for up to ;2.2 ns ~Adams method !.
FIG. 5. Plot of the magnetization arrows for field 1: ~a!beforeMxcrosses
thexaxis~0.107 ns !,~b!whenMxcrosses the xaxis~0.136 ns !, and ~c!after
Mxcrosses the xaxis~0.241 ns !.7601 J. Appl. Phys., Vol. 89, No. 11, 1 June 2001 Tsiantoset al.
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129.174.21.5 On: Thu, 18 Dec 2014 13:32:37solvers, that is nst ~Adams !/nst~BDF!, has been used. Note
that the simulation time has to be the same in order to havea fair comparison. The above mentioned method has beenproposed to approximate numerically the stiffness of a sys-tem of ordinary differential equations ~ODEs !in micromag-
netics by Tsiantos and Miles.
10Another factor that has to be
considered is the cost of each method per time step. This costper time step is important in cases that the ratio nst ~Adams !/
nst~BDF!is larger than 1. If the ratio of time steps is smaller
than 1 then the case is nonstiff. For the Adams the main costper time step is the function evaluation. However, for thecase of BDF there is some extra cost for the linear algebrainvolved. This extra cost is due to linear and nonlinear itera-tions. In general, the preconditioned case gives faster resultsin terms of the nonlinear and linear iterations. However, thepreconditioned method roughly doubles the average cost pernonlinear iteration because it computes and processes thepreconditioner.
5
III. RESULTS
The simulations show that two different reversal mecha-
nisms occur for the different fields. For field 1 there is anonuniform rotation of magnetization towards the directionof the applied field, with the magnetization at the ends rotat-ing faster than the magnetization in the center. For field 2 themagnetization at the ends and in the center rotates in oppo-site directions leading to the formation of a 360° wall after0.22nsassociatedwithapeakintheexchangeenergy.More-over, the time for the xcomponent of the magnetization to
cross the xaxis is 0.136 and 0.135 ns for field 1 and field 2,
respectively. Figures 2 and 3 give the evolution of the mag-netic components for the different fields. Furthermore, afterreversal the magnetization oscillates with decreasing ampli-tude. Figure 4 shows how the micromagnetic energy contri-butions evolve in time ~field 1 !. After the xcomponent of the
magnetization crosses zero, an energy transfer occurs be-tween the stray field energy and the Zeeman energy. Figures5 and 6 show the magnetization distribution when m
xcrosses
thexaxis~required by the definition of the problem !, as well
as at two other time moments for fields 1 and 2, respectively.
With regard to the stiffness of the problem the ratio of
the total number of time steps taken by the two solvers, thatis nst ~Adams !/nst~BDF!, is 0.784 for field 1 and 0.614 for
field 2, which means that the nonstiff method ~Adams !uses
larger time steps than the stiff method ~BDF!and conse-
quently the systems are not stiff. The total number of thetime steps taken by each method for field 1 is 11473 ~Ad-
ams!and 14628 ~BDF!. For field 2 we have that nst ~Adams !
511342 and nst ~BDF!518479. The simulation time consid-
ered for field 1 is 2.15 ns and for field 2 is 2.23 ns. Theaverage time step for the Adams method was 0.2 ps for bothfields.
The
mMAG problem No. 4 can be misleading with re-
gards to its stiffness because the Adams method takes toomany time steps. Thus, it can be thought of as a stiff case ifit will not be compared to a stiff method. The possible ex-planation for the large number of time steps is the low valueof the damping constant used,
a50.02. The low value of a
causes the magnetization to move around the effective field
so the time integrator needs very small time steps to followthe path of the magnetization.
For the ODE solver we used mixed error criterion with
absolute and relative tolerance equal to 10 e-4. Moreover,
after 2.42 ns for field 1 and 2.23 ns for field 2 the amplitudeof the oscillations of the magnetization obtains the requestednumerical accuracy.
ACKNOWLEDGEMENT
This work was supported by the Austrian Science Fund
~Project No. Y132-PHY !.
1The mMAG standard problem definitions are available at http://
www.ctcms.nist.gov/ ;rdm/mumag.html
2W. F. Brown, Jr., Micromagnetics ~Wiley, New York, 1963 !.
3D. R. Fredkin and T. R. Koehler, IEEE Trans. Magn. 26,4 1 5 ~1990!.
4C. W. Gear, Numerical Initial Value Problems in Ordinary Differential
Equations ~Prentice–Hall, Englewood Cliffs, NJ, 1971 !.
5P. N. Brown and A. C. Hindmarsh, J. Appl. Math. Comp. 31,4 0~1989!.
6Y. Saad and M. H. Schultz, SIAM J. Sci. Comput. ~USA!7,8 5 6 ~1986!.
7V. D. Tsiantos, J. J. Miles, and B. K. Middleton, 3rd European Confer-
ence on Numerical Mathematics and Advanced Applications (ENUMATH’99),Jyva¨skyla¨, 1999, edited by P. Neittaanmaki et al. ~World Scientific,
Singapore, 2000 !, pp. 743–752.
8V. D. Tsiantos, J. J. Miles, and B. K. Middleton, Comput. Math. Appl.
~submitted !.
9B. Yang and D. Fredkin, IEEE Trans. Magn. 34, 3842 ~1998!.
10V. D. Tsiantos and J. J. Miles, 16th IMACS World Congress 2000 on
Scientific Computation, Applied Mathematics and Simulation, August21–25, 2000, Lausanne, Switzerland ~unpublished !.
FIG. 6. Plot of the magnetization arrows for field 2: ~a!whenMxcrosses the
xaxis~0.135 ns !,~b!0.151 ns, and ~c!0.198 ns from the beginning of the
simulation.7602 J. Appl. Phys., Vol. 89, No. 11, 1 June 2001 Tsiantoset al.
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129.174.21.5 On: Thu, 18 Dec 2014 13:32:37 |
5.0042544.pdf | APL Mater. 9, 030903 (2021); https://doi.org/10.1063/5.0042544 9, 030903
© 2021 Author(s).Advances in magneto-ionic materials and
perspectives for their application
Cite as: APL Mater. 9, 030903 (2021); https://doi.org/10.1063/5.0042544
Submitted: 31 December 2020 . Accepted: 05 February 2021 . Published Online: 05 March 2021
M. Nichterwitz , S. Honnali ,
M. Kutuzau , S. Guo , J. Zehner ,
K. Nielsch , and
K. Leistner
COLLECTIONS
Paper published as part of the special topic on Magnetoelectric Materials, Phenomena, and Devices
This paper was selected as an Editor’s Pick
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Advances in magneto-ionic materials
and perspectives for their application
Cite as: APL Mater. 9, 030903 (2021); doi: 10.1063/5.0042544
Submitted: 31 December 2020 •Accepted: 5 February 2021 •
Published Online: 5 March 2021
M. Nichterwitz,1,2
S. Honnali,1M. Kutuzau,1
S. Guo,1J. Zehner,1,3K. Nielsch,1,3
and K. Leistner1,a)
AFFILIATIONS
1Leibniz IFW Dresden, Helmholtzstrasse 20, Dresden 01069, Germany
2Physical Chemistry, Technische Universität Dresden, Dresden 01062, Germany
3Institute of Material Science, TU Dresden, Dresden 01062, Germany
Note: This paper is part of the Special Topic on Magnetoelectric Materials, Phenomena, and Devices.
a)Author to whom correspondence should be addressed: k.leistner@ifw-dresden.de
ABSTRACT
The possibility of tuning magnetic material properties by ionic means is exciting both for basic science and, especially in view of the excellent
energy efficiency and room temperature operation, for potential applications. In this perspective, we shortly introduce the functionality of
magneto-ionic materials and focus on important recent advances in this field. We present a comparative overview of state-of-the-art magneto-
ionic materials considering the achieved magnetoelectric voltage coefficients for magnetization and coercivity and the demonstrated time
scales for magneto-ionic switching. Furthermore, the application perspectives of magneto-ionic materials in data storage and computing,
magnetic actuation, and sensing are evaluated. Finally, we propose potential research directions to push this field forward and tackle the
challenges related to future applications.
©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license
(http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0042544 .,s
I. MAGNETO-IONIC MATERIALS AS A NOVEL
APPROACH TO ENERGY-EFFICIENT MAGNETIC
SYSTEMS
Magnetic materials are important in a plethora of indus-
trial applications, ranging from large-scale hard magnets in wind
turbines and micromagnets in microelectromechanical systems
(MEMS) to magnetic nanostructures for data storage and process-
ing devices. Specific magnetic properties required for the individ-
ual applications are usually irreversibly set through the design of
phase, composition, microstructure, and shape during the fabrica-
tion process. During the operation of magnetic devices, the direc-
tion of magnetization is conventionally controlled by the applica-
tion of external magnetic fields, often realized by electromagnets or,
in nanoscale devices, by large spin polarized electric currents. For
both cases, Joule heating and associated energy dissipation present
a severe challenge. This problem continuously triggers numerous
research activities on magnetoelectric (ME) materials1–3in which
the magnetic material is controlled by the electric voltage instead of
electric current, thereby reducing energy consumption.Magneto-ionic control of magnetic materials is a novel
approach in the field of magnetoelectricity. The magnetic material
in this case is in contact with a solid or liquid electrolyte. By the
application of an external voltage, ionic motion and electrochemical
reactions are triggered which can reversibly affect the performance
of the magnetic material (Fig. 1). As the field is rapidly evolving,
very diverse material/electrolyte systems are studied, and several
denominations and categorizations, such as redox-based, electro-
chemical, ion-exchange, or magneto-ionic control of magnetism,
are in use.4–6In this perspective, we consider voltage-tunable mag-
netic materials systems in which the underlying mechanism is based
on ionic motion and electrochemical reactions as magneto-ionic
materials. The main advantage of magneto-ionic materials, in con-
trast to most other magnetoelectric materials, is that they can be
operated at room temperature and at low voltage. Since chemical
changes are involved, the non-volatile setting of a magnetic state by
voltage is possible, requiring only a small current. This is in con-
trast to the volatile magnetoelectric effects achieved by capacitive
charging via the electrochemical double layer and associated surface
electronic structure changes in similar gating architectures.7,8Thus,
APL Mater. 9, 030903 (2021); doi: 10.1063/5.0042544 9, 030903-1
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FIG. 1 . Magneto-ionic systems consist
of a magnetic material in contact with a
solid or liquid electrolyte. The magnetic
properties, such as saturation magne-
tization and coercivity, are controlled
by voltage-triggered ionic motion and
electrochemical reactions. This is
promising for a drastic energy saving
in comparison to conventional electric
current-controlled magnetic devices in
many possible applications.
magneto-ionic materials offer a route toward a reversible manipu-
lation of magnetic properties and are highly promising for ultralow
power magnetic devices.
The first reports on electrochemical voltage control of mag-
netism in ultrathin metal films dealt with oxygen-based interfacial
reduction/oxidation and associated changes in magnetic anisotropy
and coercivity.9–13Since then, numerous magneto-ionic material
systems consisting of magnetic metals and/or oxides in combina-
tion with a solid or liquid electrolyte have been investigated; an
overview of these advances until 2018/19 can be found in recent
reviews on magnetoelectric materials.4–6,14Within the past two
years, significant progress has been achieved, such as magneto-
ionic materials that function on the base of proton15–17or nitro-
gen ion18transport, the extension toward 3D magneto-ionic sys-
tems,19,20and novel magneto-ionic functionalities such as tunable
exchange bias.21,22In this expanding field, most reports still deal with
the exploration of fundamental phenomena. With the key advan-
tages of energy efficiency and room temperature operation, how-
ever, targeted research toward specific applications is expected in the
future.
In this perspective, we give an overview on important lat-
est advances in magneto-ionic research, discuss the challenges
and opportunities of magneto-ionic materials, and propose future
research strategies.
II. RECENT ADVANCES IN MAGNETO-IONIC
RESEARCH
In the field of magneto-ionic materials, significant advances
have been made within the past two years, especially regarding room
temperature operation, switching speed, reversibility, and the exten-
sion of magneto-ionic tunability to diverse magnetic phenomena
and functional materials. To illustrate the broad range of magneto-
ionic materials, Fig. 2 gives recent examples for voltage-tunable mag-
netization curves in ultrathin single metal films15[Fig. 2(a)], metal
alloy films13[Fig. 2(b)], oxide superlattices17[Fig. 2(c)], and three-
dimensional oxyhydroxide nanoplatelet structures19[Fig. 2(d)].
In the following, we will focus on important advances from the
past two years in light of the magnetic property, which is controlled
by magneto-ionic means and with respect to application-relevant
aspects, such as the switching speed and reversibility.A. Magneto-ionic control of saturation magnetization
The voltage-control of the saturation magnetization ( MS) by
magneto-ionic mechanisms has been demonstrated in a large vari-
ety of systems. To give an overview on the attainable MSvaria-
tions per defined voltage ( V) change, the magnetoelectric (ME)-
voltage coefficients, ΔMS/|ΔV|,4are plotted in Fig. 3 for selected
magneto-ionic systems and as a function of the time between the two
FIG. 2 . Examples for voltage control of magnetic properties by ion migration and
electrochemical reactions for various material systems. (a) Out-of-plane hysteresis
loops corresponding to the virgin state and the first switching cycle for the proton-
based magneto-ionic control in the Pt/Co(0.9 nm)/GdOx layer system. Reprinted
with permission from Tan et al. , Nat. Mater. 18, 35 (2019). Copyright 2019 Springer
Nature. (b) Out-of-plane hysteresis loops of an L1 0CoPt film (2.8 nm) polarized in
LiClO 4in DMC/EC at different voltages (adapted from Reichel et al.13). (c) Out-of-
plane hysteresis loops of a [(La 0.2Sr0.8MnO 3)1(SrIrO 3)1]20superlattice under ionic
liquid gating. Figure reproduced with permission from Yi et al. , Nat. Commun. 11,
902 (2020). Copyright 2020 Author(s), licensed under a Creative Commons Attri-
bution 4.0 License. (d) Voltage-controlled ON-switching of magnetization by the
electrochemical reduction of 3D FeOOH nanoplatelets in a 1M LiOH electrolyte
(adapted from Nichterwitz et al.19).
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FIG. 3 . Comparison of the ME-voltage coefficients for (a) an absolute and (b) relative variation in MSfor selected magneto-ionic systems as a function of the time elapsed
between the setting of the two different MSstates considered in ΔMS. The ME-voltage coefficients are given for the largest reversible MSchange. All data refer to room
temperature measurements, except if otherwise stated. In (a), for reasons of comparability, only studies from which ΔMSin emu cm−3could be directly extracted are depicted.
In (b), the maximum value of MSis set as the reference value to ensure comparability: ΔMS,rel= [MS1(V1)−MS2(V2)]/MS1(V1) with MS1(V1)>MS2(V2). All data refer to room
temperature measurements, if not stated differently [np—nanoporous].
voltage-induced MSstates. The data for very recent studies (from
2019 to 2020) are shown in red to highlight the current activities in
the field. It should be noted that the time does not always represent
the switching speed inherent to the material system, but sometimes
also the time required for the magnetic measurement (which might
be slower than the actual possible switching speed). Thus, the pre-
sentation in Fig. 3 gives the currently demonstrated time scales for
magneto-ionic MS-control.
Absolute values for the MSvariation per volt [Fig. 3(a)] are
mainly reported for magneto-ionic systems based on ferrites and
other transition metal oxides in thick film ( ≥100 nm) or in bulk
form.23–30Several systems, with both the liquid or solid electrolyte,
are directly derived from battery research here and measured with in
situmagnetometry.23,25,26In some cases, the magnetically active elec-
trodes are prepared by mixing transition metal oxide nanoparticles
with binders and conductive nanoparticles.23,24,26,27This procedure
optimizes the electrochemical operation, but at the same time, the
magnetic moment per volume is diluted. Figure 3(a) shows that,
so far, the largest ME-voltage coefficients for MSare in the range
of several tens of emu cm−3/volt. This seems small in comparison
to conventional magnetic materials exhibiting high MS(e.g., CoFe
alloy with MSup to 1950 emu cm−3); however, in the studied sys-
tems with small overall MS, the magneto-ionic effects are sufficientto modulate MSto a large extent. For instance, a 70% decrease in MS
is reversibly achieved by Li intercalation in ZnFe 2O4for an applied
voltage of just 1.25 V.23For most systems in Fig. 3(a), the time scales
for the magneto-ionic measurements range from minutes to hours,
which is very long with regard to applications. However, it must be
acknowledged that most studies focused on the fundamental under-
standing of the mechanisms, targeted large tunable volumes, and uti-
lized time-intensive measurements such as in situ superconducting
quantum interference device (SQUID) magnetometers.26,27
Recently, Ning et al.30reported non-volatile magneto-ionic
control of MSin epitaxial SrCo 0.5Fe0.5O3−δ(SCFO) thin films
via ionic liquid gating. Reversible and continuous control of
the magnetization up to 100 emu cm−3was demonstrated at room
temperature, which was ascribed to voltage-induced changes in
the oxygen stoichiometry. At negative voltage ( −2 V), oxygen is
extracted from the perovskite SCFO, which leads to the decrease
in the magnetic moment (80% decrease in MSafter 2 min). After
prolonged gating, the nonmagnetic brownmillerite phase forms and
thus ferromagnetic ordering is suppressed ( MS∼0 after 3 min).
The ferromagnetic state can be recovered by applying a posi-
tive voltage (+2 V) at which oxygen is reinserted. In this way,
reversible ON/OFF control of MSis obtained by alternating the
polarity of the gating voltage. The switching speed of few minutes is
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remarkable, especially in view of the relatively large film thickness of
up to 150 nm.
De Rojas et al.18proposed the use of nitrogen ions instead of
oxygen ions to improve the magneto-ionic switching speed. They
demonstrate a reversible ΔMSof 1 emu cm−3upon the electrolytic
gating of CoN (85 nm) thin films in propylene carbonate (PC) with
solvated Na+species. To reach a ΔMSof 1 emu cm−3, in this system,
voltages of ±4 V and switching times around 4 min are required.
A higher voltage of ±8 V and a longer time of 6 min are required
to achieve the same effect in Co 3O4films of the same thickness. This
difference between the magneto-ionic behavior of CoN and Co 3O4is
discussed with regard to the lower activation energy for ion diffusion
and the lower electronegativity of nitrogen compared to oxygen. In a
further study, de Rojas et al.28revealed the importance of the gating
geometry for the magneto-ionic efficiency for Co 3O4. The insertion
of a conductive buffer layer underneath the semiconducting and
magneto-ionically active Co 3O4film (130 nm) to realize a bottom
gate electrode yields faster switching and an increase in the ME-
voltage coefficients from around 0.01 emu cm−3V−1to 0.11 emu
cm−3V−1.28For the magneto-ionic control of Co 3O428and CoN,18
in comparison to Li intercalation in spinel ferrite systems,23the ME-
voltage coefficients are significantly smaller, but faster switching is
demonstrated.
The relatively scarce data for absolute ME-voltage coefficients
(withΔMSin emu cm−3) in Fig. 3(a) are connected to measurement
limitations for magneto-ionic materials, as they are often utilized in
the ultrathin film or nanoporous form. One challenge is the difficulty
to accurately determine the volume of these materials. A second
challenge is that the very small magnetic moments, as in ultrathin
films, are usually not measurable with conventional magnetome-
try, especially if in situ characterization requires further addition of
the electrolyte and cell components. Therefore, for ultrathin films,
surface-sensitive magnetic techniques based on the anomalous Hall
effect (AHE) or the magneto-optical Kerr effect (MOKE) are uti-
lized to probe relative changes in magnetization during magneto-
ionic switching.15,31,32To give a more comprehensive overview, ME-
voltage coefficients for the relative variation in MSover time are
plotted in Fig. 3(b). Here, the great variety and extended activities
regarding magneto-ionic material systems become obvious. Effec-
tive magneto-ionic control of MScan also be achieved in systems
containing metallic ferromagnets, such as Fe,33Co,34FePt,10CoPt,13
and Pd(Co).16
In the FeO x/Fe system, the magneto-ionic switching relies
on the reversible electrochemical transformation (surface oxida-
tion/reduction) between weakly magnetic iron oxide and ferromag-
netic iron metal in alkaline aqueous solution.35In this oxygen-ion
based system, Duschek et al. achieved relative variations of MSup
to 64% in continuous FeO x/Fe films (2 nm)32and almost ON/OFF
switching of MSin FeO x/Fe nanoisland geometry33for a voltage
change of just 1 V. The increased magneto-ionic effect in FeO x/Fe
nanoislands, as compared to the thin film geometry, is attributed to
the larger surface/volume ratio, which promotes the interfacial oxi-
dation/reduction reactions. Recently, a continuous voltage-triggered
ON switching of magnetism based on the reduction of paramagnetic
β-FeOOH nanoplatelets to ferromagnetic Fe [Fig. 2(d)] has been
achieved in the same electrolyte.19
The insertion and removal of oxygen is also proposed as the
origin for the voltage-induced change in MSin Co films (0.8 nm)gated via a GdO xlayer.34Here, voltage pulses of ±10 V were applied
and continuous changes in magnetization were recorded down to
a time scale of milliseconds. Di et al.12also reported variations of
MS(up to 60% MSreduction) of an ultrathin Co layer. The effects
were attributed to voltage-triggered reversible Co surface oxidation
in the liquid alkaline electrolyte. The effects occur within a very small
voltage range ( ΔVof 0.4 V) and are demonstrated in a time scale of
about 40 s.
For nanoporous Pd(Co) polarized in 1M KOH solution,
Gößler et al.16demonstrated a fully reversible voltage-induced ON
and OFF switching of magnetization for a voltage change of about
1 V. This strong magneto-ionic effect is discussed with regard to
electrochemical hydrogen sorption, which affects the magnetic cou-
pling of the Co clusters via the Ruderman–Kittel–Kasuya–Yoshida-
type interaction in the Pd matrix. In consequence, magneto-ionic
switching between the ferromagnetic and the superparamagnetic
state is achieved.
There have been remarkable advances also with regard to
the voltage-control of magnetization in perovskite transition metal
oxides by ionic mechanisms.17,36,37In superlattices comprised of
alternating one unit cell of SrIrO 3and La 0.2Sr0.8MnO 3, protons and
oxygen ions can be transferred using ionic liquid gating, which trig-
gers reversible phase transitions.17As a result, voltage-controlled
ON/OFF switching of MS[Fig. 2(c)] is demonstrated. Vasala et al.36
studied the electrochemical (de)intercalation of fluoride ions into
a two-layer La 1.3Sr1.7Mn 2O7system to tune the magnetization by
exploiting the high sensitivity of the magnetic states to the Mn oxi-
dation state and the distance between the perovskite building blocks.
They achieve a 67% change in MSfor low applied voltages ( <1 V),
which is reflected in a high ME-voltage coefficient [Fig. 3(b)]. So far,
the large magneto-ionic effects are restricted to a low temperature
(e.g., 10 K) here.
B. Magneto-ionic control of magnetic anisotropy
and coercivity
The control over magnetic anisotropy and coercivity ( HC) is
a key requirement for the design of magnetic materials in most
types of applications. Figure 4 gives an overview on ME-voltage
coefficients for HCachieved by magneto-ionic mechanisms, again
as a function of the time passed between the settings of the two
states. Most studies deal with ultrathin Co films exhibiting per-
pendicular magnetic anisotropy (PMA).9,11,12,15,38In such films, the
anisotropy and spin reorientation can be controlled by magneto-
ionic mechanisms, resulting in an ON/OFF-switchable coercivity
and remanence [Fig. 2(a)]. Recently, large magneto-ionic effects on
anisotropy and HChave also been reported for Fe films with in-plane
uniaxial anisotropy39and for typical hard magnetic materials such
as L1 0CoPt13and SmCo 5.40This is a promising sign that magneto-
ionic control can be transferred to a wide range of usable magnetic
materials in the future.
For the magneto-ionic control of anisotropy switching in ultra-
thin Co layers voltage-gated via GdO x, advances in switching speed
and reversibility are notable. Tan et al.15demonstrated the bene-
fits of protons over oxygen ions as the functional ions and achieved
fast (100 ms) and highly reversible (2000 cycles) anisotropy switch-
ing at room temperature. In previously utilized GdO x/Co structures,
which relied on oxygen ion migration, significant magneto-ionic
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FIG. 4 . Comparison of the ME-voltage coefficients for variation in HCfor selected
magneto-ionic systems as a function of the time elapsed between the settings
of the two different HCstates considered. The ME-voltage coefficients are given
for the largest reversible HCchange. All data refer to room temperature mea-
surements, if not stated different (aq.—aqueous solution, DMC/EC—dimethyl car-
bonate and ethylene carbonate, PC—propylene carbonate, YSZ—yttria-stabilized
zirconia, and np—nanoporous).
effects required high temperatures or were restricted to the edges
of the patterned electrodes.9For the proton-based mechanism, the
authors suggested that protons, generated by atmospheric water
splitting at the Au top electrode, are transported through GdO xto
the Co interface.15For this mechanism, a sufficiently humid atmo-
sphere is required. The magnetic anisotropy switching is ascribed
to the high sensitivity of the interface anisotropy of Co to adsorbed
H. In the same work, anisotropy switching is also observed for a
Au/GdO x/Pd/Co/Pd structure, which is in this case explained by the
hydrogenation of the Pd layer adjacent to the Co layer. Lee et al.41compared different proton conducting gate oxides and their effect
on the switching speed for the magneto-ionic control of Co lay-
ers. The remanence ratio approaches zero within a time scale of
>80 s for gadolinium doped ceria (GDC) and barium cerium yttrium
zirconate (BZCY), while it takes less than 1 s for yttria-stabilized
zirconia (YSZ). The Pt/Co/YSZ/Au structure is further optimized
toward a Pd/Co/Pd/YSZ/Pt architecture in which Pt is used as top
electrode to catalyze the electrochemical reaction and Pd serves as a
hydrogen loading layer as well as a protection layer between the YSZ
and the Co. Here, the fastest switching speed of <2 ms is achieved
for a thickness of 10 nm for YSZ and with a gate voltage of +6 V.
Huang et al.42recently showed that voltage-tunable PMA on
the base of oxygen ion migration can also be achieved in
Pt/Co/CoO/TiO 2(TaO x) heterojunctions. In combination with the
spin current generated from the Pt layer, this makes voltage control
of the spin-orbit torque (SOT) induced perpendicular magnetization
switching of Co possible.
Besides the substantial work on magneto-ionic control of ultra-
thin Co films with PMA, an extension to films with in-plane
anisotropy has recently been reported. Zehner et al.39achieved a
fully reversible low voltage-induced collapse of coercivity and rema-
nence in FeO x/Fe films (10 nm) with uniaxial in-plane anisotropy
during electrolytic gating at low voltage in 1M LiOH aqueous
solution [Figs. 5(a) and 5(b)]. In the initial FeO x/Fe films, Néel
wall interactions stabilize a blocked state with high coercivity.
With dedicated and angle-resolved in situ Kerr microscopy, work-
ing in combination with liquid electrolyte gating, inverse changes
in coercivity and anisotropy and a coarsening of the magnetic
microstructure [Fig. 5(c)] were probed. The quantitative analysis
of the anisotropy and domain size changes allowed us to reveal
the redox-induced change in Néel domain wall interactions and
in the microstructural domain wall-pinning sites as the origin for
the voltage-triggered magnetic deblocking. This reversible modula-
tion of local defects to tune HCgoes beyond state-of-the-art thin
film magneto-ionics in which extrinsic properties such as coer-
civity are usually changed by controlling the intrinsic magnetic
anisotropy. With this approach, voltage-assisted 180○magnetization
switching with a high energy efficiency is achieved within seconds.
Zehner et al.21further extended the magneto-ionic functionality to
FIG. 5 . Voltage-control of coercivity and magnetic domains in FeO x/Fe films with uniaxial in-plane anisotropy. (a) In-plane hard axis magnetization curve in pristine state and
at−0.02 V (oxidation) and at −1.1 V (reduction) showing the voltage-induced collapse of coercivity and remanence. (b) Angular dependence of coercivity for the pristine
(=oxidized) and the reduced state. (c) Magnetic domains in reduced and oxidized state as observed by in situ Kerr microscopy (adapted from Zehner et al.39).
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in-plane exchange bias thin films, which are important for stable and
artificial magnetization distributions.43The authors could demon-
strate voltage-controlled non-volatile programming of exchange
bias fields in FeO x/Fe/IrMn heterostructures with fully shifted hys-
teresis loops by a redox-based transition of the ferromagnetic layer.
Recent advances are reported also for the magneto-ionic
control of coercivity and magnetization in bulk-like materials.
Navarro-Senent et al.20studied mesoporous Co–Pt/CoO microdisks
(>300 nm) in PC with traces of NaOH. The voltage-driven oxygen
migration promoted a partial reduction of CoO to Co and led to
a strong decrease in coercivity ( −85%) at −10 V, which could be
partially recovered. In a subsequent study, Navarro-Senent et al.44
investigated the behavior of nanoporous Co–Pt films covered by
10 nm AlO xor HfO xin the same electrolyte. Differences in the
magneto-ionic behavior of uncoated Co–Pt films are ascribed to the
oxygen acceptor behavior of HfO xand the oxygen donor behavior
of AlO x.
For micrometer-sized SmCo 5, Yeet al.40showed a huge mod-
ulation of coercivity ( ΔHCof 1 T for ΔV<1 V) by electrochemical
hydrogenation in 1M KOH aqueous electrolyte. Even though very
long timkes are required for the largest HCchange, voltage-assisted
magnetization reversal could be demonstrated within around 3 min.
Further improvements in switching speed are proposed by the
reduction of the particle size, the introduction of additional diffusion
paths, or optimized device geometries.
III. APPLICATION PERSPECTIVES
FOR MAGNETO-IONIC MATERIALS
In the following, application perspectives for magneto-ionic
materials in the fields of memory devices, neuromorphic comput-
ing, spin-torque nano-antennas, sensors, and magnetic actuation are
discussed.
Digital memory is a crucial part in our current digital era.
The advent of internet of things, machine learning, and artificial-
intelligence (AI) based applications, such as image recognition50
or protein folding,51impose strict requirements on the processing
speed and energy efficiency of data-centric tasks. Already today, the
worldwide energy consumption of the data centers of all cloud ser-
vice providers is about 200 TW/year.52Magnetic hard disk drives
(HDD) remain a conventional way to store data in computers and
large data centers, but the electric currents required for data writing
are associated with the energy loss due to Joule heating. Also, charge
based memories, such as flash and dynamic/static random-access
memory (RAM), have high energy consumption and further device
miniaturization becomes problematic mainly because of increasing
leakage currents.53The use of magnetic elements for RAM seems
appealing due to non-volatility and fabrication compatibility with
CMOS.54Additionally, complex magnetic textures can be used for
information storage.55,56In such magnetic memory concepts, elec-
tric currents are still used to control magnetization dynamics. The
combination of magnetic memory concepts with energy-efficient
magneto-ionic control could be a route to reduce the operational
energies for next generation data storage media.
For HDD, control of coercivity or magnetization of the stor-
age media by a voltage, instead of a magnetic field, is appealing
to reduce the energy consumption. In high anisotropy magnetic
thin films, which are required for data stability in high densitymagnetic data storage, a temporary reduction of coercivity during
the writing process (magnetization switching) is desirable. So far,
this is accomplished by heat-assisted magnetic recording,57which,
however, involves the generation of heat with a laser. Here, pro-
vided that the switching speed is improved, magneto-ionic mech-
anisms to control coercivity may be a route to more energy-efficient
voltage-assisted magnetic recording.
The device structure and nanoscale ion transportation mecha-
nism in some magneto-ionic devices are comparable with those of
resistive switching devices, which are used for the non-volatile resis-
tive random access memory (RRAM).58It has been experimentally
demonstrated that device resistance can be switched among multi-
states to achieve an ultra-high density/capacity storage.59In analogy,
non-volatile ion transportation triggered by voltage could result in
progressive changes in the magnetization value in magneto-ionic
materials. Thus, we propose that the stable multi-level intermediate
states observed in many magneto-ionic materials could be extended
into a multi-state magnetic storage concept.
Current-driven magnetic domain wall motion has raised hopes
for new memory or logic devices, but so far, high threshold cur-
rent densities and defect pinning effects pose challenges to the fur-
ther development of this concept.60Magneto-ionic approaches have
already shown that domain-wall propagation fields can be tuned
through oxygen ion-based magneto-ionic control of ultrathin Co
layers61and Fe films.39Offering the advantage of lower threshold
current densities and topologically protection,62skyrmions are an
alternative candidate to domain wall memory logic devices. One
way to stabilize skyrmions is via the Dzyaloshinskii–Moriya inter-
action (DMI). By adjusting the strength of the DMI, it is possi-
ble to tune the domain wall chirality and the skyrmion’s winding
number, making it more suitable for practical applications. Interest-
ingly, with regard to potential magneto-ionic control, DMI can be
induced by oxygen chemisorption on a ferromagnetic surface.63A
recent work demonstrated that the DMI strength can be (so far irre-
versibly) tuned by oxygen-ion migration at the ferromagnet/heavy
metal (Co/Pt) interface.31Furthermore, gas phase experiments have
shown that skyrmions can be induced in ultrathin Fe upon hydro-
gen exposure.64This indicates that modulation of skyrmions by fast
proton-based magneto-ionic control may be feasible.15Fine tuning
and reversibility of skyrmion parameters with high endurance, for
example, through hydrogenation, could be favorable for the next
generation spin-orbitronic devices.
In general, the limiting factors for future applicability in
data technology might be, in the first place, the reversibility and
switching speeds. In order to compete with the state-of-the-art
technology, a stability over >1015cycles and switching speeds
between 10 ms (HDD) and 10 ns [spin transfer torque (STT) RAM]
are desirable.65
The practical applications of AI require processing of large vol-
umes of data, which, in conventional computing architecture (sep-
arate processing and storage units), is energy expensive and limited
by von Neumann bottleneck and the memory wall.66Neuro-inspired
computing chips that emulate the working principles of the biologi-
cal brain are expected to perform better and more energy efficient.67
Memristive elements are shown to emulate synaptic properties.68
Since latest magneto-ionic approaches resemble the redox memris-
tors in its operational principles, they may serve as an alternative
building block for neuromorphic hardware. An artificial synapse and
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different synaptic functions based on the magneto-ionic setting of
different magnetization states were already demonstrated in the Co
thin film.34
Magnonics is gaining a lot of popularity for wave-based com-
puting69and microwave electronic70applications. It has been pre-
dicted that terahertz (THz) spin waves can be excited by domain
wall motion induced by STT and SOT.71Large bandwidths of fre-
quency modulation can be achieved, which is important for signal
processing72and in telecommunication technologies. It was theoret-
ically demonstrated in antiferromagnets that the wavelength of the
emitted spin waves depends on material lattice constants.73One of
the limitations to excite the spin waves is the high threshold cur-
rent density in materials with strong exchange interactions. Here,
the magneto-ionic mechanism such as ion intercalation or hydro-
genation may be an approach to tune the lattice constants and the
exchange interactions to obtain the desired wavelength.
Spin-torque nano-oscillators are magnetic tunnel junctions
that emit microwave signals when a direct current induces sus-
tained magnetization precession. A frequency tuning of 50 MHz
has already been demonstrated in an electrically gated spin Hall
nano-oscillator by engineering the electronic bands at the ferromag-
netic/oxide interface.74However, an all ionic control of PMA would
potentially lead to higher modulation of damping,75thereby directly
influencing the auto-oscillation threshold current.
The opportunities to tune magnetic properties by a low voltage
are also of interest for magnetic sensors. Modern magnetic field sen-
sors rely on the giant magnetoresistance (GMR) or tunneling mag-
netoresistance (TMR) effect in functional layer stacks and are key
to the information technology and numerous scientific and indus-
trial measurement applications. Voltage control of GMR and TMR
stacks via the electric field control of the magnetic anisotropy is
proposed as a promising route to tuning and enhancing the sensi-
tivity and the linear range of magnetic sensors.76,77With magneto-
ionic approaches, ultralow voltage and non-volatile programming
of magnetic sensors may be within reach, but this remains largely
unexplored. One recent study on oxide/metal layers demonstrates a
tunable magnetoresistance (MR) and even a voltage-controlled sign
change in MR by magneto-ionic mechanisms.78
A different type of potential sensor application relates to the
interplay of electrochemistry and magnetism, which is inherent to
magneto-ionic materials. The magnetic state not only depends on
the applied voltage but also on the type of electrolyte species, pH,
and hydration state, and thus, it can be regarded as a magnetic fin-
gerprint to sense solution, solid-state, or atmospheric chemistry.
Such a direct magneto-chemical sensor would have the advantage
that functionalized magnetic particles, which are often utilized in
magneto-chemical sensing platforms,79are not required. A first
example in this direction is in situ characterization of (de)lithiation
processes in Li xCoO 2cathodes for Li ion batteries by utilizing in situ
SQUID magnetometry.80
Magnetic actuation is important in a plethora of modern tech-
nologies, including microfluidic chips,81,82(micro)robotics,83and
MEMS.84Up to now, the magnetic actuation functionality requires
an external magnetic field as the control parameter, which is usu-
ally provided by an electromagnet. This becomes problematic par-
ticularly when downscaling to the microscale, as here, high current
densities are required to achieve sufficient magnetic field strength.84
Switchable micromagnets by magneto-ionic control could presentan energy-efficient alternative to standard micro-electromagnets,
especially as the demand for high switching speed is less crucial
than for memory devices. As an example, in some magnetophoretic
devices, particle transport in a liquid above a magnetic film is real-
ized by magnetic actuation in a time scale of seconds via the modula-
tion of artificial magnetic domain patterns.85The required external
magnetic field is so far realized by electromagnets. Direct magneto-
ionic control of the magnetic domains21could potentially replace
magnetic field control in such devices in a succinct way.
To achieve sufficient stray fields for actuation, a high energy
product of the magnet is required. Thus, materials with high satu-
ration magnetization, as a precondition for a high remanence, and
sufficient HCwill be favorable here. Furthermore, magneto-ionic
materials exhibit the unique feature of different magnetic states ( HC,
remanence) being able to be set by an external voltage, and this
opens the door to a voltage-programmable actuation functionality.
IV. POTENTIAL RESEARCH DIRECTIONS
AND CONCLUDING REMARKS
While the technology readiness of this emerging topic is
still at the fundamental research levels 1 and 2 (1—basic princi-
ples observed and 2—technology concept formulated), the recent
research works clearly show that magneto-ionic control can be
extended to a broad range of material systems and magnetic func-
tionalities. In the following, we propose potential research directions
to push this field forward, especially in view of potential applications.
So far, magneto-ionic control is demonstrated for academic
material systems and the material choice has primarily been con-
ducted on the base of the targeted magneto-ionic mechanism. Now,
with the basic knowledge on magneto-ionic principles, the research
could extend toward magnetic materials, which exhibit optimized
initial magnetic properties with regard to application. Here, research
can rely on long-standing engineering approaches to design mag-
netic materials, including the optimization of the microstructure,
texture, intergranular phases, or composition in alloys. For example,
magneto-ionic control of bulk ferrites and rare earth hard magnets is
so far restricted to isotropic materials.29,40An extension to textured
materials could push forward the application potential for actua-
tion, as the accessible energy product could be increased. Further-
more, the search for magneto-ionic mechanisms to control antifer-
romagnetic layers, which are currently hot candidates for spintronic
devices, seems promising. In this regard, Mustafa et al.22recently
demonstrated voltage control of exchange bias by the intercalation of
Li in antiferromagnetic Co 3O4, placed adjacent to the ferromagnetic
Co layer.
The energy efficiency of magneto-ionic systems is the most
intriguing aspect for their application potential. Research works
on magneto-ionic control already clearly demonstrate that a large
variety of magnetic properties can be tuned by a low voltage. The
electrochemical reactions involve electric currents, but these are
orders of magnitude smaller than those required for electric cur-
rent controlled magnetic devices. In addition, as the mechanism
relies on chemical changes, the effects are often non-volatile and
thus do not require a continuous power supply. For the FeO x/Fe
system, magneto-ionic switching energies in the range of few fem-
tojoules (for a 50 nm device) are projected,39which is com-
parable to the lowest reported switching energies for magnetic
APL Mater. 9, 030903 (2021); doi: 10.1063/5.0042544 9, 030903-7
© Author(s) 2021APL Materials PERSPECTIVE scitation.org/journal/apm
tunnel junction devices.86Considering that the switching speed
is not yet optimized, this indicates that magneto-ionic mecha-
nisms can yield an improvement in the energy efficiency of sev-
eral orders of magnitude. Nevertheless, quantitative information
on the energy consumption of magneto-ionic systems is still too
scarce and should be provided more often in the future to high-
light the competitiveness with alternative approaches in magnetic
technologies.
The limitation of the switching speed, due to the required ionic
motion, is considered as one of the main challenges for the appli-
cation of magneto-ionic materials. Consequently, future research
should address this issue. For ultrathin Co films with PMA, which
is the most-studied magneto-ionic material, the optimization of the
device architecture, film thickness, and gate oxide has successfully
increased the switching speed to the range of (sub)milliseconds41
and, for small effects, also to the sub-nanosecond regime.49With
this, these systems approach the required switching speeds for
memory applications. Only for few systems besides Co, e.g., for
the FeO x/Fe system, a switching speed in the range of seconds is
reported,39which may suffice for actuation and sensor applications.
Time scales of hours to days, which occur for many magneto-ionic
systems (see Figs. 3 and 4), seem unacceptable from an application
standpoint. In many cases, actually, the maximum possible switch-
ing speed may not be known due to measurement restrictions. Thus,
a deeper understanding and an improvement of the key aspects for
the switching speed is an important research strategy. In a first step,
time-resolved magnetic measurements would offer access to the
“true” switching speed. Second, interface- and defect-engineering
could be pursued as a promising route to optimize the ion transport
pathways and thus the kinetics, similar to the successfully applied
concepts in the field of memristive systems.87
The reversibility of magneto-ionic effects is a further impor-
tant issue that defines the possible application areas. At first glance,
magneto-ionic systems without a structural phase transformation
(such as Li intercalation in spinel structure24) or with purely inter-
face effects (such as surface effects at ultrathin Co films12,15) seem
favorable here. In contrast, many magneto-ionic systems involve
a phase transformation, which exponentiates tuning possibilities
but may decrease reversibility due to strain-related fatigue or irre-
versible microstructure changes. In such systems, good reversibility
is often found at the surface/interface regions or for nanoparticu-
late morphologies.28,33,88,89Thus, the minimization of the affected
volume, e.g., via interface functionalization in nanostructured mate-
rials, seems to be a promising route to high reversibility. Further-
more, the unprecedented possibilities of magneto-ionic fine tun-
ing of material properties by voltage at room temperature, even
in the case of limited reversibility, should be acknowledged as a
novel material design approach. Indeed, non-volatile magneto-ionic
manipulation has already been proposed as the energy-efficient pat-
terning method for artificial domains in exchange bias systems21or
to set specific DMI values.31
Magneto-ionic systems often involve liquid electrolytes. There
are several advantages of liquid electrolytes over solid electrolytes,
namely, the high electric fields that are accessible due to electro-
chemical double layer formation, the possibility to infiltrate 3D
geometries, the easier characterization of solid/liquid in comparison
to solid/solid interfaces, and the more straightforward understand-
ing of mechanisms in the frame of conventional electrochemistrymodels. At the same time, the presence of a functional liquid is
a critical point for implementation in most technologies. There-
fore, research efforts should also target a transfer from solid/liquid
systems to all-solid-state magneto-ionic systems wherever possible.
Recent examples are the use of ionic gels,90hydrated oxides,15and
devices that are derived from solid state battery architectures.25,36
On a more futuristic level, solid/liquid magneto-ionic principles
may become useful in brain-inspired computing concepts91or for
magnetic actuation in microfluidic devices.85
So far, research on magneto-ionic materials focused, to a large
extent, on the basic science. Thus, in order to move from academic
research to the application of magneto-ionic materials, the develop-
ment of various laboratory demonstration prototypes is an essential
future task.
In summary, this perspective gives an overview about impor-
tant research advances in the field of magneto-ionic control of
magnetism. These include the growing diversity of magneto-ionic
materials systems and functionalities and the improvements toward
high switching speeds. We discuss various application perspec-
tives for magneto-ionic materials in memory, computing, actuation,
and sensor technologies and suggest potential research directions.
For instance, the quantification of switching energy and switching
speeds and the development of magneto-ionic prototype systems
seem important to highlight the energy saving potential and increase
the competitiveness for future applications.
AUTHORS’ CONTRIBUTIONS
M.N. and S.H. contributed equally to this work.
ACKNOWLEDGMENTS
The authors acknowledge funding by the DFG (Project No.
LE 2558 2-1). This project has received funding from the European
Union’s Horizon 2020 research and innovation programme under
the Marie Sklodowska-Curie Grant Agreement No. 861145.
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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Fast switching of magnetic fields in a magneto-optic trap
Rev. Sci. Instrum. 72, 4055 (2001); 10.1063/1.1408935
Permanent magnet film magnetooptic waveguide isolator
J. Appl. Phys. 75, 6286 (1994); 10.1063/1.355426
Magnetooptic waveguide hysteresis loops of ‘‘planar’’ magnetic garnet films
J. Appl. Phys. 66, 3342 (1989); 10.1063/1.344131
Single mode magnetooptic waveguide films
Appl. Phys. Lett. 48, 508 (1986); 10.1063/1.96489
Switching and modulation of light in magnetooptic waveguides of garnet films
Appl. Phys. Lett. 21, 394 (1972); 10.1063/1.1654427
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On: Fri, 19 Dec 2014 15:00:26Magneto-optics and motion of the magnetization in a
film-waveguide optical switch
P. K. Tien
Bell Telephone Laboratories, Holmdel, New Jersey 07733
D. P. Schinke and S. L. Blank
Bell Telephone Laboratories, Murray Hill. New Jersey 07974
(Received 6 December 1973; in final form 8 April 1974)
We discuss magnetic properties of iron-garnet films and the use of these films as film-waveguide
optical switches. Our experimental study involves the observation of magnetic domains, measurements
of Faraday rotation constants, coercive forces and anisotropy fields, and a detailed investigation of
the switching process between 0 and 300 MHz. Our theoretical study includes magneto-optics in film
waveguides, and analysis of the serpentine circuit, and an extensive calculation of the motion of the
magnetization. For low driving fields, the process of optical switching is the formation of periodic
domains and the subsequent motion of domain walls. For higher driving fields, the process is
rotation of the magnetization in unison. The switching field required for! 00'':;) modulation is severil
times the anisotropy field in the film, which can be as small as 0.1 Oe.
I. INTRODUCTION
In a recent letter, Tien et al.1 reported a novel mag
neto-optical switch which may be important as an active
element in integrated optical circuits. 2 Figure l(a) is a
photograph of this magneto-optical switch and Fig. 1(b)
is a schematic drawing of this switch. 1 It involves sev
eral novel features which include the use of an iron
garnet film as an optical waveguide and a serpentine
electric circuit for driving the motion of the magnetiza
tion. In this iron-garnet film waveguide, 2 a TE wave is
converted into a TM wave, or vice versa, because of the
magneto-optic effect (Faraday rotation). 1 By detecting
the TE and TM waves separately, switching or modula
tion of the light is achieved as the amount of conversion
varies with the motion of the magnetization. The mag
netization rotates very nearly in the plane of the film,
and the fact that the in -plane demagnetizing field is
nearly zero in the thin-film geometry makes the opera
tion very efficient. Once the driving field overcomes the
in-plane anisotropy field, which is very small in these
films, one should expectthe magnetization to rotate
freely. 1 One purpose of this paper is to answer the fol
lowing important questions: (0 Does the rotational mo
tion of the magnetization depend on the ferromagnetic
resonance? (ii) What is the frequency response of the
optical switch? (iii) What are the factors which ulti
mately limit its operation?
In this present paper, we report our recent work in
volving the observation of magnetic domains in iron
garnet films and measurements of their Faraday rota
tion constants, coercive forces, and magnetiC aniso
tropy. We have also extended the switching operation up
to 300 MHz. We have developed a theory of magneto
optics in film waveguides and analyzed the coupling of
the waveguide modes to the motion of the magnetization.
Rotations of the magnetization up to 1800 were observed
in the switching experiments, affording a unique oppor
tunity to observe the collective dynamiC behavior of fer
rimagnetic spins. These large rotations of the magneti
zation are highly nonlinear and can only be computed
numerically, Our numerical calculations agree well with
the experiments.
The iron-garnet films used in these experiments were
3059 Journal of Applied Physics, Vol. 45, No.7, July 1974 y 3GaO. 7sSco.5Fe3. 75012 deposited on (111)-oriented
Gd3Gas012 substrates. The films have only one in-plane
INPUT PRISM
COUPLER
(b) \
\ OUTPUT PRISM
COUPLER
-""",C_,."",'{E.
'L;;~~~"""'TM
MAGNETIC
FILM
FIG. 1. (a) Photograph of a film-waveguide magneto-optical
switch. The disk in the center of the photograph is a thin
sin~le-:crystalline ferrimagnetic film of Y3GaO.7,ScQ.,Fe3.75012
WhICh IS used as an optical waveguide. Two prism couplers
couple an infrared laser beam into and out of this magnetic
waveguide. Between the two prism couplers and placed in close
contact with the magnetic film is a small serpentine electric
circuit. (b) Schematic drawing of the film-waveguide magneto
optical switch in operation. As the current in the serpentine
circuit is on and off, the light beam emerging from the output
prism coupler switches between two different directions.
Copyright © 1974 American Institute of Physics 3059
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FIG. 2. Magnetic domains observed in iron-garnet film with
out any externally applied magnetic field.
easy axis, which is induced during the growth process.
In the switching experiments, the easy axis was always
placed parallel to the direction of optical propagation.
Our recent films have a saturation magnetization 41TM
= 1050 Oe, and a saturated Faraday rotation constant
g", 1500/cm at 10 152-Jlm laser wavelength. Both the in
plane coercive and anisotropy fields typically range
from 0,1 to 0,3 Oe, It is known that the best Permalloy
films used for magnetic memories have similar coer
cive and anisotropy constants but a much larger 41TM
-10000 Oe. We will show that the problem of optical
switching considered here is closely related to the flux
reversal observed in magnetic memories made of
Permalloy films,
II. EXPERIMENTAL ARRANGEMENT
The iron-garnet film shown in Fig. 1 is typically 2-3
Jlm thick and propagates several TE and TM waveguide
modes. Two rutile prisms separated typically by a dis
tance of 8 mm are used to couple a laser beam into and
out of the magnetic ftlm, In our coordinate system, the
light wave propagates along the x direction, and the film
is in the xy plane. A small serpentine circuit fabricated
by the conventional photlithographic techniques is placed
between the two prisms and it is in close contact with
the film, The rf current applied to the circuit produces
a magnetic field in the x direction, which rotates the
magnetization vector, M . We will show later theoretical
ly that the magneto-optic effect is directly proportional
to the component of M in the direction of light propaga
tion (along the x axis), We will also discuss why a ser
pentine circuit is necessary to couple TE and TM wave
guide modes which have different wave velocities in the
film waveguide. Normally in the experiment, the light
from a 10 152-Jlm He-Ne laser is coupled into the film
as a TE (or TM) waveguide mode of the order m. As the
light wave propagates in the film, it is gradually con
verted into a TM (or TE) waveguide mode of the same
order. When the waves reach the output prism which is
birefringent, the converted TM (TE) wave and the re
maining TE (TM) wave are coupled out of the film into
two spatially separate beams in space. These two beams
are separated by an angle of 20° 11'. Therefore, if the
J. Appl. Phys., Vol. 45, No.7, July 1974 TE-TM (or TM-TE) conversion in the film is com
plete at the peaks of the rf current and is zero at the
zeros of the current, we expect a light wave to emerge
from the output prism and to switch between the TM and
TE directions at the frequency of the current in the cir
cuit. It is interesting to note that even if the TE-TM
(or TM-TE) conversion is incomplete, 100% intensity
modulation is always possible for the converted TM (or
TE) beam. Such a large modulation ratio is important
in a communication system. In addition to the rf mag
netic field produced by the circuit, we often add a uni
form dc magnetic field on the order of 1 Oe, either along
the y direction or along a direction at 45 ° between the x
and y axes. The dc field ensures the return of M to the
equilibrium position at the zeros of the rf field. It also
Significantly improves the effiCiency and the frequency
response of the device.
III. EXPERIMENTAL OBSERVATIONS
It has been known for some time in the experiments
on magnetic memories that flux reversal in Permalloy
films3,4 for small switching fields is caused by three
distinct processes: domain-wall motion for switching
frequencies below 1 MHz, incoherent rotation of the
magnetization for frequencies between 1 and 10 MHz,
and rotation in unison for frequencies above 10 MHz. If,
however, the switching field is substantially larger than
the coercive field, rotation in unison is always the dom
inant process. Our experiments indicate that similar
processes are operative in iron-garnet films. In this
section, we discuss first the observation of magnetic
domains, then the measurements of the longitudinal and
transverse hystereSiS loops, and finally, optical switch
ing experiments between 60 Hz and 300 MHz.
It is difficult to observe magnetic domains when the
magnetization is in the plane of the film. We have suc
ceeded in observing them by placing the film under a
microscope and between two polarizerssuch that the
plane of the film is tilted at an angle of 45 ° from the
axis of the microscope. Because a dipping process of
liquid-phase epitaxy was used to grow these iron-garnet
FIG. 3. The photograph shows that the domains in Fig. 2 can
be wiped out completely by applying a small and uniform dc
magnetic field parallel to the film.
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FIG. 4. Periodic domains excited in iron-garnet film.
films, there are two separate films, one at the top and
the other at the bottom of the substrate, Since the mi
croscope is focused on the top film, the dark images ob
served in Figs. 2-5 are the magnetic domains of the top
film and the light ones are those of the lower film. These
dark and light images of the domains overlap one another
in these photgraphs. Figure 2 shows the domains ob
served without any applied field. The familiar reversed
domains are clearly seen at the edge of the film. The
domains were wiped out in Fig. 3 by applying a uniform
dc magnetic field. The amount of the field required for
this purpose varies with the direction, and it provides a
rough estimate for the sizes of the coercive and aniso
tropy fields. Figure 4 shows a set of periodic domains
in a pattern similar to those excited by a serpentine cir
cuit. These domains can be made to appear and disap
pear by applying avery small and uniform dc field (0.2
Oe or less) along a proper direction. We observed that
the large modulation of light obtained by moving a small
bar magnet as described previouslyl is caused by the ex
citation of these domains and the subsequent motion of
the domain wall. The formation of these periodic do
mains may be due to local anisotropy variations induced
during the growth of the film. In fact, "magnetic rip
ples" and anisotropy imperfections have been discussed5
in Permalloy films. Finally, Fig. 5 shows a domain pat
tern obtained by exerting a small pressure to the center
of the film indicating the effect of magnetostriction.
Hysteresis loops 6 are normally measured in Permalloy
films using small pickup coils. It is more convenient for
us to make these measurements by detecting the net
magnetization in a given direction from the observation
of TE to TM conversion using the arrangement shown in
Fig. 1. As will be discussed fully later, it is easy to
calculate the components of the magnetization parallel
and normal to the easy axis based on the observation of
TE and TM conversion. Being free of interference from
stray Signals, the present magneto-optical method of
measuring the hysteresis loops has considerable advan
tage over the use of pickup coils.
For the longitudinal hysteresis loop shown in Fig. 6,
}vls is the component of the magnetization parallel to the
easy axis; it varies as a uniform dc field Hs applied in
J. Appl. Phys., Vol. 45, No.7, July 1974 FIG. 5. Magnetic domains observed by applying a small amount
of pressure at the center of the film.
the same direction is varied. The field at which the
magnetization curve crosses the horizontal axis is the
coercive force He which is 0,104 Oe in Fig. 6. Similar
ly, for the transverse hysteresis loop shown in Fig. 7,
Mn and Hn are the net magnetization and the applied dc
field, respectively, in the direction normal to the easy
axis. This measurement indicates an anisotropy field
HK:::: 1. 2 Oe which is much larger than the value 0.1-0.3
Oe measured earlier by us using a torque magneto
meter.l We believe this increase in HK is due to the
pressure exerted on the film by the prism couplers. This
is consistent with our earlier observation in connection
with Fig. 5 that the magnetic domains in these films are
very sensitive to applied pressure.
The last set of experiments involves optical switching
and the study of the motion of the magnetization over a
wide range of frequencies. These experiments were per-
i
/HC = 0.104 (Oe)
-0.1 o 0.1 0.2
Hx (Oe)--
-1.0
FIG. 6. Longitudinal hysteresis loop.
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On: Fri, 19 Dec 2014 15:00:263062 Tien, Schinke, and Blank: Film-waveguide optical switch 3062
1.0
t
-1.8 -1.0
-1.0
FIG. 7. Transverse hysteresis loops. 10 I
I
I
I
I
I
. I
I :/K = 1.2 (Oel
1.8
formed under a variety of conditions, but only a typical
one will be described below. Here, again, we use the
arrangement shown in Fig, 1. The laser beam is coupled
into the film as a TE wave, and the converted TM wave
is detected by a germanium avalanche photodiode which
has a flat response up to several GHz. To boost the sig
nal-to-noise ratio, the detector is followed by amplifiers
and filters which limit a bandwidth -10 MHz. The detect
ed signal is displayed on an oscilloscope to study the
waveform of the modulation from which the TE-TM
conversion and the motion of the magnetization vector at
the switching frequency n can be calculated. A combina
tion of dc and rf currents is applied to the serpentine
circuit which produces a "switching field", Hso + Hs
sinnt, in the x direction which is the easy axis. An ad
ditional uniform dc magnetic field, which will be denoted
as the "transverse field" Hn is applied in the y direction.
The interplay involving these fields Hso, Hs' and Hn, will
be discussed fully in the theory in Sec. IV. Figure 8
shows the frequency response of the switch. The ordi
nates are the peak TM intensities, RTM, as measured at
the detector for various switching fields and frequencies.
The vertical scale is normalized by taking the TM re
sponse to be unity when the magnetization vector is fully
along the x axis by applying a large current in the ser
pentine circuit. The horizontal scale is the switching
frequency n. The measured data are shown as isolated
points and the numbers associated with these points are
the values of Hs in Oe. The data in Fig. 8 will be dis
cussed in detail in Sec. VII. Figure 9 shows an oscillo
graph for the switching operation at 300 MHz. The top
trace is the applied rf field, the middle trace is the TM
response, and the lower trace is the remaining pickup
in the scope when the laser light was blocked. The hori
zontal time scale is 2 nsec/div.
IV. MAGNETO-OPTICS IN A FILM WAVEGUIDE
In this section, we discuss the theory of magneto
optics for the case where the film waveguide consists of
J. Appl. Phys., Vol. 45, No.7, July 1974 a magnetic film on a nonmagnetic substrate. (The oppo
site situation using ordinary passive films on gyromag
netic substrates has been discussed by Wang et al.7) We
follow Landau and Lifshitz8 by setting the relative per
meability !J. = 1 and llse a dielectric tensor for the mag
netic material. The mathematics developed below is ap
plicable to other thin-film problems. After having de
rived magneto-optics in film waveguides, we proceed to
show how a serpentine circuit can provide a continuous
coupling between the TE and TM waves which have dif
ferent wave velocities.
In our coordinate system, the TM wave involves the
field components Hy' E., and Ex, and the TE wave, Ey,
Hz, and Hx' In film-waveguide theory, 2 we customarily
take a/ay = O. The curl relations of the Maxwell equa
tions for the TM and TE waves in the magnetic film are
then
and
2 I-
0::
u.i en
~l w
t::!
-.oJ <t
:E
0::
0 z aHy _. D' az -zw x'
aH. . D ---"'"= -zw . ax z'
aEx aEz . 1-1 ----=zw!J.O". az ax y'
aEy . on -=-zW!J. . az x'
1.0 x
0.9
0.8
0.7
0.6
0.5
0.4
0.3 •• • • x x 0.2 • • • • x 0.1 • • SWITCHING FIELD Hs (IN Oe)
EXPERIMENTAL THEORETICAL
DATA DATA
• 0.25 ~ X 0.50 • 0.75 ~ • 1.00 c:::J o AS MARKED 1.00
1.50
3.00
86.0
5.0 04.5
3.0
1.5
00 100 200
SWITCHING FREQUENCY n (MHz) 300 (1)
(2)
FIG. 8. The frequency response of the magneto-optical
switch: Experimental data are shown as isolated points. The
shaded areas are the results of a theoretical calculation. They
represent the range in which RTM can vary as Hso is increased
from 0 to Hs' The parameters used in the calculation are Hn
=10 Oe, HK=1.2 Oe, 471M=1050 Oe, and h=10 MHz.
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FIG. 9. Oscilloscope traces observed at the switching fre
quency [2=300 MHz.
Here the D's and E's are related through the following
tensor:
D~ n~ -io 0
ni 0 (3)
o ni
where nl is the refractive index of the film and Ilo and
(0 are the permeability and dielectric constant of the
vacuum, respectively. If the magnetization M is satu
rated along the x axis which is the direction of optical
propagation, 0 is related to the saturated Faraday rota
tion constant e as shown in Eq. (10).
Let the TM and TE waves propagate in the film ac
cording to exp (if3Mx) and exp (if3Ex), respectively, and
k == w/ c, where w is the angular frequency of the laser
and c is the velocity of light in a vacuum. We choose the
solutions of Eqs. (1) and (2) to be
(4)
The choice of Eq, (4) for the solution deserves some
comments. First, the factors exp( ± ibEz) and exp(± ibMz)
are necessary; without them, we would obtain the usual
magneto-optic effect in an infinite medium. Second, by
using exp(± ibEz) and exp(± ibMz) instead of cos(bEz) and
cos(bMz), we avoid committing ourselves to a specific
film geometry or to a specific waveguide mode. Finally,
the use of bE and bM implies that they satisfy the usual
film-waveguide equations, 2
(5)
J. Appl. Phys., Vol. 45, No.7, July 1974 where Wand mare, respectively, the thickness of the
film and the order of the waveguide mode, The <1>' shave
been defined previously, 2 Now we are able to simplify
the problem by conSidering only the waves in the magne
tic film without going through the process of matching
boundary conditions. After substituting Eqs, (3) and (4)
into Eqs, (1) and (2), we obtain, by solving for A(x) and
B (x), the two coupled linear differential equations,
2f3E oAa(x) = (kn1Y ~ B(x) exp[ ± i(bM -bE)z) exp( -i~f3x), x nl
(6)
2f3M oBex) == -f3~'; A(x) exp[ 'f i(bM -bE)z) exp(i~f3x), (7) ax nl
where
(8)
For films more than 1 Jim thick and restricting our
selves to the waveguide modes far beyond cutoff, the
field distributions of E ~ and E. for the TE (m) and TM (n)
modes are almost identical if the order of the mode
m = n, and they are almost orthogonal if m '* n. For TE
and TM waves of the same order m, we may drop the
factors exp[±i(bM -bE)z) and exp[=t=i(bM -bE)z) in Eqs, (6)
and (7). By combining Eqs. (6) and (7) we have
(9)
which will be discussed extensively below, We note that
even though f3M '* f3E, i. e" ~f3,* 0, we can reasonably ap
proximate the ratio f3M/f3E", 1 in Eq. (9).
Consider first the Simplest case for ~f3 = O. For the
magnetization uniform ally distributed over the sample
and saturated in the x direction we have
(10)
where e is the Faraday rotation constant in rad/cm. For
the presently used films, e=150o/cm or 2.62 rad/cm,
After substituting Eq. (10) into Eq, (9), we obtain
(11)
For TE to TM mode conversion starting from x == 0, the
solutions are
(12)
E;M = R(x) (f3M/ knl)A(O) sin(iJx). (13)
We note from Eqs. (12) and (13) that the Faraday rota
tion in a film waveguide is identical to that for an infinite
medium provided that ~f3 = 0 as we have assumed at the
beginning. The factor f3M/k~ in E;M is necessary to en
sure that all of the power in the TE wave is converted
into that in the TM wave over a distance corresponding
iJx= tJT.
In the second case, we again consider a uniform mag
netization saturated in the x direction, but with ~f3,* 00
By solving Eqs, (9) and (10), we obtain
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(14)
E~ = B(x) = (J3~/2 J3~/2/kn1) ({~J32 ! /12)1/2
XA(O) exp( -i~J3x) sin[ (t~J32 + /12)1 /2X J. (15)
We can easily show that when ~J3=0,. Eqs. (14) and (15)
reduce to Eqs. (12) and (13). For ~J3*0, the maximum
possible power conversion efficiency from TE to TM is
the square of the factor /1/(t~J32 + /12)1/2 appearing in E;M
in Eq. (15). For example, for a film thickness of 3.5
!lm, ~J3=0. 00045 and k=24. '7 rad/cm for the m =0
modes. Using /1= 150o/cm or 2.62 rad/cm we obtain
/12/(t~J32 + /12).., 1%. The conversion efficiency attainable
is thus very small when ~J3 * O. E;M in Eq. (15) increas
es continuously with distance x up to x = ~1T( t~J32 + /12)-1/2
which is defined as the coherence length.
To increase the conversion efficiency when ~J3 * 0, we
consider, as the third case, the use of the serpentine
circuit. In the serpentine circuit, the current flows in
the + y direction in one leg, in the -y direction in the
next leg, etc. The field produced by the circuit thus al
ternates in the + x and -x directions as the light beam
underneath the circuit goes by one leg and another. The
serpentine structure has a period P satisfying the phase
matching condition
(16)
In addition to a uniform magnetic field Hn applied in the
y direction, we feed a current 1=10[1 +sin(Ot)] into the
circuit to produce a switching field in the x direction.
We assume, for simplification, that when Ot= -~1T and
1 = 0, the magnetization is uniformly saturated in the y
direction because of H n' and when Ot = ~1T and 1 = 210, the
magnetization is saturated alternatively in the + x and
-x directions over the period of the serpentine struc
ture. Under these conditions, we can write approximate
ly, instead of Eq. (10),
(17)
where cos(~J3x) is the fundamental Fourier component
of the spatial field distribution produced by the serpen
tine circuit. Rewriting cos(~J3x) in Eq. (17) in the form
of ~[exp(i~J3x) +exp(-i~J3x)J and using the term involving
exp(i~j3x) for Eq. (6) and that involving exp( -i~J3x) for
Eq. (7), we obtain by combining Eqs. (6) and (7) .
(18)
Now the solutions are
E;E =A(x) :::A(O) cos{1 ~/1[1 + sin(Ot)] Ix},
E~=B(x) =(,B~/2,B1/2/k~)A(0) sin{1 M[l + sin(Ot)J Ix}.
(19)
By examining Eq. (19), we find that the serpentine cir
cuit provides a continuous coupling between the TE and
J. Appl. Phys., Vol. 45, No.7, July 1974 TM waves despite ~J3*0, and consequently, the TE to
TM conversion can be complete if Ot = ~1T and /1x = ~1T.
Moreover, the amplitude modulation of the TM wave is
always 100%. For a small /1x, we may approximate
sin{1 ~/1[1 +sin(Ot)]lx} by I ~8[1 + sin(Ot)] Ix; the amplitude
of the TM wave then varies linearly with the current in
the circuit.
V. MOTION OF MAGNETIZATION
We now consider a more complicated problem in fer
romagnetism involving the collective behavior of spins.
Although the serpentine circuit produces a switching
field which varies both in time and space, the spacing
between any two neighboring legs of the serpentine struc
ture is fairly large, ranging typically from several
tenths of a millimeter to several millimeters. Since
these spacings are too large for the excitation of the
spin waves, we are still in the regime of the uniform
magneto static mode. 9 It is then only necessary to single
out a small area under anyone leg of the serpentine
structure in which the motion of the magnetization may
be considered uniform. Moreover, because of the use of
the serpentine circuit, the effect of ~j3 needs not to be
considered. Hence, we consider in this section a small
sampling area of the film. Within that area, ~J3 can be
considered to be zero and the magnetization to be uni
form. For optical switching, a switching field somewhat
larger than or comparable to the anisotropy field is ap
plied. Hence, the process considered is that involving
the rotation of the magnetization in unison.
Let us formulate the problem more explicitly. The
motion of the magnetization should follow the Gilbert
equation10 which is a modification of the well-known
Landau-Lifshitz equation. The motion is under the in
fluence of a switching field H$[l +sin(Ot)] and an aniso
tropy field Hk; both of which lie parallel to the easy x
axis. In addition, a transverse dc magnetic field Hn is
applied in the y direction. For H$ .., 1 Oe, the calculated
ferromagnetic resonance produced by the dc part of the
switching field is -134 MHz. The switching frequency
considered here ranging from 0 to 300 MHz can thus be
above or below the resonsnce. We will calculate the mo
tion of the magnetization under the above conditions us
ipg the method similar to that of Smith,6 Conger and
Essig,l1 Olsen and Pohm,12 or Gyorgy.13 Our problem is
a more complicated one since it involves a time-depen
dent switching field.
We start from the model proposed by Gilbert, which
is
-=-lrl(MXH)+- MX- . dM Q ( dM)
dt M dt (20)
The equation removes the inconsistencies found in the
Landau-Lifshitz equation for the heavily damped case.
According to Gillette and Oshima14 and Smith, 6 Eq. (20)
can be reduced to the form
~ =-lrl(MXH) -~[MX(MXH)]. (21)
In Eqs. (20) and (21), t is the real time and
(22)
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On: Fri, 19 Dec 2014 15:00:263065 Tien, Schinke, and Blank: Film-waveguide optical switch 3065
IRON GARNET
FILM z
y
FIG. 10. A spherical coordinate system used to calculate the
motion of the magnetizatioii.
where
(23)
The gyro magnetic ratio I yl is 208 MHz/Oe. The quanti
ty X is the damping constant originally proposed by
Landau and Lifshitz, expressed here in MHz and related
to the Gilbert damping constant O! through Eq. (23),
To simplify the discussion, a spherical coordinate
system defined below will be used. In Fig. 10, we
choose M in the r direction and a z axis normal to the
film. Since M. stays very nearly in the plane of the film,
it is convenient to choose 1/! as the angle between M and
its projection to the xy plane, and cp as the angle between
this projection and the x axis. Again the easy axis of the
film is along the x axis. Let the free energy of the mag
netic system be E. Then the generalized force F and the
torque T applied to the magnetization are, respectively,
F=-vE,
T=rxF.
We have
at 1 a[ T = (M xH) = cp --1/! --. a1/! cos1/! acp (24)
(25)
(26)
Here r, 1/!, andcp are the unit vectors of the spherical
coordinate system according to our convention.
The free energy of our magnetic system is
[=Ksin2cp -H)l + sin(Ot)]M cos1/! coscp
(27)
The first term of Eq. (27) is the anisotropy energy and
K is the anisotropy constant. The second and third terms
represent magnetic energies of the external fields ap
plied in the x and y directions, respectively. The last
term represents the demagnetizing energy when M is
out of the xy plane. Even though 1/! is very small, the
term is significant because of the large demagnetizing
J. Appl. Phys., Vol. 45, No.7, July 1974 factor in the z direction, and it does play an important
role in the dynamic behavior. It is convenient to use di
mensionless quantities:
(28)
hs=H/H K•
By substituting Eqs. (24)-(28) into Eq. (21) we obtain
two coupled equations of motion,
~~= IYIMh K(Ltan1/!+ !: Sin\b)
Xh (! sin2cp 1 dL)
- K ~·2 cos21/! -cos1/! dcp , (29)
d1/! = I Y I Mh (! sin2cp _ dL)
d1" K 2 cos1/! dcp
I· 21T ) -XhK ~L sin1/! +1?K sin21/! . (30)
where
and
~~ = -hJ 1 + sin(Ot) J sincp + hr coscp.
Equations (29) and (30) are important and they will be
used for numerical calculation in Sec. VI.
VI. NUMERICAL CALCULATION
We first divide both sides of Eqs. (29) and (30) by Q
and then convert 1" into the real time t. The resultant
difference equations are
where
and
Fl = L sin1/! + !1T sin 21f;
K (31)
(32)
(33)
(34)
(35)
USing ~(Ot) = O. 021T for each step of a numerical calcula
tion, we divide a full cycle of switching into 100 steps.
The motion of the magnetization is obtained by numeri
cally integrating ~cp and ~1/! step by step. The initial val
ues of cp and 1/! are computed from the equilibrium posi
tion of M at Ot = 0; they are
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On: Fri, 19 Dec 2014 15:00:263066 Tien, Schinke, and Blank: Film-waveguide optical switch 3066
1.0.
0..9
0..8
0..7
0..6
0..5
0..4
i 0..3
0..2
-s-0..1
~ 0.
'-'I -0..1
-0..2
-0.3
-0..4
-0..5
-0..6
-0..7
-0..8
-0..9
-1.0. ~
t-
1.\.1
0. ~ ~ If
~ II
--
III It.I (0) X=1MHz
1
217'
1.0. r-=:::======::::=:--T-=-=-----.-==--, 0..9
t 0..8
0..7
-S-0..6
:g 0..5
'-' 0..4
1 0..3
0..2
0..1 (b)X = 5MHz
o.~------------~~------------~------~
0. 217' ~nt
1.0. L==::::=====:r--=.~----.---, 0..9
t 0..8
0..7
-S-0..6
:g 0..5
'-' 0..4
0..3
0..2
0..1 (e) >.. = 50.MHz
O'o.~------------~-------------L----~ ~ nt 217'
FIG. 11. The curves of cos¢' vs Ot are calculated using 0 = 10
MHz. HK'" 1. 2 Oe, 471'M=1050 Oe, Hn=1.0 Oe, Hs=1.0 Oe,
and a switching field in the form Hs[l+sin(Ot)1. In (a) A=l
MHz. the magnetization oscillates spontaneously. This type
of oscillation is completely damped out in (h) A = 5 MHz and in
(e) A= 50 MHz.
(36)
l/!(O) =0. (37)
It is important to note that if both C/>(O) and 1/>(0) were
zero, it would take an infinite time to start the first
move. The role of Hn is to prevent M from being dead
locked at the x axis. This has the effect of shortening
the switching time and thus improving the frequency re
sponse of the device. We also note in Eqs, (31) and (32)
that both ~C/> and ~l/! are inversely proportional to the
switching frequency n, and thus a higher value of n
means more steps are needed for the integration to
achieve a same amount of motion in C/> and l/!. Stated
more specifically, when the switching frequency increas
es, the time available to the magnetization for a full
swing between the x and y axes becomes correspondingly
shorter. It is then necessary to apply a larger switching
field and, consequently, to exert a larger magnetic tor
que for the rotation of the magnetization.
J. Appl. Phys., Vol. 45, No.7, July 1974 VII. RESULTS OF NUMERICAL CALCULATION
We made extensive numerical calculations using Eqs.
(31) and (32). The results obtained for various forms of
the switching fields are discussed fully below. Although
the motion of the magnetization is complex and highly
nonlinear, it is seen that the roles played by HK, Hso,
Hs, and Hn follow simple rules.
We choose to represent the motion of the magnetiza
tion and the associated magneto-optical effect by two
different sets of curves:
(i) The plot of cosC/> vs nt for nt from 0 to 211'. Here,
we show the motion of M in the xy plane during the en
tire cycle of switching.
(ii) The TE to TM (or TM to TE) conversion. This is
directly proportional to I Mx I or I cosC/> I. (Note that the
absolute values of Mx and cosC/> are used.) Thus,
I cosC/> I max and I cosC/> I min are proportional to the maxi
mum and minimum amplitudes of the TM signal, respec
tively. The quantity RTM = (I cosC/> I ~ax -I cosC/> I !In) is then
the normalized TM response measured by the detector
as described in Sec. III in connecting with Fig. 8, and
the curves, RTM vs n, represent the frequency response
of the optical switch.
First, we study the .effect of the ferromagnetic reso
nance. USing a switching field in the form Hs[l + sin(nt)]
and the parameters n=10 MHz, HK=1.2 Oe, Hs=1.0
Oe, Hn = 1. 0 Oe, and 411'M = 1050 Oe, we plot cosC/> vs
Qt in Figs. l1(a)-l1(c) for the damping constants .\ = 1,
5, and 50 MHz, respectively . For .\ = 1 MHz, the reso
nance is underdamped, and M oscillates spontaneously
at a frequency somewhat smaller than the calculated re
sonance frequency Qo, but about 10 times the switching
frequency n. Here no is computed by ignoring Hn and we
consider the dc field in the x direction only. Thus,
(38)
We should point out that when the amplitude of the rf
field is comparable to the dc field such as these in our
case, the ferromagnetic resonance is not well defined.
The oscillation described above is completely damped
out at A? 5 MHz as seen from Figs. 11(b) and l1(c). We
note also that a tenfold increase of .\ from A = 5-50 MHz
merely depresses the motion of M by -40%, The iron
garnet films used here have a .\ ranging from 5 to 20
MHz. Hence, the ferromagnetic resonance does not play
any significant role in the switching operation. These
res'ults are interesting since it is a frequent notion,
though a misleading one, that the optical magnetoswitch
ing depends critically on the ferromagnetic resonance
frequency and linewidth.
Next, we plot, in Fig. 12, RTM vs n for H.=l and 3
Oe. Surprisingly, the two curves in the figure are al
most identical in spite of a threefold increase in H •• To
understand this, we consider below the roles played by
Hs and Hn in the motion of the magnetization. It is obvi
ous that the most efficient way to switch M between the
x and y axes is to use a field which alternates between
the x and y directions. For this we need two mutually
perpendicular rf coils. However, the important applica
tion of the switch is in the modulation of light, and we
prefer, for practical purposes, to have one dc field Hn
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On: Fri, 19 Dec 2014 15:00:263067 Tien, Schinke, and Blank: Film-waveguide optical switch 3067
1.0,---...,---,----,--,--,---,--,----,
c 'e -e-.8
N ..
~~i'6
N-e-.4 .. o u
~
.. .2
~
f-a::
o A = 10MHz
Hn=I.00e
4vM = 1050 Oe
HK= 1.2 Oe
Hs(1+sinntl
40 80 120 160 200 240 280
SWITCHING FREQUENCY n (MHz)
~
FIG. 12. The curves of RTM vs 0 are calculated for Hs = 1. 0
and 3.0 Oe, respectively. The other parameters used in the
calculation are HK= 1. 2 Oe, Hn = 1. 0 Oe, 4rrM = 1050 Oe, and
71.=10 MHz. We again use a switching field in the form
Hs[1 + sin (Ot) I.
along the y direction and one rf switching field H. along
the x direction. In that case, M can be pulled toward the
y axis by the transverse field Hn, or it is pushed to the
y axis when the switching field Hs is negative along the
positive x axis. By using a switching field in the form
Hs[l +sin(Ot)], which is never negative, we rely com
pletely on the effect of Hn' It is then clear that one can
not improve the performance of the switch by increasing
H. alone, as indicated in Fig. 12.
We can, indeed, improve RTM by raising H. and Hn
simultaneously as shown in Fig. 13. Here, RTM is plot
ted for Hs ==Hn = 1. 0, 1. 5, 3.0, and 4.0 Oe. A response
of nearly 100% efficiency is seen between 0 and 300 MHz
for Hs==Hn=6.0 Oe (about 5 times the anisotropy fielq).
To illustrate the waveforms of the responses, coscp is
plotted vs Ot in Figs. 14(a)-14(d) for Hs =Hn = 3.0 Oe
and for 0=40, 100, 200, and 300 MHz, respectively.
Instead of increasing Hs and Hn together to enhance the
motion of the magnetization as described above, we can
achieve the same result by using a fixed Hn. and by apply-
c
'E
-e-.8
N
OJ
~ii'6
-e-.4
N
~ ()
• ~ fa: .2 SWITCHING FIELD:
Hs (1 +sin n t I
Hn"HS;HK=t.20e
A =10 MHz
20 60 100 140 180 220 260 300
SWITCHING FREQUENCY.(l(MHzl
FIG. 13. The curves of R™vs 0 are calculated for Hs=Hn
=1.0,1,5,3.0, and6.00e, respectively. We again use
4rrM= 1050 Oe, HK= 1. 2 Oe, 71.= 10 MHz, and a switching field
in the form Hs[1 + sin (Ot) I.
J. Appl. Phys., Vol. 45, No.7, July 1974 1.0 r----;:;~;:::;;:_--------__,
08
-9-0.6
en
804 Hs(1+sinnt)
Hs=Hn=3.0 Oe
A = 10MHz
(a)n = 40MHz 0.2
°O~----------L-----~~~--~--~
7T 27T
1.0 .------..-~----------------------.
08
-9-0.6
en
804
0.2 (b) n=100MHz
O~-----------~------~~~--~ o 7T
1.0,...-------:::_-"""""--------,
0.8
-9-0.6
en
8 0.4
0.2 (e) n=200MHz
°O~------~------~--~ 7T 27T
1.01------=:::===::::-------,
0.8
-9-0.6
en
80.4
0.2
°o~-----------L---------~~-~ 7T 27T (d) n =300MHz
nt---+
FIG. 14. The curves of coscp vs Ot are computed under the
same conditions shown in Fig. 13 for the case H.=H,,=3.0 De
and for (a) 0=40 MHz, (b) 0=100 MHz, (c) 0=200 MHz, and
(d) Q = 300 MHz.
ing a switching field in the form Hso + H. sin(Ot). In this
case, for Hso <Hs' the switching field is negative during
a part of the switching cycle. When it is negative, it as
sists Hn in moving M toward the y axis. During the other
part of the switching cycle, when the switching field is
positive, it attracts M toward the x axis against the
force of H". However, by using too small of a value of
Hso, the magnetization could swing a full 1800 from the
+x to -x axis, and then RTM would have a frequency
twice the switching frequency. In order to avoid this in
the experiments described in Sec. ill, we adjust the val
ue of H.o in each experiment so that it is smaller than
Hs' yet large enough to restrict the motion of M within
the first quadrant of the xy plane. The data obtained in
these experiments are isolated points in Fig. 8. To com
pare them with the theory, we calculate the range in
which RTM can vary as Hso is increased from 0 to H •• We
made three sets of the calculations for Hs = 1. 0, 1. 5,
and 3. 0, respectively. They are presented by three
shaded areas in Fig. 8. In each set of the calculation,
H. and Hn are held constant. The other parameters used
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On: Fri, 19 Dec 2014 15:00:263068 Tien, Schinke, and Blank: Film-waveguide optical switch 3068
in the calculations are H"=1.0 Oe, HK=1.2 Oe, 41TM
= 1050 Oe, and ,\ = 10 MHz. It is interesting that the
three shaded areas share one common lower boundary
at which Hsn=Hs' We also find that the theory agrees
well with the experiments. For those experimental points
involving Hs« HK, the switching process may not be ro
tation in unison and we should not expect the present
theory to be applicable.
There are many different ways to operate this mag
neto-optical switch. We have described only a few of
these to illustrate the basic principles involved. For ex
ample, H" or Hs can be applied along a direction at 45 °
between the x and y axes. A modulator can be construct
ed so that the magnetization rotates 180 ° instead of 90 ° .
We have also made calculations using a rf field along the
y axis and a dc field along the x axis. The results thus
obtained are similar to those described above.
VIII. CONCLUSIONS
We have reported a detailed study of an iron-garnet
film -waveguide magneto -optical switch, Our experimen
tal work has included measurements of magnetic pro
perties of iron-garnet films, observation of magnetic
domains, and experiments on the optical switch from
0-300 MHz. Our theOretical work has included a theory
of magneto-optics in optical waveguides, an analysis of
the serpentine circuit, and an extensive calculation for
the motion of the magnetization. As a result of this
study, the basic processes involved in magneto-optical
switching are better understood. Our important conclu
sions are as follows:
(1) For low driving fields, the optical switching in
volves periodic magnetic domains and domain-wall mo
tion, but for higher driving fields, the basic process is
rotation of the magnetization in unison.
(ii) For films having a damping constant of 1 MHz or
less, the magnetization vector oscillates spontaneously
upon application of the switching field, but for films
having larger damping constants, this type of oscillation
does not exist and the motion of the magnetization does
not depend on ferromagnetic resonance.
(iii) As the switching frequency increases, less time
is available for the magnetization to perform a 90 ° ro
tation, and consequently, a larger driving field is needed
to speed up the rotation. The upper limit for the switch
ing frequency is well above 1 GHz. It comes only when
the driving field H. approaches 41TM.
(iv) The iron-garnet film used for optical Switching
should have (1) a low in-plane magnetic anisotropy, (2)
J. Appl. Phys., Vol. 45, No.7, July 1974 a magnetic damping constant between 5 and 20 MHz, and
(3) low optical absorption.
For the baseband modulation between 0-300 MHz, a
driving field of about five times the anisotropy field is
required. The anisotropy field in iron-garnet films used
in the experiments varies from less than O. 3 to 1,2 Oe
depending on the pressure exerted on the film by the
prism c'oupler (because of the magnetostrictive effect).
Recently, materials of much larger Faraday rotation
(GdPr2Fes012, B=1125°/cm) have been developed. is
Using these new materials, efficient magneto-optical
switches having a size of about 1 mm2 and operating at
1. 06 /J.m laser wavelength can be constructed. Moreover,
the iron-garnet films discussed here exhibit the same
magnetic properties as the Permalloy films and are thus
excellent materials for magnetic memory. By combining
magnetic memory with optical switching, we expect a
number of interesting thin-film devices for integrated
optical circuits. In particular, the Single-material op
tical fibers recently developed in Bell Laboratories ex
hibit very low transmission losses near 1.1 /J.m light
wavelength. The iron-garnet films which are transparent
from 1. 1 to 5.0 f.J.m can be important in the applications
of optical communication.
ACKNOWLED(3MENTS
The authors wish to thank Dr. C.K.N. Patel for crit
ical reading of the manuscript, and Dr. R. Wolfe and
Dr. R.C. Le Craw for valuable discussions.
lP.K. Tien, R.J. Martin, R. Wolfe, R.C. LeCraw, andS.L.
Blank, Appl. Phys. Lett. 21, 394 (1972).
2P.K. Tien, Appl. Opt. 10, 2395 (1971).
3A. V. Pohm and E. N. Mitchell, IRE Trans. Electron.
Comput. EC-9, 308 (1960).
4R. F. Soohoo, Magnetic Thin Films (Harper & Row, New
York. 1965).
5See, for example, Ref. 4, Chap. 5.
6D.O. Smith, J. Appl. Phys. 29, 264 (1958).
7S. Wang, M.L. Shah, andJ.D. Crow, IEEE J. Quantum
Electron. QE-8, 212 (1972).
sL. Landau and E. Lifshitz, P\lys. Z. Sowjetunion 8, 153
(1935).
9L.R. Walker, Phys. Rev. 105, 390 (1957).
lOT. L. Gibert, Phys. Rev. 100, A 1243 (1955).
11R.L. CongerandF.C."Essig, Phys. Rev. 104, 915 (1956).
12C.D. OlsonandA.V. Pohm, J. Appl. Phys. 29,274 (1958).
13E.M. Gyorgy, J. Appl. Pb,ys. 29, 283 (1958).
14p.R. Gillette and K. Oshima, J. Appl. Phys. 29, 529
(1958) .
15S.H. Wemple, J.F. Sillion, Jr., L.G. Van Uitert, andW.H.
Grodkiewicz, Appl. Phys. Lett. 22, 331 (1973).
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1.4883297.pdf | Current-driven domain wall mobility in polycrystalline Permalloy nanowires: A
numerical study
J. Leliaert, B. Van de Wiele, A. Vansteenkiste, L. Laurson, G. Durin, L. Dupré, and B. Van Waeyenberge
Citation: Journal of Applied Physics 115, 233903 (2014); doi: 10.1063/1.4883297
View online: http://dx.doi.org/10.1063/1.4883297
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/23?ver=pdfcov
Published by the AIP Publishing
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130.64.175.185 On: Fri, 05 Dec 2014 17:31:08Current-driven domain wall mobility in polycrystalline Permalloy nanowires:
A numerical study
J. Leliaert,1,2,a)B. Van de Wiele,1A. Vansteenkiste,2L. Laurson,3G. Durin,4,5L. Dupr /C19e,1
and B. Van Waeyenberge2
1Department of Electrical Energy, Systems and Automation, Ghent University, Sint-Pietersnieuwstraat 41,
9000 Gent, Belgium
2Department of Solid State Science, Ghent University, Krijgslaan 281/S1, 9000 Gent, Belgium
3COMP Centre of Excellence and Helsinki Institute of Physics, Department of Applied Physics,
Aalto University School of Science, P.O. Box 11100, FI-00076 AALTO, Finland
4Istituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91, 10135 Torino, Italy
5ISI Foundation, Via Alassio 11/c, 10126, Torino, Italy
(Received 7 April 2014; accepted 2 June 2014; published online 17 June 2014)
A complete understanding of domain wall motion in magnetic nanowires is required to enable future
nanowire based spintronics devices to work reliably. The production process dictates that the samplesare polycrystalline. In this contribution, we present a method to investigate the effects of material
grains on domain wall motion using the GPU-based micromagnetic software package MuMax3. We
use this method to study current-driven vortex domain wall motion in polycrystalline Permalloynanowires and find that the influence of material grains is fourfold: an extrinsic pinning at low
current densities, an increasing effective damping with disorder strength, shifts in the Walker
breakdown current density, and the possibility of the vortex core to switch polarity at grainboundaries.
VC2014 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4883297 ]
I. INTRODUCTION
A large number of future spintronics devices1–3are
based on the controlled movement of domain walls through
magnetic nanowires. To achieve this, a thorough under-standing of domain wall mobility in real nanowires is of
paramount importance. Research has mainly focused on do-
main wall motion in perfect nanowires
4,5or nanowires with
edge roughness.6,7However, it has recently been recog-
nized that disorder distributed throughout the whole wire,
as present in any real material, can have an important effecton the domain wall mobility.
8–10Due to the imperfect fabri-
cation process, distributed disorder exists on the atomistic
scale (interstitials, vacancies, dislocations, etc.) and on thelevel of the material grains. Grain properties as grain size,
thickness, etc., can vary, while the grain boundary corre-
sponds with a misfit of the lattice structure of neighbouringgrains. Numerical
10and experimental11–13investigations
show that distributed disord er gives rise to local pinning
potentials. For a magnetic vortex in Permalloy, the poten-tials have a depth of 1 to 5 eV,
11,12,14and an interaction
range approximately equal to the vortex core diameter since
the measured potential well is convolved with the vortexcore profile.
11,13Despite some controversy regarding the
nature of the measured disorder,14a link with the grain
structure of the material is suspected.11,13In this contribu-
tion, a method is presented to simulate polycrystalline
materials in a computationally efficient way, and the influ-
ence on the current-driven mobility of vortex domain wallsis investigated.II. METHODS
We perform micromagnetic simulations using the GPU-
based micromagnetic software package MuMax3,15which
solves the Landau-Lifshitz equation16with spin-transfer-tor-
que (STT) contributions17
@m
@t¼/C0c0m/C2Beffþam/C2@m
@t
/C0u/C1r½/C138 mþbm/C2u/C1r½/C138 m: (1)
Here, mis the space and time varying magnetization
vector field, with a fixed amplitude jmj¼1. Furthermore, c0
denotes the gyromagnetic ratio 1.7595 /C21011rad/Ts and ais
the dimensionless Gilbert damping constant. The effectivefield B
effis the derivative with respect to mof the energy
density /C15with contributions of the exchange, anisotropy,
Zeeman, and demagnetizing energy18
Beff¼/C01
c0Ms@/C15
@m: (2)
The last two terms of equation (1)are the STT terms,
taking into account the effects of a spin-polarized current
running through the nanowire. The first STT-term represents
the adiabatic interactions of the conduction electrons withthe local magnetization, while the second term is a smaller
non-adiabatic contribution with size b. Furthermore, uhas
the dimensions of a velocity
u¼glBP
2eMsJ; (3)
with gthe Land /C19e factor, lBthe Bohr magneton, ethe elec-
tron charge, and Pthe polarization of the current density J.19 a)Electronic mail: jonathan.leliaert@ugent.be
0021-8979/2014/115(23)/233903/6/$30.00 VC2014 AIP Publishing LLC 115, 233903-1JOURNAL OF APPLIED PHYSICS 115, 233903 (2014)
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130.64.175.185 On: Fri, 05 Dec 2014 17:31:08In our simulations, we consider head-to-head magnetic
domain walls. To efficiently simulate the domain wall
motion in infinitely long magnetic nanowires, the computa-tion is restricted to a window centered around the moving
domain wall. To this end, the demagnetizing field introduced
by the magnetic charges at the edges of the window is com-pensated by an opposite external field. The magnetization
dynamics are evaluated by timestepping equation (1). Every
timestep the simulation window is shifted to keep the domainwall centered, i.e., to keep the average magnetization compo-
nent along the nanowire axis close to zero.
The grain structure of the polycrystalline material is
implemented using a Voronoi tesselation, in which each
Voronoi cell represents a grain. This approach enables one to
define both edge roughness
6,20and material grains.21
Subdividing the material in grains starts with defining ran-
domly distributed points (Voronoi centers) across the simula-
tion geometry. A Voronoi cell consists of all points closestto a common Voronoi center. To cover the infinitely long
simulation geometry, we virtually divide the nanowire into a
grid of square tiles, sufficiently large so we can expect atleast a few Voronoi centers per tile. Poisson statistics are
used to determine the number of centers in the tile, while
their positions are uniformly distributed over the tile using arandom number generator with seed based on the tile index.
This way we can map each discretization cell in the moving
computational domain to a Voronoi cell without explicitlystoring the complete tesselation along the wire. Indeed, when
shifting the computational domain, new grains can be
inserted from the sides based on the tile index. Also, as thesimulation window might sometimes move backwards, this
enables grains that previously left the simulation window to
re-enter. Our implementation is sketched in Fig. 1, and an
example is shown in Fig. 2.
Having subdivided the geometry, one can vary the local
material parameters in and between the grains. This way,grain dependent anisotropy directions can represent the dif-
ferent lattice orientations in grains. However, Permalloy is
specifically designed to minimize the influence of anisotropyon the magnetization dynamics. The other possibilities are
grain dependent saturation magnetization representing thick-
ness variations between grains
9and reduced exchange stiff-
ness at the grain boundaries representing a reduced magnetic
coupling between neighbouring grains.21It is shown that
both approaches to implement the influence of materialgrains on the magnetization dynamics can give rise to static
pinning potentials corresponding to those that are experi-
mentally found for distributed disorder.
10In Sec. III,w e
investigate the influence of grains on the domain wall
motion. To discriminate between the influence of thickness
variations and reduced mutual exchange coupling, we treatthem separately, while in real materials both effects are
expected to play a role simultaneously.
We consider Permalloy nanowires of thickness 10 nm
and width 400 nm discretized in finite difference cells of size
3.1255 /C23.125 /C210 nm
3. The simulated time frame is
500 ns while the moving window around the domain wall is1200 nm wide, and shifts with the domain wall. Material pa-
rameters typical for Permalloy are used: exchange stiffness13/C210
/C012J/m, saturation magnetization 860 /C2103A/m, and
Gilbert damping a¼0.01 and 0.02. The average diameter of
the Voronoi cells is 10 nm, which corresponds to the thick-
ness of the nanowire. First, the influence of the grain bounda-
ries is studied by reducing the exchange stiffness from 100%to 30% of the original value in steps of 10%. Second, the
influence of grain thickness fluctuations is studied by varying
the saturation magnetization within the different grains. Amaximum deviation Dof the average saturation magnetiza-
tion M
sis considered from 0% to 25% in steps of 5%. In a
given simulation, the saturation magnetization for each grainis taken randomly from the set fM
s/C0D;Ms/C0D=2;Ms;
MsþD=2;MsþDg.
The domain wall motion is driven by spin-polarized
currents. While some experiments based on domain wall
motion22–25report values for the degree of non-adiabaticity
b/C25a, it was recently shown38that the scheme used to
extract bis compromised by the effects of disorder.
Therefore, we follow experiments suggesting that b>afor
Permalloy26–33and use b¼2a.
The depth of the pinning potentials caused by disorder10
is an order of magnitude larger than an energy kBTat
T¼300 K ( kBis the Boltzmann constant). Therefore, thermal
effects on the domain wall mobility are negligible and the
nanowires are simulated at a temperature of 0 K.
III. RESULTS AND DISCUSSION
A. Non-disordered nanowires
To understand the influence of material grains on do-
main wall motion, we first consider the dynamics in a nano-
wire without disorder. The considered cross-sectionaldimensions of the nanowire dictate that the static equilibrium
domain wall configuration is a vortex wall.
34A vortex wallFIG. 1. The different Voronoi centers (black cells) are generated within the
different tiles, with a random generator that uses the tile indices as a seed.
For each discretization cell, the Voronoi cell to which it belongs is deter-
mined by looking for the closest Voronoi center in its own (dark grey) andall neighbouring (light grey) tiles. In this way also Voronoi centers outside
the simulation window (blue) are found.
FIG. 2. An example of a nanowire with dimensions 200 nm /C21600 nm sub-
divided into Voronoi cells (grayscale) with average diameter 20 nm.233903-2 Leliaert et al. J. Appl. Phys. 115, 233903 (2014)
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130.64.175.185 On: Fri, 05 Dec 2014 17:31:08(Fig. 4) is a domain wall in which the magnetization rotates
around a vortex core with out-of-plane magnetization. Figure 3
shows that for a non-disordered wire, i.e., without reductionin exchange coupling or thickness variations, two linear ve-
locity vs. applied current regimes are separated by the
Walker breakdown (WB). Below the Walker breakdown, thevortex domain wall transforms into a transverse domain wall
that moves along the wire without changing its shape, result-
ing in a constant velocity which is linearly dependent on theapplied current (Fig. 4). Above the Walker breakdown, the
transverse wall is no longer stable and a periodic motionwith successive transformations from vortex to transverse
domain wall structure and vice versa takes place. Since vor-
tex walls move at lower velocity than transverse domainwalls for the same applied current, this results in a sudden
drop in domain wall velocity at the Walker breakdown.
Above the Walker breakdown, the velocity increases againwith growing current densities (Fig. 3).
B. Polycrystalline nanowires
We initialize the magnetization as a transverse domain
wall, as this is the only stable state in non-disordered nano-
wires below the Walker breakdown at non-zero current den-sities. The mobility of this domain wall is simulated with
different reductions in exchange coupling between the
grains, as shown in Fig. 3. Figure 5shows similar mobility
curves for grains simulated as variations in the saturation
magnetization. Both Figs. 3and5show that at low current
densities, the disorder is able to pin the magnetic domainwall. This extrinsic pinning mechanism
8gets stronger for
larger reductions in the exchange coupling and larger varia-
tions in saturation magnetization as the depth of the corre-sponding pinning potential increases.
10
Below the Walker breakdown and above the depinning
threshold, the velocity vdepends linearly on the current den-
sity J. From the analytical one-dimensional model,35it is
known that the slope of this curve is proportional to b=a.
Disorder in the nanowires makes the medium through whichthe domain wall moves more viscous, giving rise to a larger
effective damping parameter
8aeff¼aintþaext.aintis the
Gilbert damping constant that dictates how fast energy isirreversibly dissipated to the lattice while a
extarises from
energy transfer to other modes by the scattering processes
within the magnetic system,36for example, the emission of
spin waves. To determine aext,w efi t b=aeffto the slope of
the mobility curves as a function of the reduction in the
exchange coupling or the variation in the saturation magnet-ization for a
int¼0.01 and aint¼0.02. Fig. 6shows in both
cases a similar increase in aextfor a growing average pinningFIG. 3. Velocity vs. applied current density of vortex domain walls in poly-
crystalline nanowires with different reductions of the exchange stiffness at
the grain boundaries. For increasing exchange reductions, a larger extrinsicpinning takes place, the slope of the mobility curve is reduced, and the
Walker breakdown is shifted towards lower current densities.
FIG. 4. Vortex domain wall motion in a non-disordered nanowire. Below
the WB, the vortex wall transforms into a transverse wall with fixed shape
although this is not the equilibrium state in the absence of an applied current.
Above the Walker breakdown, the vortex wall periodically transforms
between vortex and transverse walls.FIG. 5. Velocity vs. applied current density of vortex domain walls in poly-crystalline nanowires with different variations in the saturation magnetiza-
tion. For larger reductions, an extrinsic pinning takes place and the slope ofthe mobility curve lowers.233903-3 Leliaert et al. J. Appl. Phys. 115, 233903 (2014)
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130.64.175.185 On: Fri, 05 Dec 2014 17:31:08strength of the grains. This observation justifies the splitup
ofaeffinto a constant intrinsic part, and an extrinsic part
depending on the disorder in the magnetic system. The rela-
tive effect of disorder becomes increasingly important in sys-tems with a small intrinsic damping. To further investigate
the extrinsic damping, we vary the average grain diameter
and determine a
eff. The results are shown in Fig. 7. We see
that aeffand thus aextincrease for larger grain diameters,
which is in correspondence with experimental results.37This
is explained as follows. Every grain gives rise to a potentialwell which contains a discrete spectrum of energy levels.
When the domain wall enters a grain, it excites spin waves.
When the wavelength of these spin waves corresponds toone of the energy levels, the spin wave remains in the poten-
tial well of the grain and damps out. In larger grains, more
energy levels get excited and the resulting damping is larger.
In Ref. 10, it was shown that a single defect simulated
as either a region with reduced saturation magnetization or
exchange coupling can give rise to equivalent potentialwells. However, depending on the simulation approach, the
collective influence of the ensemble of grains on the domain
wall dynamics differs: only grains simulated with a reducedmutual exchange coupling have a large effect at and above
the Walker breakdown. Below the Walker breakdown, the
domain wall is of the transverse type. At the Walker break-down, a vortex core is nucleated at the wire edge. A reduc-
tion in exchange coupling facilitates the nucleation of the
vortex core, explaining the reduction in Walker breakdowncurrent density in Fig. 3. While above the Walker break-
down, in a non-disordered wire, periodic domain wall trans-
formations (transverse to vortex domain wall and vice versa)take place, the reduced mutual exchange coupling enables
the vortex core to switch polarization at a grain boundary
and hence follow a trajectory near the middle of the wire, asillustrated in Fig. 8. Here, the vortex domain wall never
transforms into the transverse domain wall explaining the
reduced domain wall velocity above the Walker breakdown.This motion resembles the motion of a vortex wall in a non-
disordered wire for b¼a, which is elaborated in more detail
in Ref. 38. Both phenomena originate from the large effect
of local exchange stiffness variations on the vortex core sta-
bility. On the contrary, the effect on the core stability of var-
iations in the saturation magnetization is much weaker.Consequently, the effect on the Walker breakdown current
density is negligible and we do not observe vortex core
switching. Hence, also the domain wall velocity above theWalker breakdown is hardly affected (Fig. 5).
IV. CONCLUSION
In conclusion, we have presented a computationally effi-
cient method to simulate polycrystalline nanowires and have
employed it to investigate the influence of material grains oncurrent driven domain wall motion. We discriminate
between the influence of thickness variations and reduced
mutual coupling between grains and find that the influenceof grains on the domain wall dynamics is fourfold. First, an
extrinsic pinning regime at low current densities appears.
Second, under the Walker breakdown, an effective dampingparameter a
efflowers the slope of the mobility curves. This
damping parameter consists of a constant intrinsic part andFIG. 6. aextas function of the disorder
strength for two different implementa-
tions of grains: a reduction in exchange
coupling at the grain boundaries or a
variation in the saturation magnetiza-
tion between different grains. aextwas
calculated from mobility curves for
aint¼0.01(red crosses) and aint¼0.02
(blue circles).
FIG. 7. Effective damping ( aeff) for different average grain sizes. All values
were determined for aint¼0.02 and DMs¼15%.
FIG. 8. The trajectory of a vortex wall moving through a polycrystalline Permalloy nanowire, 400 nm wide and 10 nm thick. The core is driven by a current
density of 17 /C21012A/m2which is larger than the Walker breakdown. The exchange coupling at the grain boundaries is reduced to 40% of its original value.
The white/black colors indicate a positive/negative vortex core polarization. The core switches its polarization at a grain boundary and consequen tly changes
its transverse propagation direction. As a result, the vortex core stays in the body of the nanowire and the domain wall never transforms into a transve rse wall.233903-4 Leliaert et al. J. Appl. Phys. 115, 233903 (2014)
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130.64.175.185 On: Fri, 05 Dec 2014 17:31:08an extrinsic part which depends on the disorder strength.
Third, only the reduced exchange coupling between grains
facilitates the nucleation of a vortex core, resulting in a lowerWalker breakdown current density and, fourth, the vortex
core is able to switch its polarization at a grain boundary.
Therefore, above the Walker breakdown, the vortex domainwall does not transform into a transverse domain wall and
the vortex core continues to move in the body of the nano-
wire. This work complements earlier studies with edgeroughness, which also has an influence on the mobility of the
domain walls.
6
We presented specific case studies of current driven do-
main wall motion in Permalloy nanowires. However, the
code is freely available15and can be used to investigate the
influence of material grains in other micromagnetic systems.In further research, it could be applied to the influence of
grains on domain wall motion in perpendicular magnetic ani-
sotropy (PMA) materials
39where very large damping is
measured ( aeff¼0.15 in Ref. 40). This might be due to an
extrinsic damping originating in the polycrystalline structure
of the samples.
ACKNOWLEDGMENTS
J.L. thanks M. Dvornik for fruitful discussions. This work
was supported by the Flanders Re search Foundation (B.V.d.W.
a n dA . V . ) ,t h eA c a d e m yo fF i n l a n dt h r o u g ha nA c a d e m y
Research Fellowship (L.L., Project No. 268302), and through
the Centres of Excellence Program (L.L., Project No. 251748),Progetto Premiale MIUR-INRIM “Nanotecnologie per la met-
rologia elettromagnetica” (G.D.), and MIUR-PRIN 2010-11
Project2010ECA8P3 “DyNanoMag” (G.D.)
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1.4985850.pdf | Tunneling magnetoresistance of perpendicular CoFeB-based junctions with exchange
bias
Orestis Manos , Alexander Böhnke , Panagiota Bougiatioti , Robin Klett , Karsten Rott , Alessia Niesen , Jan-Michael
Schmalhorst , and Günter Reiss
Citation: Journal of Applied Physics 122, 103904 (2017); doi: 10.1063/1.4985850
View online: http://dx.doi.org/10.1063/1.4985850
View Table of Contents: http://aip.scitation.org/toc/jap/122/10
Published by the American Institute of Physics
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Orestis Manos,a)Alexander B€ohnke, Panagiota Bougiatioti, Robin Klett, Karsten Rott,
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Center for Spinelectronic Materials and Devices, Department of Physics, Bielefeld University,
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(Received 1 June 2017; accepted 21 August 2017; published online 13 September 2017)
Recently, magnetic tunnel junctions with perpendicular magnetized electrodes combined with
exchange bias films have attracted great interest. In this paper, we examine the tunnel
magnetoresistance of Ta/Pd/IrMn/Co-Fe/Ta/Co-Fe-B/MgO/Co-Fe-B/capping/Pd magnetic tunneljunctions dependent on the capping layer, i.e., Hf or Ta. In these stacks, perpendicular exchange
bias fields of /C0500 Oe along with perpendicular magnetic anisotropy are combined. A tunnel
magnetoresistance of (47.2 61.4)% for the Hf-capped sample was determined compared to the Ta
one (42.6 60.7)% at room temperature. Interestingly, this observation is correlated with the higher
boron absorption of Hf compared to Ta, which prevents the suppression of the D
1channel and leads
to higher tunnel magnetoresistance values. Furthermore, the temperature dependent coercivities ofthe soft electrodes of both samples are mainly described by the Stoner-Wohlfarth model including
thermal fluctuations. Slight deviations at low temperatures can be attributed to a torque on the soft
electrode which is generated by the pinned magnetic layer system. Published by AIP Publishing.
[http://dx.doi.org/10.1063/1.4985850 ]
I. INTRODUCTION
Magnetic tunnel junctions (MTJs) are the backbone of
modern spintronics. MTJs with a fully epitaxial (001) MgObarrier sandwiched by (001) bcc ferromagnetic electrodes,such as Fe, Co, and CoFe, were first theoretically predictedto show high tunnel magnetoresistance (TMR) of several
100% as a consequence of the coherent tunneling of D
1elec-
trons.1–3The experimentally discovered large TMR ampli-
tude of in-plane magnetized MTJs with a crystalline MgObarrier rendered a major breakthrough for these materials.
4,5
Nevertheless, for memory applications, the interest rap-
idly changes towards out-of-plane magnetized systems.MTJs with perpendicular magnetic anisotropy (PMA) haveseveral advantages as compared with their in-plane counter-parts. First, an increasing density of memory cells on a wafercan be realized since no elliptical shape is required to stabi-lize the anisotropy direction.
6Furthermore, the PMA energy
is usually much larger than the energy related to the shapeanisotropy that can be obtained in planar MTJs, allowing
long memory retention at a small size.
7Additionally, for a
given retention time, the critical current density to writeinformation by Spin Transfer Torque (STT) switching isstrongly reduced, provided that Gilbert damping remains lowenough.
8However, neighboring MTJs in a memory array as
well as the reference layer of the STT-switched MTJ will bemagnetically disturbed. This is of major importance sinceeven after a large number of STT switching events, the mag-netic states of the MTJs do not “creep” either to some inter-mediate state or completely reverse. One distinct advantage
of MTJs with exchange bias (EB) layers is the robustness of
the reference magnetization against such perturbation.
9Although several investigations on MTJs with EB have
been reported,4,10,11the combination of perpendicular MTJs
(p-MTJs) with perpendicular exchange bias (PEB) is still
challenging. Here, we address this issue by investigating
stacks of p-MTJs with PEB using MnIr/CoFe and CoFeBelectrodes and varying the capping of the soft electrode.
There are two primary requirements for p-MTJs with
PEB from the magnetic aspect. First, the pinned part of the
junction must have large PEB along with low coercivity ( H
c)
in order to prevent the simultaneous switching of both elec-trodes. Second, the stacks of the soft and pinned electrodemust show high PMA to ensure that the relative orientationof the electrodes’ magnetizations can be parallel (low resis-tance) or antiparallel (high resistance) in the perpendiculardirection. Moreover, the bottom part of the junction is pre-ferred for the pinned part because MnIr acts as an additional
seed layer that promotes the (111) texture of the subsequent
ferromagnetic layer and therefore higher PMA as van Dijkenet al. reported.
12
Although the CoFeB/MgO films directly grown on a
MnIr layer exhibit relatively high PEB, one of their major
disadvantages is the insufficient PMA. Zhang et al.13showed
that the introduction of an interlayer of CoFe/Ta at the anti-ferromagnet (AF)/ferromagnet (FM) interface leads to alarge PEB with large PMA. Furthermore, the presence ofBoron (B) can influence the stack twofold. First, the crystal-lization of CoFeB is inextricably connected to the B diffu-sion within the CoFeB layer and to the neighboring layers.In particular, a high B concentration leads to poor crystalli-
zation and consequently to small PMA. On the other hand,
the presence of B at the interface of CoFeB/MgO is detrimen-tal to TMR
14because it suppresses the conductance through
the band of D1symmetry, which is known to be responsible
for high TMR in epitaxial CoFe/MgO/CoFe (001).15 a)Electronic mail: omanos@physik.uni-bielefeld.de
0021-8979/2017/122(10)/103904/5/$30.00 Published by AIP Publishing. 122, 103904-1JOURNAL OF APPLIED PHYSICS 122, 103904 (2017)
Therefore, in order to reduce the concentration of B in
CoFeB and at the CoFeB/MgO interface, one option is toinsert a B absorber material in close vicinity to the CoFeB.
16
We verify in this work that the introduction of a material
with larger B absorption than Ta (such as Hf) enhances the
PMA of the free electrode and leads to larger TMR.
II. PREPARATION
The films were deposited on thermally oxidized Si
wafers at room temperature (RT) by DC magnetron sputter-ing, at an Ar pressure of P ¼2/C210
/C03mbar. The following
two types of samples were prepared and investigated:
(1) Ta(4)/Pd(2)/Mn 83Ir17(8)/Co 50Fe50(1)/Ta(0.6)Co 40Fe40B20
(0.8)/MgO(2)/Co 40Fe40B20(1.2)Hf(5)/Pd(3)
(2) Ta(4)/Pd(2)/Mn 83Ir17(8)/Co 50Fe50(1)/Ta(0.6)Co 40Fe40B20
(0.8)/MgO(2)/Co 40Fe40B20(1.2)Ta(3)/Pd(3)
where the number in parentheses is the nominal thickness of
each layer in nm. The layer thicknesses stemmed from an
optimization process of a series of films and were chosen forfurther investigation due to the large PEB and PMA of thesestacks. It is known that a strong (111) texture of MnIr leads
to enhanced PEB.
17For that reason, we used the buffer
layer Ta(4)/Pd(2) to induce a strong (111) texture.13Ta, Pd,
Co40Fe40B20;Co50Fe50, Hf, Mn 83Ir17, and MgO films were
deposited from elemental and composite targets. The purity
of all targets was 99.9% or higher. All samples were
annealed at 280/C14C for 60 min in vacuum ( <3/C210/C07mbar)
with a magnetic field of 7 kOe applied perpendicular to thefilm plane, in order to achieve the required coherent (001)-
textured bcc crystal structure and induce the PEB.
In the post-annealing procedure, the amorphous CoFeB
electrode crystallization starts at the CoFeB/MgO interfaces,
templated by the (001) texture in the crystalline MgO tunnel
barrier layer.
16,18Perpendicular hysteresis loops were
recorded using the magnetooptical Kerr effect (MOKE). Forsimplicity, in the rest of this paper, the films Co
40Fe40B20;
Co50Fe50,a n dM n 83Ir17will be symbolized as CoFeB, CoFe,
and MnIr, respectively. On the annealed samples, circularMTJ pillars with diameters of 120 nm, 140 nm, 240 nm, and480 nm were patterned by e-beam lithography and were
etched by Ar-ions until only the Ta/Pd layers remained as the
common bottom lead. After etching, 150 nm of Ta
2O5were
deposited to insulate the MTJs followed by a liftoff procedure.In a subsequent patterning process, individual gold contact
pads were placed on top of each MTJ and a large gold elec-
trode was placed at the edge of the common bottom contact.All measurements were performed by a conventional twoprobe technique. Additionally, for the temperature dependent
experiments, we used a closed cycle helium cryostat obtained
from Cryogenic Ltd. in a temperature range of 1.8–300 K.
III. RESULTS AND DISCUSSION
Figure 1presents the individual loops of the soft and the
pinned electrodes of the Hf- and Ta-capped samples. In par-
ticular, Figs. 1(a) and1(b) illustrate the hysteresis loops of
the soft electrodes of MgO/CoFeB/Hf/Pd (red) and MgO/CoFeB/Ta/Pd (green), respectively. Additionally, Figs. 1(c)and1(d) show the corresponding loops of the pinned elec-
trode Ta/Pd/MnIr/CoFe/Ta/CoFeB/MgO (blue). It is worth
mentioning that the pinned part of both stacks is composed
of the same sequence of materials. Furthermore, the individ-ual films were annealed under the same conditions as the full
stack, according to the description in the preparation part.
From Fig. 1, it is found that the soft and pinned electrodes
for both samples present strong PMA, while the correspond-
ing pinned ones display an EB field of H
ex¼/C0500 Oe.
Figure 2shows the perpendicular magnetic hysteresis
loops of the total stack for the Hf (red)- and Ta (green)-
capped samples, respectively. The major loops are presented
in Figs. 2(a)and2(b), while Figs. 2(c)and2(d) illustrate the
corresponding minor ones. In Figs. 2(a) and2(b), two dis-
tinct magnetic steps are observable, which correspond to the
soft and pinned electrodes. As it is expected from the indi-vidual loops, the full MTJ stacks present an EB field of
H
ex¼/C0500 Oe, along with PMA. It is crucial to be notedFIG. 1. Normalized hysteresis loops of the individual electrodes for the
(a) and (c) Hf-capped and (b) and (d) Ta-capped films, acquired via
MOKE at RT. Corresponding soft (a) MgO(2)/CoFeB(1.2)/Hf(5)/Pd(3) (red),
(b) MgO(2)/CoFeB(1.2)/Ta(3)/Pd(3) (green), and pinned (c) and (d) Ta(4)/
Pd(2)/MnIr(8)/CoFe(1)/Ta(0.6)/CoFeB(0.8)/MgO(2) (blue) electrodes.
FIG. 2. Normalized (a) and (b) major and (c) and (d) minor perpendicular
ð?Þhysteresis loops of Hf (red)- and Ta (green)-capped films, respectively
(MOKE at RT).103904-2 Manos et al. J. Appl. Phys. 122, 103904 (2017)that the EB field is in a direction opposite to the applied field
during annealing.
Moreover, for both films [cf. Figs. 2(c) and2(d)], we
observe a shift of the minor loops with respect to zero mag-
netic field. This asymmetry of the minor loop in a magnetic
hysteresis measurement unveils the dipolar interactions
between the soft and pinned electrodes.19The coupling
strength and character (ferromagnetic or antiferromagnetic)can be determined by the coupling constant Jwhich is calcu-
lated by the formula J¼l
0/C1Hs/C1Ms/C1t, where l0is the per-
meability in free space, Hsis the magnetic shift of the minor
loop, Msis the saturation magnetization, and tis the ferro-
magnetic thickness, respectively. It is worth noting that the
calculated Msand the magnetic dead layer ( tDL) for both
samples are determined from a series of films where the
thickness of CoFeB in the soft electrode varies. The Msand
tDLvalues for the Hf(Ta)-capped samples are found to be
Ms¼(1140613) emu/ccm [ Ms¼(1121613) emu/ccm] as
shown in Fig. 3(a) and tDL¼0:93 nm ( tDL¼0:98 nm),
respectively. The obtained values for Msare in good agreement
with previous reports.19In addition, the magnetic shift for the
Hf(Ta)-capped is identified to be Hs¼22 Oe ( Hs¼20 Oe),
and consequently, Jis found to be J¼(5.1960.32) merg/cm2
(J¼(4.5360.33) merg/cm2), respectively, as shown in Fig.
3(b). The positive value of Jfor both samples reflects the anti-
ferromagnetic character of coupling of both electrodes. It is
already reported20,21that the alignment of the magnetizations
of two ferromagnetic layers separated by a non-magnetic
spacer prefers such a type of antiferromagnetic coupling when
the PMA in the system is relatively large, which promotes the
magnetic volume charges (MVC) to have a dominant contribu-tion in the determination of coupling between the two ferro-
magnetic layers. In particular, a relatively strong PMA may
reduce the contribution of magnetic surface charges which
favor the ferromagnetic coupling, while, at the same time, it
promotes the MVC which introduces strong antiparallel cou-
pling between the ferromagnetic layers.
A further characteristic to be pointed out is the differ-
ence between the PMA of the soft electrodes of both sam-
ples. Figures 3(c)and3(d)show the anisotropy fields H
Kand
the uniaxial magnetic anisotropy energy density Ku, for bothsamples. HKcorresponds to the minimum field strength
applied perpendicular to the easy axis that is able to force the
magnetization to become perpendicular to the easy axis. Ku
is calculated from22
K¼Kb/C0M2
s
2l0þKs
tCoFeB; (1)
where Kis the perpendicular anisotropy energy density, Kb
is the bulk crystalline anisotropy, Ksis the interfacial anisot-
ropy, and tCoFeB is the corresponding thickness of the CoFeB
layer. The termKs
tCoFeBcorresponds to Kufor each sample, and
Kbis found to be negligible. Consequently, the larger Kuand
HKvalues for the Hf-capped sample reflect the significantly
larger PMA of the soft electrode compared to the one capped
with Ta. This behaviour is in agreement with previous inves-
tigations of Hf- and Ta-capped CoFeB/MgO stacks23and
can be attributed to the fact that Hf is a better B absorber
than Ta. Furthermore, Hf promotes the crystallization of
CoFe(B) in the bcc structure with (001) texture, whichconsequently leads to substantially higher PMA of the soft
electrode.
Figure 4summarizes the results of the TMR at RT for
both samples. In Figs. 4(a)and4(b), three representative major
TMR loops are displayed as a function of the perpendicular
magnetic field for the Hf- and Ta-capped samples, respec-tively, acquired at different bias voltages ( V
bias¼/C0120ðredÞ;
20ðgreenÞ;120ðblueÞmV). From similar loops, at several
bias voltages, we calculated the TMR ratio illustrated in Fig.4(c). After evaluating the results for six MTJs, an average
TMR value is presented in Fig. 4(d) forV
bias¼10 mV for
both samples. It is clearly observed that the Hf-capped samplehas a higher TMR ratio of (47.2 61.4)% compared to the Ta
one (42.6 60.7)% at RT. This is consistent with the claim of
Burton et al.
15that the presence of B at the CoFeB/MgO inter-
face suppresses the coherent tunneling in the D1band, leading
to the reduction of TMR. Thus, preventing the presence of B at
the interface should enhance the TMR in these junctions.
FIG. 3. (a) Saturation magnetization ( Ms). (b) Coupling constant ( J). (c)
Anisotropy field ( HK). (d) Uniaxial magnetic anisotropy energy Kufor the
Hf (red)- and Ta (green)-capped films at RT.FIG. 4. (a) and (b) Representative major TMR loops of the Hf (upper left)-
and Ta (upper right)-capped samples for Vbias¼–120 (red), 20 (green), and
120 (blue) mV, respectively. (c) Bias dependence of TMR for Hf (red)- and
Ta (green)-capped films. (d) Average TMR of six contacts acquired at
Vbias¼10 mV for Hf (red)- and Ta (green)-capped films.103904-3 Manos et al. J. Appl. Phys. 122, 103904 (2017)Moreover, this is in agreement with the fact that Hf is a better
B absorber material than Ta, which can be, e.g., concluded
from the calculated values of metal Boride enthalpies.16The
predicted formation enthalpies24of Hf (Ta) borides that may
be anticipated within a typical MTJ are DHHfB¼/C095 kJ/
mol, (DHTaB¼/C078 kJ/mol), DHHf2B¼/C067 kJ/mol, ( DHTa2B
¼/C056 kJ/mol), DHHfB 2¼/C095 kJ/mol, and ( DHTaB 2¼/C083
kJ/mol), respectively. This underpins that Hf will lead to astronger absorption of B than Ta.
Figures 5(a) and5(b) present the dependencies of the
TMR on the external perpendicular field for Hf (Ta)-cappedsamples at different temperatures T¼50 (20), 100 (100),
and 300 (300) K for V
bias¼20 (60) mV, respectively. In
Figs. 5(c)and5(d), the Hcvalues of the soft electrodes of the
Hf- and Ta-capped samples, which were extracted from thecorresponding minor TMR loops (not shown), are plotted as
a function of T
1=2. The temperature dependent behavior of
Hcfor both samples can be described by the Stoner-
Wohlfarth (SW) model25under thermal fluctuations. In this
model, the temperature dependence of Hcis given by26
Hc¼Hc01/C0T
TB/C18/C191=2"#
; (2)
where TBis the blocking temperature and Hc0is the coercivity
at 0 K. The extracted fitting parameters for the Hf (Ta)-capped
samples are: Hc0¼(1.8860.14) kOe [ Hc0¼(1.8460.10)
kOe], and TB¼318.4 K ( TB¼289.2 K), respectively. For
both samples, the experimentally observed values of Hcare in
reasonable agreement with the values predicted using Eq. (2).
However, some slight deviations are observed especially atlow temperatures. One reason could be the interaction of thesoft electrode with the reference system that is also tempera-
ture dependent and prefers the antiparallel state, thereby
adding an extra torque to the soft layers’ magnetization.Another option is a magnetization reversal via domain wallnucleation and movement, which could induce an exponentialdependence of H
con T. If only one or more mechanisms are
responsible for the experimental results will be investigated infurther experiments.
IV. CONCLUSION
In summary, we investigated the magneto-transport
properties of p-MTJs with PEB stacks Ta/Pd/IrMn/CoFe/Ta/CoFeB/MgO/CoFeB/Hf/Pd and Ta/Pd/IrMn/CoFe/Ta/CoFeB/
MgO/CoFeB/Ta/Pd. The Hf- and Ta-capped stacks showed a
PEB of H
ex¼/C0500 Oe along with PMA having TMR values
of (47.2 61.4)% and (42.6 60.7)% at RT, respectively. The
larger PMA and TMR values for the Hf- compared to the Ta-
capped sample were attributed to the enhanced B absorption
of Hf. Additionally, the temperature dependence of the Hcof
the soft electrodes was described by the Stoner-Wolframmodel, while the observed slight deviation from the model for
both samples was interpreted qualitatively by an additional
torque from the interactions occurring between the AF/FMdouble layer and the soft electrode.
ACKNOWLEDGMENTS
The authors would like to thank HARFIR and the
Deutsche Forschungsgemeinschaft (DFG, Contract No.RE1052/32) for the financial support.
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1.4942173.pdf | Developments in stochastic coupled cluster theory: The initiator approximation and
application to the uniform electron gas
James S. Spencer and Alex J. W. Thom,
Citation: The Journal of Chemical Physics 144, 084108 (2016); doi: 10.1063/1.4942173
View online: http://dx.doi.org/10.1063/1.4942173
View Table of Contents: http://aip.scitation.org/toc/jcp/144/8
Published by the American Institute of Physics
Articles you may be interested in
Linked coupled cluster Monte Carlo
The Journal of Chemical Physics 144, 044111044111 (2016); 10.1063/1.4940317THE JOURNAL OF CHEMICAL PHYSICS 144, 084108 (2016)
Developments in stochastic coupled cluster theory: The initiator
approximation and application to the uniform electron gas
James S. Spencer1and Alex J. W. Thom2,a)
1Department of Physics and Department of Materials, Imperial College London, Exhibition Road,
London SW7 2AZ, United Kingdom
2University Chemical Laboratory, Lensfield Road, Cambridge CB2 1EW, United Kingdom
and Department of Chemistry, Imperial College London, Exhibition Road, London SW7 2AZ, United Kingdom
(Received 17 November 2015; accepted 4 February 2016; published online 24 February 2016)
We describe further details of the stochastic coupled cluster method and a diagnostic of such
calculations, the shoulder height, akin to the plateau found in full configuration interaction quantum
Monte Carlo. We describe an initiator modification to stochastic coupled cluster theory and show
that initiator calculations can at times be extrapolated to the unbiased limit. We apply this method to
the 3D 14-electron uniform electron gas and present complete basis set limit values of the coupled
cluster singles and doubles (CCSD) and previously unattainable coupled cluster singles and doubles
with perturbative triples (CCSDT) correlation energies for up to rs=2, showing a requirement to
include triple excitations to accurately calculate energies at high densities. C2016 AIP Publishing
LLC. [http: //dx.doi.org /10.1063 /1.4942173]
I. INTRODUCTION
In a recent letter,1we described a novel way of formulating
coupled cluster theory stochastically. The previous exposition
was limited to calculations on a single processor, but
was shown capable, within arbitrarily small error bars,
of reproducing exact coupled cluster results for a range
of molecular systems while requiring fewer computational
resources than the full exact calculations.
Coupled cluster theory, though born in the field of nuclear
quantum mechanics, has become astoundingly successful in
the field of quantum chemistry. The pioneering work of
ˇCížek and Paldus2,3brought it to the quantum chemists’
attention, and its use has grown such as to be the “gold-
standard” of quantum chemistry.4Its success stems, in essence,
from the ability of CCSD(T) (Coupled Cluster Singles and
Doubles with perturbative Triples)5to reproduce energetics
and properties of a wide range of molecules to “chemical
accuracy,” i.e., within 1 kJ mol−1, and relative ease of
formulation, allowing it to be implemented in a range of widely
used computational chemistry software. It has not, however,
become ubiquitous within electronic structure theory owing
to a high scaling ( O(N7)for CCSD(T)61) with number of
electrons N. Numerous local approximations6,7have reduced
this scaling, even to the level of linear with system size,8
though these have yet to become widespread, and often the
onset of the reduced scaling is only in the regime of large
systems. For greater accuracy, one can turn to higher levels
of truncation of the theory, with full triples, CCSDT, or full
triples and quadruples, CCSDTQ, having scalings O(N8)and
O(N10), respectively. This higher accuracy is at the cost of
their being vastly more complicated to implement, and their
scaling makes them of little current use beyond benchmarking
a)Electronic address: ajwt3@cam.ac.ukthe energetics of very small molecules. It is in going beyond
this regime that we believe the novel stochastic coupled
cluster theory to be of great use owing to both its simplicity
of implementation and parallelizability.
Within the paradigm provided by stochastic methods,
there has been a great resurgence in their applications to
quantum chemical methods. The highly acclaimed work
of Booth, Alavi, and co-workers9,10in formulating Full
Configuration Interaction Quantum Monte Carlo (FCIQMC)
as a stochastic implementation of FCI was the impetus for
genesis of the stochastic formulation of coupled cluster theory,
and the common elements of the methods allow algorithmic
developments to be easily transferred between them. In
particular, we shall focus on the “initiator approximation”
of Cleland et al.11–13which has allowed vastly larger systems
to be studied than conventional diagonalization and both work
on new properties (explicit correlation,14density matrices,15
forces,16and excited states17,18) and new systems (from the
uniform electron gas19,20and the Hubbard model,21to real
crystalline solids22,23).
In this paper, we elucidate more algorithmic details of
the Coupled Cluster Monte Carlo (CCMC) method, as well
the inclusion of the “initiator approximation” based upon an
analogous development within FCIQMC.11These advances
allow the application of high levels of coupled cluster theory
to problems much larger than previously possible with full
coupled cluster theory. In Section II, we describe the basic
algorithm in detail and provide examples of the Monte
Carlo steps involved and denote this algorithm CCMC. The
population dynamics of CCMC is explored in Section III,
followed by more details about the critical plateau height in
Section IV in which we propose “shoulder plots” as a measure
of the critical di fficulty of a system and show how this varies
in some molecular systems. Section V introduces the initiator
approximation for CCMC and Section VI details how the
0021-9606/2016/144(8)/084108/12/$30.00 144, 084108-1 ©2016 AIP Publishing LLC
084108-2 J. S. Spencer and A. J. W. Thom J. Chem. Phys. 144, 084108 (2016)
systematic error inherent to the initiator approximation can be
extrapolated to zero. The behaviour of the CCMC method is
explored in these sections using the neon atom and nitrogen
molecule, going to six-fold excitations (i.e., CCSDTQ56). The
use of quantum chemistry methods in extended systems has
been the subject of much recent interest24–28and as a further
demonstration of the method, we calculate the CCSD and
CCSDT energies of the uniform electron gas in Section VII.
We find a substantial impact from triple excitations at realistic
densities. Finally, we o ffer some concluding remarks and
future perspective in Section VIII.
II. COUPLED CLUSTER MONTE CARLO
The key to the success of Monte Carlo techniques is the
ability to reduce computational e ffort by converting sums and
integrals over extremely large spaces into a series of discrete
samples which approximate the full calculation to arbitrary
accuracy with increasing numbers of samples. In this section,
we cast the coupled cluster equations in such a form as can
be sampled with Monte Carlo techniques and show how by
parameterization as discrete objects in excitation space, the
coupled cluster equations may be easily approximated and
solved.
The space in which we shall represent single reference
coupled cluster theory is that of excitors which act with respect
to a reference Slater determinant. Given an orthonormal set
of 2 Mspin-orbitals, we partition them into a set of N
which are occupied in the reference determinant, denoted
φi,φj,...,φ n, and 2 M−Nunoccupied, or virtual orbitals,
denotedφa,φb,...,φ f. The complete space of N-electron
Slater determinants in this basis has size(2M
N)
and will be
denoted by Dmwhere mis an N-vector listing the orbitals
occupied in a given determinant. Given the occupied /virtual
partitioning, we may also represent all possible determinants
with to respect the reference (which we will denote D0) by
listing the occupied orbitals removed and virtual orbitals added
to the determinant, such that Dab
i jrepresents the determinant
where iand jin the reference have been replaced by a
and b, respectively. Further, we shall denote by ˆ aab
i jthe
excitation operator orexcitor which performs this process,
Dab
i j=ˆaab
i jD0. We consider only those excitation operatorswhich are normal ordered (keeping a consistent ordering of
orbitals within each set of creation and annihilation operators),
as permutations of occupied or virtual indices will merely lead
to a potential sign change, and when excitors are combined,
these sign changes are taken into account. The excitors behave
such that it is not possible to excite from or to an orbital
multiple times, e.g., ˆ aa
iˆab
i=0 and ˆ aa
iˆaa
j=0, and excitors
with repeated indices such as ˆ aab
iiare not considered because
they are identically zero. By applying each of the(2M
N)
possible excitors to the reference determinant, the whole of
determinant space may be generated, and there is a one-to-one
correspondence between excitors and determinants; we may
therefore denote the excitors by the determinant they would
create, Di=ˆaiD0.
These excitors can be used to parameterize all possible
N-particle wavefunctions in this basis in a number of ways.
A simple example would be ΨFCI=
iciˆaiD0, which would
correspond a full configuration interaction type wavefunction
where each determinant has its own coe fficient in the
expansion. Coupled cluster theory uses instead an exponential
ansatz for the wavefunction, ΨCC=eˆTD0, where we define
ˆT=
itiˆai. This seemingly complicated parameterization is
used owing to its desirable property of remaining size-
consistent even if the sum of excitors is restricted to a limited
level of excitation. In essence, this is due to the fact that despite
a truncation, the exponential ensures that the wavefunction
can contain contributions from determinants at all excitation
levels.
To determine the parameters {ti}, the projected
Schrödinger equation is solved. The total number of
determinants available in the Hilbert space is larger than
the number of excitors if working in a truncated excitation
space. As there are the same number of excitation amplitudes
as there are excitors, we choose projection determinants which
come from a single excitor, i.e., for all mwithin our set of
truncated excitors, {ˆam}, we solve
⟨Dm|ˆH−E|ΨCC⟩=0. (1)
In general, this may be expressed in an iterative form,
beginning with a guess for all {ti}andEand iterating until
convergence. The complexity in this arises from the expansion
of the exponential,
eˆTD0=1+
itiˆai+1
2
ijtitjˆaiˆaj+1
3!
ijktitjtkˆaiˆajˆak+···D0. (2)
Instead of explicitly rearranging Equation (1) (or the equiv-
alent more convenient equations, ⟨Dm|e−ˆT(ˆH−E)|ΨCC⟩=0)
in an iterative form, which is the means by which many
conventional implementations work, we note that solutions to
the coupled cluster equations must also satisfy
⟨Dm|1−δτ(ˆH−E)|ΨCC⟩=⟨Dm|ΨCC⟩, (3)whereδτis some small positive number,62giving a form
reminiscent of projector methods. This is now almost in the
form of an iterative method,
⟨Dm|ΨCC⟩−δτ⟨Dm|ˆH−E|ΨCC⟩=⟨Dm|ΨCC⟩, (4)
except that the right hand side has components from many
different tiamplitudes, so it is not clear how to perform the084108-3 J. S. Spencer and A. J. W. Thom J. Chem. Phys. 144, 084108 (2016)
iteration. Noting that the projection of ΨCCon to a single
determinant within our projection space always contains the
amplitude of the excitor for that determinant plus terms
involving multiple amplitudes,
⟨Dm|eˆTD0⟩=tm+O(ˆT2), (5)
we may cancel out the higher order terms identically on the
left and right hand sides of (4), leaving
tm−δτ⟨Dm|ˆH−E|ΨCC⟩=tm. (6)
We may write this as an iteration from time τtoτ+δτ,
tm(τ)−δτ⟨Dm|ˆH−E|ΨCC(τ)⟩=tm(τ+δτ). (7)
The solutions to the coupled cluster equations will also be
solutions of this iteration procedure,63though the evaluation
of the second term in (7) is not trivial. Indeed, it turns out to
be easier to sample this term. The most straightforward way
of sampling these equations involves storing all tiamplitudes
as real numbers and performing two stochastic processes.
The first is sampling ΨCC, by selecting randomly from
all possible clusters in (2), e.g., titjˆaiˆajD0. Once selected,
the cluster is collapsed to form determinant, titjˆaiˆajD0
=titjˆanD0=titjDn. This process may involve some sign
changes or even result in zero (if iandjexcite from the
same occupied or to the same virtual orbitals).
The second stochastic process is the sampling of
the action of the Hamiltonian. For each Dngenerated
from sampling ΨCC, we could enumerate all possible Dm,
evaluating ⟨Dm|ˆH−E|Dn⟩and updating tmaccordingly. As
the Hamiltonian only connects up to single and double
excitations from a determinant, we may instead sample
this process by randomly picking Dmas a single or double
excitation of Dnand updating tmappropriately. While we have
found such an algorithm which stores all tnto well reproduce
small coupled cluster calculations, it is not e fficient, as many
clusters selected have extremely small amplitudes, requiring
large numbers of samples.
A more e fficient route is to discretize the amplitudes
as has been done in the FCIQMC method.9In analogy to
Anderson,29we shall represent amplitudes of excitors by sets
ofexcitor particles orexcips .64Each excip represents a unit
weight and is given a positive or negative sign and located at an
excitor, ˆ an. As the simulation proceeds, excips are created or
destroyed at excitors according to some simple rules, and the
mean (averaged over a number of iterations) signed number
of excips at a given excitor will be taken to represent its
(unnormalized) amplitude.
To sample the action of (7), we make very similar steps
to the algorithm using real numbers just described. First, a
cluster out of ΨCCis selected randomly and collapsed into
a determinant Dn: since ˆTis represented by populations of
excips at di fferent excitors, a cluster is formed by randomly
selecting a number of excips (which could be zero). Clusters
inΨCCformed from excitors with no excips at them would
have no amplitude and so may be safely ignored. The cluster
is collapsed by taking the product of the excitors at which
each of the chosen excips is located to give determinant Dn.
The amplitude of the cluster is given by the products of
populations of excips at each of the excitors in the cluster. Asthe spawning probability must be proportional to this, we have
chosen to select each cluster with a probability proportional
to its amplitude and make one spawning attempt per cluster.
Then, much as in FCIQMC, we follow the following steps:
1.Spawning . From each Dngenerated, we pick a random
connected single or double excitation, Dm, and create a
new excip there with probability proportional to |δτHmn|
and appropriate sign.
2.Birth /death . From each Dngenerated, with probability
proportional to |δτ(Hnn−E)|, we create an opposite signed
excip at ˆ an. Here, Ehas yet to be defined, but may be taken
as an arbitrary constant. Its later role will be to control
the population growth. Because Dnmay have been formed
from a product of excitors, it is probable that there are no
excips at the excitor ˆ an, and so, in contrast to FCIQMC, we
must create an oppositely signed excip, rather than simply
killing an existing one. If the cluster consisted of a single
excip, the opposite signed excip will cancel this out.
3.Annihilation . Finally, after these two processes have been
done for as many clusters as required to be sampled, the
new list of excips is sorted and any opposite signed pairs
of excips on the same excitor are removed.
In deterministic coupled cluster theory, Eis a parameter which
iteratively converges to the coupled cluster energy. Here, we
shall replace it with a parameter Swhich will shift the diagonal
elements of the Hamiltonian and may be updated periodically
after a number of cycles to allow the total population of excips
to be controlled.9,30
A. Normalization
A careful reader may have noticed that above formalism
introduces an imbalance in the description of the reference
and the excitor space. Indeed, the prescription of intermediate
normalization, where the overlap of the reference with the
coupled cluster wavefunction is fixed at unity, makes it rather
difficult to satisfy (4) for Dm=D0in terms of an iteration.
Instead, it is convenient to introduce a further variable, N0, to
act as the normalization constant,
|ΨCC⟩=N0eˆT
N0|D0⟩. (8)
The inclusion of N0within the exponential allows the dis-
cretization of ˆTto produce e ffectively fractional populations
on excitors which is essential for the exponential to converge.
With this normalization, the number of excips at the reference
is given by N0, and can thus be varied in the iteration satisfying
(4). In this manuscript, we will refer to these particles on the
reference as excips, though they are not strictly the same
and cannot be used as a constituent excip of a cluster; as
such they are (all) always in a cluster as they normalize it
multiplicatively.
B. Energy estimators
There are two independent estimators of the coupled
cluster energy available in this formalism. First, the time-
averaged value of the shift is, at convergence, an estimator for084108-4 J. S. Spencer and A. J. W. Thom J. Chem. Phys. 144, 084108 (2016)
the energy. Second, the energy may be estimated by projection
as
Eproj=⟨D0|ˆH|ΨCC⟩
⟨D0|ΨCC⟩. (9)
This expression could be computed exactly, as the contribu-
tions from only singly and doubly excited determinants are
relevant, but would require an iteration through all possible
pairs of single excitors to evaluate their contribution to the
doubly excited determinants. Instead, we sample both of these
contributions while sampling the cluster wavefunction.
C. Sampling
Sampling forms the core of this method, and the
implementational details of this are crucial to an e fficient
calculation. Here, we describe the di fferent sampling schemes
used in our implementation. Though we have not performed
exhaustive tests on di fferent sampling methods, these have
appeared to be the most e ffective in our calculations.
1.Sampling the wavefunction . At each timestep, we must
sample the exponential expansion of the wavefunction,
and we do this by randomly selecting a cluster—a single
term consisting of a product of specific amplitudes and
excitors—from the expansion of the exponential. We have
found that setting the number of samples taken to be the
same as the number of excips leads to stable simulations.
If the number of samples does not scale at least linearly
with the number of excips, the selection probabilities of the
clusters become increasingly small and lead to population
blooms which destabilize the simulation.
We select the size of cluster, s, to be generated so that
the probability of selecting a cluster with sexcitors decays
exponentially with s. At present, we fix these probabilities
at the beginning, with the probability decaying by a factor
of1
2for each excitor added. Each cluster is generated by
selecting sexcips randomly from the list of excips, and
we have found it best to bias the selection so that the
probability of selecting an excip is proportional to the total
number of excips at that excitor.
2.Sampling the spawning . In principle the spawning step
samples the action of the Hamiltonian upon the collapsed
cluster. A full sampling would enumerate all connected
determinants and attempt to spawn on each one-by-one.
We have followed the route of Booth et al.9in choosing a
minimalist sampling of a single attempted spawn from each
cluster. This comes with the drawback that as the number
of connected determinants increases (say with system size
or basis), the timestep must be correspondingly reduced
to keep this step stable. Some brief tests of increasing the
number of spawnings from each cluster indicate this has
the same e ffect as dividing the timestep by this number
(reducing the e ffective timestep).
3.Sampling the projected energy . The explicit evaluation
of the numerator of (9) is complicated by pairs
of single-excitations. Instead, each time a cluster is
generated, its contribution to the numerator is accumulated
(appropriately unbiased by its generation probability).D. A representative example
Let us consider a spin-free system with occupied orbitals
i,j,kand virtual orbitals a,b,c. In addition to the reference,
there are 19 possible excitors, which we shall denote with
subscripts and superscripts. For example,ab
ik, denotes the
excitor which excites electrons from iandkto orbitals aand
b, respectively.
Let us suppose we are performing coupled cluster theory
truncated at singles and doubles (a rather futile truncation
since there is only one triple excitation), i.e., CCSD. At time
τin the simulation, we have reached the situation at the top
of Fig. 1(a), with 20 excips, 10 being at the reference. We
shall accumulate various generation probabilities, pX, through
this example which will be used to unbias our sampling. We
sample with the same number of samples as excips (hence
psel=20 for this timestep). We shall also adopt a scheme
where the likelihood of forming a cluster is exponentially
decaying with probability1
2. Since the largest cluster with any
effect is of size three, we stop at that point and add in the
remaining probabilities of selection for larger clusters to that
level. Let us also suppose that we have not yet modified the
shift from its default S=EHF. The following four examples of
samples and their evolution are illustrated in Figs. 1(a)–1(d).
(a) We decide with probability psize=1
2to select a cluster with
zero excips. There is only one choice of zero-cluster, so we
setpclust=1. The action of this on the reference is merely
ˆ1|D0⟩=|D0⟩, and it has total amplitude A=N0. There are
eighteen possible single and double excitations available,
and we pick Dab
ikwith pexcit=1
18. ˆaab
ik|D0⟩=|Dab
ik⟩, so no
sign change will be needed. We create a new excip at
ab
ikwith probability δτ|⟨Dab
ik|ˆH|D0⟩A|/(pselpsizepclustpexcit)
and with sign of −sgn(⟨Dab
ik|ˆH|D0⟩). Let us say we create
a negative excip atab
ik. Death occurs with probability
|(⟨D0|ˆH|D0⟩−S)δτA|/(pselpsizepclust). Here, as EHF=S,
there is no death.
(b) We decide with probability psize=1
8to pick a cluster of
size 2.−c
jis selected with probability p1=3
10and+ab
ik
is selected with probability p2=1
10, so pclust=(2!)p1p2,
the 2! accounting for the number of ways in which
the cluster could have been chosen. The amplitude of
the cluster is given by A=N0−3
N01
N0. The cluster ˆ ac
jˆaab
ik
collapses to−|Dabc
i jk⟩when applied to the reference. We
first spawn, picking |Dc
j⟩with probability pexcit=1
18.
ˆac
j|D0⟩=−|Dc
j⟩, so we will pick up an extra negative
if we spawn there, which we do with probability
δτ|⟨Dc
j|ˆH|Dabc
i jk⟩A|/(pselpsizepclustpexcit). The signs take
a little more care. Let us say that ⟨Db
j|ˆH|Dabc
i jk⟩is
negative, giving a collective five negatives, resulting at
a negative excip atc
j. Death would occur with probability
|(⟨Dabc
i jk|ˆH|Dabc
i jk⟩−S)δτA|/(pselpsizepclust), except that in
CCSD, we do not store excips for triples, so it is ignored.
(c) We decide with probability psize=1
8to pick a cluster of
size 2.−c
jis selected with probability p1=3
10and+b
iis
selected with probability p2=2
10, giving pclust=(2!)p1p2.
The amplitude is A=N0−3
N02
N0. The cluster ˆ ac
jˆab
icollapses
to|Dbc
i j⟩when applied to the reference. We pick |Db
j⟩with
probability pexcit=1
18. ˆab
j|D0⟩=−|Db
j⟩, so we will pick084108-5 J. S. Spencer and A. J. W. Thom J. Chem. Phys. 144, 084108 (2016)
up an extra negative for spawning, which has probability
δτ|⟨Db
j|ˆH|Dbc
i j⟩A|/(pselpsizepclustpexcit). A positive value of
⟨Db
j|ˆH|Dbc
i j⟩results in a negative excip being spawned
atb
j. Death occurs with probability |(⟨Dbc
i j|ˆH|Dbc
i j⟩
−S)δτA|/(pselpsizepclust), creating a positive excip atbc
i j
because Ais negative.
(d) We decide with probability psize=1
4to pick a cluster of
size 1. +b
iis selected with probability p1=2
10, giving
FIG. 1. A representative example of the cluster selection, spawning, and
death steps of the CCMC algorithm as explained in the main text. Rounded
blue boxes represent excitors, and rectangular red boxes represent determi-
nants. (a) A size-0 cluster spawns. (b) A size-2 cluster spawns. (c) A size-2
cluster spawns and dies. (d) A size-1 cluster aborts a spawn outside the
truncation level and dies. (e) The post-annihilation populations.pclust=(1!)p1. The amplitude is A=N02
N0. The cluster
ˆab
icollapses to |Db
i⟩when applied to the reference.
We pick spawning excitation |Dabc
i jk⟩with probability
pexcit=1
18. This is a triple excitation and outside our
truncation, so it is discarded. Death occurs with probability
|(⟨Db
i|ˆH|Db
i⟩−S)δτA|/(pselpsizepclust), creating a negative
excip atb
i.
Let us assume that all the other samplings result in no death
or spawning. The resultant list of new excips is merged with
the main list, annihilation occurs, and a new step begins
(Fig. 1(e)). We note that in this implementation, much of the
amplitude information cancels between Aandpclust, resulting
a relatively e fficient sampling. This is an important detail as it
means that spawning probabilities do not decay rapidly with
increasing system size with therefore similar e ffects on the
required timestep.
III. POPULATION DYNAMICS
The sign problem and its impact on the population
dynamics of such algorithms have been previously explored
in the context of FCIQMC.31The sign problem in CCMC
is closely related to that in FCIQMC but is complicated by
the introduction of additional potential sign changes in the
algorithm from the requirement of normal ordering of orbitals
and the non-linear parameterization of the wavefunction.
Nonetheless, the population dynamics in CCMC has the
same macroscopic behaviour as in FCIQMC and follows the
following phases, as shown in Fig. 2:
1.Initial growth . We beginning with a population of excips
at the reference. By initially setting S=EHF, we ensure
that the excips at the reference do not die. Gradually, the
excitor space is populated with excips spreading outwards
from the reference. As time progresses and the populations
grow, annihilation events become more common.
2.Annihilation plateau . At a system-dependent number of
excips, annihilation events balance spawning and death
events (i.e., spawning and death events result in approxi-
mately equal number of positive and negative excips), and
the population ceases to grow, but remains approximately
constant. The relative signs of the populations at excitors
where there are many excips changes over this time until it
resembles the sign structure appropriate to the ground state
wavefunction (as parameterized by excitation amplitudes).
3.Exponential growth . The sign structure of the excips is such
that spawning once again dominates annihilation, and the
total number of excips grows exponentially with growth
rate proportional to S−ECC, where ECCis the coupled
cluster energy.
4.Population control .Sis now dynamically modified to
control the population and limit growth, with more negative
values of Sreducing population growth.
Before the annihilation plateau, the total rate of growth
is faster than that at the reference, and if one attempts to
control the population growth with the shift, the population
at the reference decreases to zero, causing the simulation to084108-6 J. S. Spencer and A. J. W. Thom J. Chem. Phys. 144, 084108 (2016)
FIG. 2. Ne cc-pVQZ CCSDTQ calculations starting with di fferent initial
particle numbers at the reference and di fferent timesteps. (a): With a carefully
chosen low timestep and initial population, a plateau is visible. An increased
timestep and initial population overshoot the plateau but have a shoulder.
The lower panel shows a maximum of the particle ratio at the position of
the shoulder and plateau. (b): “Shoulder plots” allow shoulder height to be
read o ffeasily and calculations compared.
fail. Below the plateau, in common with FCIQMC, we also
expect the sign structure of the solution to be wrong,31and so
the plateau height becomes a crucial feature in the usability
of stochastic coupled cluster.
We note that, as we begin simulations with a relatively
large population of excips at the reference, often the rate
of annihilation does not exactly match that of growth and
the population growth moves through a shoulder rather than a
plateau (see Figure 2). This is caused by e ffective overshooting
the plateau phase by starting with more excips. To determine
whether the system has reached the region of exponential
growth, we note that in this region, the population of at all
excitors should grow exponentially. A suitable alternative is
therefore to check whether the population at the reference is
growing at the same rate as that of the total population. Once
this has been satisfied, the shift, S, may be allowed to vary to
control the population.
IV. THE PLATEAU
The stability of a CCMC calculation requires reaching a
plateau or shoulder in the particle growth. Discerning sucha feature by eye is not an easy task, so we sought a more
rigorous definition. Before the plateau, a shift based on the
(faster) total particle growth rate would cause a decrease in the
population at the reference and likely lead the calculation to
fail as this population tends to zero. The post-plateau stability
arises from the fact that the growth rate of reference particles
is at least as fast as those of the total particles in the system,
and so turning on a shift will slow these rates equally and the
populations will stabilize. On this basis, we suggest that the
ratio of total particles to reference particles should be a useful
measure of these rates of growth, and we shall call this the
particle ratio . In Figure 2, we plot a typical CCMC calculation
showing a plateau and a second calculation on the same
system with a di fferent timestep and larger initial population.
The calculation with larger initial population overshoots the
plateau, but still reaches a stable growth phase. The particle
ratio shows clearly the location of the plateau by moving from
increasing to downward; the overshooting calculation shows
the same behaviour corresponding the shoulder in its growth.
From some experience with comparing such plots, it
proves convenient to plot the particle ratio not against the
progress in imaginary time but against the total number of
particles, both axes being logarithmic. In such a plot, the
plateau appears as a sharp downward kink in an otherwise
apparently linear progress. The usefulness of such “shoulder”
plots in comparing di fferent calculations is explained in
Figure 2(b). There is still some potential uncertainty as to
the definition of the shoulder height, but from experience, it
has proven to be useful to define this to be at the maximum of
the shoulder plot, and this should be taken as an upper limit
of the plateau height of the system. Such a definition is still
subject to stochastic noise, and so in this paper, we have quoted
values which are the mean of the total excip populations with
the ten largest particle ratios and used the standard deviation
of these as a measure of uncertainty. Calculations with total
particle numbers before the shoulder are often unstable, and
calculations after the shoulder are usually stable, though the
stochastic nature of such calculations removes total certainty
about this. Above the shoulder, the particle ratio gradually
decreases with increasing Nex, becoming constant for large
N0, as fluctuations in near-zero amplitudes become a smaller
part of the total population.34A stable population (due to
the control from the variable energy shift) appears in the
shoulder plot as a tight blob at the end of the line due
to (comparatively) small stochastic fluctuations in both the
population on the reference and the total population.
A histogram approach has been previously used to
automate detection of the plateau in FCIQMC;21the shoulder
plot is equally amenable to automation and both approaches
give similar values for simulations with clear plateaus.
Shoulder plots are more reliable when the plateau is shorter in
duration or tends towards a shoulder in the population growth.
As shown in the original CCMC paper,1the heights
of plateaux vary with both the size of Hilbert space and
differing levels of correlation within a molecule; system-
specific behaviour is also seen for FCIQMC calculations.9,21
Figure 3 shows the variation of plateau height with basis,
excitation, and bond length for the N 2molecule, showing
broadly that the fraction of the Hilbert space required is of084108-7 J. S. Spencer and A. J. W. Thom J. Chem. Phys. 144, 084108 (2016)
FIG. 3. Shoulder heights for the N 2molecule at di fferent bond lengths,
basis sets, and truncations. All electrons were correlated, and Lzsymmetry
enforced. Hilbert space sizes are shown by the horizontal lines.
a similar magnitude for di fferent excitation levels and basis
sets and increases as the N–N bond is broken, though staying
below 27%.
To see the e ffects of di fferent basis sets, the neon atom,
with only 10 electrons, is more amenable to study, and Figure 4
shows examples of shoulder plots for increasingly large bases.
Plateau heights are given in Table I and decrease as a fraction
of the Hilbert space with truncation level, heading to less
than 1% of the complete Hilbert space size for larger bases
with quadruples truncation. Whilst for CCSD, therefore, the
saving is relatively small, it is for higher excitation levels that
the Monte Carlo sampling considerably reduces the storage
required.
V. THE INITIATOR METHOD
With the insight that the plateau is excited only when
the particles achieve a sign-coherent structure, Cleland et al.
introduced what they call the “initiator method”11to the
FCIQMC algorithm (denoted i-FCIQMC). In its mature
version, it avoids the e ffects of sign-incoherence by restricting
spawning. If a determinant has no walkers present, then the
algorithm only allows a walker to be spawned there if there
are more than a critical number, nadd, of walkers present at the
FIG. 4. All-electron CCSDTQ calculations on the neon atom using Dun-
ning’s cc-pVXZ basis sets.32,33The shoulder is clearly discernible in each
line, and its height increases with basis set cardinality. A vertical descent from
the maximum is given by a clearly defined plateau, and a chevron-like peak
is characteristic of a shoulder. As seen for cc-pVTZ, calculations started with
more excips (purple) lead to a shoulder which can overestimate the position
of a plateau, though reducing the initial number of excips (light blue) can also
lead to overshoot and the shoulder plot becomes concave (cf. Fig. 2(b) (green
line) where a well-defined plateau produces a vertical section of the shoulder
plot). Calculations were started with 102or 103excips at the reference (which
can be read o ffas where each line would cross Nex/N0=100).
determinant from which it was spawned. This modification of
the algorithm dramatically alters the dynamics, removing the
plateau phase and greatly stabilizing the calculations. This is,
however, at the price of introducing a bias into the algorithm,
dependent on the number of walkers. This is manifested as an
error in the energy which systematically decreases with the
number of walkers (becoming zero in the large walker limit).
Owing to the great similarity between stochastic coupled
cluster theory and FCIQMC, it is expected that the initiator
method will confer similar benefits in stochastic coupled
cluster theory. As spawning occurs from clusters of excips
rather than single walkers, the initiator method requires some
modification. The obvious choices are for a cluster to be
allowed to initiate (i.e., spawn onto an unoccupied excitor) if
either one or all of its component excips come from an excitor
whose population is >nadd. We denote these the partial and
full initiator methods, respectively. A representative sample of
the behaviour of each of these schemes is shown in Figure 5.
From this and with the ideas of maintaining sign-coherence,
we have settled upon requiring all excips within a cluster to
TABLE I. All-electron Ne Hilbert space (H.S.) sizes and shoulder heights. Hilbert space sizes were determined by Monte Carlo and only significant figures are
shown. Lzsymmetry was enforced except in the case of cc-pV6Z. Shoulder error estimates in the last significant digit are shown in parentheses.
Truncation SD SDT SDTQ SDTQ5 SDTQ56
Basis H.S. Shoulder H.S. Shoulder H.S. Shoulder H.S. Shoulder H.S. Shoulder
cc-pVDZ 400 33 (1) 4 680 194 (2) 30 654 594 (4) 113 550 1610 (20) 259 460 2660 (10)
cc-pVTZ 1 940 160 (30) 60 710 2900 (300) 1.032 ×1062.27 (1)×1041.021×1071.90 (5)×1046.143×1079.6 (2)×105
cc-pVQZ 6 710 290 (10) 4.128 ×1057100 (200) 1.412 ×1072.39 (6)×1052.81×1083.7 (1)×106
cc-pV5Z 17 100 2500 (400) 1.815 ×1067.9 (2)×1041.068×1081.57 (6)×1063.66×109
cc-pV6Z 82 400 2.2 (3) ×1041.636×1072.3 (3)×1051.731×1091.45 (3)×107084108-8 J. S. Spencer and A. J. W. Thom J. Chem. Phys. 144, 084108 (2016)
FIG. 5. Ne cc-pVQZ CCSDTQ calculations without initiator and with partial
and full initiator methods. The full initiator (yellow) method has a signifi-
cantly lower shoulder height and is stable at even 100 000 excips whereas the
partial (red) is unstable when the shift is engaged at 240 000 excips (before the
plateau) and has a similar profile (green) to the non-initiator (blue) method.
come from excitors with populations >naddbefore allowing it
to initiate (the full initiator method).
As the initiator calculation reduces the amount of
spawning, it can be thought of as generally reducing the
particle noise within a calculation. To investigate the e ffect of
this, we note that in the regime of large numbers of particles,
the variance of the numerator and denominator of the projected
energy should decrease with 1 /Nex. More formally, denoting
the numerator of the projected energy at a given timestep to be
Eproj,numer (τ), to enable comparison of di fferent calculations,
we may calculate
s2=Var[Eproj,numer (τ)]τ⟨Nex(τ)⟩τ
⟨Eproj,numer (τ)⟩2τ, (10)
which we shall call the normalized relative variance of the
projected energy. This is a substantial component of the
eventual error estimate for a given calculation (though the
actual errors are calculated through a blocking analysis35).
A calculation with double the value of s2will have to be
run for approximately twice as long to achieve the same
estimate of error, and so it should be as low as possible to
reduce the computational resources required in a calculation.
For N 2in a cc-pCVDZ basis, we show s2for both standard
and initiator calculations in Figure 6. Around equilibrium,
it can be seen that the variance is significantly lower for
initiator calculations, and so using the initiator method should
dramatically reduce the number of steps required to achieve a
given error bar.
In Figure 7, we show data from a typical set of initiator
calculations using CCSDT on the neon atom in a cc-pV6Z
basis. As the population increases, the correlation energy
gradually converges to the true correlation energy. In this
case, the population required for the initiator approximation’s
bias to have decayed is of the order of 5 ×105psips, compared
to a shoulder height of 2 .3(3)×105psips, providing no benefit
in reducing the number of psips needed. In all initiator CCMC
calculations we have tried on systems in this paper (on the
neon atom, N 2molecule, and Uniform Electron Gas (UEG)),
this pattern is repeated, and we see that a population of the
order of the shoulder height is required to converge the initiator
FIG. 6. The normalized relative variance of the projected energy, s2(see
text for definition), for the N 2molecule in a cc-pCVDZ basis. Calculations
were performed with about 70 000 excips (for rNN≤2.4 Å) and 200 000 oth-
erwise. The initiator calculations show a generally reduced variance around
equilibrium, leading to fewer steps being required to converge to a given error
estimate.
FIG. 7. CCSDT initiator energies and extrapolations for Ne cc-pV6Z. Plotted
are the raw iCCMC energies (blue), Ei(Nex), and various fits. The curve
(green) was fit using all data to the form Ei(Nex)=p(Nex)q+Ei(∞), varying
p,q, and Ei(∞), resulting in an estimate of Ei(∞)=0.363 13 (7)which
closely matches the exact Ecorr=0.363 206 46 from MOLPRO /MRCC. Fits
for fewer data points are denoted Ei(∞)in the legend, and included data
points up to the value of Nexplotted, for varying p,q, and Ei(∞), as well
as varying pandEi(∞)and fixing q=−1, giving Ei(∞)=0.363 17 (4). The
shoulder for this system is at 2 .3(3)×105excips.
error. This is in contrast to that in FCIQMC where the initiator
adaptation substantially helps the sign problem allowing much
smaller populations to be used, and unfortunately we do not
see any similar reductions in CCMC. The reduction in variance
is, however, still beneficial.
VI. EXTRAPOLATION
In previous studies of the initiator approximation for
FCIQMC, the initiator error has been seen to decay as total
particle number increases, but there has been little known
about the form of this decay. In Appendix A, we show that
the initiator error is expected to fall o ffas the inverse of the
number of particles in iFCIQMC, and we shall use a similar
form to extrapolate results for initiator CCMC (iCCMC),
writing the energy as a function of number of excips, Nex,084108-9 J. S. Spencer and A. J. W. Thom J. Chem. Phys. 144, 084108 (2016)
Ei(Nex)=pNq
ex+Ei(∞),
where we will allow p,q, and Ei(∞)to be variable parameters.
To test the form, we have performed CCSDT calculations
on neon in a cc-pV6Z basis set, where the shoulder lies
at 2.3(3)×105excips out of a total space of 1 .64×107
excitors, and we have been able to perform the exact CCSDT
calculation with MOLPRO36,37and MRCC.38For this system,
iCCMC calculations were performed for increasing total excip
populations, and projected energy estimators and errors are
determined using an automated blocking script.65As noted in
Appendix B, biases are introduced in calculations in which
the shift varies greatly, and we have kept only calculations
in which the standard deviation of the shift is lower than
0.05Eh. For each population, projected energy estimators for
that and lower populations were used in a non-linear least
squares fit performed with gnuplot.39In Figure 7, we show
fits with pandqallowed to vary, as well as fixing q=−1 (as
justified in Appendix B). The latter leads to considerably more
stable fitting as we avoid overfitting the data. It shows that
data from Nex<25 000 provide estimates of Ei(∞)matching
Ecorrwithin error bars (albeit large ones), and we have used
this method as the basis of our later extrapolations. The least
squares fits produce error estimates on the values of the fitted
variables, and we have used these as is as error estimates for
the extrapolated values, but caution that they are likely to be
under-estimates of the errors.
VII. THE UNIFORM ELECTRON GAS
Whilst the neon atom provides a convenient single-
reference system to test the scaling of shoulders with basis,
there should be little surprise that the CCSDTQ correlation
energy is within a millihartree of the CCSDT even at the
cc-pV6Z basis. In this section, we turn attention to the
three dimensional uniform electron gas whose Wigner–Seitz
radius, rsprovides a convenient tunable parameter to di fferent
regimes of correlation, which has been relatively little studied
with coupled cluster theory. The analytical work of Bishop
et al.40,41provides approximations for the CCSD energy in
the thermodynamic limit, but for finite systems, the principal
means of calculation of the energy has been through either
FIG. 8. The shoulder height for CCSDT on the 3D UEG for systems with M
plane waves.TABLE II. Correlation energies for the 14-electron uniform electron gas.
Extrapolated initiator stochastic CC results were obtained by basis-set ex-
trapolation from stochastic coupled cluster calculations. FCIQMC results are
from Ref. 43.
rs ECCSD ECCSDT EFCIQMC
0.1−0.6639 (1) −0.6659 (2)
0.5−0.5897 (1) −0.5965 (2) −0.5969 (3)
1.0−0.5155 (3)a−0.5317 (3) −0.5325 (4)
2.0−0.4094 (1) −0.4354 (4)b−0.4447 (4)
aForrs=1, Shepherd et al. have found ECCSD =−0.5152 (5)(Fig. 7 of Ref. 48).
bOwing to non-converging initiator energies, non-initiator stochastic CC energies were
used in the basis set extrapolation.
fixed-node quantum Monte Carlo42or FCIQMC20,43when
extrapolated to the complete basis set limit. Coupled cluster
work on such systems has been on extremely small bases44
or limited to CCSD45or CCSD(T).46Without extensive
parallelism (which is to be the subject of a future paper),
we have kept the system sizes small, concentrating on the
closed-shell 14-electron gas in plane wave bases. We have
used a spherical energy cuto ffand denote the number of
plane waves M. FCIQMC, CCSD, and CCSD(T) results are
available in such systems,43,46,47along with a well-tested
methodology to extrapolate to the complete basis set limit.48
We begin by studying the CCSDT shoulder heights for
the UEG as a function of rs. Figure 8 shows that these rise
significantly with rs, of the order of r3
s, as well as increasing
with the plane wave cuto ffenergy.
For values of rs≤2, initiator CCSD and CCSDT
calculations were performed followed by linear fits against
1/MforM≥179, and the M=∞limit was extracted. Results
are in Table II. We have also investigated the convergence of
the initiator method for these systems, as shown in Figures 9
and 10. Whilst the convergence of the extrapolation for rs=1
in Figure 9 appears to well-match the exact result, we find the
same methodology predicts a result many standard deviations
away from the true correlation energy for rs=2 (Figure 10).
This non-convergent behaviour of the initiator approximation
may be owing to changes in the nature of the correlation
with increasing rs, as the UEG is known to become more
multireference with increasing rs. We therefore would caution
FIG. 9. CCSDT initiator extrapolation for the 14-electron 3D uniform elec-
tron gas with M=925 and rs=1.084108-10 J. S. Spencer and A. J. W. Thom J. Chem. Phys. 144, 084108 (2016)
FIG. 10. CCSDT initiator extrapolation for the 14-electron 3D uniform
electron gas with M=925 and rs=2. Though the raw initiator energies
do not appear to be converging, the extrapolation does appear to converge
to a lower energy than the true correlation energy, Ecorr, which was cal-
culated with non-initiator CCSDTMC. The shoulder height in the latter is
663 000(15 000).
against the use of initiator extrapolation in such cases where
there is not a clear sign of convergence.
VIII. CONCLUSION
We have investigated the stability of stochastic coupled
cluster calculations and suggest that the particle ratio (between
the total excip population and the population on the reference)
is a convenient measure of the progress of a stochastic coupled
cluster calculation. The particle ratio exhibits a maximum as
a calculation progresses through the growth phase, and this
maximum signals the critical number of particles required for a
stable calculation and indicates a “shoulder height” equivalent
to the plateau height in FCIQMC. As a fraction of the truncated
Hilbert space of a coupled cluster calculation, this e ffective
plateau height remains a similar order of magnitude (1%–10%)
as basis set is increased and decreases as the truncation level
increases, allowing an estimate of the upper bound storage
required of a CCMC calculation.
We have investigated two possible implementations of
the initiator approximation in CCMC, and of the most stable,
we conclude that the initiator approximation performs a
similar function to that in FCIQMC and stabilizes calculations
with low numbers of excips at the expense of introducing a
systematic initiator error. We have demonstrated that this error,
which in all cases, we have investigated causes a systematic
lowering of the energy, decays as N−1
excipand this can be used as
a basis for extrapolation of the true correlation energy. Unlike
FCIQMC, we have found that the initiator error does not decay
away quickly in comparison to the critical stable population
indicated by “shoulder plots.” In both the 3D uniform electron
gas and Ne atoms in large bases, we show that the particle
population require to reduce the initiator error su fficiently is
of the order of the shoulder height. Therefore, the reductions
in particle population, which in FCIQMC can amount to many
orders of magnitude, do not appear to be present in stochastic
coupled cluster theory.
However, the benefits of the initiator approximation are
still manifold. With careful extrapolation, it is possible toperform calculations which are not possible without use
of the initiator approximation. Additionally, the initiator
approximation dramatically reduces the variance of the
projected energy within a calculation, and thus by its use,
the required calculation times are greatly reduced. There is
reason to be optimistic as there is much which can be tuned
in the stochastic coupled cluster algorithm. In particular,
the role of the population of particles on the reference,
which acts as a normalization, is critical in the stability
of calculations and in the projected energy. With further
algorithmic modifications and optimization of parameters, the
convergence of the initiator energy with particle number will
likely become faster allowing yet larger calculations to be
performed.
Finally, we demonstrate that this method is capable of
calculating coupled cluster energies of the uniform electron
gas up to rs=2. Studies beyond this will require significantly
more computational resources, but are still possible. Of
particular note is the vital contribution of triple excitations
even at these modest values of rs, and we hope to make a
more detailed study of this in the future.
ACKNOWLEDGMENTS
We are grateful to Ms. Ruth Franklin and Mr. William
Vigor for discussions on the algorithms. A.J.W.T. thanks
the Royal Society for a University Research Fellowship and
Imperial College London for a Junior Research Fellowship,
where this work was started. J.S.S. acknowledges the
research environment provided by the Thomas Young Centre
under Grant No. TYC-101. Calculations were performed
on the Imperial College High Performance Computing
facilities49and the University of Cambridge High Performance
Computing facilities using the NECI50and the HANDE51,52
codes. Molecular Orbital integrals were generated with
a modified version of Q-Chem,53and for Ne cc-pV6Z,
MOLPRO.36Raw and analysed data are available in
Refs. 54 and 55. We use an automated iterative blocking
algorithm35,56–58to accurately estimate the stochastic error in
all CCMC calculations presented in this paper. Figures were
plotted using matplotlib.59
APPENDIX A: INITIATOR EXTRAPOLATION
We derive a form for the convergence of the initiator
approximation with psip population in full configuration
interaction quantum Monte Carlo. We expect that, while
CCMC behaviour will be more complicated, it should follow
the same general form.
Let us denote the true ground state wavefunction ψ0
with energy E0and impose intermediate normalization
on it such that ⟨D0|ψ0⟩=1. Consider the instantaneous
wavefunction at a given timestep, ψ(τ). We can expect
it to consist of some amount of the ground state and a
difference term, ψ(τ)=C(τ)ψ0+∆(τ), where we can specify
that ⟨D0|∆(τ)⟩=0.In a converged simulation, Cis the
population of the reference determinant and is proportional to
the total number of particles. The projected energy is given
by084108-11 J. S. Spencer and A. J. W. Thom J. Chem. Phys. 144, 084108 (2016)
E(τ)=⟨D0|ˆH|ψ(τ)⟩
⟨D0|ψ(τ)⟩=⟨D0|ˆH|ψ(τ)⟩
C(τ). (A1)
In an unbiased FCIQMC calculation ⟨∆(τ)⟩τ=0 (where
⟨⟩τindicates the long time average over τ). In moving to the
initiator modification, we introduce a systematic bias in the
dynamics, resulting in a ⟨∆(τ)⟩τwhich no longer vanishes. For
a given value of the initiator threshold and value of C, there
will be a maximum component of ∆(τ), i.e., someδ, such that
for all determinants i,|⟨Di|∆(τ)⟩|< δ. Let us assume that any
dependence δhas on Cis at most of order C0(i.e., constant),
as a higher power would lead to the initiator approximation
not converging with increasing psip population.
Explicitly evaluating the projected energy,
E(τ)=⟨D0|ˆH|C(τ)ψ0+∆(τ)⟩
C(τ)(A2)
=C(τ)⟨D0|ˆH|ψ0⟩
C(τ)+
i,0⟨D0|ˆH|Di⟩⟨Di|∆(τ)⟩
C(τ)(A3)
≤E0+
iH0iδ
C(τ). (A4)
Denoting the maximal value of H0iasη, if there are Ndoub
double excitations of the reference, we find
|E(τ)−E0|≤Ndoubηδ
C(τ). (A5)
Since Ndoubis constant and C(τ)scales linearly with Npsip,
we may attempt to fit to (Npsip)−1as an upper bound. We note,
however, that there is no guarantee of monotonic convergence
behaviour, so fitting may well not succeed.
APPENDIX B: POPULATION CONTROL EFFECTS
Appendix A contains the assumption that the initiator
approximation is the cause of all systematic errors in the
simulation. This is not strictly true; the e ffect of population
control is well understood to introduce a small bias which
decays with increasing population.30,60We briefly investigate
here the e ffect of population control on the convergence of
the initiator error. Any population control bias need not be the
same in both CCMC and iCCMC calculations with the same
population owing to the impact on sampling.
The shift is updated every Atimesteps during the
population control phase according to9,30
S(τ)=S(τ−Aδτ)−γ
Aδτln(Nex(τ)
Nex(τ−Aδτ))
,(B1)
whereγis a (typically small) damping parameter. The
variance in the shift (which is independent of the length of a
simulation) is a measure of the population fluctuations caused
by requiring the population to be approximately constant.
Figs. 11 and 12 show the e ffect of population control on the
14-electron UEG at rs=1; Fig. 12 also contains data for
the equivalent CCMC calculations at populations above the
plateau. All simulations converge to within stochastic error
at sufficiently large populations (Fig. 11). For the initiator
method, above about 70 000 excips, each line has statistically
equivalent energy but shows a discernible bias, decaying with
increasing population. From the insensitivity to population
FIG. 11. iCCMC convergence studies of CCSDT for the 14-electron 3D
uniform electron gas at rs=1 using M=925 plane waves for di fferent
population control parameters (top) and the corresponding variance in the
shift observed during each calculation (bottom). δτ=10−4in all calculations.
control parameters, we infer that this is likely to be initiator
error. The separation of population control error from initiator
error is not trivial and requires a large number of calculations,60
but we do not see significant population control error in the
non-initiator results in Fig. 12.
The extrapolation also requires that data from simulations
using smaller populations of excips be monotonically
FIG. 12. CCMC and iCCMC calculations of CCSDT for the 14-electron
3D uniform electron gas at rs=1 using M=925 plane waves as a function
of population for di fferent population control parameters. δτ=10−4in all
calculations. Dotted lines represent calculations for which the shift variance
is greater than 0.002. The iCCMC calculations shown are a more detailed
view of the top panel in Fig. 11.084108-12 J. S. Spencer and A. J. W. Thom J. Chem. Phys. 144, 084108 (2016)
convergent, and whilst this is far from a comprehensive study,
we observe there to be relatively smooth convergence in all
cases for calculations where σ2
Shift<0.002 (Fig. 12), and we
therefore note that extrapolation requires very careful choices
of population control parameters γandAto achieve this.
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1.5048627.pdf | The effect of polarizable environment on two-photon absorption cross sections
characterized by the equation-of-motion coupled-cluster singles and doubles
method combined with the effective fragment potential approach
Kaushik D. Nanda , and Anna I. Krylov
Citation: J. Chem. Phys. 149, 164109 (2018); doi: 10.1063/1.5048627
View online: https://doi.org/10.1063/1.5048627
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The Journal of Chemical Physics 149, 180901 (2018); 10.1063/1.5052551THE JOURNAL OF CHEMICAL PHYSICS 149, 164109 (2018)
The effect of polarizable environment on two-photon absorption cross
sections characterized by the equation-of-motion coupled-cluster
singles and doubles method combined with the effective fragment
potential approach
Kaushik D. Nanda and Anna I. Krylov
Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482, USA
(Received 16 July 2018; accepted 1 October 2018; published online 29 October 2018)
We report an extension of a hybrid polarizable embedding method incorporating solvent effects in
the calculations of two-photon absorption (2PA) cross sections. We employ the equation-of-motion
coupled-cluster singles and doubles method for excitation energies (EOM-EE-CCSD) for the quantum
region and the effective fragment potential (EFP) method for the classical region. We also introduce
a rigorous metric based on 2PA transition densities for assessing the domain of applicability of
QM/MM (quantum mechanics/molecular mechanics) schemes for calculating 2PA cross sections.
We investigate the impact of the environment on the 2PA cross sections of low-lying transitions in
microhydrated clusters of para -nitroaniline, thymine, and the deprotonated anionic chromophore of
photoactive yellow protein (PYPb). We assess the performance of EOM-EE-CCSD/EFP by comparing
the 2PA cross sections against full QM calculations as well as against the non-polarizable QM/MM
electrostatic embedding approach. We demonstrate that the performance of QM/EFP improves when
few explicit solvent molecules are included in the QM subsystem. We correlate the errors in the 2PA
cross sections with the errors in the key electronic properties—identified by the analysis of 2PA natural
transition orbitals and 2PA transition densities—such as excitation energies, transition moments,
and dipole-moment differences between the initial and final states. Finally, using aqueous PYPb,
we investigate the convergence of 2PA cross sections to bulk values. Published by AIP Publishing.
https://doi.org/10.1063/1.5048627
I. INTRODUCTION
Electronic structure methods enable highly accurate cal-
culations of molecular structures, energetics, and properties;
however, the domain of applicability of ab initio methods in
terms of the size of a system is limited by their steep scaling and
large computational costs. Despite significant progress in com-
puter hardware and algorithms, full quantum-chemical mod-
eling of condensed-phase systems remains elusive. This lim-
itation can be circumvented by using multi-scale approaches
combining a high-level quantum mechanical (QM) descrip-
tion of a system (i.e., a solvated chromophore) with a more
approximate treatment, such as classical molecular mechan-
ics (MM), of the environment. Among many different fla-
vors, explicit solvent approaches are particularly attractive
because, in contrast to implicit solvent models (e.g., polar-
izable continuum), they are capable of describing specific
interactions such as solvent-solute hydrogen bonding and salt
bridges.
Numerous hybrid multi-scale methods1differ in their
description of the interactions between the QM subsystem
(treated at a high level of theory) and the environment. In
the most commonly used QM/MM approach, classical force
fields are used to describe the environment (hence, MM) and
the interaction between the QM and MM parts is described
by electrostatic embedding.2This basic QM/MM strategy hasbeen successfully used for describing both the ground- and
excited-state energetics and properties of large systems.3–6The
QM/MM approach can be improved by using polarizable force
fields in which the charge distributions in the MM part can
respond to the changes in the electronic distribution in the QM
part.
The quality of force fields (whether polarizable or not)
is essential for obtaining reliable electronic properties of the
QM subsystem. Force fields are commonly derived using
empirical parameterization aiming at reproducing particular
observables. A more rigorous description can be achieved by
using force fields derived from first principles, such as in the
effective fragment potential (EFP) approach.7–11In the EFP
method, the MM subsystem is fragmented into smaller sub-
units (fragments), and the force-field parameters of each of
these fragments are obtained from ab initio calculations. The
fragments are frozen (i.e., their structures do not change),
which enables using a fixed set of parameters computed
once. In principle, these parameters can be computed “on
the fly,” but this, of course, leads to increased computational
costs. The QM/EFP scheme has been shown to yield accu-
rate solvatochromic shifts, ionization and detachment ener-
gies, electron-attachment energies, and redox potentials in
a variety of solvated systems including protein-bound chro-
mophores.12–19QM/EFP is similar to the polarizable embed-
ding (PE) approach.20–23The description of electrostatics and
0021-9606/2018/149(16)/164109/14/ $30.00 149, 164109-1 Published by AIP Publishing.
164109-2 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)
polarization is nearly identical in QM/EFP and PE; QM/EFP
also includes (non-empirical) terms describing dispersion and
exchange-repulsion interactions.
Higher-order response properties such as two-photon
absorption (2PA) cross sections are particularly sensitive to
the environment.24For example, the 2PA cross sections of red
fluorescent proteins with identical DsRed-like chromophores
(DsRed, mCherry, and mStrawberry) differ by a factor of
five.25As pointed out in a recent validation study, the inclu-
sion of solvent effects on 2PA cross sections is critically
important for unambiguous comparison between theory and
experiment.26,27
Nonlinear response properties are formally given by sum-
over-states expressions, which depend on the wave functions
and excitation energies of allelectronic states of the sys-
tem. Thus, in contrast to the one-photon transition moments
expressed as the matrix elements between the initial and the
final states only, the quality of the description of nonlinear
properties, such as 2PA cross sections, depends on the quality
of the representation of the entire spectrum by a model Hamil-
tonian. This is the primary reason why modeling 2PA cross
sections is challenging.
The QM/MM approach has a number of limitations that
make reliable calculations of 2PA cross sections difficult. First,
QM/MM schemes with only one chromophore in the QM part
cannot describe delocalized states, such as those in molecular
aggregates28or charge-transfer (CT) states delocalized over
several solvent molecules.29This issue can be partially reme-
died by a careful choice of the QM subsystem, for example,
by including some solvent molecules. Second, small errors in
the QM/MM excitation energies (present in the denominator
of the formal expression) and transition dipole moments can
accumulate leading to large errors in the calculated 2PA cross
sections. Third, the quality of computed 2PA cross sections is
sensitive to the degree of electronic correlation included in the
model Hamiltonian.
In the context of 2PA calculations, the EFP method has
been employed in combination with time-dependent density
functional theory (TDDFT)30but not with high-level wave-
function methods such as the equation-of-motion coupled-
cluster singles and doubles method for excitation energies
(EOM-EE-CCSD).31–33Coupled-cluster methods have been
employed within the polarizable embedding scheme in the
calculations of 2PA cross sections.22In this article, we
extend our EOM-CCSD implementation of the 2PA cross sec-
tions34–36to include the effect of solvent environment via
QM/EFP embedding. We assess the quality of the EOM-
EE-CCSD/EFP 2PA cross sections (i.e., computed with the
EOM-EE-CCSD wave functions embedded in the polariz-
able environment of EFP fragments) relative to the full
EOM-EE-CCSD calculations and identify the main sources
of errors. We consider low-lying transitions in small micro-
solvated clusters of three chromophores: para -nitroaniline
(pNA), thymine, and the deprotonated anionic chromophore
of photoactive yellow protein (PYPb). To analyze the role of
solvent polarization, we compare the EOM-EE-CCSD/EFP
2PA cross sections for the microhydrated clusters with
the results from two non-polarizable QM/MM schemes:
(a) EOM-EE-CCSD/(EFP without polarization effects) and(b) EOM-EE-CCSD/MM, wherein the solvent molecules are
represented by CHARMM22 point charges37,38(we denote
these calculations as QM/CHARMM). To assess the effect of
bulk solvation, we calculate the 2PA cross sections of the PYPb
chromophore in water and investigate the convergence of the
results with respect to the QM size by including an increased
number of water molecules in the QM subsystem.
II. THEORY
A. The QM/EFP scheme
In the QM/EFP embedding scheme, the Hamiltonian of
the whole QM/MM system is written as
Hfull=HQM+HMM+HQM=MM, (1)
where HQMandHMMdescribe the QM and MM parts, respec-
tively, and HQM/MMdescribes the interaction between these
two subsystems. In the QM/EFP approach, the MM system
is broken into effective fragments and its energy, EMM, is
computed as the sum of many-body electrostatic, polariza-
tion, dispersion, and exchange-repulsion interactions between
the fragments
EMM=Eelec+Epol+Edisp+Eexch. (2)
In calculations of solvated species, each solvent molecule is
treated as an EFP fragment. The EFP method can also be
applied to macromolecules via a fragmentation scheme that
can deal with breaking covalent bonds.17
In polar and hydrogen-bonded solvents such as the ones
studied here, electrostatics and polarization are the leading
terms for inter-fragment interactions. The electrostatic inter-
action energy, Eelec, is computed using Stone’s distributed
multipole analysis39–41by placing permanent multipoles (up
to octopoles) at the atoms and bond midpoints. Only the inter-
actions of the induced dipoles with the permanent multipoles
and the induced dipoles on other fragments are included in the
calculation of Epol. A detailed description of the dispersion
and exchange-repulsion terms as well as the damping func-
tions needed to correct the artificial charge-penetration effects
and to prevent “polarization catastrophe” at close distances
between the permanent and induced multipoles can be found
elsewhere.9,10,42–44
The QM system is embedded in the polarizable environ-
ment such that the electrostatic and polarization interactions
between the QM and MM subsystems are included explicitly
in the one-electron part of HQM,
HQM
pq=h pjH0+Velec+Vpolj qi, (3)
where s are the molecular orbitals. Velecis the Coulomb
potential due to the nuclear charges and permanent multi-
poles of the fragments, whereas Vpolis the potential due to
the induced multipoles. The induced dipoles on each frag-
ment are calculated self-consistently with each other and with
the Hartree-Fock (HF) wave function using a two-level itera-
tive scheme. The converged induced dipoles of the reference
HF determinant are fixed during subsequent calculations of
the ground- and excited-state wave-function amplitudes (e.g.,
Tamplitudes in CC and excitation amplitudes in EOM-CC).164109-3 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)
The contribution to the polarization energy due to the interac-
tions of the induced dipoles with the QM electronic density,
Epol,QM/MM, is included in the state energies. Since the induced
multipoles calculated at the HF level are kept fixed in the calcu-
lations of the excited-state wave functions, their contribution
to the excitation energies is the same for all excited states.
In other words, beyond the HF level, the polarizable environ-
ment is not allowed to respond to the change in the electronic
wave function. State-specific energy corrections due to the
response of the polarizable environment to different electron
distributions of each state can be computed perturbatively12
and were shown to improve the quality of computed energy
differences.13,14,16–19Here, we omit these corrections because
this state-specific treatment cannot be easily incorporated into
the response equations solved in 2PA calculations. In the cur-
rent implementation,12the dispersion and exchange-repulsion
interactions between the QM and MM systems ( Edisp,QM/MM
andEexch,QM/MM) are treated approximately, as interactions
between the EFP fragments, and are added to the state ener-
gies; thus, they do not contribute to wave functions, excitation
energies, and other transition properties. The total energies of
the ground and excited states and the excitation energy can be
written as
EQM=MM
gr =h grjH0+Velec+Vpol
HFj gri+EMM+Epol,QM=MM
HF
+Edisp,QM=MM+Eexch,QM=MM, (4)
EQM=MM
ex =h exjH0+Velec+Vpol
HFj exi+EMM+Epol,QM=MM
HF
+Edisp,QM=MM+Eexch,QM=MM, (5)
and
QM=MM
ex,gr=h exjH0+Velec+Vpol
HFj exi
h grjH0+Velec+Vpol
HFj gri, (6)
where grand exare the ground- and excited-state wave
functions. Note that in CIS (configuration interaction singles),
TDDFT, and EOM-CC calculations, excitation energies are
computed directly and not as energy differences from state-
specific calculations.
B. The EOM-EE-CCSD/EFP method
Our choice of the electronic-structure method for the QM
system is EOM-EE-CCSD31,45,46in which the excited-state
wave functions are given by
j i=ReTj0i. (7)
Here, 0is the reference determinant and eT|0iis the
CCSD wave function. The CCSD cluster operator Tand the
EOM-CCSD operator Rcomprise single and double excitation
operators
ˆT=ˆT1+ˆT2and ˆR=ˆR0+ˆR1+ˆR2. (8)
The amplitudes of the operator ˆTsatisfy the following equa-
tions:
hj¯Hj0i=0, (9)
where ¯H=e THeTis the similarity-transformed Hamiltonian
anddenote singly and doubly excited determinants. In the
EOM/EFP scheme, the one-particle component of the Hamil-
tonian His given by Eq. (3). The EOM amplitudes Rkand thecorresponding energies Ekfor EOM-CCSD states are obtained
by diagonalizing ¯Hin the space of the reference, singly, and
doubly excited determinants
¯HR k=EkRk. (10)
Since ¯His non-Hermitian, its left ( h k| =h0Lk|) and right
(| ki= |Rk0i) eigenstates are not Hermitian conjugates, but
form a biorthonormal set
h0Lm Rn0i=mn. (11)
An important feature of the EOM-CCSD/EFP scheme is that
it preserves the biorthonormality of the target EOM states
because the induced multipoles are computed self-consistently
at the HF step and kept fixed during the calculations of the
CCSD and EOM amplitudes. This biorthogonality is impor-
tant for computing transition properties such as 1PA and 2PA
transition moments.
C. 2PA within the EOM-EE-CCSD/EFP scheme
We compute the 2PA transition moments and cross sec-
tions for EOM-CCSD wave functions following the approach
described in Refs. 34 and 36. Formally, the 2PA transi-
tion moments are given by the following sum-over-states
expressions:
Mxy
f i(!1,!2)= X
n h fjyj nih njxj ii
ni !1
+h fjxj nih njyj ii
ni !2!
(12)
and
Mxy
i f( !1, !2)= X
n h ijxj nih njyj fi
ni !1
+h ijyj nih njxj fi
ni !2!
. (13)
Here,!1and!2are the frequencies of the two absorbed
photons satisfying the 2PA resonance condition !1+!2
=
fiwith
ni=En Ei. To derive expressions for the
EOM-CCSD 2PA transition moments, we replace the dipole-
moment operator, a, with the similarity-transformed dipole-
moment operator, ¯ a=e TaeT, and the wave functions,
h k|s and | kis, with the EOM-CCSD target states, h0Lk|s
and | Rk0is, in the above expressions. Then, we recast these
expressions into the numerically and formally equivalent
form
Mxy
f i(!1,!2)=
h0Lfj¯yjX!1,x
ii+h˜X!1,x
fj¯yjRi0i
(14)
and
Mxy
i f( !1, !2)=
h0Lij¯xjX !2,y
fi+h˜X !2,y
ij¯xjRf0i
,
(15)
where Xand ˜Xare the first-order response wave functions
computed by solving auxiliary response equations. This refor-
mulation of the sum-over-states expressions leads to practical
expressions of the 2PA moments in terms of just the zeroth-
and first-order wave functions of the initial and the final states.
The response equations are
˜X!1,x
k(¯H Ek+!1)=h0Lkj¯xjIi (16)164109-4 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)
and
(¯H Ek !1)X!1,x
k=hIj¯xjRk0i, (17)
where Is denote the Slater determinants from the target-
state manifold (e.g., the reference, singly excited, and doubly
excited determinants for EOM-EE-CCSD).
D. Summary of the algorithm
The EOM-EE-CCSD/EFP method is implemented in
the Q-Chem quantum-chemistry package.47,48The procedure
for calculating 2PA cross sections involves the following
steps:
1. First, using a standard procedure,9,10all MM solvent
molecules are replaced by rigid effective fragments
placed at the same position and orientation as the original
solvent molecules. The QM subsystem is now in the field
of these effective fragments.
2. An iterative self-consistent field procedure is then
employed such that the induced dipoles on each EFP
fragments are self-consistently iterated with the HF wave
function and with induced dipoles of other EFP frag-
ments. The converged induced dipoles are then kept fixed
in the subsequent steps of the calculation.
3. The CCSD T- and-amplitudes and EOM-EE-CCSD
R- and L-amplitudes and energies are then computed
using the HF wave function and orbitals calculated in
the second step. Using these amplitudes, the CCSD
and EOM-EE-CCSD state and transition densities are
formed for calculating the state and transition dipole
moments.
4. In the 2PA step, the excitation energies without the state-
specific corrections (due to the polarization response of
the environment to the wave functions of the ground
and excited states) are used to determine the photon
frequencies according to the 2PA resonance condition.
The procedure for calculating the 2PA cross sections is
described in Refs. 34 and 36.
E. Characterization of 1PA and 2PA transitions
in terms of natural transition orbitals
Valuable insight into solvent effects can be gained by
understanding electronic transitions (1PA, 2PA, etc.) in terms
of molecular orbitals. This entails mapping the changes
in electronic density induced by excitation onto pairs of
molecular orbitals. For 1PA transitions, such maps are pro-
vided by the reduced transition one-particle density matrices
(1PDMs),
,f
f ig
pq=h fjˆpyˆqj ii, (18)
where ˆ pyand ˆqcreate and annihilate electrons in molecular
orbitals pand q, respectively. Importantly,
is related to
the experimental observables, e.g., the 1PA transition moment
is given by
a
f i=h fjaj ii=X
pqf
f ig
pqa
pq. (19)
The elements of transition 1PDMs can be interpreted as
the amplitudes of an excitation operator that generates the finalstate from the initial correlated state
f=X
pqf
f ig
pqˆpyˆq i+higher excitations . (20)
The squared norm of
,
jj
jj2=X
pq
2
pq, (21)
gives the weight of one-electron character of the transition and
provides a bound to the respective expectation values by virtue
of the Cauchy-Schwarz inequality.49,50
Transition 1PDM can be used to define an exciton’s wave
function ( exc), which is a two-particle quantity that contains
essential information about the changes in electron density
upon the transition51–54
exc(rH,rP)=X
pqf
f ig
pq p(rH) q(rP), (22)
where rHandrPare the hole and particle (electron) coordi-
nates (using rH=rP=r, excis reduced to the transition
density).
Equations (20) and (22) allow one to interpret the indi-
vidual elements of
as weights of particular configura-
tions. For example, using localized orbitals, one can express
the probability of finding the exciton in a spatial domain
Das
1
jj
f ijj2X
p2D,q2Df
f ig2
pq. (23)
As discussed in Subsection II F, this reasoning can be extended
to quantify the extent of charge-transfer character.55,56
Further simplification in exciton’s representation can be
achieved by applying singular-value decomposition (SVD) to
. The SVD procedure reduces the description of electronic
excitation to one-electron transitions between pairs of orbitals
called natural transition orbitals54,56–59(NTOs). Using NTOs,
the exciton’s wave function can be written as
exc(rP,rH)=X
KK˜ K,P(rP)˜ K,H(rH), (24)
where ˜ K,Pand˜ K,Hare the particle and hole orbitals obtained
by SVD of
andKare the corresponding singular values.
Usually, only a few Ks are significant, giving rise to a simple
interpretation of excited-state characters in terms of excitations
between hole and particle orbitals. By using NTOs, one can
express the inter-state matrix elements between many-body
wave functions in terms of matrix elements between orbitals,
facilitating the connection between molecular orbital theory
and experimental observables
a
f i=h fjaj ii=X
KKh˜ K,Pjaj˜ K,Hi. (25)
A detailed description of the NTO analysis and its implemen-
tation in Q-Chem is given in Refs. 54 and 55.
We recently extended the concepts of transition 1PDMs
and NTOs to 2PA transitions.36In addition to enabling
visualization of a 2PA transition in terms of molecular
orbitals, the 2PA NTO analysis provides insight into the
character of the virtual state involved and the type of 2PA
transition.164109-5 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)
F. Conditions necessary for using fragment approach
for describing 2PA transitions in the condensed phase
Let us now discuss the domain of applicability of the frag-
ment approaches for modeling 2PA. With an aim to provide
a metric quantifying whether the conditions needed for the
QM/EFP treatment are met by a particular 2PA transition, we
invoke the NTO analysis of the respective perturbed 1PDM.
As per Eq. (25), the 2PA moments can be expressed in terms
of matrix elements between the respective hole and particle
orbitals
Mab
f i(!1,!2)=X
Ka
Kh˜ a
K,Pjbj˜ a
K,Hi, (26)
where ˜ a
K,Pand ˜ a
K,Hare the Kth particle and hole NTOs for
the 2PA a-component ( a,b=x,y,z) transition 1PDM with a
singular value of a
K.
Since the NTOs are linear combinations of atomic orbitals
( s), i.e.,
˜ K=X
dK
, (27)
Eq. (26) can be expressed in the basis of s as
Mab
f i(!1,!2)=X
K,,a
KdK,P
dK,H
h jbj i. (28)
The corresponding 2PA a-component transition 1PDM in the
atomic orbitals basis is
f
a
f ig
=X
Ka
KdK,P
dK,H
. (29)
By partitioning the QM system into the chromophore (A) and
solvent (B), we can partition the above 1PDM into the four
parts,A A
a
f i,A B
a
f i,B A
a
f i, andB B
a
f i, whereC D
represents the contribution of the hole residing on fragment D
and a particle residing on fragment C
C D
a
f i=X
2C,2Df
a
f ig
. (30)
The probability of finding the hole on fragment D and the
particle on fragment C is now given by
C DQa
f i=1
jj
a
f ijj2X
2C,2DfC D
a
f ig2
=jjC D
a
f ijj2
jj
a
f ijj2. (31)
The four probabilities corresponding to the four fragment
1PDMs give a measure of the charge-transfer contributions
within and across fragments into the 2PA transition moment.
This is similar to the charge-transfer (CT) numbers used to
quantify the extent of charge transfer in electronic transi-
tions.54,55Breaking the system into a QM solute and an MM
solvent does not introduce large errors only when the proba-
bility of finding both the hole and particle is predominantly on
the solute, i.e., when
A AQa
f i1. (32)
In other words, the performance of fragment approaches such
as QM/EFP deteriorates with increasing contributions of the
solvent to the 2PA transition 1PDMs.We computed these CT numbers for the 2PA transitions
in microhydrated clusters studied here as well as for the rep-
resentative snapshots from solvated PYP simulations. The
analysis reveals that the charge-transfer probabilities between
the chromophore and the solvent are low (A BQa
f i+B AQa
f i
<0.01). The probability of finding both the hole and particle
NTOs on the solvent molecules is even lower. This indicates
that the QM/MM fragmenting of these systems should not
introduce large errors into the 2PA cross sections because the
2PA transition is predominantly localized on the chromophore.
This encouraging finding is far from obvious as in contrast to
1PA transitions for which only the initial and final states need
to be localized on the solute, one could expect a noticeable
charge-transfer character in the virtual states.
III. COMPUTATIONAL DETAILS
All calculations were carried out using the Q-Chem
quantum-chemistry package.47,48We used the B3LYP/6-
31+Goptimized geometry for pNA, the RI-MP2/cc-pVTZ
geometry for thymine, and the CCSD/6-31+Ggeometry for
PYPb in the calculations of bare chromophores. The struc-
tures used in the QM/EFP calculations for the microsol-
vated clusters of pNA, thymine, and PYPb were taken from
Refs. 12, 60, and 61, respectively; they are shown in Fig. 1.
We used the EOM-EE-CCSD/6-31+Glevel of theory within
three different QM/MM schemes in which the MM subsys-
tem was represented by the (1) EFP fragments, (2) EFP
fragments without polarization, and (3) CHARMM22 point
charges. Core electrons were kept frozen in all calculations.
In each of these systems, we characterized the lowest sym-
metric 2PA transition considering degenerate photons. For
pNA, we also considered non-degenerate photons, the first
photon having a frequency of 25 400 cm 1(3.1492 eV).
We used the TheoDORE package53for computing the
FIG. 1. The studied microhydrated clusters of pNA, thymine, and PYPb.164109-6 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)
charge-transfer probabilities following the 2PA NTO calcu-
lations with Q-Chem.
In the QM/EFP calculations, all solvent water molecules
were replaced by the standard effective fragments placed
at the same positions and orientations as the initial solvent
molecules.9,10This means that the Cartesian geometries of the
initial structures and the QM/EFP model clusters are slightly
different. To eliminate the ambiguity due to slight geometry
change, we used these QM/EFP geometries in all calculations
of the microsolvated clusters, meaning that the structures of the
water molecules in extended QM subsystems are exactly the
same as in the effective fragments. All Cartesian geometries
are provided in the supplementary material.
To model bulk solvation, we performed molecular dynam-
ics (MD) simulations with a PYPb chromophore and one
sodium cation in a periodic box of 1717 water molecules using
the NAMD software.62We used the CHARMM22 parameters
for water and the sodium cation; the parameters for PYPb are
given in the supplementary material. After the initial minimiza-
tion for 20 ps (with 2 fs time step), we equilibrated the system
for 2 ns (with 1 fs time step) in an NPT ensemble at 298 K and
1 atm and followed up with a production run of 1 ns. From this
trajectory, we picked 21 snapshots separated by 50 ps for the
2PA calculations using EOM-EE-CCSD/6-31+G//EFP (the
pdb files for the snapshots are given in the supplementary
material). Along the equilibrium trajectory, the sodium cation
is more than 10 Å from the PYPb chromophore. In the 2PA
calculations, we included only the chromophore and water
molecules.
To investigate the effect of the QM size on the computed
2PA cross section, we systematically extended our QM subsys-
tem by including explicit water molecules at the phenolate and
carboxylic acid ends of PYPb. In these calculations, the struc-
tures of QM water molecules were identical to the structures in
the MD snapshots. Using these snapshots, we also performed
CIS/6-31+G//EFP calculations to investigate the convergence
of the key components—excitation energies, transition dipole
moments, and dipole-moment differences between the ground
and excited states. In these CIS/EFP calculations, the structures
of the model systems were obtained by using the standard EFP
procedure9,10on the MD snapshots, meaning that the structures
of water molecules in the QM system are slightly different from
the structures in the MD snapshots.
IV. RESULTS AND DISCUSSION
A. Molecular orbital framework for 2PA transitions
in pNA, thymine, and PYP chromophore
To understand the effect of microsolvation on the 2PA
transitions of the three benchmark molecules, we begin by
characterizing these transitions in the bare chromophores. We
aim to understand the nature of solute-solvent interactions by
analyzing the orbital character of the transitions. We calculated
the 2PA NTOs using our wave-function analysis toolkit.36The
leading NTO pairs and the molecular orientations are shown
in Fig. 2.
In our calculations, the pNA molecule is in the xzplane,
with the zaxis along the dipole moment. The lowest symmet-
ric 2PA transition has a large microscopic cross section (see
FIG. 2. Dominant NTOs (isovalue = 0.05) for the (a) z-component transition
1PDM of the lowest symmetric 2PA transition in pNA with degenerate and
non-degenerate photons, (b) x- and y-component transition 1PDMs of the low-
est 2PA transition in thymine with degenerate photons, and (c) x-component
1PDM of the lowest 2PA transition in the PYPb chromophore with degenerate
photons.
Table I) and is dominated by the zzcomponent of the 2PA tran-
sition moment. The NTO analysis of the zcomponent 1PDM
shows that this transition is well described by just one NTO pair
with a large singular value. This NTO pair (shown in Fig. 2)
reveals the intramolecular charge-transfer character: upon this
transition, the dipole moment increases due to the shift in
the electronic density towards the electron-withdrawing nitro
group. A more detailed analysis of this transition can be found
in Ref. 36. The charge-transfer character is further supported
by analyzing the participation ratios ( PRNTO) computed for the
z-component 1PDM and its component !DMs (see Ref. 36 for
definitions). Whereas the PRNTOfor the z-component 1PDM
is1-2 for both degenerate and non-degenerate cases, the
PRNTOs for its component !DMs are>20. This is a signature
of intramolecular charge-transfer transitions. Intramolecular
charge-transfer 2PA transitions are driven by the change in the
dipole moment upon excitation ( h f|a| fi h i|a| ii), as
given by an approximate expression for the 2PA transition
moment
Mab
f i(!1,!2) h fjbj ii
!1
h fjaj fi h ijaj ii
h fjaj ii
!2
h fjbj fi h ijbj ii
,
(33)
which reduces to
Maa
f i(!) 2h fjaj ii
!
h fjaj fi h ijaj ii
(34)
for degenerate photons and diagonal elements of the 2PA
transition-moment tensor.36,63,64Equation (34) also shows that164109-7 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)
TABLE I. 2PA cross sections with degenerate and nondegenerate photons computed with EOM-EE-CCSD/
6-31+G//MM for the microsolvated clusters of para -nitroaniline. % errors relative to the full QM approach
in parentheses. (
in eVs, all other quantities in a.u.)
System QM MM
f 2PA 2PA
!1=!2!1,!2
pNA pNA 4.55 0.47 4543 6296
pNA + 2H 2O pNA + 2H 2O 4.49 0.44 4848 7012
pNA + H 2O EFP 4.49 0.44 4961(2.3) 7182(2)
pNA EFP 4.48 0.45 5059(4.4) 7339(5)
pNA EFP (no pol) 4.50 0.45 4990(2.9) 7181(2)
pNA CHARMM 4.49 0.45 5018(3.5) 7236(3)
pNA + 4H 2O pNA + 4H 2O 4.46 0.43 4920 7237
pNA + H 2O EFP 4.46 0.45 5161(4.9) 7590(5)
pNA EFP 4.45 0.45 5286(7.4) 7843(8)
pNA EFP (no pol) 4.48 0.45 5143(4.5) 7506(5)
pNA CHARMM 4.50 0.45 4994(1.5) 7166( 1)
pNA + 6H 2O pNA + 6H 2O 4.46 0.43 4884 7159
pNA EFP 4.48 0.45 5206(6.6) 7575(6)
pNA EFP (no pol) 4.51 0.45 5025(2.9) 7168(0)
pNA CHARMM 4.51 0.45 5051(3.4) 7230(1)
in this case, the 2PA transition moment is proportional to
the transition dipole moment ( h f|a| ii) and inversely pro-
portional to the excitation energy (
fi=!1+!2). Thus,
understanding the difference in the permanent dipole moments
of the ground and excited states, transition dipole moment,
and excitation energy in the microsolvated clusters is the key
to understanding how solvent impacts 2PA cross sections.
For example, bulk solvation stabilizes the excited state more
than the ground state because of a larger excited-state dipole
moment, resulting in the lowering of the excitation energy rel-
ative to the gas phase. However, in microsolvated clusters, the
change in the excitation energy depends on the relative ori-
entation of the solute molecules and the extent to which they
stabilize or destabilize the hole and particle orbitals.
In our calculations, thymine ( Cs) is in the xyplane. The
lowest A0transition with degenerate photons has a small
microscopic cross section dominated by the Mxxand Myy
components. The NTO analyses of the x- and y-components
of the respective 1PDMs show that each of these compo-
nent transitions can be described by a single pair of NTOs,
which indicates that the transition is accompanied by some
intramolecular charge transfer, away from the oxygen of the
carbonyl group across the methyl group. This is further val-
idated by the PRNTOs for these 1PDMs ( 1-2) and their
component !DMs (>20). So, this 2PA transition can also be
explained by Eq. (34) and the changes in its 2PA cross section
in microhydrated clusters can be explained by understanding
the corresponding changes in the transition moments, differ-
ences in permanent dipoles, and excitation energy. The shift in
electronic density increases the dipole moment of the molecule
in the excited state, which suggests red shift in aqueous
solution.
PYPb also has Cssymmetry and is in the xyplane in our
calculations. For this chromophore, the lowest symmetric 2PA
transition with degenerate photons has a large microscopic2PA cross section with a dominant Mxxcomponent. The 2PA
NTO analysis for the x-component 2PA 1PDM gives two dom-
inant NTO pairs, which represent the opposing channels of
electronic density flow upon the 2PA transition. The domi-
nant channel reveals charge transfer from the carboxylic acid
to the phenolate and the second channel reveals the charge
transfer from the phenolate to the carboxylic acid. The latter
channel corresponds to the orbital character of the 1PA transi-
tion between these states shown in the supplementary material
(Fig. S1). Thus, the dominant NTO pair indicates that the vir-
tual state is dominated by the initial state, which has a larger
electronic dipole than the excited state.65By contrast, in the
studied transition in pNA, the character of the virtual state
is dominated by the final state with a larger dipole moment
than the ground state and only one NTO pair. Quantitative
analysis of the PRNTOs for this 1PDM (1-2) and its compo-
nent!DMs (>30) also shows that this 2PA transition in PYPb
has some intramolecular charge-transfer character. Moreover,
the respective transition dipole moment is also large. Equa-
tion (34), therefore, explains the large 2PA cross section for
this transition.
B. Microsolvated p-nitroaniline clusters
In all microhydrated clusters of pNA, the excitation energy
calculated with the full QM approach is slightly lower than
that of the bare chromophore. This indicates that the water
molecules, which are clustered around the nitro group, stabi-
lize the particle orbital of the lowest symmetric 2PA transition
more than the hole orbital. Moreover, the character of the
dominant 2PA NTOs for this transition does not change upon
microsolvation. This is expected for weakly interacting solute-
solvent systems. The change in the oscillator strength is also
small. Table S1 in the supplementary material indicates that
the difference in permanent dipole moments of the ground and164109-8 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)
excited states increases, which along with the slight decrease
in the excitation energy increases the 2PA cross section by
up to 8% for degenerate photons and 15% for non-degenerate
photons in accordance with Eq. (33). Across the three clus-
ters, the full QM cross section increases from dihydrates
to tetrahydrates and decreases slightly from tetrahydrates to
hexahydrates.
The QM/EFP scheme with just the pNA molecule in QM
reproduces this trend but slightly exaggerates the changes
across these clusters with errors of about 2%-9% relative
to the respective full QM cross sections (Table I). Omitting
the polarization contributions of the EFP fragments in the
QM/EFP calculations results in smaller errors (2%-5%) rel-
ative to the full QM cross sections. This approach, however,
yields slightly smaller cross sections for the 2PA transition
with non-degenerate photons for the hexahydrate than the
dihydrate. Similarly, the QM/CHARMM approach also yields
smaller errors relative to the full QM cross sections across the
three clusters but fails to reproduce the trend obtained with the
full QM calculation. In fact, QM/CHARMM shows a decrease
in the cross sections for both sets of photons from dihydrate to
the tetrahydrate and an increase from tetrahydrate to hexahy-
drate. Yet, the small differences between QM/EFP, QM/EFP
with polarization turned off, and QM/CHARMM preclude us
from arguing in favor of any of these approaches. Such small
differences also indicate that for these systems the impact
of intermolecular polarization on 2PA cross sections is not
large.
We also report the QM/EFP results for the dihydrate and
tetrahydrate clusters with the QM subsystem comprising the
pNA and its nearest hydrogen-bonded water. These calcula-
tions result in smaller errors of 2%-5% relative to the full
QM cross sections. Similar reduction of the QM/EFP errors
upon inclusion of the hydrogen-bonded waters in the QM sys-
tem was reported for the core-ionization energies of solvated
glycine.19
The errors in the 2PA cross sections for the three QM/MM
schemes relative to the full QM results can be traced back to
the slightly smaller errors in the excitation energies, transition
moments, and dipole-moment differences with the QM/EFP
scheme as tabulated in the supplementary material (Table S1).
We note that the errors in the excitation energies for these three
QM/MM approaches relative to the full QM results are very
small (<1%), but the errors in the transition dipole moments
and dipole-moment differences are slightly larger (up to 2%-
3% for both).
C. Microsolvated thymine clusters
The orbital character of the lowest A02PA transition in
thymine does not change upon microsolvation. In the full QM
calculation, the microsolvation changes the excitation energy
by up to 3%, the oscillator strength by up to 14%, and the 2PA
cross sections by up to 20% (we note that the magnitude of
the microscopic cross section is small). In the full QM cal-
culation, the excitation energy of this transition decreases in
T1–T2 relative to the bare thymine (see Table II), but in the T3
cluster, the excitation energy is higher than in bare thymine.
The difference in trends can be attributed to the following
structural feature: only in the T3 cluster, the water molecule isTABLE II. 2PA cross sections with degenerate photons computed with EOM-
EE-CCSD/6-31+G//MM for the microsolvated clusters of thymine. % errors
relative to the full QM approach in parentheses. (
in eVs, all other quantities
in a.u.)
System QM MM
f2PA
T T 5.64 0.24 104
T1 T1 5.57 0.26 118
T EFP 5.58 0.25 104( 12)
T EFP (no pol) 5.58 0.25 105( 11)
T CHARMM 5.57 0.25 106( 10)
T2 T2 5.51 0.21 114
T EFP 5.50 0.21 109( 4)
T EFP (no pol) 5.52 0.22 109( 4)
T CHARMM 5.53 0.22 108( 5)
T3 T3 5.68 0.23 89
T EFP 5.69 0.23 91(2)
T EFP (no pol) 5.67 0.23 94(6)
T CHARMM 5.67 0.23 95(7)
T11 T11 5.54 0.27 122
T EFP 5.55 0.25 102( 16)
T EFP (no pol) 5.57 0.25 102( 16)
T CHARMM 5.56 0.25 104( 15)
T12 T12 5.49 0.24 120
˜T1 EFP 5.48 0.24 117( 3)
˜T2 EFP 5.50 0.23 106( 12)
T EFP 5.49 0.23 103( 14)
T EFP (no pol) 5.51 0.23 103( 14)
T CHARMM 5.50 0.23 103( 14)
T112 T112 5.46 0.25 125
˜T11 EFP 5.45 0.25 121( 3)
˜T2 EFP 5.47 0.23 104( 17)
T EFP 5.47 0.24 101( 19)
T EFP (no pol) 5.49 0.24 101( 19)
T CHARMM 5.48 0.24 101( 19)
near the carbonyl group that is across from the methyl group,
thereby stabilizing the hole orbital shown in Fig. 2(b). All
three QM/MM schemes reproduce this trend in the excitation
energy across the clusters with negligible errors ( <1%) relative
to the full QM calculation. The 2PA cross section decreases
from T1 to T2 to gas-phase thymine to T3. All three QM/MM
schemes, however, yield smaller 2PA cross sections for the T1
cluster than for the T2 cluster, with comparable errors. For
the T3 cluster, QM/EFP (2% error) does slightly better than
the other two schemes (6%-7% error). The errors in the 2PA
cross sections relative to the full QM calculations can be traced
back to even smaller errors in the transition moments and
dipole-moment differences between the ground and excited
states (Table S2). For thymine clusters, these differences across
the three QM/MM schemes are small. The larger excitation
energies and smaller dipole-moment differences between the
ground and excited states of thymine compared to pNA and
PYPb explains why these thymine systems have smaller 2PA
cross sections.
The results for the thymine dihydrates and trihy-
drates computed with the three QM/MM schemes are also164109-9 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)
comparable, with errors ranging between 11% and 20% rela-
tive to the full QM calculation. This indicates that polarization
has a negligible effect on the 2PA cross sections in thymine
clusters. Similar to our observation for the pNA clusters, inclu-
sion of explicit water molecules in the QM subsystem for
thymine polyhydrates reduces the errors of the QM/EFP cal-
culation. In terms of the structure, the T12 dihydrate can be
viewed as a superposition of T1 and T2 monohydrate clusters.
Considering the large QM/EFP errors in the 2PA cross section
for the T1 cluster, the inclusion of the ˜T1 fragment (the tilde
indicates that the ˜T1 fragment and T1 cluster are structurally
similar) in the QM subsystem reduces the error in the 2PA
cross sections from 14% to about 3% for the T12 cluster. Sim-
ilarly, the structure of the T112 trihydrate can be described as
a superposition of the T11 dihydrate and the T2 monohydrate.
The QM/EFP errors in the 2PA cross section relative to the full
QM calculation are reduced from 19% to 3% when the ˜T11
fragment (structurally similar to the T11 cluster) of T112 is
included in QM.
D. Microsolvated clusters of the anionic
PYP chromophore
The lowest excited state of the phenolate chromophore
of PYP (PYPb) has a large 2PA cross section (Table III).
In the microsolvated clusters, the 2PA cross section for this
transition increases significantly (up to 110% in the PYPb-
WPWP1cluster). We also note that the variations in the exci-
tation energy for this transition are large across the three
clusters. In the PYPb-W P1and PYPb-W PWP1clusters, the
water molecules are hydrogen-bonded with the phenolate oxy-
gen. This interaction stabilizes the hole NTO of the transi-
tion (see Fig. S1 of the supplementary material) more than
the particle NTO and results in an increase in the excita-
tion energy. In the PYPb-W CWP1cluster, water molecules
make hydrogen bonds with both the phenolate and the car-
boxylic acid group. This stabilizes both the hole and particle
NTOs to a similar extent, and the excitation energy does not
change significantly in this cluster relative to the bare PYPb
chromophore.The three QM/MM schemes show small errors ( <2%)
in the excitation energy relative to the full QM calcula-
tion. The QM/EFP errors are slightly smaller than those
of the other two non-polarizable QM/MM schemes. This is
expected for a charged chromophore for which mutual polar-
ization is more pronounced. Interestingly, QM/EFP overes-
timates the excitation energies for all clusters, whereas the
other two QM/MM schemes underestimate the excitation
energies.
In the PYPb-W P1and PYPb-W CWP1clusters, the
QM/EFP errors in the 2PA cross sections relative to the full QM
calculation are smaller than the errors of the other two QM/MM
schemes, which again signifies that solvent polarization plays
an important role in these clusters. The errors for the PYPb-
WPWP1cluster are comparable across all three schemes. The
errors in the 2PA cross sections can be traced back to smaller
errors in the transition moments and electronic dipole differ-
ences between the ground and excited states given in Table S3.
Interestingly, QM/EFP yields more accurate dipole-moment
differences than the other two schemes but performs poorly
for the transition moments.
In Sec. IV E, we investigate the effect of bulk solvation
on PYPb and show that the accuracy of the 2PA cross sec-
tions can be significantly improved by including nearest water
molecules in the QM subsystem.
E. Anionic PYP chromophore in aqueous solution
Table S4 in the supplementary material and Fig. 3 show
the excitation energies and microscopic 2PA cross sections
with degenerate photons for the 21 MD snapshots of PYPb in
solution computed with QM/EFP. We use the following nota-
tions for calculations with different QM subsystems: (a) PYPb
means that only PYPb is included in QM, (b) PYPb+ means
that PYPb plus all water molecules within 2.5 Å of the phe-
nolate oxygen are included, (c) +PYPb includes PYPb plus all
water molecules within 2.5 Å of the carboxylic acid group,
(d) +PYPb+ includes PYPb plus all water molecules within
2.5 Å of the phenolate oxygen and the carboxylic acid group,
TABLE III. 2PA cross sections with degenerate photons computed with EOM-EE-CCSD/6-31+G//MM for the
microsolvated clusters of PYPb. % errors relative to the full QM calculation in parentheses.
System QM MM
f 2PA
PYPb PYPb 3.21 1.06 9578
PYPb-W P1 PYPb-W P1 3.25 1.01 16862
PYPb EFP 3.26 0.99 15961( 5)
PYPb EFP (no pol) 3.23 1.00 15353( 9)
PYPb CHARMM 3.22 1.00 15291( 9)
PYPb-W CWP1 PYPb-W CWP1 3.21 1.07 16878
PYPb EFP 3.22 1.02 15177( 10)
PYPb EFP (no pol) 3.20 1.02 14469( 14)
PYPb CHARMM 3.20 1.01 14500( 14)
PYPb-W PWP1 PYPb-W PWP1 3.37 0.98 20095
PYPb EFP 3.41 0.94 18155( 10)
PYPb EFP (no pol) 3.32 0.96 18105( 10)
PYPb CHARMM 3.31 0.97 18212( 9)164109-10 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)
FIG. 3. (a) Excitation energies, (b) 1PA transition moments, (c) dipole-moment differences between the ground and excited states, and (d) microscopic 2PA
cross sections for the 21 snapshots with varying size of the QM subsystem. The horizontal lines indicate average properties for PYPb (blue), PYPb+ (red), and
+PYPb+ (black) calculations across the 21 snapshots. The shaded area indicates the standard deviation for the +PYPb+ calculations across the 21 snapshots.
(e) PYPb++ includes PYPb plus all water molecules within
3.5 Å of the phenolate oxygen, and (f) PYPb+++ includes
PYPb plus all water molecules within 4.0 Å of the phenolate
oxygen.
With just the PYPb chromophore in QM, the average
2PA cross section across the snapshots shows about 28%
increase relative to the bare chromophore (Table III) with
a large standard deviation. With an exception of snapshot
#15, the 2PA cross sections are higher than the reference
gas-phase value. As expected, the excitation energies across
all snapshots increase relative to the bare chromophore. The
transition moments and dipole-moment differences between
the initial and the final states are given in Table S4 (sup-
plementary material). We note that the transition moments,
in general, decrease, while the dipole-moment differences
increase, the latter overcompensating the former resulting
in higher 2PA cross sections in solution relative to the gas
phase.
The QM/EFP results for the microsolvated clusters show
that 2PA cross sections are sensitive to the choice of the QM
subsystem and that the inclusion of solvent molecules in QM
causes a noticeable change in the cross section. We carried
out a similar analysis for the PYPb chromophore in solution,
wherein we extend the QM subsystem by including more water
molecules in the QM part. First, we included the first solva-
tion shell (defined by the cutoff distance of 2.5 Å) around
the phenolate oxygen (denoted by PYPb+). We see that the
PYPb+ 2PA cross sections increase significantly (increase in
the average is15%) and excitation energies decrease (by
up to 0.11 eV) across the snapshots relative to the QM/EFPresults with just the chromophore in QM. On the other hand, the
inclusion of the first solvation shell of water molecules around
the carboxylic acid group (denoted by +PYPb) increases the
2PA cross sections to a smaller extent than the PYPb+ calcu-
lations in five of the six snapshots for which we conducted
such calculations. In general, the excitation energies increase
slightly (up to 0.02 eV) in the +PYPb calculations rela-
tive to the QM/EFP calculations with just the chromophore
in QM.
We note that the 2PA cross sections for the +PYPb+
QM/EFP calculations, wherein the QM subsystem includes
the first solvation shells around both the phenolate oxy-
gen and the carboxylic acid group, can be approximated
by adding the differences between the PYPb+ and PYPb
results to the differences between the +PYPb and PYPb
results
+PYPb +2PA=PYPb2PA+PYPb +2PA PYPb2PA
++PYPb2PA PYPb2PA
. (35)
The increase in the average 2PA cross section across snapshots
for the +PYPb+ calculations is 22%. These changes in the
2PA cross sections between QM/EFP calculations with differ-
ent QM subsystems can be traced back to the corresponding
changes in the transition moments and dipole-moment differ-
ences between the ground and excited states given in Table S4
(supplementary material).
Since the changes due to the inclusion of water molecules
in QM at the two ends of the PYPb chromophore are approx-
imately additive, we test if the inclusion of the first solvation164109-11 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)
shell in QM gives sufficiently converged results by includ-
ing water molecules beyond the first solvation shell only at
the phenolate end. For snapshots #13 and #17, we include all
water molecules within 4.0 Å from the phenolate oxygen. For
snapshot #13, the excitation energy and 2PA cross section for
the PYPb+++ calculations show less than a 1% change rela-
tive to the results for the PYPb+ calculation with just the first
solvation shell in QM. The change in the transition moments
from PYPb+ to PYPb+++ for this snapshot is positive, which
cancels the negative change in the dipole-moment difference
leading to an overall smaller change in the 2PA cross section.
For snapshot #17, the change in the 2PA cross section from
PYPb+ to PYPb++ (water molecules within 3.5 Å) is about
5%. However, the 2PA cross section change by <1% from
PYPb++ to PYPb+++ calculations. The transition moments
and dipole-moment differences also show negligible changes
from PYPb++ to PYPb+++. These calculations clearly
suggest that the convergence of the 2PA cross sections with
the size of QM subsystem depends on the convergence of the
excitation energies, transition moments, and dipole-moment
differences.
Given the above results, a crucial question is then: How
fast do these properties converge with the size of the QM sub-
system? In other words, how large the QM solvation shell
around the chromophore should be for the converged bulk
results? While we could not carry out such convergence stud-
ies with EOM-EE-CCSD, we can investigate the convergence
of the key electronic properties contributing to the 2PA cross
sections using a lower-level method. Towards this goal, we
investigated the convergence of the excitation energies, tran-
sition moments, and dipole-moment differences by comput-
ing them with the CIS/EFP scheme for all snapshots (the
results for snapshots #13 and #17 are presented in Fig. 4).
For this analysis, we gradually extend the QM subsystem by
including more and more water molecules in QM using the
EFP-modified geometries (as in the case of the microhydrated
clusters, this strategy allows us to compare the properties of dif-
ferent embedded QM subsystems using identical structures).
In all model structures, the chromophore is oriented in the same
way such that the x-component of the 1PA transition moment
and dipole-moment difference always dominates. We approxi-
mate the corresponding Mxxmoment according to Eq. (34) and
then estimate the error in the 2PA cross section in terms of the
error in this 2PA transition moment as a function of the QM
size. The use of the EFP-modified extended QM subsystem
does not introduce significant differences in the behavior of
the excitation energies, total transition moments, and dipole-
moment differences in our analysis. The differences in these
properties for these two EFP-modified snapshots across dif-
ferent embedded QM subsystems (given in Table S5 of the
supplementary material) are small when compared to the cor-
responding values for embedded QM subsystems constructed
with the geometries taken from the MD snapshots (given in
Table S6 of the supplementary material for snapshots #13 and
#17).
We expand the QM subsystem by adding waters within
a cutoff distance from either the phenolate oxygen or the
carboxylic acid group of the chromophore. We vary the
cutoff distance from 0 Å (chromophore only) to 5 Å.
FIG. 4. CIS/6-31+G//EFP calculations for snapshots #13 and #17 showing
the behavior of key electronic properties as a function of the size of the QM sub-
system. (a) The number of water molecules as a function of the cutoff distance
from the phenolate oxygen or the carboxylic acid group, (b) excitation ener-
gies, (c) x-component transition moments, (d) x-component dipole-moment
differences between the ground and excited states, and (e) xx-component 2PA
transition moments computed using Eq. (34) as functions of the number of
QM water molecules.
For snapshots #13 and #17, the maximum number of
water molecules included in QM was 54 and 43, respec-
tively [Fig. 4(a)]. The cutoff distance of 2.5 Å gives the164109-12 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)
structure corresponding to the +PYPb+ calculations discussed
above.
Figure 4(b) shows the behavior of the excitation energy
as a function of the QM water molecules. In both snapshots,
we observe an initial steep drop from the calculation with
the bare chromophore in QM to the +PYPb+ calculation.
The excitation-energy curve for snapshot #17 plateaus after
about 20 water molecules ( 4 Å cutoff) are included in QM.
By contrast, this curve for snapshot #13 shows a slow grad-
ual decrease and plateaus when about 42 water molecules
(4.6 Å cutoff) are included in the QM. Including the first
solvation shell (+PYPb+ calculations) around the two ends
of PYPb reduces the error in the excitation energy from
3.5% to 2.6% and 1.6% to 0.8% for snapshots #13 and #17,
respectively, relative to the calculations with the largest QM
subsystems.
Figure 4(c) shows the behavior of the x-component tran-
sition moment as a function of QM water molecules. After an
initial steep increase upon addition of the first solvation shell,
the changes are much smaller for both snapshots. The errors
in the +PYPb+ calculations are 1.8% and 0.4% for snapshots
#13 and #17, respectively, relative to the calculations with the
largest QM subsystems.
Figure 4(d) shows the behavior of the x-component
dipole-moment difference as a function of QM water
molecules. As for the excitation energies, the dipole-moment
difference for snapshot #17 is well converged after the inclu-
sion of 20 water molecules in the QM subsystem. However,
the convergence is slower for snapshot #13 for which conver-
gence is reached only after 45 molecules are included in QM.
Whereas the absolute errors relative to the calculation with the
largest QM subsystem are small for snapshot #17 ( <2.5%),
the corresponding errors are slightly larger for snapshot #13
(up to 6.2%). Including the first solvation shell decreases the
absolute errors for snapshot #17 from 0.9% to 0.1%, while the
absolute errors increase for snapshot #13 from 4.3% to 5.5%.
Figure 4(e) shows the approximate Mxxtransition
moment, calculated using Eq. (34), as a function of the number
of QM water molecules. Similar to the behavior of the excita-
tion energy and dipole-moment difference, the xx-component
2PA transition moment for snapshot #17 changes by no more
than 3.3% after the inclusion of 20 water molecules. On the
other hand, the change is larger and convergence is slower
(after inclusion of 45 water molecules in QM) for snapshot
#13. This is highlighted in that the absolute errors in +PYPb+
estimates for this 2PA transition moment are 6.2% for snapshot
#13 as against 0.3% for snapshot #17. Since the approximate
error in the 2PA cross sections is twice the error in the dom-
inant Mxxcomponent, for snapshot #13, we estimate that our
best EOM-EE-CCSD 2PA cross section (+PYPb+ calculation)
is within12.5% from the converged bulk value.
Figure S2, Table S5, and the discussion in the supple-
mentary material provide the CIS/EFP error analysis for all
21 snapshots as a function of the cutoff distance. The errors
in the average microscopic 2PA cross sections for PYP and
+PYP+ calculations are expressed in terms of the errors in
theMxxtransition moments relative to the converged values
[Eq. (S1)]. These errors are 12.6% and 3.3%, respectively.
This clearly shows that the errors in the average 2PA crosssections relative to the converged values drop significantly
when the water molecules in the first solvation shell are treated
at the QM level. Thus, we estimate that our best bulk-averaged
value of the EOM-CCSD 2PA cross section (+PYPb+ calcu-
lation) is about 3-4% below the converged result. The analysis
of the convergence of 2PA cross sections with respect to the
size of the QM subsystem shows that the first solvation shell
provides a good balance between computational cost and con-
verged results for the final averaged spectra, despite slower
convergence for individual snapshots. A similar conclusion
for another polarizable embedding QM/MM scheme has been
reported in Ref. 66 for converged X-ray absorption spectra
averaged over multiple snapshots.
V. CONCLUSIONS
We extended the EOM-CCSD method of calculating 2PA
cross sections to condensed-phase calculations, wherein the
quantum system is embedded in the polarizable environment
represented by EFP fragments. We also presented a met-
ric based on transition 1PDMs that helps us to assess the
applicability of a particular QM/MM embedding scheme for
studying 2PA transitions.
We analyzed the solvent effects on 2PA cross sections in
microhydrated clusters of pNA, thymine, and the PYPb chro-
mophore. We benchmarked the EOM-EE-CCSD/EFP results
against full EOM-EE-CCSD calculations for these systems.
When only the chromophore is included in the QM subsys-
tem, the errors in the EOM-EE-CCSD/EFP 2PA cross sections
relative to the full EOM-EE-CCSD treatment are <20%. The
EOM-EE-CCSD/EFP method captures the main trends in the
2PA cross sections as the full QM calculations. A system-
atic inclusion of explicit solvent molecules in QM reveals
that the calculated 2PA cross sections are sensitive to the
QM size. When nearest water molecule(s) are added to the
QM subsystem, the EOM-EE-CCSD/EFP errors drop down to
5%.
In the EOM-EE-CCSD/EFP method, we turned off the
contribution of the polarization effects and analyzed its
impact on the 2PA cross sections. We find that polariza-
tion has a large impact on the EOM-EE-CCSD/EFP 2PA
cross sections of (anionic) PYPb clusters but not for clus-
ters of thymine and pNA. We also compared our results
with the EOM-EE-CCSD/MM approach, wherein the sol-
vent molecules are represented by CHARMM charges. While
we do not see significant differences in the results with
these three QM/MM schemes for thymine and pNA clus-
ters, EOM-EE-CCSD/EFP performs better for the PYPb
clusters.
We also performed NTO analyses of the 2PA transitions
in order to understand their orbital character and to obtain
insight into the impact of the solvent. The analyses revealed
the intramolecular charge-transfer character of these 2PA tran-
sitions, which then allowed us to evaluate the 2PA transition
moments from three key components: excitation energies, 1PA
transition moments, and dipole-moment differences between
the ground and excited states. The errors in the EOM-EE-
CCSD/EFP 2PA cross sections relative to the full EOM-EE-
CCSD results were then traced back to the errors in the164109-13 K. D. Nanda and A. I. Krylov J. Chem. Phys. 149, 164109 (2018)
excitation energies, 1PA transition moments, and dipole-
moment differences.
We have also reported the EOM-EE-CCSD/EFP 2PA
cross section of the lowest symmetric transition in PYPb
in solution. Our analysis shows that inclusion of few water
molecules in the QM subsystem reduces the errors in the exci-
tation energies, 1PA transition moments, and dipole-moment
differences significantly relative to the converged values with
extended QM subsystems. Consequently, the inclusion of
water molecules in the QM improves the calculated 2PA cross
section significantly.
SUPPLEMENTARY MATERIAL
See supplementary material for (1) 1PA NTOs for the
studied transitions in pNA, thymine, and PYPb in the gas
phase; (2) key quantities such as excitation energies, transition
moments, dipole-moment differences, and 2PA cross sections
for the microhydrated clusters and aqueous PYPb calculated
with EOM-EE-CCSD/EFP and CIS/EFP methods; (3) force-
field parameters for PYPb; (4) Cartesian coordinates; and (5)
a separate .txt file containing pdbs for the 21 MD snapshots of
PYPb in solution.
ACKNOWLEDGMENTS
A.I.K. acknowledges support by the U.S. National Sci-
ence Foundation (Grant No. CHE-1566428). K.D.N. acknowl-
edges helpful discussions with Dr. Ilya Kaliman on EFP
and Dr. Atanu Acharya and Tirthendu Sen on NAMD
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1.2910724.pdf | Influence of interlayer magnetostatic coupling on the ferromagnetic resonance
properties of lithographically patterned ferromagnetic trilayers
Y. Nozaki, K. Tateishi, S. Taharazako, S. Yoshimura, and K. Matsuyama
Citation: Applied Physics Letters 92, 161903 (2008); doi: 10.1063/1.2910724
View online: http://dx.doi.org/10.1063/1.2910724
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/92/16?ver=pdfcov
Published by the AIP Publishing
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137.189.170.231 On: Sun, 21 Dec 2014 23:43:15Influence of interlayer magnetostatic coupling on the ferromagnetic
resonance properties of lithographically patterned ferromagnetic trilayers
Y . Nozaki,a/H20850K. T ateishi, S. T aharazako, S. Yoshimura, and K. Matsuyama
Department of Electronics, Kyushu University, Fukuoka 819-0395, Japan
/H20849Received 26 February 2008; accepted 26 March 2008; published online 23 April 2008 /H20850
The influence of magnetostatic coupling in a Ni 81Fe19/Cu /Ni81Fe19elliptical element on the
ferromagnetic resonance /H20849FMR /H20850properties was experimentally and numerically investigated.
Discontinuous variations in the FMR frequencies were observed when the relative orientation ofmagnetization was switched between parallel and antiparallel. Furthermore, the trilayer elementexhibits field-insensitive FMR properties for magnetization of antiparallel orientation. This isattributable to the conflicted magnetic field dependence of the FMR frequency in the top and bottomNi
81Fe19layers with antiparallel magnetization. © 2008 American Institute of Physics .
/H20851DOI: 10.1063/1.2910724 /H20852
Microstructured metallic ferromagnets are promising
candidates for efficient microwave absorbers in monolithicmicrowave integrated circuit, because the ferromagneticresonance /H20849FMR /H20850frequencies f
FMRcan be widely tuned in
the range of several tens of gigahertz by application of ex-ternal magnetic fields.
1–3However, in order to keep the ab-
sorption frequency, an adequate magnetic field must be con-tinuously applied during operation. This is inconvenient forapplication in low electric-power-consuming devices, suchas radio-frequency identification tags.
In order to resolve this problem, a method to control the
f
FMRis proposed using magnetostatic coupling fields in pat-
terned magnetic multilayers. As the relative orientation ofmagnetization in the multilayer changes from parallel /H20849P/H20850to
antiparallel /H20849AP /H20850, the polarity of the fringe field varies from
negative to positive with respect to the magnetization in eachmagnetic layer. Consequently, the f
FMR of the patterned
multilayer can be discontinuously increased by switching themagnetic configuration from P to AP, because the positivefringe field in AP enhances the f
FMR.
The FMR properties of an elliptical element consisting
of Py/Cu/Py /H20849Py=Permalloy=Ni 81Fe19/H20850were investigated by
on-chip FMR measurements4–6and also micromagnetic
simulations. The effect of fringe-field coupling on the FMRproperties is quantitatively examined by comparing the fielddependence of f
FMRin the P and AP configurations. In the
case of AP, the relative orientations of magnetization withrespect to the external magnetic field are different in the topand bottom magnetic layers. As a consequence, the top andbottom magnetic layers will exhibit different f
FMRat a non-
zero external field if there is no magnetostatic coupling. It isinteresting how such a conflict in the field dependence of
f
FMR affects the FMR properties of a magnetostatically
coupled trilayer element with AP configuration.
The FMR properties of lithographically patterned Py/
Cu/Py trilayers have been investigated by means of the trans-mission parameter S
21measurements of a microfabricated
coplanar waveguide /H20849CPW /H20850embedded with the trilayer ele-
ments, as schematically shown in Fig. 1/H20849a/H20850. Figure 1/H20849b/H20850
shows a scanning electron micrograph of the CPW embed-ded with Py/Cu/Py elliptical elements with a lateral size of1.2/H110030.3
/H9262m2. The thickness of both the top and bottom Py
layers was 18 nm. The Cu spacer thickness was always9 nm, which is adequately thick to make the orange peel/H20849Néel /H20850coupling between the top and bottom Py layers
negligible.
7
The CPW was fabricated on a glass substrate by lift-off
of an electron-beam evaporated bilayer consisting of Ti/H208495n m /H20850/Au /H2084950 nm /H20850. The lift-off patterns were produced by
means of electron-beam lithography. The strip width and the
strip to ground-plane spacing of the CPW were 1.3 and0.5
/H9262m, respectively. The Py/Cu/Py elements were also fab-
ricated on the strip by means of the lift-off method in com-bination with electron-beam evaporation and electron-beamlithography.
To identify the effect of interlayer fringe-field coupling
on the FMR property of the trilayer element, the influence ofmagnetostatic interactions between the elements must beminimized. In this study, the spacing distance between thetrilayer elements was set at 0.8
/H9262m. It is numerically evalu-
ated that an 18 nm thick Py plate with a lateral size of 1.2/H110030.3
/H9262m2produces a stray field of 1.5 Oe at the body center
of an adjacent element with a spacing distance of 0.8 /H9262m.
The CPW embedded with trilayer elements is connected to avector network analyzer using an on-chip probing systemand is placed into the gap of an electromagnet that can gen-erate a maximum in-plane magnetic field of 1.4 kOe. Thepower level of microwaves injected for the S-parameter mea-
surements was a constant −5 dBm, which produces an acmagnetic field with an amplitude of 6.2 Oe along the shortaxis of the elliptical element. The amplitude of the micro-wave absorption caused by the FMR in the trilayer elementson the strip is expected to be in the order of 10
−3dB. In order
to detect such a small attenuation requires the subtraction of
field-independent background signals, based on themicrowave-transmitting properties of the bare CPW withoutPy/Cu/Py elements. Therefore, in this study, the amplitude ofS
21was normalized to the high-field saturated S21at 1.4 kOe
where the magnetization of the element is completely satu-rated, i.e., a single domain configuration with little end-domain structure is expected.
Figure 2/H20849a/H20850shows a gray scale plot of the amplitude of
S
21measured by sweeping the external magnetic field Hbias
from 1.3 to −1.3 kOe. In this study, the Hbiaswas alwaysa/H20850Electronic mail: nozaki@ed.kyushu-u.ac.jp.APPLIED PHYSICS LETTERS 92, 161903 /H208492008 /H20850
0003-6951/2008/92 /H2084916/H20850/161903/3/$23.00 © 2008 American Institute of Physics 92, 161903-1
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
137.189.170.231 On: Sun, 21 Dec 2014 23:43:15applied along the easy direction of the element, i.e., the x
direction in Fig. 1/H20849b/H20850. The red contrast indicates the strong
absorption of microwaves caused by the FMR in the Py/Cu/Py elements on the CPW. The inset of Fig. 2/H20849a/H20850shows the
peak profile of S
21measured at 240 Oe as a function of mi-
crowave frequency transmitted through the CPW. The peakmicrowave absorption frequency of 8.5 GHz corresponds tothef
FMRof the elements at 240 Oe. The variation of fFMR
during the magnetization reversal process in the Py/Cu/Py
elements is shown in Fig. 2/H20849b/H20850. Figure 2/H20849c/H20850shows the detail
of field variation in fFMR in the field range from 100 to
−360 Oe. The fFMRrapidly increases at −12 and −293 Oe,
indicating the rapid change in the magnetic configurations ofthe trilayer elements. In the case of P configuration, which isexpected for both region I /H20849−12 Oe /H11021H
bias/H110211.3 kOe /H20850and
region III /H20849−1.3 kOe /H11021Hbias/H11021−293 Oe /H20850, as indicated by the
red solid lines in Fig. 2/H20849b/H20850, the magnetic field dependence of
fFMRcan be well fitted with the general formula for fFMR
by accounting for the shape anisotropy of the elliptical
element. It is interesting that the field variation in fFMR
observed in region II is obviously smaller than that for the
P configuration.Figure 3/H20849a/H20850shows the minor hysteresis loop of fFMR. The
sweep direction of the Hbiaswas reversed before reaching the
magnetic field at which the second discontinuous increase of
fFMRappears. Figures 3/H20849b/H20850and3/H20849c/H20850show the FMR spectrum
at the remanent states, swept from 350 Oe /H20851point /H20849A/H20850in Fig.
3/H20849a/H20850/H20852and then back from −200 Oe /H20851point /H20849B/H20850in Fig. 3/H20849a/H20850/H20852,
respectively. The fFMRat the remanent state demagnetized
from region II is 8.1 GHz, which is 1.3 GHz higher than thatfor the remanent state with P magnetization. Furthermore,the peak intensity of microwave absorption is twice as largeas that for the P configuration.
To confirm that these characteristic FMR properties in
region II are attributed to the formation of AP configuration,the temporal variation of magnetization was numericallyevaluated by application of an ac field along the hard axis,using the three-dimensional micromagnetics code
OOMMF
supplied by the National Institute of Standards andTechnology.
8The lateral size of the element and the thick-
ness of Py and Cu layers used for the simulation werematched with the experimental ones. The rounded shape inthe ends of the element, observed in the scanning electronmicrograph /H20851Fig. 1/H20849b/H20850/H20852, was also simulated in the calcula-
tions.
For these simulations, the trilayer element was dis-
cretized into a uniform rectangular grid with a size of 20/H1100320/H110034.5 nm
3. The strength of exchange coupling /H20849A=1.3
/H1100310−11J/m/H20850, saturation magnetization /H20849Ms=1.0 T /H20850, and
crystalline anisotropy /H20849Ku=0/H20850used in the simulations are
typical for Py. The Gilbert damping constant /H9251was set to
0.01, which is consistent with the range of values previouslymeasured in Py films.
9,10Figure 4/H20849a/H20850shows the magnetic
domains of the top and bottom Py layers at the remanentstate calculated for P and AP. The end domain, caused by astrong demagnetizing effect near the ends of the element, isobviously shrunk as the AP configuration is formed, becausethe fringe field from the neighboring Py layer is positivelyapplied to the magnetization. The longitudinal component ofthe normalized magnetization M
x/Msin each magnetic layer
is increased from 0.71 to 0.95 as the magnetic configurationchanges from P to AP. The uniformity of the magnetic do-main configuration in each magnetic layer will enhance theQfactor of the FMR spectra.
FIG. 1. /H20849a/H20850Schematic configuration of the experimental setup and /H20849b/H20850scan-
ning electron micrograph of Py/Cu/Py elliptical elements with lateral size of1.2/H110030.3
/H9262m2, fabricated on a 1.3 /H9262m wide CPW.
FIG. 2. /H20849Color online /H20850/H20849a/H20850Gray-scale plot of S21measured by sweeping the
Hbiasfrom +1.3 to –1.3 kOe. The inset shows the frequency variation in the
transmission coefficient S21, measured at an external magnetic field of
240 Oe. /H20849b/H20850Magnetic-field dependence of the FMR frequency. The region
where the discontinuous increase of fFMRappeared during the magnetization
reversal process is magnified in /H20849c/H20850.
FIG. 3. /H20849Color online /H20850/H20849a/H20850Minor hysteresis loop of the FMR frequency.
Filled and open circles indicate the fFMRmeasured by sweeping the Hbias
from 350 to −200 Oe and back from −200 to 350 Oe, respectively. /H20849b/H20850
FMR spectra measured at the remanent state for decreasing magnetic field/H20849A/H20850, and /H20849c/H20850that for increasing magnetic field /H20849B/H20850.161903-2 Nozaki et al. Appl. Phys. Lett. 92, 161903 /H208492008 /H20850
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
137.189.170.231 On: Sun, 21 Dec 2014 23:43:15Figure 4/H20849b/H20850shows the amplitude of the magnetization
oscillations as a function of the frequency of the ac hard-axisfield calculated for P and AP. The amplitude of the ac mag-netic field is the same as the experimental condition/H208496.2 Oe /H20850. Magnetization oscillation along the in-plane hard
axis is maximized at 7.7 and 9.3 GHz for P and AP, respec-
tively. The increase of f
FMR, caused by changing the mag-
netic configuration from P to AP, is approximately consistentwith the experimentally observed increase of f
FMR at the
remanent state for region II. However, in the cases of both Pand AP, the calculated f
FMRis approximately 1 GHz higher
than the experimental values. This discrepancy is probablyattributed to the suppression of effective shape anisotropy ofthe element due to surface oxidation. In the experiment, therewas no capping layer applied on the surface of the trilayerelement, so that a nonmagnetic dead layer may be formednear the surface of the top Py layer due to oxidation. Theasymmetry of the switching field from AP to P in the majorand minor hysteresis loops of f
FMRsuggests that there is a
discrepancy in the coercive fields for the top and bottom Pylayers. It is believed that this result supports the surface oxi-dation of the top Py layer, which results in the suppression ofshape anisotropy of the element. It was numerically evalu-ated that the f
FMRfor P and AP are decreased to 7.4 and
8.8 GHz, respectively, assuming the magnetic dead layerwith a thickness of 4.5 nm /H20849not shown /H20850.
The peak amplitude of magnetization oscillation for AP
is 3.6 times larger than that for P. The spatial distributions ofthe oscillating amplitude at the peak frequencies for P andAP configurations are shown in the inset of Fig. 4/H20849b/H20850. The
blue and red areas show large precessional motions of mag-netization, although they are out of phase with a phase dif-ference of 180°. The white regions have no magnetic activ-ity. In both cases, ripple domains are formed along thelongitudinal direction of the element. Comparing the spatialdistributions of magnetization response between P and AP, aninhomogeneous magnetization distribution is emphasized for
the P configuration, which has large end domains. Further-
more, out of phase precessions appear in the vicinity of thepattern ends. These are attributable to the distributed effec-tive fields in the trilayer element with the P configuration.Suppression of the resonant peak amplitude for P is causedby the inhomogeneous spatial distribution of the magnetiza-tion response.
Figure 4/H20849c/H20850shows the variation of f
FMRwith the increase
in the Hbiascalculated for the P and AP configurations. The
increasing ratios of fFMRat 100 Oe are 16.4% and 0.52%,
with respect to the fFMRat the remanent states for P and AP
configurations, respectively. These values are approximatelyconsistent with the experimental values /H2084912.6% for P and
2.7% for AP /H20850. The small variation in f
FMRwith the magnetic
field observed for AP is associated with the competitive fielddependence of f
FMRin the top and bottom Py layers which
are magnetostatically coupled with each other. From thesenumerical results for the AP configuration, i.e., the higherFMR frequency at the remanent state and a field-insensitive
f
FMRfeature, it can be concluded that the AP configuration is
realized at region II in Fig. 2/H20849c/H20850
In summary, the FMR properties of Py/Cu/Py elliptical
element were examined by using the on-chip FMR measure-ments. The FMR frequency at the remanent state can bediscontinuously changed by switching between P to AP con-figurations of magnetization. The difference between theFMR frequencies measured for the P and AP configurationsis approximately consistent with the shift in FMR frequencydue to the fringe field coupling, as evaluated from the micro-magnetic simulation using
OOMMF . The magnetic field varia-
tion in the FMR frequency is suppressed as the magneticconfiguration changes from P to AP. The field insensitivity inthe FMR frequency for the AP configuration is attributable tothe conflict in the field dependence of the FMR frequenciesfor the top and bottom Py layers.
This study was supported by the Industrial Technology
Research Grant Program in 2005 from the New Energy andIndustrial Technology Development Organization /H20849NEDO /H20850
of Japan.
1C. S. Tsai, J. Su, and C. C. Lee, IEEE Trans. Magn. 35, 3178 /H208491999 /H20850.
2N. Cramer, D. Lucic, R. E. Camley, and Z. Celinski, J. Appl. Phys. 87,
6911 /H208492000 /H20850.
3B. Kuanr, Z. Celinski, and R. E. Camley, Appl. Phys. Lett. 83, 3969
/H208492003 /H20850.
4G. Counil, J. Kim, T. Devolder, C. Chappert, K. Shigeto, and Y. Otani, J.
Appl. Phys. 95, 5646 /H208492004 /H20850.
5F. Giesen, J. Podbielski, T. Korn, M. Steiner, A. van Staa, and F. Grundler,
Appl. Phys. Lett. 86, 112510 /H208492005 /H20850.
6Y. Nozaki, K. Tateishi, S. Taharazako, M. Ohta, S. Yoshimura, and K.
Matsuyama, Appl. Phys. Lett. 91, 122505 /H208492007 /H20850.
7B. D. Schrag, A. Anguelouch, S. Ingvarsson, G. Xiao, Y. Lu, P. L. Trouil-
loud, A. Gupta, R. A. Wanner, W. J. Gallagher, P. M. Rice, and S. S. P.Parkin, Appl. Phys. Lett. 77, 2373 /H208492000 /H20850.
8See http://math.nist.gov/oommf/.
9W. K. Hiebert, A. Stankiewicz, and M. R. Freeman, Phys. Rev. Lett. 79,
1134 /H208491997 /H20850.
10M. Covington, T. M. Crawford, and G. J. Parker, Phys. Rev. Lett. 89,
237202 /H208492002 /H20850.
FIG. 4. /H20849Color online /H20850/H20849a/H20850Magnetic domain configurations of the top and
bottom Py layers at the remanent state calculated for P and AP configura-tions. /H20849b/H20850Frequency dependence of the amplitude of magnetization oscilla-
tion. Spatial distributions of oscillating amplitude of magnetization at reso-nance frequencies are also depicted. /H20849c/H20850Comparison of the magnetic field
variations in f
FMRcalculated for P and AP configurations.161903-3 Nozaki et al. Appl. Phys. Lett. 92, 161903 /H208492008 /H20850
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
137.189.170.231 On: Sun, 21 Dec 2014 23:43:15 |
1.5093371.pdf | APL Mater. 7, 081114 (2019); https://doi.org/10.1063/1.5093371 7, 081114
© 2019 Author(s).Route to form skyrmions in soft magnetic
films
Cite as: APL Mater. 7, 081114 (2019); https://doi.org/10.1063/1.5093371
Submitted: 20 February 2019 . Accepted: 28 July 2019 . Published Online: 16 August 2019
D. Navas
, R. V. Verba
, A. Hierro-Rodriguez
, S. A. Bunyaev
, X. Zhou , A. O. Adeyeye
, O. V.
Dobrovolskiy
, B. A. Ivanov
, K. Y. Guslienko
, and G. N. Kakazei
ARTICLES YOU MAY BE INTERESTED IN
Perspective: Magnetic skyrmions—Overview of recent progress in an active research field
Journal of Applied Physics 124, 240901 (2018); https://doi.org/10.1063/1.5048972APL Materials ARTICLE scitation.org/journal/apm
Route to form skyrmions in soft magnetic films
Cite as: APL Mater. 7, 081114 (2019); doi: 10.1063/1.5093371
Submitted: 20 February 2019 •Accepted: 28 July 2019 •
Published Online: 16 August 2019
D. Navas,1
R. V. Verba,2
A. Hierro-Rodriguez,1,3
S. A. Bunyaev,1
X. Zhou,4A. O. Adeyeye,4
O. V. Dobrovolskiy,5,6
B. A. Ivanov,2,7
K. Y. Guslienko,8,9
and G. N. Kakazei1,a)
AFFILIATIONS
1Institute of Physics for Advanced Materials, Nanotechnology and Photonics (IFIMUP)/Departamento de Física e Astronomia,
Universidade do Porto, 4169-007 Porto, Portugal
2Institute of Magnetism, National Academy of Sciences of Ukraine, 03142 Kyiv, Ukraine
3School of Physics and Astronomy, University of Glasgow, G12 8QQ Glasgow, United Kingdom
4Department of Electrical and Computer Engineering, National University of Singapore, 117583, Singapore
5Physikalisches Institut, Goethe University, 60438 Frankfurt am Main, Germany
6Department of Physics, V. Karazin National University, 61077 Kharkiv, Ukraine
7National University of Science and Technology, “MISiS”, Moscow 119049, Russian Federation
8Depto. Física de Materiales, Universidad del País Vasco, UPV/EHU, 20018 San Sebastián, Spain
9IKERBASQUE, The Basque Foundation for Science, 48013 Bilbao, Spain
a)Author to whom correspondence should be addressed: gleb.kakazei@fc.up.pt
ABSTRACT
Magnetic skyrmions which are topologically nontrivial magnetization configurations have attracted much attention recently due to their
potential applications in information recording and signal processing. Conventionally, magnetic skyrmions are stabilized by chiral bulk
or interfacial Dzyaloshinskii-Moriya interaction (DMI) in noncentrosymmetric B20 bulk crystals (at low temperatures) or ultrathin mag-
netic films with out-of-plane magnetic anisotropy (at room temperature), respectively. The skyrmion stability in the ultrathin films relies
on a delicate balance of their material parameters that are hard to control experimentally. Here, we propose an alternate approach to
stabilize a skyrmion in ferromagnetic media by modifying its surroundings in order to create strong dipolar fields of the radial sym-
metry. We demonstrate that artificial magnetic skyrmions can be stabilized even in a simple media such as a continuous soft ferro-
magnetic film, provided that it is coupled to a hard magnetic antidot matrix by exchange and dipolar interactions, without any DMI.
Néel skyrmions, either isolated or arranged in a 2D array with a high packing density, can be stabilized using antidot as small as
40 nm in diameter for soft magnetic films made of Permalloy. When the antidot diameter is increased, the skyrmion configuration
transforms into a curled one, becoming an intermediate between the Néel and Bloch skyrmions. In addition to skyrmions, the con-
sidered nanostructure supports the formation of nontopological magnetic solitons that may be regarded as skyrmions with a reversed
core.
©2019 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license
(http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5093371 .,s
I. INTRODUCTION
The development of new data storage technologies that com-
bine ultrahigh areal density and low power consumption is one of
the highest priorities of modern nanoscience. Recently, topologically
stabilized magnetization configurations—magnetic skyrmions—
were proposed as candidates for the implementation of the next gen-
eration of memory and logic devices.1Magnetic skyrmions whichare two-dimensional spin textures with sizes ranging from a few
to several hundred nanometers have been the focus of interest of
researchers during the last decade. Hexagonal skyrmion lattices
were discovered in noncentrosymmetric crystals with B20 struc-
ture2–4and in ultrathin Fe/Ir(111) films,5,6at low temperatures.
Then, the concept of individual skyrmion stabilization in ultrathin
films at room temperature by interface induced Dzyaloshinskii-
Moriya interaction (DMI) was suggested by Fert et al. ,7,8resulting in
APL Mater. 7, 081114 (2019); doi: 10.1063/1.5093371 7, 081114-1
© Author(s) 2019APL Materials ARTICLE scitation.org/journal/apm
observation of such skyrmions in ultrathin multilayer films and dots
of Co/Pt, Ir/Co/Pt, etc.9–13Both kinds of chiral skyrmions are stabi-
lized due to the presence of the relativistic Dzyaloshinskii-Moriya
interaction, bulk DMI14,15in the former and interface DMI16in
the latter case. The main requirements for application of nanoscale
skyrmions in information processing and spintronic devices are
skyrmion stability at room temperature and without an external
magnetic field. The lattices of B20 skyrmions are stable only at low
temperatures and high magnetic fields, which makes their use in any
device applications difficult.
It was found recently that magnetic skyrmions can be driven
by a spin-transfer torque mechanism at a very low current den-
sity.17This enables devices with much smaller power consumption
and faster processing. The effect has been demonstrated at low tem-
peratures in both chiral bulk magnetic structures18–20and ultrathin
films.21,22
Recently, the controllable creation, annihilation, and manip-
ulation of interfacial-DMI induced nanoscale skyrmions at room
temperature were experimentally demonstrated.9–13,23However, the
room temperature skyrmion stability in the ultrathin films relies
on a delicate balance of the exchange interaction, uniaxial mag-
netic anisotropy, and DMI. These parameters are poorly controlled
for layer thicknesses of and below 1 nm. Thus, skyrmions in such
films are typically metastable, i.e., they can be easily destroyed by
thermal agitation or weak external fields.24Alternatively, it was
demonstrated that Bloch skyrmions can be stabilized in the absence
of DMI either in artificial crystals formed by the combination of
nanodot arrays and perpendicularly magnetized films25or in sub-
micron dots with moderate perpendicular magnetic anisotropy.26,27
Also, skyrmion lattices at room temperature in the absence of exter-
nal magnetic field, stabilized by a competition between the intrin-
sic exchange, magnetocrystalline anisotropy, and dipolar interac-
tion, were experimentally observed in a multiferroic Ni 2MnGa
single crystals with inversion symmetry.28However, relatively thick
(at least few nanometers) films of soft magnetic materials were
never considered as a medium for skyrmion formation. Their main
practical advantage is much lower magnetization damping in com-
parison with mentioned ultrathin multilayers (e.g., for Ni 80Fe20,
the Gilbert damping constant α≤0.01) that allows us to consider
microwave devices on their basis. As expected, the stabilization
of skyrmions in such objects requires principally new underlying
mechanisms.
In this work, by means of analytical theory and micromagnetic
simulations, we demonstrate the route to obtain stable magnetic
skyrmions and their dense arrays in soft ferromagnetic continuous
films without DMI. Our main idea is to create skyrmions by means
of a strong stray dipolar field generated by a patterned hard mag-
netic layer near the soft magnetic film in a hybrid bilayer structure.
Recently, we have used a similar approach to overcome the lim-
its of vortex formation in soft ferromagnetic nanodots by dipolarly
coupling them to the antidot matrix.29Here, we show that the com-
peting exchange and dipolar interactions in the patterned nanos-
tructures can lead to the stabilization of magnetic topological soliton
states, including chiral Néel and curled skyrmions, as well as their
nontopological counterparts.
The nanostructure under consideration is shown in Fig. 1.
It consists of a continuous soft ferromagnetic film of thickness
tSL, which is in contact with a patterned hard magnetic layer of
FIG. 1 . (a) Schematic of the considered nanostructure, consisting of a soft mag-
netic layer underneath of a hard magnetic antidot matrix with perpendicular mag-
netization. (b) Cross section showing the distribution of the stray magnetic field
created by the hard layer.
thickness tHL, having a circular hole (an antidot) with diameter d.
The hard layer (HL) possesses sufficient perpendicular magnetic
anisotropy for the magnetization to be in the saturated out-of-
plane state at zero external field. Since the soft and hard mag-
netic layers are in direct contact, a nonzero interlayer exchange
interaction exists between them. The antidot can be isolated,
or a two-dimensional antidot lattice can be formed in the film
plane.
II. ANALYTICAL THEORY
A. Model and magnetic energy of the nanostructure
To understand which magnetization configurations can be sta-
bilized in the studied nanostructure, we consider the magnetic
energy of the system as a functional of its magnetic state.30We
assume that the magnetization in the hard layer is uniform and is
directed perpendicularly to the film plane, MHL=pM HLez, where
p=±1 corresponds to the magnetization direction “up” or “down,”
MHLis the saturation magnetization of the hard layer, and ezis the
out-of-plane unit vector. This assumption is valid if the perpendic-
ular magnetic anisotropy of the hard layer is sufficiently strong (see
supplementary material, SM #2 for details). In addition, we assume
that the thickness of the soft layer is of the order of or smaller than
the material exchange length so that the magnetization distribu-
tion of the soft layer can be considered uniform along the thickness
z-coordinate. In the following, we consider the case of an isolated
antidot.
The magnetic energy of the soft layer can be written as
W[MSL(ρ)]=∫wd2ρ, where ρis the two-dimensional radius-
vector in the soft layer plane and wis the energy density. The
latter is comprised by the nonuniform exchange ( wex), Zeeman
(wZ), dipolar ( wdip), and interlayer exchange ( wIL) contributions.
For derivation of the explicit expressions for magnetic energy den-
sity, we describe the magnetization of the soft layer in terms
of the spherical azimuthal and polar angles θandφso that
MSL=MSL(sinθcosφ, sinθsinφ, cosθ). Then, the nonuniform
APL Mater. 7, 081114 (2019); doi: 10.1063/1.5093371 7, 081114-2
© Author(s) 2019APL Materials ARTICLE scitation.org/journal/apm
exchange energy is
wex=tSLASL((∇θ)2+(∇φ)2sin2θ), (1)
where ASLis the exchange stiffness of the soft layer. The energy of
the interlayer exchange is equal to
wIL=−pJcosθΘ(ρ/R−1), (2)
where Θ(x) is the Heaviside step function31andJis the interlayer
exchange strength.
The term wZdescribes the energy of the magnetization in an
external magnetic field. In the considered case, the magnetic field
consists of two contributions: the applied field, which is assumed
to be perpendicular to the film plane ( Be=Be,zez), and the stray
fieldBadproduced by the hard layer with an antidot. The stray field
is radially symmetric and has two components, perpendicular to
the film plane component Bad,zand a radial component Bad,ρ(see
supplementary material, SM #1 for details). The first one is approxi-
mately constant in the soft film region under the antidot and changes
its sign near the antidot border [see Fig. 2(a)]. The radial compo-
nent increases with the distance from the antidot center and peaks
at its border before decaying toward zero. Using the in-plane polar
coordinate system ( ρ,χ), the Zeeman energy density is expressed
as
wZ=−tSLMSL(Bzcosθ+Bρsinθcos(φ−χ)). (3)
The dipolar energy is produced by nonzero magnetostatic
charges, which have two contributions—surface charges M⋅ezand
volume charges ( ∇⋅M). In thin films, the first contribution can be
approximated as the energy density,
wdip,s=tSLμ0M2
SL
2cos2θ, (4)
FIG. 2 . (a) Normalized profile of the stray field, created by the hard magnetic layer
with an antidot, in the soft magnetic one (micromagnetic calculations). The antidot
aspect ratio tHL/R= 0.75. (b) Schematic magnetization profiles of topological and
nontopological solitons expected in the nanostructure.i.e., it describes the in-plane shape anisotropy of a thin ferromag-
netic film. For calculation of the volume contribution, we assume
that the solution we are searching for has a radial symmetry, θ=θ(ρ)
andφ(ρ,χ)=χ+ψ(ρ). This is a natural assumption because of the
radial symmetry of the nanostructure. Then, the density of volume
charges∇⋅M=MSLρ−1∂ρ[ρsinθ(ρ)cosψ(ρ)]depends only on the
radial coordinate ρ.This means that the corresponding dipolar field
has only a radial component, Bdip,ρ(ρ)=μ0∫Gρρ(ρ,ρ′)Mρ(ρ′)d2ρ′,
where Gρρis theρρ-component of the tensor magnetostatic Green’s
function, defined in supplementary material, SM #1, averaged over
the film thickness. In the case φ=χ±π/2, the volume magnetic
charges and corresponding demagnetizing field are identically equal
to zero. The density of the volume contribution to the dipolar energy
density is equal to
wdip,v=−tSLMSL
2sinθcos(φ−χ)Bdip,ρ. (5)
B. Soliton structure
Stable and metastable magnetization distributions of the soft
layer correspond to minima of the energy W[MSL(ρ)]. Away from
the antidot, the magnetization distribution is determined by com-
peting dipolar and interlayer exchange interactions. If the interlayer
exchange energy ( J) is strong enough, at least J>tSLμ0M2
SL(see
details in supplementary material, SM #2), the soft layer has per-
pendicular magnetization. It is easy to satisfy this inequality for high
quality soft/hard interfaces.
Magnetization distributions that minimize the energy
W[MSL(ρ)] are the solutions of the corresponding Euler-Lagrange
equation. The equation for the azimuthal magnetization angle φ(ρ)
has the following form:
λ2
SL∇(sin2θ∇φ)=1
μ0MSL(Bρ+Bdip,ρ[θ,φ])sinθsin(φ−χ), (6)
whereλSL=√
2ASL/μ0M2
SLis the exchange length of the soft layer
material (λSL≈5 nm for Permalloy). It is clear that Eq. (6) has
exact solutions φ=χorφ=χ+π, which correspond to the radial
direction of the in-plane magnetization component, as is in Néel
skyrmion configurations. Simultaneously, it follows from Eq. (6)
that the function φ=χ±π/2 is not a solution, although in this
case Bdip,ρ[θ,φ] = 0. Thus, we cannot expect the formation of
magnetic solitons with an exact Bloch-like structure, for example,
Bloch skyrmions. Instead, Eq. (6) has a solution in the form φ=χ
+ψ(ρ), which describes a magnetization configuration with a
complex curling in-plane component. The Néel-like magnetization
configuration is favored by the Zeeman term (the magnetization
direction is parallel to the radial stray field), while the dipolar
term promotes the formation of curled magnetization distribution
(to minimize the volume “magnetic charges”). Therefore, one can
expect a transition from the Néel-like solitons to the curled solitons
when the role of the dipolar energy increases in comparison with the
Zeeman energy.
Next, we consider the properties of the function θ(ρ) in the case
of Néel-like solitons with φ=χ, which can be realized for the hard
layer polarization p= +1 (so that the in-plane magnetization compo-
nent is parallel to the radial stray field). Then, the magnetic energy
density is reduced to
APL Mater. 7, 081114 (2019); doi: 10.1063/1.5093371 7, 081114-3
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w=tSLASL⎛
⎝(dθ
dρ)2
+(1
ρ2−1
λ2
SL)sin2θ⎞
⎠
−tSLMSL(Bzcosθ+Bρsinθ)+wdip,v+wIL, (7)
where Bρ>0. Equation (7) determines the properties of the func-
tionθ(ρ), namely,θ(∞)→0, and sinθcosθ→Cρ,dθ/dρ→Catρ
→0 (see supplementary material, SM #3 for details). These condi-
tions are the same as for two-dimensional solitons in easy-axis fer-
romagnetic films.32Accounting that the soliton topological charge
is proportional to cos θ(0)−cosθ(∞),30the solitons can be either
topological, when θ=π−Cρatρ→0, so that the magnetization
direction in the core and that far from the core at ρ→∞are oppo-
site, or nontopological, with θ=Cρatρ→0. In the case when the
external magnetic field almost compensates the z-component of the
antidot stray field, the second term in Eq. (7) stimulates the forma-
tion of an in-plane magnetization, θ=π/2 atρ>λSL. Moreover,
the radial component of the stray field also promotes the magne-
tization to lie in the soft layer plane and even leads to a decrease in
the soliton core size (see supplementary material, SM #4). Thus, one
can expect the following structure of magnetic solitons: a core with
out-of-plane magnetization in the center of the antidot followed by
an in-plane magnetized part and the transition to the magnetiza-
tion direction θ= 0 near the antidot edge, as depicted schemati-
cally in Fig. 2(b). It should be noted that in the considered case of
the compensated out-of-plane stray field, the energy of a soliton
does not depend on the core polarization. Thus, both the topo-
logical soliton (skyrmion) and the nontopological soliton can exist
simultaneously.
III. MICROMAGNETIC SIMULATIONS
In order to validate the analytical calculations and provide a
deeper insight into possible magnetization distributions in the stud-
ied hybrid nanostructure (Fig. 1), we performed a set of micro-
magnetic simulations, using the MuMax3 micromagnetic simula-
tion code.33We used Ni 80Fe20(Permalloy, saturation magnetization
MSL= 8.1×105A/m, exchange stiffness ASL= 1.05×10−11J/m)
for the soft ferromagnetic layer, which has low damping ( α≤0.01)
and, therefore, good microwave properties. For the material of hard
layer (HL) with antidots, we have explored different materials with
saturation magnetization varied from MHL= 4×105A/m (Co/Pd
multilayers) to MHL= 1×106A/m (Fe/Pt multilayers). Our simula-
tions showed qualitatively the same results within the studied range
of the magnetic and geometrical parameters. However, higher values
of the saturation magnetization of the hard layer allow for the for-
mation of magnetic soliton configurations at smaller matrix thick-
nesses tHLand for smaller antidot diameters ddue to the stronger
stray magnetic field generated by the antidot matrix. For this reason,
here we focused our attention on the study of the magnetic behav-
ior of patterned nanostructures with a hard layer of large saturation
magnetization: we used the material parameters of Fe/Pt multilayers
with MHL= 1000 kA/m, exchange stiffness AHL= 2×10−11J/m,
and perpendicular anisotropy constant Ku= 1×106J/m3.34The
thickness of the Permalloy layer was fixed at 3 nm, while the matrix
thickness, antidot diameter, and distance between them were sys-
tematically varied. The cell size was fixed to (2 ×2×1) nm3. We also
performed simulations with different cell sizes along the z-direction
to verify that there are no simulation artifacts in the calculations ofthe exchange coupling between the layers. The interlayer exchange
coupling between the layers was introduced as the volume exchange
interaction with the exchange stiffness being the mean value of stiff-
ness of soft and hard layers, AIL= (ASL+AHL)/2. This approach is
often used in micromagnetic simulations and was confirmed exper-
imentally, i.e., for a CoPd/NiFe multilayer system.35Formally, it
is equivalent to the interlayer coupling with the constant J= (ASL
+AHL)/(aSL+aHL)≈70 mJ/m2, where ais the lattice constant of
the corresponding layer. The effects of interantidot interactions were
studied by the application of the periodic boundary condition with
the unit cell 2 ×2 and 4×4 antidots.
In the simulations, a strong perpendicular external magnetic
field of 1.2 T was applied first in order to completely saturate the
hybrid nanostructure. After saturation, the applied magnetic field
was gradually reduced to zero, and magnetization configuration was
found at each step of the field decrease by the energy minimization
starting from previous one (damping constants were set αSL=αHL
= 0.1 to speed up simulations). A sufficiently large perpendicular
anisotropy in the hard layer prevents matrix reversal in zero and
moderate negative fields.
First, we consider the case of an isolated antidot. Different mag-
netization configurations of the soft layer, observed at remanence
for a fixed thickness of the hard layer tHL= 20 nm and different
antidot diameters, are shown in Fig. 3. In the case of very small
antidots, d≤30 nm, the remanent state is a quasi-single-domain
(SL) with an almost completely in-plane magnetization in the region
below the antidot [Fig. 3(a)]. When the diameter is increased, we
observe different soliton magnetization configurations. In the range
d= 40–60 nm, the remanent state is a Néel skyrmion [Fig. 3(b)].
Ford= 75–150 nm, we observe the formation of a nontopologi-
cal counterpart of the Néel skyrmion, which has the same direction
of magnetization in the core and away from the antidot, separated
by a region with in-plane magnetization pointing in a radial direc-
tion [Fig. 3(c)]. Finally, in perfect agreement with analytical predic-
tions, the in-plane part of the magnetization distribution becomes
curled with a further increase in antidot diameter and the soliton
becomes an intermediate state between the Néel and Bloch soli-
tons [Fig. 3(e)]. It is clear that the polar angle of magnetization
varies with the distance from the soliton core. This is related to
the spatial dependence of the radial component of the stray field
generated by the antidot matrix. Note that the magnetization con-
figuration of the soft layer is determined by the polarity of the hard
layer, chosen to be p= +1. If it is reversed, then the magnetization
of the soft layer is reversed too. In particular, the Néel skyrmion
becomes of inward structure, instead of the outward one shown
in Fig. 3.
Note that in the case of zero external magnetic field, the per-
pendicular component Bzof the stray field is not compensated. This
removes the energy degeneracy of the skyrmion and its nontopo-
logical counterpart and results in an increase in the skyrmion core
size and a decrease in the core size of the nontopological solitons.
It happens because in the former case the core direction is paral-
lel to the total field, while in the latter it is antiparallel [compare
Figs. 3(c) and 3(d)]. In addition, the uncompensated perpendic-
ular stray field leads to a weak tilt of the “in-plane” part of the
magnetic soliton from perfect in-plane direction. Of course, when
the stray field is compensated by an external field, this tilt dis-
appears and the solitons acquire a structure shown schematically
APL Mater. 7, 081114 (2019); doi: 10.1063/1.5093371 7, 081114-4
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FIG. 3 . Different remanent states of the soft layer in zero bias field: top—in-plane view, bottom— x-zcentral cross section. (a) Quasi-single-domain state (antidot diameter d
= 30 nm). [(b) and (d)] Néel skyrmion ( d= 50 nm and 75 nm, respectively). (c) Néel nontopological soliton ( d= 75 nm). (e) Curled nontopological soliton ( d= 200 nm). (f)
Curled skyrmion ( d= 200 nm). The soft film thickness is 3 nm, and the hard layer thickness is 20 nm. Note different scales for (a)–(d) and (e) and (f) correspondently.
in Fig. 2(b) with degenerated skyrmion and nontopological soliton
configurations.
The observation of skyrmions or nontopological solitons at
remanence is directly related with the strength of the out-of-plane
stray field of the antidot matrix. This effect is clearly seen in the
simulated minor hysteresis loops, shown in Fig. 4. These loops were
obtained by the application of an external magnetic field, which was
varied from +1.2 T to −0.5 T and then back to +1.2 T, thus avoid-
ing the magnetization reversal of the matrix. The hysteresis loops
are asymmetric with respect to the external field. This asymmetry
is more pronounced for smaller antidot diameters and, naturally,
for thicker hard layers. Therefore, for smaller values of the anti-
dot diameter (e.g., for d= 50 nm), both core reversals, from up to
down and back, occur at positive external fields. This results in theexistence of only one stable configuration at remanence, with the
core polarity opposite to the matrix magnetization, i.e., skyrmion
configuration [Fig. 4(a)]. On the contrary, for larger antidot diam-
eters (e.g., for d= 100 nm), the core reversal from up to down occurs
at a relatively small negative field [Fig. 4(b)]. Therefore, there are
two stable magnetic states at remanence [Figs. 3(c) and 3(d) and
Figs. 3(e) and 3(f)]. While the nontopological soliton is naturally
formed when the perpendicular field is reduced to zero from positive
saturation, it can be transformed into the skyrmion state by applying
a small negative field. However, to transform the skyrmion back into
the nontopological soliton, it is necessary to apply a large positive
field.
Our results are summarized in Fig. 5 in the form of a phase
diagram of the different magnetization configurations of the soft
FIG. 4 . Simulated minor hysteresis loop of nanostructure
under a perpendicular external field (without reversal of the
hard layer). For better vertical resolution, only the magneti-
zation of the soft layer in the region twice larger in diameter
than antidot is accounted. tHL= 20 nm, d= 50 nm (a) and d
= 100 nm (b).
APL Mater. 7, 081114 (2019); doi: 10.1063/1.5093371 7, 081114-5
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FIG. 5 . Phase diagram of remanent states of the soft magnetic film after perpendic-
ular saturation. SD—quasi-single-domain state, Sk—skyrmion. The shaded area
corresponds to the region where both the skyrmion and nontopological soliton
states are stable at zero bias field. Outside this region, the nontopological soliton
state is unstable.
magnetic layer at remanence. The quasi-single-domain (SD) state is
observed in the range of small antidots and thin hard layers. The
soliton states appear at remanence above a certain critical antidot
diameter for a given matrix thickness. Naturally, this critical diame-
ter becomes smaller when the thickness of the hard layer increases
since a thicker matrix creates a stronger stray field. In our sim-
ulations, we observed the Néel skyrmions in antidots as small as
40 nm in diameter. It should be emphasized that this size is much
smaller than the characteristic sizes for which other topologically
nontrivial configurations—magnetic vortices—are observed in iso-
lated magnetic nanodots of the same diameter (see, e.g., Ref. 36).
Thus, our simulations confirm the crucial role of the stray field
generated by the antidot matrix in decreasing the soliton core size
and stabilizing nontrivial magnetization configurations in smaller
objects. It is clear from Fig. 5 that the Néel skyrmion is stable in
the range of smaller antidot diameters and thicker hard layers. In
the opposite range, the stray field from the hard matrix is smaller
and the role of the demagnetizing energy of the soft layer increases,
promoting the formation of curled skyrmions. The diagram also
shows the region of bistability, where both skyrmion and nontopo-
logical soliton states with opposite core polarities are stable at zero
field. This bistability at remanence requires larger antidots when
the thickness of the hard layer increases because the perpendicular
stray field from the matrix increases with the ratio tHL/dincreas-
ing. Such a situation is impossible in ferromagnetic materials with
bulk or interfacial Dzyaloshinskii-Moriya interaction, in which only
skyrmions can be stabilized, and an attempt to reverse the skyrmion
core leads to the disappearance of nontrivial magnetization
configurations.
In all the simulated data presented above, the thickness of the
soft layer was fixed to tSL= 3 nm. We have performed more sim-
ulations where tSLwas varied. These simulations showed almost
no change in the phase diagram of the remanent magnetization
states (Fig. 5). This insensitivity is related to a combination of thedominant role of the exchange interaction, the stray field from the
hard layer, and dipolar contributions from surface magnetic charges
in the determination of the magnetization distributions in the soft
layer. All these contributions have the same dependence on the soft
layer thickness [see Eqs. (2), (3), and (5)]. Only the term correspond-
ing to the volume magnetic charges has a different dependence on
the soft layer thickness [Eq. (6)]. Its effect is rather weak, especially
in the determination of the critical size of the antidot supporting
skyrmion formation (see supplementary material, SM #4). The only
requirement on the soft layer thickness is that it should be smaller
than a critical value (see calculations in supplementary material,
SM #2). Above this critical value, the exchange interaction between
hard and soft layers becomes insufficient to stabilize out-of-plane
magnetization in the second one at positions away from the anti-
dot. In this case, the inhomogeneous magnetization configurations
exist in the soft magnetic layer, forming a kind of exchange spring.37
For the studied hybrid magnetic system, it happens for Permalloy
thicknesses above 5 nm.
Finally, the influence of the antidot periodicity on the soliton
configuration was studied by varying the distance between antidots
arranged into square arrays. For this purpose, we fixed tHL= 30 nm
and d= 40 nm, which is the smallest antidot diameter and hard
layer thickness combination allowing the existence of the skyrmions,
and vary the antidot periodicity from almost the isolated case to a
very closely packed array of 60 nm periodicity. Figure 6 shows the
evolution of magnetic configuration at the remanent state for differ-
ent antidot periods. The Néel skyrmion for the antidot arrays with
period≥100 nm and the Néel nontopological soliton for the period
in the range 70–90 nm can be seen in Figs. 6(a) and 6(b), respectively.
Finally, at periods of 60 nm and below, the in-plane quasi-single-
domain state is formed, as seen in Fig. 6(c). The decrease in antidot
lattice period affects the soft magnetic layer configuration stability
in a way similar as does the decrease in the hard layer thickness.
Both changes lead to the reduction of the stray field density inside
the antidot. However, in a broad region of the antidot lattice peri-
ods, only the polarity of the soliton is affected by a decrease in the
perpendicular stray field. At very high packing densities, the radial
stray field becomes weak enough to support the formation of mag-
netic soliton states. For antidot lattices of other geometries (e.g.,
honeycomb), we expect similar results since the main impact is pro-
duced by a reduction of stray fields and not by interaction (either
exchange or dipolar) between neighbor solitons. Indeed, contribu-
tion from the exchange interaction is negligible since the nearest
solitons are separated by uniformly magnetized regions with the
size much larger than the exchange length. Also, the dipolar inter-
action between soliton cores decreases rapidly with the intercore
distance and for the 3 nm thick core became insignificant at few
tens of nanometers.38–40Therefore, the only difference between var-
ious lattice geometries will be in the critical lattice constant at which
the skyrmion state loses its stability. The mentioned weakness of
intersoliton interaction leads to the possibility to switch the mag-
netic configuration of individual elements in a dense square array
(period 70–90 nm) from the Néel nontopological soliton to the
Néel skyrmion by applying locally the relatively small negative per-
pendicular field. This property allows us to consider the proposed
system for applications in information storage as recording media
and in magnonics as reconfigurable two-dimensional magnonic
crystals.41
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FIG. 6 . Remanent magnetization states
of the soft magnetic film, coupled to a
30 nm thick antidot matrix with antidots
of diameter 40 nm and different periods
of the square antidot lattice: (a) 100 nm
antidot lattice period (Néel skyrmion
state), (b) 80 nm period (Néel nontopo-
logical state), (c) 60 nm period (quasi-
single-domain state).
IV. CONCLUSIONS
We have demonstrated a novel method to achieve mag-
netic skyrmion configurations in soft ferromagnetic films with-
out Dzyaloshinskii-Moriya interaction. Our approach is based on
using a hybrid nanostructure, where a soft magnetic layer is cou-
pled by dipolar and interlayer exchange interactions to an out-
of-plane magnetized hard layer having an antidot or antidot lat-
tice. In this nanostructure, interlayer exchange is responsible for
the soft layer magnetization direction away from the antidot
(out-of-plane direction). Simultaneously, the radial component of
the stray magnetic field generated by the antidot plays a crucial
role in the formation of inhomogeneous magnetization configura-
tions in the soft magnetic layer near the antidot—the radial Néel
skyrmions.
We showed by means of micromagnetic simulations that the
proposed patterned nanostructure allows for the realization of topo-
logically nontrivial magnetization configurations for antidots as
small as 40 nm in diameter (for a Permalloy soft magnetic layer).
Depending on the material and geometric parameters, it is possi-
ble to achieve the formation of stable Néel solitons (skyrmions or
their nontopological counterparts) at remanence, or curled solitons
with a complex magnetization distribution, resembling an interme-
diate state between the Néel and Bloch skyrmions. The formation
of the curled solitons is a result of the competing demagnetizing
and Zeeman energy contributions to the stray field created by the
antidot matrix. The curled skyrmions are realized in the case of rel-
atively thin hard layers and large antidot diameters, while smaller
antidots and thicker hard layers support the formation of Néel
skyrmions.
The proposed nanostructure also allows for the formation of
a two-dimensional skyrmion lattice with a high packing density.
When the separation between antidots becomes smaller than their
sizes, the skyrmion configuration in the soft layer is suppressed
in favor of a quasi-single-domain state. Our findings open a wayfor investigation of the magnetic skyrmions in soft magnetic mate-
rials with good high-frequency properties due to low magnetiza-
tion damping and their possible applications in microwave and
information storage devices.
SUPPLEMENTARY MATERIAL
See supplementary material for different aspects of analytical
theory that are described in detail: calculation of the stray field from
a hard magnetic layer with an antidot (SM #1), condition on the
interlayer exchange (SM #2), properties of the function θ(ρ) deter-
mining the magnetization profile of the soft magnetic layer (SM #3),
and estimation of the soliton core radius (SM #4).
ACKNOWLEDGMENTS
The Portuguese team acknowledges the Network of Extreme
Conditions Laboratories-NECL and Portuguese Foundation of Sci-
ence and Technology (FCT) support through Project Nos. NORTE-
01-0145-FEDER-022096, MIT-EXPL/IRA/0012/2017, POCI-0145-
FEDER-030085 (NOVAMAG), PTDC/FIS-MAC/31302/2017, EXPL
/IF/00541/2015 (S.A.B.), and Grant No. SFRH/BPD/90471/2012
(A.H.-R.). Work at IMag was supported by the Ministry of
Education and Science of Ukraine (Project No. 0118U004007).
K.Y.G. acknowledges support from IKERBASQUE (the Basque
Foundation for Science) and the Spanish MINECO, Grant No.
FIS2016-78591-C3-3-R. R.V.V. B.A.I., K.Y.G., and O.V.D. acknowl-
edge the support from the European Union Horizon 2020 Research
and Innovation Programme under Marie Sklodowska-Curie, Grant
Agreement No. 644348. A.H.-R. acknowledges the support from
Spanish MINECO under Project Ref. No. FIS2016-76058-C4-4-R
and from European Union’s Horizon 2020 research and innova-
tion program under the Marie Skłodowska-Curie Action Reference
No. H2020-MSCA-IF-2016-746958. B.A.I. was supported by the
Program of NUST “MISiS” (Grant No. K2-2017-005), implemented
APL Mater. 7, 081114 (2019); doi: 10.1063/1.5093371 7, 081114-7
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by a governmental decree dated 16th of March 2013, No. 211. G.N.K.
and O.V.D. acknowledge the support from European Cooperation in
Science and Technology (COST), Project No. CA16218 “NANOCO-
HYBRI”. A.O.A. was supported by the Ministry of Education, Sin-
gapore, under Research Project No. R-263-000-C61-112. A.O.A. is a
member of Singapore Spintronics Consortium (SG-SPIN).
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© Author(s) 2019 |
1.4928727.pdf | Static property and current-driven precession of 2π−vortex in nano-disk with
Dzyaloshinskii-Moriya interaction
Xianyin Liu , Qiyuan Zhu , Senfu Zhang , Qingfang Liu , and Jianbo Wang
Citation: AIP Advances 5, 087137 (2015);
View online: https://doi.org/10.1063/1.4928727
View Table of Contents: http://aip.scitation.org/toc/adv/5/8
Published by the American Institute of Physics
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Static property and current-driven precession of 2 π-vortex
in nano-disk with Dzyaloshinskii-Moriya interaction
Xianyin Liu,1Qiyuan Zhu,1Senfu Zhang,1Qingfang Liu,1,a
and Jianbo Wang1,2,a
1Key Laboratory for Magnetism and Magnetic Materials of Ministry of Education, Lanzhou
University, Lanzhou, 730000, People’s Republic of China
2Key Laboratory for Special Function Materials and Structure Design, Ministry of Education,
Lanzhou University, Lanzhou, 730000, People’s Republic of China
(Received 15 June 2015; accepted 5 August 2015; published online 13 August 2015)
An interesting type of skyrmion-like spin texture, 2 π-vortex, is obtained in a thin
nano-disk with Dzyaloshinskii-Moriya interaction. We have simulated the existence
of 2π-vortex by micromagnetic method. Furthermore, the spin polarized current is
introduced in order to drive the motion of 2 π-vortex in a nano-disk with diameter
2R=140 nm. When the current density matches with the current injection area,
2π-vortex soon reaches a stable precession (3 ∼4 ns). The relationship between the
precession frequency of 2 π-vortex and the current density is almost linear. It may
have potential use in spin torque nano-oscillators. C2015 Author(s). All article con-
tent, except where otherwise noted, is licensed under a Creative Commons Attribution
3.0 Unported License. [http: //dx.doi.org /10.1063 /1.4928727]
Dzyaloshinskii-Moriya interaction (DMI)1–3is an additional term in the exchange interaction.
It is firstly found in helical magnets with B20 structure, such as MnSi,4FeCo5and FeGe,6and
also occurs at the interface in the pseudomorphic Fe monolayer on Ir (111):7according to the
material system, DMI is divided into two categories.8–11For magnetic system, the introduction of
DMI can lead to many curious phenomena. The influences of DMI on both domain walls9,12and
skyrmions have caused extensive interest. Skyrmion induced by DMI has aroused wide concern in
experiment13and micromagnetic simulations.14–16Skyrmion can be move at a relatively low current
density, which can be used as the track memory.17,18A interesting type of skyrmion-like spin texture
can occur in a thin nano-disk with DMI, and it can be considered as one skyrmion in the center
of anti-skyrmion (skyrmion-anti-skyrmion pair)19or one skyrmion core with a circle spin stripe.20
Fig. 1(a) shows a typical distribution of the magnetization for skyrmion-like spin texture. Along
the radial direction of the nano-disk, the rotation of magnetization from the center to the boundary
is 2π, and it is called 2 π-vortex.21The magnetization distribution for 2 π-vortex is di fferent from
that of skyrmion or ferromagnetic /anti-ferromagnetic state. 2 π-vortex is first found in the ultrathin
nano-disk with DMI by using variation calculus,21,22and it in ferromagnetic thin films has been
generated with the help of an ultrashort laser pulse.19However, there are few publications focusing
on the behaviors of 2 π-vortex driven by current.
In this article, we have simulated the existence of 2 π-vortex by energy minimization methods.
Furthermore, the spin polarized current is introduced in order to drive the motion of 2 π-vortex in a
2R=140 nm nano-disk. And the relationship between the precession frequency of 2 π-vortex and
the current density has been studied. The preliminary results show that 2 π-vortex can oscillate in the
case of current driving. When the current density matches with the current injection area, 2 π-vortex
reaches the stable precession. The range of the precession frequency is wide (2.0 to 3.3 GHz for
2R=140 nm) under low current density control without magnetic field. When the current is intro-
duced, the precession of 2 π-vortex will soon reach stable (3 ∼4 ns), which is much faster than the
vortex and skyrmion.23Therefore, it may have potential use in spin torque nano-oscillators.
aE-mail address: liuqf@lzu.edu.cn (Qingfang Liu) wangjb@lzu.edu.cn (Jianbo Wang)
2158-3226/2015/5(8)/087137/6 5, 087137-1 ©Author(s) 2015
087137-2 Liu et al. AIP Advances 5, 087137 (2015)
FIG. 1. (a) A typical distribution of the magnetization for the 2 π-vortex. (b) Schematic diagram of the magnetic ellipse
nanopillar. (c) Schematic diagram of current-injection device and the red area represents where current is injected.
A spin transfer torque is considered as shown in Fig 1(b): a polarizer layer, a nonmagnetic
spacer and a free layer. The free layer is located in xy-plane. The polarizer layer is fully fixed along
thexaxis. The free layer is chosen as a disk, and the diameter of the disk 2Rvaries from 60 to
180 nm with increment of 20 nm. The thickness of the free layer is 0.4 nm and the mesh size is
0.5×0.5×0.4 nm3. When the oscillation of 2 π-vortex driven by a spin polarized current is studied,
the schematic diagram of current injection device is shown in Fig. 1(c). And the red area represents
where current is injected. The diameter 2 ris 40 to 100 nm with increment of 10 nm. The micromag-
netic parameters are chosen as:15the saturation magnetization Ms=580 kA /m; the exchange sti ff-
ness constant A=15 pJ /m and the perpendicular magneto crystalline anisotropy K=0.8 MJ/m3.
The behavior of 2 π-vortex with spin polarized current is described by the Landau-Lifshitz-Gilbert
(LLG) equation with the spin transfer e ffect:
dm
dt=−|γ|m×Heff+α(
m×dm
dt)
+|γ|βε m×mp×m−|γ|βϵ′m×mp. (1)
Where mis the unit vector of the local magnetization, mpis a unit vector parallels the magnetization
direction of the polarizer layer. Heffis the e ffective field. ϵ′is the secondary spin transfer term.
β= }
µ0e J
tMs,ε=PΛ2
(Λ2+1)+(Λ2−1)(m·mp), where Pis the spin polarization rate, tis the film thickness,
and the parameter Λis the mismatch between spacer and ferromagnetic resistance. For the magne-
tization dynamics, the damping factor α=0.03. We used the spin polarization rate P=0.4, the
parameterΛ=2 and the secondary spin transfer term ϵ′=0.1. The Oersted field induced by current
is not considered in simulations. The e ffective field of the system is consisting of the exchange field,
the anisotropy field, the demagnetization field and the DMI e ffective field. The DMI e ffective field is
described by:
⃗HDMI=2D
µ0Ms[(⃗∇·⃗m)
ˆ z−⃗∇mz] (2)
The summation is performed on the neighbor pairs. The DMI parameter Dvaries from 1 to
6 mJ/m2. The simulation was carried out using the Object Oriented Micromagnetic Framework
(OOMMF)24including DMI module.22
In the case of ultrathin nano-disk, the total micromagnetic energy density is taken by:
εtot=A*
,(∂m
∂x)2
+(∂m
∂y)2
+
-−Kum2
z+εd+εDMI (3)087137-3 Liu et al. AIP Advances 5, 087137 (2015)
Where Kuis the e ffective anisotropy constant, which is defined as: Ku=K−2πM2
s;εdis the
demagnetization energy density, εDMIis the energy density functional of DMI. And it is taken by:
εDMI=tD(
mx∂mz
∂x−mz∂mx
∂x)
+(
my∂mz
∂y−mz∂my
∂y)
(4)
Where tis the film thickness. Integrating the energy density in the whole film and using variation
calculus, the minimum energy can be 2 π-vortex.
Fig. 2(a) shows the minimum energy state in nano-disks with di fferent DMI parameter D
and diameter 2R. When D≤2 mJ/m2, the minimum energy state is ferromagnetic state. When
D=3 mJ/m2, the minimum energy state is skyrmion. In this case, the size of nano-disks has no
effect on the steady state. For D=4 mJ/m2, the minimum energy state is skyrmion for the diam-
eter2R=60 nm. When the diameter 2R≥80nm, the minimum energy state is 2 π-vortex. When
FIG. 2. (a) Relaxed states in nano-disk with di fferent Dand2R. (b)The corresponding skyrmion number destiny of the
ferromagnetic state, skyrmion and 2 π-vortex. (c) The diameters 2Rs1and2Rs2as functions of 2R.087137-4 Liu et al. AIP Advances 5, 087137 (2015)
D=5 mJ/m2, the minimum energy state is 2 π-vortex for the diameter 2R≤160 nm. However, the
minimum energy state is more complex state (3 π-vortex or others) for the diameter 2R=180 nm.
ForD=6 mJ/m2, the minimum energy state is 2 π-vortex for the diameter 2R≤100 nm. When the
diameter 2R≥120 nm, the minimum energy state is more complex state. Anyhow, when the Dand
2Ris suitable, 2 π-vortex may occur in real nano-disk even without the influence of the magnetic
field or others.
In order to understand the di fference among the ferromagnetic /anti-ferromagnetic state, skyrmion
and 2π-vortex, the skyrmion number Sis defined as equation25S=1
4π
m·(∂m
∂x×∂m
∂y)dxdy. Fig 2(b)
shows the skyrmion number destiny of the ferromagnetic state, skyrmion and 2 π-vortex, respectively.
For a ferromagnetic structure, S=0. For a skyrmion, S=±1. For 2 π-vortex, the skyrmion number
S=S1+S2, where S1is the skyrmion number of the skyrmion core and S2is the anti-skyrmion. Be-
cause the unit vectors of the two skyrmion are opposite, the skyrmion number of 2 π-vortex S=0.
Although, the skyrmion number of 2 π-vortex is same as that of the ferromagnetic state, the skyrmion
number destiny is di fferent from that of the ferromagnetic state. Therefore, 2 π-vortex shows di fferent
behavior as compared with skyrmion or ferromagnetic /anti-ferromagnetic state, and may display
some new feature.
The size of 2 π-vortex varies with the diameter of nano-disk. Fig 2(c) shows the diameter of
the skyrmion core 2Rs1and the diameter of the circle spin stripe 2Rs2as functions of 2Rwith DMI
parameter D=4 mJ/m2. As 2Rincreases from 80 to 180 nm, the diameter of the skyrmion core
2Rs1increases from 10 to 63 nm, and the diameter of the circle spin stripe 2Rs1increases from 40
to 130 nm. Therefore, the size of nano-disk 2 π-vortex in a 2-dimensional film of the chiral magnets
is induced by DMI and the circular boundary. When the diameter of nano-disk 2R=140 nm, the
diameter of the skyrmion core 2Rs1=35 nm and the diameter of the skyrmion core 2Rs2=91 nm,
which is chosen to study the dynamics of 2 π-vortex.
We studied the current-induced oscillate of 2 π-vortex in a 2R=140 nm nano-disk with DMI
parameter D=4 mJ/m2. In most cases, the current injection area 2ris chosen to 40 – 80 nm or
100 nm, and 2 π-vortex would be damped oscillate as shown in Fig 3(a) ( 2r=60 nm). However,
2π-vortex reaches the stable precession, when the current injection area 2r=90 nm. Fig 3(b) shows
thatmxandmztakes on a disordered state at the first 3 ns, and then reach a stable oscillation state.
FIG. 3. The m xand m zcharacteristics with time at 2r=60 nm (a) and 2r=90 nm (b). (c) Snapshots of some significant
magnetization configurations. (d) The trajectory of the skyrmion core center. the inset is the position as a function of time.087137-5 Liu et al. AIP Advances 5, 087137 (2015)
Snapshots of some significant magnetization configurations are shown in Fig 3(c). After introducing
a spin-polarized current, the position of the skyrmion core shifts and the proportion of the skyrmion
decreases to a certain value in 3 ns. Simultaneously, the circle spin stripe becomes irregular, and the
proportion of spin stripe is gradually reduced. After 3 ns, 2 π-vortex reaches a stable precession. The
precession is a complex oscillator, including the skyrmion core position precession, the skyrmion
core size precession and the spin stripe. These precessions have the same frequency, so our major
investigations have focus on the precession of the core position. In the stable precession, the trajec-
tory of the skyrmion core center is restricted in a small ellipse area. The equilibrium position of
oscillation is not at the geometric center of the nano-disk, and it is about 12 nm from o ff-center. The
trajectory of the skyrmion core center is shown in Fig 3(d), and the inset of Fig 3(d) is the position
as a function of time. The value of 2Rs1and2Rs2are fluctuant, which are smaller than those of stable
states. The spin stripe is an irregular ring with a small bulge at the edge, whose position has the
same periodic precession as the skyrmion core.
Fig. 4(a) shows the resonant frequencies of two cases with and without considering the first
3 ns. By comparing the two figures, the two peaks, which are marked by black arrows, are from
the first 3 ns unstable oscillations; and the other three peaks, which are marked by red arrows, are
from the stable oscillation. In this study, we focus on the first precession frequency of 2 π-vortex.
Fig 4(b) shows the first precession frequency of 2 π-vortex as a function of the current density J.
AsJincreases from 3 ×1010to 6.75×1010A/m2, the first precession frequency increases from 2.0
to 3.3 GHz. The relationship between the first precession frequency and current density is almost
linear. When the current injection area 2 r=90 nm for D=4 mJ/m2, the greater the current density,
the faster the speeds of both the skyrmion core and the spin stripe. And the precession frequency
of 2π-vortex increases with the speed of the skyrmion core and the spin stripe. The current density
FIG. 4. (a) The FFT results with and without first 3ns. (b) The first precession frequency as a function of the current density.087137-6 Liu et al. AIP Advances 5, 087137 (2015)
between 3×1010and 6.75×1010A/m2is continuously controlled, and therefore the first precession
frequency of 2 π-vortex between 2.0 and 3.3 GHz is also continuously controlled. When the current
density J<3×1010A/m2, 2π-vortex would be damped oscillation. The current-supplied energy is
completely used to o ffset the damping; therefore, 2 π-vortex thus finally reaches a steady state. For
the current density J>6.75×1010A/m2, the precession becomes complicated, and the precession
frequency has a plurality of values. The current-supplied energy not only o ffsets damping, but also
makes skyrmion severe deformation, which is the cause of the complex resonance frequency.
When the diameter of the nano-disk changes, the matching of the current injection area will
change, such as: For 2 R=160 nm, the 2 r=110 nm; for 2 R=120 nm, the 2 r=80∼100 nm; but
2R=100 nm, 2 π-vortex cannot reach the stable precession. When the current density is constant,
the first precession frequency of 2 π-vortex decreases with the diameter of nano-disk increasing.
We have calculated the existence conditions of 2 π-vortex in the nano-disk. When the nano-disk
Dand2Ris suitable, 2 π-vortex may occur even without the influence of the magnetic field or
others. The diameter of the skyrmion core 2Rs1and the diameter of the circle spin stripe 2Rs2
increases with that of the nano-disk. Then, the oscillation of 2 π-vortex driven by a spin polarized
current is studied by micromagnetic simulations. For 2R=140 nm nano-disk, we selected the
introduction of current region of diameter 2r=90 nm. In this case, 2 π-vortex can stabilize the
precession in 3 ∼4 ns. The precession frequency of 2 π-vortex can be controlled by current density
from 2.0 to 3.3 GHz. Thus, it may have a potential use in spin torque nano-oscillators.
ACKNOWLEDGMENTS
This work is supported by the National Basic Research Program of China (2012CB933101),
the NSF of China (Grant Nos. 51171075 and 51371092), and PCSIRT (Grant No. IRT1251).
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1.5137837.pdf | Appl. Phys. Lett. 116, 072403 (2020); https://doi.org/10.1063/1.5137837 116, 072403
© 2020 Author(s).Reduced spin torque nano-oscillator
linewidth using irradiation
Cite as: Appl. Phys. Lett. 116, 072403 (2020); https://doi.org/10.1063/1.5137837
Submitted: 14 November 2019 . Accepted: 29 January 2020 . Published Online: 19 February 2020
Sheng Jiang
, Roman Khymyn
, Sunjae Chung
, Tuan Quang Le
, Liza Herrera Diez
, Afshin
Houshang
, Mohammad Zahedinejad
, Dafiné Ravelosona
, and Johan Åkerman
COLLECTIONS
This paper was selected as Featured
Reduced spin torque nano-oscillator linewidth
using Heþirradiation
Cite as: Appl. Phys. Lett. 116, 072403 (2020); doi: 10.1063/1.5137837
Submitted: 14 November 2019 .Accepted: 29 January 2020 .
Published Online: 19 February 2020
Sheng Jiang,1,2
Roman Khymyn,2
Sunjae Chung,1,3
Tuan Quang Le,1,2
Liza Herrera Diez,4
Afshin Houshang,2,5
Mohammad Zahedinejad,2
Dafin /C19eRavelosona,4,6
and Johan A˚kerman1,2,5,a)
AFFILIATIONS
1Department of Applied Physics, School of Engineering Sciences, KTH Royal Institute of Technology, Electrum 229, SE-16440 Kista,
Sweden
2Department of Physics, University of Gothenburg, 412 96, Gothenburg, Sweden
3Department of Physics Education, Korea National University of Education, Cheongju 28173, Korea
4Centre de Nanosciences et de Nanotechnologies, CNRS, Universit /C19e Paris Saclay, 91120 Palaiseau, France
5NanOsc AB, Kista 164 40, Sweden
6Spin-Ion Technologies, 10 boulevard Thomas Gobert, 91120 Palaiseau, France
a)Author to whom correspondence should be addressed: johan.akerman@physics.gu.se
ABSTRACT
We demonstrate an approach for improving the spectral linewidth of a spin torque nano-oscillator (STNO). Using Heþion irradiation, we
tune the perpendicular magnetic anisotropy (PMA) of the STNO free layer such that its easy axis is gradually varied from strongly
out-of-plane to moderate in-plane. As the PMA impacts the non-linearity Nof the STNO, we can, in this way, control the threshold current,
the current tunability of the frequency, and, in particular, the STNO linewidth, which dramatically improves by two orders of magnitude.Our results are in good agreement with the theory for nonlinear auto-oscillators, confirm theoretical predictions of the role of the nonlinearity,
and demonstrate a straightforward path toward improving the microwave properties of STNOs.
Published under license by AIP Publishing. https://doi.org/10.1063/1.5137837
Spin-torque nano-oscillators (STNOs) are among the most
promising candidates for nanoscale broadband microwave genera-tors
1–6and detectors.7–9STNOs can generate broadband microwave
frequencies ranging from hundreds of MHz to subterahertz,10–13con-
trolled by both magnetic fields and dc currents.5,14Moreover, the
device size can be reduced to a few tens of nanometers, which is ofgreat opportunity for industrial applications. They can also host arange of magnetodynamical spin wave modes, such as propagatingspin waves of different orders,
15,16and magnetodynamical solitons,
such as spin wave bullets15and droplets.3
However, the applicability of these devices has suffered from their
low power emission and large linewidth. Nonlinear auto-oscillatortheory
17–20explains the large linewidth as a consequence of the strong
nonlinearity N, i.e., the dependence of the microwave frequency on its
precession amplitude. Ncan be controlled not only by the magnitude
and direction of the magnetic field14,21–25but also by the magnetic
properties of the free layer of the STNO, such as the magnetic anisot-ropy and the effective magnetization.
20For instance, in an easy-plane
free layer, Nchanges gradually from positive to negative values as thedirection of the magnetic field rotates from out-of-plane to in-
plane.17,20A few experimental studies have corroborated22,26that the
linewidth can be minimized when Napproaches zero at the critical
angle of magnetic field. This shows a way to improve the linewidth byselectively reducing N.
Whereas all previous studies aimed at minimizing Nfocused
only on varying the direction and magnitude of the magnetic field ona single device, this minimal Ncan only be achieved in a narrow
range of conditions, limited generating frequency, and will require acomplicated design for applications as microwave generators. In ourwork, we therefore study systematically how Nis affected by the
strength of perpendicular magnetic anisotropy (PMA; H
k) in a set of
nanocontact (NC) STNOs with a [Co/Pd]/Cu/[Co/Ni] spin valvestructure. To engineer the PMA, we utilize He
þirradiation to modify
the interfaces of the [Co/Ni] multilayer where the PMA originatesfrom and is sensitive to.
27–30We show how Ncan be continuously
tuned as Hkis controlled by Heþirradiation fluence in otherwise
identical devices. Most importantly, the linewidth is dramaticallyimproved at moderate H
kvalues, where N! 0. Finally, we show
Appl. Phys. Lett. 116, 072403 (2020); doi: 10.1063/1.5137837 116, 072403-1
Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplexcellent agreement of our experimental results with nonlinear auto-
oscillator theory.20
The STNO devices were fabricated from all-perpendicular (all-
PMA) [Co/Pd]/Cu/[Co/Ni]31,32and orthogonal [Co/Pd]/Cu/NiFe
spin valves (SVs). The full stack consists of a Ta (5)/Cu (15)/Ta (5)/Pd(3) seed layer and an all-PMA [Co (0.5)/Pd (1.0)] /C25/Cu (7)/[Co (0.3)/
Ni (0.9)] /C24/Co (0.3) or orthogonal [Co (0.5)/Pd (1.0)] /C25/Cu (7)/
Ni
80Fe20(4.5) SV with a Cu(3)/Pd(3) capping layer (here, NiFe served
as a reference with negligible PMA), sputtered onto a thermally oxi-dized 100 mm Si wafer (numbers in parentheses are layer thicknessesin nanometers). The deposited stacks were first patterned into8lm/C220lm mesas using photolithography and ion-milling etching,
followed by chemical vapor deposition (CVD) of an insulating 40-nm-
thick SiO
2film. Electron beam lithography and reactive ion etching
were used to open NCs (with a nominal radius of RNC35 nm) through
SiO 2in the center of each mesa. The processed wafer was then cut into
different pieces for Heþirradiation with the fluence Fvarying from 6
to 20/C21014Heþ/cm2. This was done by using a commercial Spin-Ion
technologies Heþbeam facility with high uniform ion irradiation on
centimeter scale. Fabrication was completed using lift-off lithographyand deposition of a Cu (500 nm)/Au (100 nm) top electrode in a singlerun with all irradiated pieces. Our protocol hence ensures that all otherproperties, except the He
þfluence, are identical from device to device.
We used our custom-built probe station for static and microwavecharacterization. The details can be found in Refs. 33and34.
To accurately determine the effective magnetization, l
0Meff
¼l0Ms/C0l0Hkof the [Co/Ni] free layers, spin-torque ferromagnetic
resonance (ST-FMR)35–39measurements were performed on the Heþ-
irradiated STNOs ( supplementary material ). The fluence information
and the obtained l0Meffare presented in Table I . The value of Meff
(Hk) increases (decreases) as the fluence increases.32Here, the NiFe
free layer is used as a reference for a larger Meffsample.
InFig. 1 , we compare the calculated FMR frequency, fFMR,u s i n g
the measured Meff, with the microwave signals generated from the
STNO devices. The inset in Fig. 1 shows a typical power spectral
density (PSD) for F¼10/C21014Heþ/cm2. All PSD spectra are well
fitted with a Lorentzian function, and the extracted frequency fvs
magnetic field is presented in Fig. 1 with different symbols for each
d i f f e r e n tfl u e n c e .W en o t et h a tt h e r ea r ed o u b l ep e a k sa ta r o u n d0 . 9Tfields for F¼6/C210
14Heþ/cm2. This double peak phenomenon
is infrequent and occurs randomly between devices. It is likely anapparent effect of mode hopping between two closely spaced auto-oscillation modes. The sputtered [Co/Ni] films are polycrystalline, andit is possible that grain boundaries underneath the nanocontact canimpact the auto-oscillation modes from device to device and lead to
slightly different local modes. All data show a quasi-linear dependenceon the magnetic field, and fextends to lower values as M
eff(Hk)
increases (decreases). This behavior is consistent with the calculated
value of the FMR frequency fFMR, plotted as dashed lines in Fig. 1 .T h e
overall trends of fFMRare in good agreement with the auto-oscillation
f. The difference between the calculated fFMRand the measured auto-
oscillation fis a direct measure of the nonlinearity of the magnetiza-
tion precession,5,15,24,40which is discussed in detail below.
We now turn to the current-induced frequency tunability.
Figures 2(a)–2(e) show fvs dc current Idcatl0H¼0:72 T; fdepends
linearly on Idcat all values of Meff. The current-induced frequency
tunability df=dIdcis extracted from the slopes of the linear fits, plot-
ted as each dashed line in Figs. 2(a)–2(e) .df=dIdcvsMeffis then
plotted in Fig. 2(f) . We found that (i) df=dIdcdecreases from
0.50 GHz/mA for nonirradiated [Co/Ni] to /C00.13 GHz/mA for
NiFe as Meffincreases (or Hkdecreases) and (ii) the sign of df=dIdc
changes from positive (for [Co/Ni]) to negative (for NiFe), consis-
tent with the easy axis transition from out-of-plane for [Co/Ni] toin-plane for NiFe.
We carried out detailed measurements at different magnetic
fields to further understand the behavior of df=dI
dc.Figure 3(a) shows
one example of extracted fvsIdcat different fields, ranging from 0.37
to 1.12 T in steps of 0.05 T, for F¼6/C21014Heþ/cm2. All data show
clear linear dependencies on Idc. Here, we would like to define one
numerical relation about the tunability, df=df¼Ithðdf=dIdcÞ,t oc o m -
pare our experimental results directly with theory, where f¼Idc=Ithis
the dimensionless supercriticality parameter20andIthis the threshold
current. Ithw a se x t r a c t e df r o mp l o t so fi n v e r s ep o w e r1 =PvsIdc
(supplementary material ). After having obtained all Ithand df=dIdc
for different Meffvalues ;df=dfis shown as solid dots in Fig. 3(b) .
df=dffor different Meffvalues shows a decreasing behavior with the
increasing magnetic field, a trend that becomes weaker for the highest
irradiation and for NiFe. The overall df=dfdecreases as Meff(Hk)TABLE I. Sample structure information and the calculated effective magnetization
l0Meffof the free layer ([Co/Ni] or NiFe) for various Heþ-irradiation fluences. l0Meff
are measured by ST-FMR ( supplementary material ).
Structure Fluence ( /C21014Heþ/cm2) l0Meff(T)
[Co/Pd]/Cu/[Co/Ni] 0 /C00.68
[Co/Pd]/Cu/[Co/Ni] 6 /C00.44
[Co/Pd]/Cu/[Co/Ni] 10 /C00.14
[Co/Pd]/Cu/[Co/Ni] 20 0.03[Co/Pd]/Cu/NiFe … 0.98
FIG. 1. Auto-oscillation frequency vs in-plane magnetic field for various irradiated
STNOs with RNC¼35 nm. The dashed lines are the calculated FMR frequencies
fFMR, based on the values of l0Meffobtained from ST-FMR measurements. Inset: A
typical power spectral density (PSD) of an STNO with F¼10/C21014Heþ/cm2at
Idc¼/C0 14 mA.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 116, 072403 (2020); doi: 10.1063/1.5137837 116, 072403-2
Published under license by AIP Publishingincreases (decreases). It is very close to zero when l0Meff/C250f o r
F¼20/C21014Heþ/cm2.T h es i g no f df=dffor NiFe is even negative.
To understand the behavior of tunability vs Meff(Hk) from Heþ-
irradiated STNOs, we considered the nonlinear auto-oscillator theoryof Slavin and Tiberkevich,
19Slavin and Tiberkevich,20Slavin and
Tiberkevich,40and Slavin and Kabos,41which was derived for univer-
sal auto-oscillation systems and has proven to be consistent with theLandau–Lifshitz–Gilbert–Slonczewski (LLGS) equation.
20This theory
allows us to describe the experimental observation analytically. The
auto-oscillation frequency fgenerated from an STNO is expressed as
fðIdcÞ¼fFMRþN
2pP;P¼jcj2¼f/C01
fþQ; (1)
where Nis the nonlinearity factor, Pandcare the normalized power
and amplitude of the stationary precession, and Qis the nonlinear
damping coefficient. From Eq. (1), the frequency shift is mainly
decided by the nonlinearity N. Taking the derivation of Eq. (1),df=df
is derived as
df
df¼Ithdf
dIdc¼N
2p1þQ
fþQðÞ2: (2)
The nonlinear frequency shift coefficient Nfor STNOs domi-
nates the frequency tunability and may be positive, zero, or negative,
depending on the magnetic field direction and magnetic anisotropy oft h ef r e el a y e ri nS T N O s .
To explain the experimental observations using this analytical
theory, we derive Nwith our experimental conditions. The nonlinear-
ity is expressed as
40N¼/C0xHxMxHþxM=4 ðÞ
x0xHþxM=2 ðÞ; (3)
xH¼cH;
xM¼4pcMeff;
x0¼cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
xHxHþxM ðÞp
:8
>><
>>:(4)
We note that Eqs. (3)and(4)are valid for the magnetization of
the free layer being aligned to the magnetic field direction. Utilizing
Eqs.(3)and(4),w ec a l c u l a t e df=df(/N ), where fand Q are used as
fitting parameters for all data in Fig. 3(b) , and we find reasonable good
agreements with 1.5 for fand 3.0 for Q, respectively. All the calculated
results are shown as the solid lines alongside the experimental results
inFig. 3(b) . It should be noted that the theoretical calculation coin-
cides with experimental results in the overall trend although there are
discrepancies between experiment and theory. One reason for these
discrepancies is that the theory does not take into account the current-induced Joule heating and Oersted fields that are present in the experi-
ments. Besides, the calculated nonlinearity Ncan also explain the
frequency difference between the calculated f
FMRand the generated
microwave frequency finFig. 1 .D u et ot h en e g a t i v ev a l u eo f N(or
df=df) for NiFe, fis expected to be lower than fFMR, as predicted in
Eq.(1)and consistent with our experimental observations in Fig. 1 .
This auto-oscillation mode is often characterized as a localized
bullet.14,15,40In contrast, Nis positive for easy out-of-plane [Co/Ni],
and so f>fFMRinFig. 1 .14,31,40In this case, the auto-oscillating modeFIG. 3. (a) Extracted auto-oscillation frequency fvsIdcat different magnetic fields
forF¼6/C21014Heþ/cm2. Some minor frequency jumps at l0H¼0:87 T are
shown as rectangular boxes, possibly due to film inhomogeneities generating differ-ent dynamical behaviors. (b) df=df[i.e., I
thðdf=dIdcÞ] vs magnetic field, where Ithis
extracted from the intercept of the inverse power of the auto-oscillation signals and
df=dIdcare the slopes of the linear fits of frequency as Idc>Ith. The solid lines are
the theoretical calculation from Eqs. (2)–(4) .
FIG. 2. (a)–(e) PSD vs Idcin STNOs with different irradiated fluences at
l0H¼0:72 T. The red dotted line represents the linear fits of the auto-oscillation
frequency. Dfstands for the minimal linewidth. (f) slope df=dIdcvsl0Meffextracted
from the fits of (a)–(e).Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 116, 072403 (2020); doi: 10.1063/1.5137837 116, 072403-3
Published under license by AIP Publishingis a propagating spin-wave.14,42,43All these experimental observations
confirm the theoretical predictions.
Furthermore, according to nonlinear auto-oscillator theory, the
linewidth Dfcan be expressed as20
Df¼CþkBT
EðPÞ1þN
Ceff/C18/C192"#
; (5)
where kBis the Boltzmann constant and Tis the temperature. Cþand
E(P) are the damping function and time-averaged oscillation energy as
a function of the power P,r e s p e c t i v e l y . Ceffis the effective damping. In
Eq.(5),Dfexhibits a quadratic dependence on the nonlinearity N.T o
compare with our experimental results, we extracted Dffrom the data
inFigs. 2(a)–2(e) ,a ss h o w ni n Fig. 4 .Figure 4(a) presents representa-
tive spectra, exhibiting that Dfdecreases with fluences. The overall Df
inFig. 4(b) was indeed dramatically improved by about two orders of
magnitude as Ndecreases (as Meffincreases), and it reaches a lowest
value for l0Meff¼0:03 T, where N! 0.Dfagain increases for the
NiFe free layer when Nbecomes moderately negative.
There are also contributions due to the variation of other param-
eters, such as Cþ;Ceff,a n d E(P). The values CþandCeffare propor-
tional to the damping a,44which only changes by about a factor of 2 at
F¼20/C21014Heþ/cm2as compared to the non-irradiated sample.32
The oscillation energy E(P) can be assumed20,44asEðPÞ/Vsinh,
where Vis the volume of the auto-oscillating mode and his the preces-
sional angle. This precessional angle/amplitude depends on the linearand nonlinear magnetic damping. According to theory [Eq. (24) inRef.20], the stationary amplitude can be expressed as a function of
supercriticality fand nonlinear damping Q,a si ti sw r i t t e ni nE q . (1).
We can fit the data in Fig. 3(b) with only one set of parameters
(f¼1:5,Q¼3), which indicates that the amplitude does not change
significantly with He
þirradiation. As for the volume V,i td e p e n d so n
the mode profile. We performed micromagnetic simulations for the
two limiting cases of high and low Meff(supplementary material ). The
profiles of the auto-oscillatory modes in both cases are spin wavespropagating out of the NC area. We observe only a minor difference
between the two cases, which consists of a slightly more ellipticalprofile for the lower M
eff. Therefore, the effective volume of the modes
does not significantly change with Heþirradiation and, thus, does not
significantly affect the linewidth of the auto-oscillations. We hence
conclude that the hundredfold improvement of the linewidth is mainly
due to the changes in non-linearity.
The excellent agreement between our experimental results and
theory confirms that the linewidth can be minimized intentionally bycontrolling the nonlinearity in general and, in particular, tuning it tozero. When the PMA compensates the demagnetization field, the non-linearity identically equals zero regardless of the external conditions.We can therefore minimize Dfby choosing free layer materials with
l
0Meff!0. We emphasize that our study hence offers a universal
path to solving one of the key issues in utilizing STNOs as microwavegenerators. As for the generated microwave power—another keydrawback of this type of microwave generator—we did not observe animprovement in this study, mainly due to the slight degradation inmagnetoresistance (MR) values.
32We expect that the power can
instead be dramatically improved using magnetic tunnel junction-
based STNOs, where MR can be over two orders of magnitude greater
than that in spin valve-based STNOs.2,16,45
In conclusion, we present a systematic study of the variation of
nonlinearity against PMA in STNOs. By using Heþirradiation to con-
tinuously tune the PMA of the [Co/Ni] free layer, the nonlinearity N
(along with the frequency tunability df=dIdc) shows a continuous
decreasing trend as Hk(Meff) decreases (increases). As a consequence
of this decreasing nonlinearity, we achieve an approximately hundred-fold improvement in the linewidth. Our experimental observations arein excellent agreement with nonlinear auto-oscillator theory. This sys-tematic study not only verifies the theoretical prediction but also offersa route to improving the linewidth—selecting a free layer’s materialwhose M
eff!0, which is of great importance for commercializing
microwave generators.
See the supplementary material for ST-FMR measurements to
extract the effective magnetization l0Meff, current-induced tunability
df=dIdcand threshold current Ith, and micromagnetic simulations.
This work was supported by the Swedish Foundation for
Strategic Research (SSF), the Swedish Research Council (VR), andthe Knut and Alice Wallenberg Foundation (KAW). Additionalsupport for this work was provided by the European ResearchCouncil (ERC) under the European Community’s Seventh
Framework Programme (No. FP/2007-2013)/ERC Grant No.
307144 “MUSTANG.”
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Published under license by AIP Publishing |
5.0005764.pdf | Appl. Phys. Lett. 116, 172403 (2020); https://doi.org/10.1063/5.0005764 116, 172403
© 2020 Author(s).Generation, electric detection, and orbital-
angular momentum tunneling of twisted
magnons
Cite as: Appl. Phys. Lett. 116, 172403 (2020); https://doi.org/10.1063/5.0005764
Submitted: 24 February 2020 . Accepted: 10 April 2020 . Published Online: 27 April 2020
Min Chen , Alexander F. Schäffer
, Jamal Berakdar
, and Chenglong Jia
Generation, electric detection, and orbital-angular
momentum tunneling of twisted magnons
Cite as: Appl. Phys. Lett. 116, 172403 (2020); doi: 10.1063/5.0005764
Submitted: 24 February 2020 .Accepted: 10 April 2020 .
Published Online: 27 April 2020
Min Chen,1Alexander F. Sch €affer,2
Jamal Berakdar,2
and Chenglong Jia1,a)
AFFILIATIONS
1Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou University, Lanzhou 730000, China
2Institut f €ur Physik, Martin-Luther-Universit €at Halle-Wittenberg, 06099 Halle (Saale), Germany
a)Author to whom correspondence should be addressed: cljia@lzu.edu.cn
ABSTRACT
A scheme for generating twisted magnons that carry orbital angular momentum in ferromagnetic nanodisks is presented. The topological
signature of these eigenmode excitations entails particular features in the associated spin pumping currents. The latter is electrically
detectable and can be used to identify these magnons. Considering two disks coupled via the dipole interaction, angular momentumtunneling is demonstrated. The predictions are based on a transparent analytical model and are confirmed by full numerical simulations.As the orbital angular momentum of the magnon is robust to damping, the current findings endorse the potential of twisted magnons fortwo-dimensional planar integrated spin-wave circuits.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0005764
Spin-waves (or their quanta magnons) are the low-energy collec-
tive excitations of ordered magnetic systems and are considered as a
potential medium for information exchange with several advantages
such as nanoscale integration and low energy consumption.
1–3In
recent years, spin-wave based devices such as logic gates,4,5filters,6
waveguides,7,8diodes,9beamsplitters,10,11and multiplexors12have
been proposed and designed. An important factor that affects the fidel-
ity of information transmitted as magnonic signals is magnetic
(Gilbert) damping that leads to a decaying magnon density. Recently,
we pointed to additional types of twisted spin waves that carry a defi-
nite amount of orbital angular momentum.13,14This orbital angular
momentum is measurable and protected against the damping, and,
hence, it is worthwhile considering it as a carrier of information. In
this Letter, we show that twisted spin-waves can be excited in ferro-
magnetic (FM) nanodisks. The nontrivial topology, i.e., the spatial and
temporal configurations of spin-waves, can be electrically read out,exploiting spin pumping effects and the inverse spin Hall effect
(ISHE). Another interesting issue is the tunneling of orbital angular
momentum between two disks coupled through the dipole–dipole
interaction, and the tunneling is evidenced by pumped DC/AC spin
currents and spin accumulation.
Disk-shaped FM was studied intensely theoretically and experi-
mentally in the past, for example, in the context of whispering gallery
modes.
15Here, we consider a disk with perpendicular magnetization
M(this direction is taken as the z-axis), whose radius Ris chosen to besmall enough so that the exchange interaction is more prominent than
the dipole interaction. By introducing a dimensionless unit vector field
m¼M=Ms,w i t h Msbeing the saturation magnetization, the total
Lagrangian density of the FM nanodisk reads
L ¼ /C0Að @lmÞ2þK zm2
z; (1)
where Ais the exchange stiffness and Kzdescribes the uniaxial mag-
netic anisotropy. Linearizing Eq. (1), we arrive at the Euler–Lagrange
equation that determines the spin-wave dynamics,
/C0A @2
lwmþK zwm¼0; (2)
withl¼x;yandwm¼mx/C0imy. The above equation shows the
non-diffractive Bessel solutions in cylindrical coordinates r!ðq;/Þ
for spin-wave excitations,
wmðr;tÞ/J‘ðk‘;nqÞexpði‘//C0ixtÞ; (3)
where ‘2Z.J‘ðxÞis the Bessel function of the first kind with order ‘.
k‘;nis determined from the boundary condition and is the nth root of
the Bessel prime function dJ‘ðqÞ=dqjq¼Runder the Neumann bound-
ary conditions. The eigenfrequency is x¼A k2
‘;nþK z.
Similar to the twisted spin-waves in a FM nanowire, we define a
pseudo Poynting-like vector as
P:¼Jz
l¼A = wmðr;tÞ@lw?
mðr;tÞ/C2/C3
¼/C0 A n‘‘
qe/; (4)
Appl. Phys. Lett. 116, 172403 (2020); doi: 10.1063/5.0005764 116, 172403-1
Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplwhere Jz
lis the z-polarized (magnon) spin current along the lspatial
direction and n‘is the spin-wave density. It is then straightforward to
introduce a canonical orbital angular moment of the spin-waves as
L¼r/C2P.O n efi n d st h a t ‘¼ez/C1hLiby averaging Lover the whole
disk. Thus, this ‘can be identified as the topological charge of spin-
waves, and it is an intrinsic damping-resistant parameter, as shown
below. Furthermore, in terms of the ISHE, these topological spin-
waves can be detected electrically by measuring the pumped spin cur-rents. In general, here, we have two types of pumped spin currents(I
s/C24m/C2@tm):16(i) the x/C0andy-polarized AC spin currents are
Ix
s/C24/C0 @tmy//C0xcosðxt/C0‘/Þ; (5)
Iy
s/C24@tmx//C0xsinðxt/C0‘/Þ; (6)
respectively. (ii) The z-polarized DC spin current is
Iz
s/C24=wm@tw?
m/C2/C3
/x: (7)
Clearly, the integral over the whole nanodisk implies Ix
s¼Iys/C170,
and only Iz
sremains, which means that now both AC and DC signals
do not depend on the internal (spatial) phase structure of the topologi-
cal spin-waves (and hence, no dependence on ‘is present). However,
these spatial inhomogeneities can be detected by the angular-resolvedISHE. For example, given that j‘j¼1, the left and right half-disks pos-
sess exactly opposite AC signals, as supported by the full-fledged
numerical simulations presented below.
Considering the SO(2) rotational symmetry of the FM disk, we
e x p e c tt h a tt h ed i p o l ei n t e r a c t i o nd o e sn o td e s t r o yt h ea b o v ee i g e n m o d eexcitations but would lift the degeneracy between the 6‘modes.
17
To this end, we conduct micromagnetic simulation using the
GPU-accelerated, open-source software package mumax320for an
experimentally realistic nanodisk made of ferromagnetic alloy FeB. Inthe simulations, a FeB disk with the diameter of 120 nm and a thick-
ness of 2 nm is selected to have an exchange interaction that is stronger
than the dipole interaction. The saturation magnetization isM
s¼1:375/C2106A/m, the exchange stiffness constant is A¼ 0:9
/C210/C011J/m, the uniaxial magnetic anisotropy along the zdirection
isKz¼1:05/C2106J/m, and the Gilbert damping parameter is
a¼0:005.17For the following simulation results, a cell of 2 /C22
/C22n m3within the typical range of exchange interactions is used to
discretize the FeB disk. To launch the desired spin-waves, we excite
the nanodisk locally by applying a spatially inhomogeneous radio fre-
quency (rf) magnetic field. As discussed in Refs. 17–19 ,d i f f e r e n t
‘-eigenmodes can be excited by appropriate spatial configurations of
the rf fields. In the present study, we focus on the topologically non-
trivial ‘¼61 eigenmodes for a clear demonstration.
First, a sinc function of the rf magnetic fields, ByðtÞ
¼BðyÞsincðxtÞey,w i t h x¼60 GHz and BðyÞ¼61:5m Ti nt h eu p
(down) half-disk, is superimposed to launch spin-waves. As shown in
Fig. 1 ,t h e ‘¼61 eigenmodes are indeed excited in the FeB nanodisk
and the dispersion relation of spin-waves, given by the two-dimensionalfast Fourier transform (FFT) from the space-and-time domain ðq;tÞis
split into many subbands because of the Neumann boundary condition
used in the simulations. The twofold degeneracy of 6‘eigenmodes is
lifted by the dynamic dipole interaction. Using the discrete-time FFT,the ferromagnetic resonance spectra are obtained, in which the two low-est eigenfrequencies with the wavenumber k
‘;n¼k61;0are identified as
x/C0¼3:61 GHz and xþ¼3:79 GHz, respectively.Next, the same spatial configuration but with a single-frequency
sinusoidal microwave, B6
yðtÞ¼BðyÞsinðx6tÞey, is applied to excite
a single spin-wave eigenmode in the FeB nanodisk. Snapshots in
Figs. 2(a) and 2(b) show excellent agreement with the theoretical
formula, Eq. (3)of‘¼61a n d n¼0. After turning off the externally
applied microwave, the finite Gilbert damping leads to a time-decaying spin-wave density [ Fig. 2(c) ]. However, when evaluating the
topological charge ‘from numerically integrating orbital angular
moment hLi, we find that ‘is damping resistant and is conserved
during the spin-wave evolution [ Fig. 2(d) ].
To investigate the spin pumping effect, we take the lowest eigen-
frequency ( x
/C0) excitations as a demonstration mode. As shown in
Fig. 3(a) , the numerical spin-wave density is well described by the
Bessel function with ‘¼/C01. The numerically calculated z-polarized
spin currents over both the left ( hIz
siL) and right ( hIz
siR) half-disks
possess DC behavior [ Fig. 3(b) ]. The angular dependence of hIz
sðqÞi
inFigs. 3(c) and 3(d) is determined by the function cos 2 ‘/.
FIG. 1. FFT showing the excited ferromagnetic resonance spectra of the FeB nano-
disk. Insets: (Left) snapshot of the mxcomponent of spin-wave eigenmodes with
‘¼61. (Right) Dispersion relation of spin-waves, in which the dashed lines corre-
spond to the eigen wavevector k61;n.
FIG. 2. (a) and (b) Snapshots of the in-plane nonequilibrium magnetization configu-
ration of the ‘¼61 eigenmodes. (c) and (d) Time resolved amplitude of m xat the
point (90, 90) nm and the averaged orbital angular moment hLi, respectively.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 116, 172403 (2020); doi: 10.1063/5.0005764 116, 172403-2
Published under license by AIP PublishingConcerning the pumped AC spin current, hIx
siandhIy
sias depicted
inFig. 4 , the integral over both the left and right half-disks shows a
time-oscillating behavior with frequency x/C0but with a phase differ-
ence ‘p. Clearly, as expected, the topology ( ‘) of the spin-wave
eigenmodes can be extracted using the spin-pumping method.
Another important issue is the transmission and interference of
spin-waves in two coupled FM disks. It has been shown that the spin-wave coupling efficiency depends on both the geometry of magnonicstructures and the characteristics of the spin-wave modes.
21–23Here,
we explicitly demonstrate these important effects by exciting only oneof the two point-adjacent FeB disks, meaning that disk I is exposed
to the single-frequency microwave B/C0
yðtÞ. As depicted in Fig. 5(a) ,
the Bessel eigenmode with topological charge ‘¼/C01i se x c i t e di n
disk I and is then transmitted into disk II. Furthermore, as demon-strated in Fig. 5(b) , the point-like inter-disk exchange interaction
is ignorable and the transmission of spin waves is solely mediated
by the inter-disk dipole interaction. However, the pumped spin
current signals show that the amplitude of spin-waves in disk II is
strongly reduced and there is a p=2 phase shift between two disks.
Compared to the pumped spin currents of a single FeB disk in
Figs. 3 and 4, we notice that the eigenfrequency of spin-waves is
slightly modified by the dipole interaction and the z-polarized spin
current hI
z
sibecomes a DC þAC mixed signal with a doubled fre-
quency of the x-polarized AC spin current.
In conclusion, we studied analytically and numerically the spin-
wave excitations in ferromagnetic nanodisks. Using the micromagnetic
simulation program mumax3 , we showed that twisted spin-wave
eigenmodes can be launched individually and can be detected based
on the inverse spin Hall effect and methods sensing spin accumula-
tions. The topological charge of spin-waves is protected against damp-ing and has a well-defined signature in the DC/AC pumped spin
current. Furthermore, the topological charge can be exchanged
between two dipole-coupled nanodisks. We find, moreover that the
topological charge of spin-waves is robust against reasonable disk
shape variations that do not destroy the topology. For instance, the
twisted magnons can be launched in ellipsoid, square, and rectangle
structures as long as the spin-wave modes are dominated by theexchange interaction rather than the dipole interaction. Our results
add an additional twist to the topological spin-wave and their use for
information transfer and processing.
FIG. 3. (a) Snapshot of spin current density hIz
siof‘¼/C0 1 eigenmode. (b) The
integral z-polarized spin density over the left and right half-disks, hIz
siL(dark blue
line) and hIz
siR(dark yellow line), respectively. (c) and (d) Angular resolved hIz
si
along the circle with the radius q¼Randq¼R=2 respectively.
FIG. 4. The integral (a) x- and (b) y-polarized spin current density over the left
(dark blue line) and right (dark yellow line) half-disks.
FIG. 5. Transmission of spin-waves between two coupled FeB disks. Note that only
disk I is excited locally by the applied rf magnetic field B/C0
yðtÞ. (a) shows a snapshot
of the y-component of magnetization m y. (b) The integral x- and z-polarized spin
density averaged over disk I (dark blue line) and disk II (dark yellow line) with orwithout point-like inter-disk exchange interaction.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 116, 172403 (2020); doi: 10.1063/5.0005764 116, 172403-3
Published under license by AIP PublishingThis work was supported by the National Natural Science
Foundation of China (Nos. 91963201 and 11834005), the GermanResearch Foundation (Nos. SFB 762, and SFB TRR 227), and theProgram for Changjiang Scholars and Innovative Research Team in
University (No. IRT-16R35).
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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Published under license by AIP Publishing |
5.0008386.pdf | J. Chem. Phys. 152, 224111 (2020); https://doi.org/10.1063/5.0008386 152, 224111
© 2020 Author(s).Reactive molecular dynamics simulation for
isotope-exchange reactions in H/D systems:
ReaxFFHD development
Cite as: J. Chem. Phys. 152, 224111 (2020); https://doi.org/10.1063/5.0008386
Submitted: 23 March 2020 . Accepted: 11 May 2020 . Published Online: 12 June 2020
Mohammad Ebrahim Izadi
, Ali Maghari
, Weiwei Zhang
, and Adri C. T. van Duin
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of Chemical PhysicsARTICLE scitation.org/journal/jcp
Reactive molecular dynamics simulation
for isotope-exchange reactions in H/D systems:
ReaxFF HDdevelopment
Cite as: J. Chem. Phys. 152, 224111 (2020); doi: 10.1063/5.0008386
Submitted: 23 March 2020 •Accepted: 11 May 2020 •
Published Online: 12 June 2020
Mohammad Ebrahim Izadi,1
Ali Maghari,1,a)
Weiwei Zhang,2
and Adri C. T. van Duin2
AFFILIATIONS
1Department of Physical Chemistry, School of Chemistry, College of Science, University of Tehran, Tehran, Iran
2Department of Mechanical and Nuclear Engineering, The Pennsylvania State University, University Park,
Pennsylvania 16802, USA
a)Author to whom correspondence should be addressed: maghari@ut.ac.ir. Tel.:+98-21-6111-3307. Fax: +98-21-6640-5141
ABSTRACT
To investigate the chemical isotope-exchange reactions within a system composed of a mixture of hydrogen and deuterium (H/D) in the
plasma media, the ReaxFF HDpotential was parameterized against an appropriate quantum mechanics (QM)-based training set. These QM
data involve structures and energies related to bond dissociation, angle distortion, and an exchange reaction of the tri-atomic molecu-
lar ions, H 3+, D 3+, H 2D+, and D 2H+, produced in the hydrogen plasma. Using the ReaxFF HDpotential, a range of reactive molecular
dynamics simulations were performed on different mixtures of H/D systems. Analysis of the reactions involved in the production of these
tri-atomic molecular ions was carried out over 1 ns simulations. The results show that the ReaxFF HDpotential can properly model isotope-
exchange reactions of tri-atomic molecular ions and that it also has a perfect transferability to reactions taking place in these systems. In
our simulations, we observed some intermediate molecules (H 2, D 2, and HD) that undergo secondary reactions to form the tri-atomic
molecular ions as the most likely products in the hydrogen plasma. Moreover, there remains a preference for D in the produced molecu-
lar ions, which is related to the lower zero-point energy of the D-enriched species, showing the isotope effects at the heart of the ReaxFF HD
potential.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0008386 .,s
I. INTRODUCTION
The positively charged tri-atomic molecular hydrogen/
deuterium ions (H 3+, D 3+, H 2D+, and D 2H+) have attracted the in-
terest of chemical theorists,1,2spectroscopists,3–5and astronomers6,7
owing to their apparent simplicity and great importance in astro-
chemical environments.8Using a liquid-nitrogen-cooled multiple-
reflection discharge cell along with a difference-frequency laser
system, the spectrum of H 3+molecular ion was first detected
by Oka in 1980.9In fact, in a discharge cell filled with hydro-
gen, the hydrogen plasma, H 3+is the most abundant ion. This
molecular-ion system has a density of two electrons distributed
around three nuclei bonded to each other in a three angular
configuration.10However, the neutralized tri-atomic molecules of
hydrogen/deuterium (H 3, D 3, H 2D, and D 2H), having a linearstructure in the electronic ground state, are nonbonding or
unstable.11
Until now, theoretical methods have consistently endeavored to
achieve ever more accurate potential energy surfaces (PESs) for H 3+
at 0 K and subsequent calculations of the rovibrational states.2,8,9,12
So far, however, there has been no detailed investigation of chem-
ical reactions within the hydrogen discharge cell using molecular
dynamics (MD) simulations. Moreover, far too little attention has
been paid to isotopes within the framework of MD simulations.
Therefore, the aim of this work is to investigate the isotope-exchange
reactions in the H 2/D2plasma, with focused study on the production
of H 3+and its alternative isotopic mixtures (D 3+, H 2D+, and D 2H+)
using reactive molecular dynamics (RMD) simulations.
ReaxFF has been developed by van Duin et al. since 200113
and can be used as an effective means to track chemical reactions
J. Chem. Phys. 152, 224111 (2020); doi: 10.1063/5.0008386 152, 224111-1
Published under license by AIP PublishingThe Journal
of Chemical PhysicsARTICLE scitation.org/journal/jcp
within the framework of MD simulations. ReaxFF allows the calcu-
lation of partial atomic charges even for large systems, compared to
expensive quantum mechanical (QM) calculations. ReaxFF param-
eterization is based on QM calculations specifically using the DFT
method.13Up to now, the ReaxFF potential has been parameter-
ized for a large body of the periodic table in order to simulate
several chemical reactions and chemical processes including hydro-
carbon combustion,13,14silicon and silicon oxide systems,15,16high
energy materials,17,18production of nanostructures and their behav-
iors such as carbon nanotubes,19fullerene,20,21MoS 2layers,22metal-
lic and oxide nanoparticles,23,24proteins25,26and DNA27strands,
metal oxides (TiO 228and ZnO29), mineral complexes,30and water.31
It should be remembered that apart from the 2018 study by Zhang
et al. ,32there is little work on the incorporation of isotopes within
the ReaxFF potential. Therefore, the present investigation aims to
parameterize the ReaxFF potential to investigate isotope-exchange
reactions in the production of tri-atomic molecular hydrogen ions
and their isotopic mixtures under a discharge condition.
II. COMPUTATIONAL METHODOLOGY
A. QM calculations
It has been shown that for H 4+and H 3+complex ions, the most
reliable QM method is the QCISD (quadratic configuration interac-
tion with single and double excitation for correcting size-consistency
errors) level of theory with the basis set of cc-pVQZ.33Therefore,
first-principles calculations based on QCISD theory were performed
to provide all QM data in a training set. These calculations were car-
ried out using GAUSSIAN 09 software package.34QCISD/cc-pVQZ
calculations were carried out for the dissociation of two and three
H–H bonds in H 3+molecular ion converting to H 2+ H+and 2H +
H+fragments, respectively. Also, the energies related to an exchange
reaction (H′+ H 3+→H2H′++ H) are obtained at the same level and
incorporated within the training set. In addition, to introduce H 3+
configuration, angle bend energies were also calculated and inserted
into the training set. In this study, except for the exchange reaction
(H′+ H 3+→H2H′++ H), which has to be in doublet state, other
QCISD calculations were calculated on singlet systems and also the
QM data related to non-isotope and isotope mixtures have been
obtained from electronic energies and electronic energies plus ZPE
corrections, respectively.
B. The ReaxFF force field
In the ReaxFF potential, there are two sets of potential func-
tions, valence and nonbonding interactions.13,15The general form of
the ReaxFF potential functions is
Esystem=Ebond +Eunder +Eover+Elp+Eval+Etor
+EvdWaals +ECoulomb , (1)
in which the partial energy contributions are made up of bond,
under-coordination, over-coordination, lone pair, valence angle,
torsion angle, van der Waals, and Coulomb energies, respectively.
Except for the nonbonding van der Waals and Coulomb terms, the
other terms are connectivity dependent in such a way that their
energy contributions approach zero upon bond dissociation. In the
ReaxFF potential, there is a direct correlation between bond energyand bond order (BO). In addition, the bond order is associated with
the bond length, so the bond energy is related to inter-atomic dis-
tances. Consequently, the potential energy may refrain from dis-
continuity while chemical bonds are forming or breaking during
chemical reactions. Because of the reactive nature of the ReaxFF
potential, the connectivity of each atom remains updated at each
time step. The chemical environment of the atoms in molecules is
altered by chemical reactions and, as a result, it is vital to update
atomic partial charges over the RMD simulations. To this end, the
ReaxFF potential is equipped with the electronegativity equilibra-
tion method (EEM).35According to this method, the atomic par-
tial charges are driven by some connectivity dependent functions.
In the ReaxFF potential function, nonbonding interactions are cal-
culated between all pairs in a system, ignoring the connectivity of
atoms.13,15
C. RMD simulation setup
For the discharge cell simulation, a periodic cubic box of 100 Å
length containing 400 H/D atoms was built. It was assumed that H 2
and D 2molecules can be atomized/ionized within the discharge cell
using an external electric field. In fact, at the early stage of this pro-
cess, after the electric discharge, the temperature and pressure are
high and the non-equilibrium condition exists. After condensation
reactions (conversion of H and D atoms to tri-atomic molecules),
the pressure and temperature decrease gradually upon equilibrium
condition. The initial pressure is set at 8 atm, and the equilibra-
tion stage at the starting point does not exist. To investigate the
effect of D concentration on products, various D fractions have been
chosen, including 0, 0.2, 0.4, 0.5, 0.6, 0.8, and 1.0. In each case, 15
different initial configurations, various initial random positions and
velocities, were taken into account to allow a reasonable trajectory
sampling in the gaseous phase. In addition, the total charge of the
system has been chosen to be +130 and the geometry-dependent
EEM is likely to make the charge delocalize. However, it should be
noted that during the simulations, the atomic partial charges did
not change significantly over the chemical reactions, so they are not
important for the reactions of this study. In fact, according to QCISD
calculations, the atomic partial charge for triatomic molecular ions
is +0.33, and in our simulations, it is +0.32 for every atom. More-
over, because of the EEM, there is a very tiny fluctuation only on the
third decimal. Therefore, we give up following the partial charges
in this system because they remain unimportant. Then, RMD sim-
ulations were carried out for each system for 1 ns at a temperature
of 150 K controlled using a Berendsen thermostat36with a 100 fs
damping constant, and an average was taken on the results of these
simulations. For instance, the number of products, number of reac-
tions, and percentage were averaged over 15 simulations. The cur-
rent simulations were performed using a velocity Verlet algorithm
and a time step of 0.25 fs in the canonical ensemble (NVT). All
of these simulations were carried out using the ReaxFF standalone
code.
III. RESULTS AND DISCUSSION
A. ReaxFF force field development
To seek the molecular dynamics of hydrogen plasma,
ReaxFF CHO14force field was re-parameterized using a training set
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composed of bond lengths, valence angles, and exchange reaction
energies. The parameterization of the ReaxFF potential was carried
out using a well-established one-parameter search technique imple-
mented by van Duin et al.37within the ReaxFF standalone code and
subsequently the newly optimized force field named ReaxFF HD.
To fit the H 3+bond energy curves, the parameterization of the
ReaxFF HDpotential against QCISD calculations was conducted for
the following reactions:
(a) H+
3→H2+ H+,
(b) H+
3→2H + H+.
Thereafter, H–H bond dissociation curves resulting from the
ReaxFF HDpotential were compared with the QM data, as shown in
Figs. 1(a) and 1(b).
To optimize the valence angle parameters for a H 3+molecu-
lar ion, the QCISD angle bend energies of the triangular H 3+con-
verting to a linear molecular ion were calculated and introduced to
ReaxFF HDdevelopment. Figure 2 shows a comparison between the
ReaxFF HDand QCISD angle bend energies.
Using the QCISD calculations, the ReaxFF HDpotential was
trained on the following exchange reaction:
(c) H′+ H+
3→H2H′++ H.
Consequently, the ReaxFF HDpotential resulted in an improved bar-
rier height of 3.80 kcal/mol, which is comparable to the literature
value (QCISD) of 3.44 kcal/mol (Fig. 3). As shown in Fig. 3, the reac-
tion passes through a transition state of a tetra-atomic molecular ion
(H4+) considered within the training set. Actually, the larger molec-
ular ions (such as H 4+and H 5+) are not very popular within the
hydrogen plasma/interstellar medium, and also they remain beyond
FIG. 2 . QCISD and ReaxFF HDdata for valence angle energies of H 3+against the
H–H–H valence angle.
the objective of this study. Similarly, to extend the ReaxFF HDpoten-
tial for deuterium, parameters for D were chosen simply by dupli-
cating the H parameters, and afterward, the D mass was inserted
in the place of H mass. Thereafter, keeping the H atom parameters
FIG. 1 . The comparison between the ReaxFF HDand QCISD energies against interatomic distance in the H 3+molecular ion converting to H 2+ H+(a) and 2H + H+(b)
fragments.
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FIG. 3 . The path of the H′+ H 3+→H + H′H2+exchange reaction and comparison
between the ReaxFF HDand QCISD relative energies that resulted from the doublet
state of the system. For more convenience, the reactants are set at zero energy.
fixed, we subsequently refitted the D–D and H–D bond and angle
parameters in such a way as to obtain distinct energy curves for
each isotopic mixture of tri-atomic molecular ions (H 2D+, D 2H+,
and D 3+).According to the literature,12,38the D-containing bonds (D–
X) have a lower zero-point energy (ZPE) relative to the alterna-
tive H-containing bonds (H–X); in addition, the nuclei tunneling
effect (non-Born–Oppenheimer solvation of the Schrödinger equa-
tion) leads to the separation of their potential energy curves. To the
best of our knowledge,39D-containing bonds have potential energy
curves stronger than alternative non-isotope bonds, meaning that
H–D potential energy curve lies below H–H and D–D lies below the
H–D. Therefore, to insert both effects together within the ReaxFF HD
potential, a new training set containing energy differences between
the non-isotope and various isotope mixtures has been produced.
Then, scaling of potential energy for isotopic mixtures has been
done by inserting these energy differences within this training set
(refer to the supplementary material for more details). We expect
that both factors are included in the ReaxFF force field training to
study the isotope effect of hydrogen, and it has been demonstrated
that this strategy is reliable on the basis of our previous studies on the
comparison of heavy and light water.32Then, using one-parameter
search technique implemented within the ReaxFF standalone code,
the ReaxFF HDpotential was re-parameterized for the deuterium iso-
tope, and the results are depicted in Figs. 4(a), 4(b), and 5. In Figs. 4
and 5, the energy contributions related to isotope effects have been
scaled and inserted into the curves associated with isotope mixtures;
as a matter of fact, the potential curve of H 3+is introduced from pre-
vious fittings on the QCISD level of theory. As a result, isotopic mix-
tures of tri-atomic molecular ions have been developed by mimick-
ing the trend of H 3+potential energy curves. As can be deduced from
Fig. 4, the substitution of hydrogen by deuterium leads to a decline in
the potential energy of the corresponding molecular ion, and subse-
quently, it increases the dissociation energy of D-containing bonds.
On the other hand, converting the triangular configuration to the
linear configuration is associated with one bond cleavage within the
tri-atomic molecular ions (e.g., the reaction inserted into Fig. 5).
FIG. 4 . The comparison among H–H, H–D and D–D interactions in the trihydrogen cation H 3+and its mixed isotopic ions using the ReaxFF HDpotential. Also, the QCISD data
are included. The energies related to the isotope mixtures contain contributions of both ZPEs and quantum tunneling effects.
J. Chem. Phys. 152, 224111 (2020); doi: 10.1063/5.0008386 152, 224111-4
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FIG. 5 . Calculated valence angle energies of H 3+, D3+, D2H+, and H 2D+molecu-
lar ions using the ReaxFF HDpotential vs H–H–H, D–D–D, H–D–D, and D–H–H
angles, respectively. Also, the QCISD data are included. Contributions of both
ZPEs and quantum tunneling effects have been incorporated in the energies
related to the isotope mixtures.
Moreover, according to the QM calculations, the triangular structure
of H 3+molecular ion is far more stable than its linear structure. As a
result, as shown in Fig. 5, the deformation from the triangular struc-
ture to the linear configuration needs more energy relative to the
typical bending energy in molecular chemistry. Moreover, often, the
more deuterium is in the structure, the greater the energy required
for the deformation.
B. Molecular dynamics simulations
To test the ReaxFF HDpotential energy for simulating chemical
exchange reactions and also for investigating the D-enhancement, a
series of RMD simulations were carried out at compositions ranging
from pure hydrogen to pure deuterium.
1. Chemical exchange reactions
To find the reactions in the system, we postprocessed tra-
jectory files containing atomic and molecular IDs. The atomic
IDs are constant and unique numbers for all atoms during the
whole simulation. However, the molecular IDs will be dynamic
while the chemical reactions dynamically proceed. In fact, atoms
will belong to a molecule if they have bond orders above the
minimum value 0.3, which is similar for all types of the bonds
within the system. To clarify the performance of this crite-
rion, we have plotted the bond order (BO) vs inter-atomic dis-
tance of one of the H-H bonds within the H 3+molecular ionusing the ReaxFF HDforce field, and the results are shown in Fig. S3
of the supplementary material. As it can be inferred from Fig. S3,
BOs less than 0.3 are related to inter-atomic distances larger than
1.62 Å, and based on energy curves in Figs. 1 and 4 at distances
>1.62 Å, the PES is around the dissociation tail. So, the criterion
for molecular recognition is reliable. Therefore, based on this cri-
terion, atoms belonging to a molecule have a similar molecular ID.
As a result, at certain intervals, every 0.25 ps, we have dealt with
a list of atomic and molecular IDs. In this study, each trajectory
file consists of 4000 frames, and accordingly, every two consecutive
frames have been compared to find the reactions taken place, in the
other words, to find which molecular ID each atomic ID belongs
to. We have analyzed these data by writing a BASH@FORTRAN
script. Using this code, 18 possible reactions over the simulations
were tracked at X D= 0.5, and consequently the percentage of these
reactions is shown in Table I. The percentage of each kind of
reaction that took place is calculated according to the following
equation:
A%=No. of reaction A
No. of all counted reactions×100. (2)
To express the variations among the measurements, the standard
deviation of the percentage related to each reaction is added next to
the percentage values. As indicated in Table I, the chemical exchange
reactions (reactions 4, 5, 6, and 7) can be simulated by the ReaxFF HD
potential. Moreover, the exchange reactions involved in the produc-
tion of tri-atomic molecular ions (reactions 4 and 5) have much
TABLE I . The chemical reactions tracked during 1 ns simulation at every 0.25 ps inter-
val at the concentration of X D= 0.5 and the temperature of 150 K using the ReaxFF HD
potential. In the place of H and H′, one could consider D and D′, respectively. In
fact, these reactions involve all possible isotope mixtures of di-, tri-, and tetra-atomic
species. In the first column, the numbers with the superscript −1 represent the
inverse reactions of similar numbers without the superscript.
Row Reactions Percentage
1 H 2+ H+→H3+7.40±0.72
2 2H + H+→H3+6.45±0.65
3 H 2++ H 2→H3++ H 2.65 ±0.20
4 H 2+ H′
3+→H2H′++ H′H 23.14 ±1.85
5 H 3++ H′→H2H′++ H 14.90 ±2.02
6 H 2+ H′+→HH′+ H+4.93±0.48
7 H 2++ H′
2→HH′++ HH′1.36±0.37
8 H 4++ H→H2+ H 3+3.049 ±0.51
9 H 4+→H3++ H 2.66 ±0.70
10 H 4+→H2++ H 2 0.238 ±0.09
11 H 2→H + H 6.82 ±1.11
1−1H3+→H2+ H+6.69±0.73
2−1H3+→H++ H + H 6.23 ±0.68
3−1H3++ H→H2++ H 2 2.06±0.40
8−1H3++ H 2→H4++ H 2.77 ±0.42
9−1H3++ H→H4+1.72±0.53
10−1H2++ H 2→H4+0.13±0.03
11−1H + H→H2 6.78±0.46
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higher percentages relative to the ones involved in the production
of diatomic molecules (reactions 6 and 7). Therefore, the ReaxFF HD
potential can effectively simulate the isotope-exchange reactions of
tri-atomic molecular ions within a hydrogen plasma according to the
literature.7,33,37
The ReaxFF DHpotential was parameterized using only one type
of exchange reaction (reaction 5), whereas other exchange reactions
(reactions 4, 6, and 7 in Table I) were characterized during the simu-
lations. In addition, six more chemical reactions (reactions 3, 3−1,
8, 8−1, 10, and 10−1), along with the reactions introduced to the
training set (reactions 1, 2, 5, 9, and 9−1), may be characterized.
Therefore, the ReaxFF HDpotential has good transferability to reac-
tions taking place in the hydrogen plasma. As mentioned in the
literature,37,40,41in interstellar media, there are stellar winds com-
posed of ionized plasma (mainly protons and electrons). Within this
media, H 2+and then H+species are two major ions participating in
reactions,
(d) H+
2+ H 2→H+
3+ H,
(e1)H2+ H+→H+
2+ H,
(e2)H+
2+ H 2→H+
3+ H.
As a result, H 3+is not only the most abundantly produced molec-
ular ion in interplanetary space, but one with a vital role in inter-
stellar chemistry. In fact, this molecular ion undergoes the follow-
ing exchange reactions to form substantial chemicals in interstellar
media:
(f)H+
3+ HD→H2D++ H 2,
(g)H+
3+ X→HX++ H 2, where X =CH 4or C 2H2.
The f-type reactions lead to D-enhancement in tri-atomic molecular
ions, while the g-type reactions are the major destruction mecha-
nism of H 3+and its isotope mixtures. On the other hand, the likely
reactions of these molecular ions have been investigated using QM
calculations and experimental facilities.5,7,33,41
Accordingly, in addition to the d–g reactions, chemical reac-
tions such as
(h)H+
3+ X→H2X++ H, where X =H or D,
(i)H+
3+ H′
2→H2H′++ H′, where H′=H or D,
(j)D++ H 2→H++ HD,
and
(k)H++ D 2→D++ HD,
could be observed in the plasma media. It should be noted that the
h reaction proceeds through an H 4+complex as transition state.
By comparing the reactions in Table I and d–f as well as h–kreactions, one may realize that the ReaxFF HDpotential can prop-
erly cover a wide range of chemical reactions taking place within the
hydrogen/deuterium plasma.
To compare the relative stability of the reactants and prod-
ucts, the internal energy ΔErof some of the reactions (Table I)
has been reported in Table II. Also, for the fifth reaction, the
exchange reaction inserted in the training set, the activation ener-
gies E a(kcal/mol) are added into Table II (see the footnote). In
Table II, for each type of the reactions, shown in Table I, sev-
eral possible isotopic mixtures are considered. As can be seen, all
of the reactions in which the tri-atomic molecular ions are pro-
duced from smaller fragments (atoms or di-atomic species) are
exothermic, which means that the tri-atomic molecular ions are
TABLE II . Internal energy ΔEr(kcal/mol) of the reactions considered in this study.
The tetra-atomic molecular ions are not given in this table, since they remain beyond
the scope of this study and also the ReaxFF HDpotential had not developed for these
reactions.
Row Reactions ΔEr(kcal/mol)
1 H 2+ H+→H3+−35.70
HD + H+→H2D+−36.59
D2+ H+→D2H+−40.66
D2+ D+→D3+−54.03
2 2H + H+→H3+−114.42
H + D + H+→H2D+−115.42
2D + H+→D2H+−125.10
2D + D+→D3+−138.46
3 H 2++ H 2→H3++ H −30.70
H2++ HD→H2D++ H −31.45
H2++ D 2→H2D++ D −25.68
H2++ D 2→D2H++ H −35.36
D2++ D 2→D3++ D −42.70
4 H 3++ HD→H2D++ H 2 −50.46
H3++ D 2→H2D++ HD −44.81
H3++ D 2→D2H++ H 2 −54.36
H2D++ HD→D2H++ H 2 −59.05
H2D++ D 2→D2H++ HD −53.41
H2D++ D 2→D3+ H 2 −66.65
D2H++ HD→D3+ H 2 −62.71
D2H++ D 2→D3+ HD −57.07
5aH3++ D→H2D++ H −26.99
bH2D++ D→D2H++ H −35.58
cD2H++ D→D3++ H −39.24
6 H 2+ D+→HD + H+−23.37
D2+ H+→HD + D+−17.34
7 H 2++ D 2→2HD −94.18
11 H + H →H2 −104.96
H + D→HD −105.09
D + D→D2 −110.69
aActivation energies E acalculated for the reactions of type 5: Ea = 1.79 kcal/mol.
bActivation energies E acalculated for the reactions of type 5: Ea = 6.12 kcal/mol.
cActivation energies E acalculated for the reactions of type 5: Ea = 25.62 kcal/mol.
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energetically more favorable than smaller fragments in the plasma
medium. Within the rows 1 and 2, by comparing similar types of
the reactions, it can be inferred that more deuterium within these
reactions releases more energy and produces tri-atomic molecular
ions richer in deuterium. In the reactions of type 3, there is a simi-
lar trend except for the third reaction; in this reaction, it seems that
the cleavage of D 2molecule needs more energy than H 2/HD (sec-
ond/fourth reactions), and so this reaction remains less exothermic
than the second/fourth ones (the reader could make similar con-
clusions for other analog exceptions within Table II). In the fourth
and fifth rows, the internal energies of isotope-exchange reactions
among tri-atomic molecular ions are reported. Over the force field
development procedure, only the exchange reaction of type 5 was
inserted within the training set. It was expected that in Table I, this
reaction has a far larger percentage, but unexpectedly the percent-
age of the fourth reaction remains higher that of the fifth. According
to Table II, it is revealed that the much higher probability and the
higher exothermicity of the fourth reaction may result in such an
unexpected percentage. Regarding the exchange reactions of type 5,
the activation energy of H′+ H 3+→H + H′H2+(where H′= H)
is 3.80 kcal/mol (Fig. 3) while for H′= D, it is 1.79 kcal/mol,
meaning that the deuterium exchange should be much faster
than the hydrogen self-exchange. However, for the second and
third reactions (type 5), the activation energies are 6.12 kcal/mol
and 25.62 kcal/mol, respectively. It seems the deuterium mixture
of tri-atomic molecular ions (H 2D+and D 2H+) remains more
robust than H 3+resulting in more activation energy over exchange
reactions.
To show the time evolution of the bi- and tri-atomic molecu-
lar ions during the simulations, the number of these species vs time
(ns) is shown in Fig. 6 for X D= 0.5 and for the other concentra-
tions in Fig. S1 of the supplementary material. It was observed that
the intermediate molecules of H 2, D 2, and HD undergo secondary
reactions to form tri-atomic molecular ions (Fig. 6). In fact, based
on the ReaxFF simulation, these diatomic molecules are not stable
in the plasma media and undergo secondary reactions to form the
most stable molecules (Table II), the tri-atomic molecular ions, as
expected within the hydrogen plasma results.12,42
2. D-enhancement
Table III shows the comparison between the D and H frac-
tions in reactants (the first and second columns) and in all tri-
atomic molecular ions produced after 1 ns simulation at different
concentrations (the third and fourth columns). In fact, at the end
FIG. 6 . Time evolution of the bi- and tri-atomic molecules observed during 1 ns
NVT-RMD simulation using the ReaxFF HDpotential at the temperature of 150 K
and at the concentration of X D= 0.5.
of the simulations (for each concentration starting from 15 differ-
ent configurations so leading to 15 various results), the tri-atomic
molecular ions and accordingly H and D atoms belonging to these
molecules have been counted. Then, those 15 fractions calculated
and the average ones along with the corresponding standard devia-
tions are reported in Table III. As can be deduced from Table III and
Fig. 6 (the dashed lines), at all concentrations, the D fraction in the
tri-atomic molecular ions lies above the H fraction. Because of the
lower ZPE in the D–X bond relative to the H–X bond, it is expected
that the tri-atomic molecular ions have a higher tendency for D
rather than H to produce molecules with lower energies (Table II).
As a result, according to Table III and Fig. 6, this phenomenon
is properly predicted by the ReaxFF HDpotential. Moreover, Fig. 7
shows a qualitative comparison between experimental results43and
TABLE III . Deuterium and hydrogen fraction in reactants and the tri-atomic molecular ions produced after 1 ns NVT-RMD
simulation at 150 K temperature.
XD(reactants) X H(reactants) X D(tri-atomic ions) X H(tri-atomic ions)
0.2 0.8 0.2057 ±0.0099 0.7943 ±0.0099
0.4 0.6 0.4114 ±0.0114 0.5886 ±0.0114
0.5 0.5 0.5150 ±0.0097 0.4850 ±0.0097
0.6 0.4 0.6058 ±0.0088 0.3942 ±0.0088
0.8 0.2 0.8087 ±0.0123 0.1913 ±0.0123
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FIG. 7 . Comparison between the fraction of the tri-atomic molecular ions produced
at the end of the simulations (after 1 ns; solid lines) and the reproduced curves
from Ref. 43 (dashed lines) against the D fraction in the reactants.
the results obtained from ReaxFF simulations (referring to Fig. S2
of the supplementary material, it seems that the system reached
the equilibrium in less than 1 ns simulation). Figure 7 plots
the comparison of the fraction curves of tri-atomic molecular
ions against the D fraction within reactants (X D). In this fig-
ure, the experimental data are represented by the dashed lines
relating to the right axis and the results form this study by the
solid lines relating to the left axis. The fractions that resulted
from this study are averaged over 15 different simulations men-
tioned in Sec. II C, and all of them gained at the end of the
simulations.
As can be inferred, although the ReaxFF HDresult differs from
the experimental results to some extent, the predicted production of
tri-atomic molecular ions generally follows the same trend. More-
over, it should be noticed that in the experimental work,43the
authors did not determine the stage of their experiment to which
the intensities of tri-atomic ions are related. Once an electric dis-
charge happens inside a hydrogen-filled chamber, there is a strong
non-equilibrium state in terms of chemical, mechanical, and ther-
mal up to reaching the equilibrium condition. One may catego-
rize a plasma process into several stages including before electric
discharge, during the electric pulse, and after electric pules, and
there are specific processes during each stage. Therefore, the results
highly depend on which stage is the objective of a given study. Our
focus has been on a short time of the initial stage of the hydro-
gen plasma formation, after the electric discharge, and we wanted
to show how tri-atomic molecular ions will be formed at this stage.
Maybe the observed differences among the curves in Fig. 7 have
come from the different objective stage of these two works. It is
noteworthy that the line strengths (from spectroscopy data) are
proportional to the concentration. In addition, there remaincomplicated details within the experimental point of view for the
production of hydrogen plasma, and taking all of the components
into account within the MD simulation framework is unclear. For
example, both in interstellar media and in the discharge chamber,
there are convection, consistent external electric field (or cosmic
rays in interstellar media), and free electrons during the production
of tri-atomic molecular ions, all of which could contribute to the
chemical exchange reactions and D-enhancement of the tri-atomic
molecular ions. Although it was impossible to consider most of
these items within the framework of MD simulation, the ReaxFF HD
potential can qualitatively predict D-enhancement feature over the
simulations.
3. Percentage of the products
To find the major and minor products of these simulations,
the average percentage of mono-, di-, tri-, and tetra-atomic species
have been calculated after 1 ns simulation at X D= 0.2, 0.4, 0.5, 0.6,
and 0.8 concentrations. The percentage of these species has been
calculated based on the sum of the number of atom(s) (H and D)
belonging to each species over 15 different configurations per each
concentration. Afterward, the average percentage of various frag-
ments has been derived for the aforementioned concentrations.
These results along with the corresponding standard deviations are
demonstrated in Table IV. According to Table IV, based on the
conditions of this study, there is one major product, tri-atomic
molecular ions, and three by-products, mono-, di-, and tetra-atomic
species. According to the literature, in the interstellar/hydrogen
plasma medium, H 3+remains the most likely molecular ion while
H4+complex ion has rarely been detected in this medium, so the
percentage of the products that resulted from these simulations are
qualitatively comparable with the literature.6,9,44In addition, accord-
ing to Tables I and IV, these complex-ions, H 4+, have a very low
percentage among reactions and final products, respectively. Also,
from Table I, it can be inferred that the sum of the H 4+produc-
tion reactions (reactions 8, 9, and 10) have a lower percentage than
the destruction ones (reactions 8−1, 9−1, and 10−1). Therefore, these
complex ions neither are important in the interstellar/hydrogen
plasma medium nor are worthwhile in the results of this
study.
4. Configuration of the molecular ions
The equilibrium configurations of the produced tri-atomic
molecular ions at the end of simulations are schematically rep-
resented in Fig. 8. Also, from the literature,33,43the bond length
and bond angle of H 3+and D 2H+ions are inserted in Fig. 8. As
shown, the produced configurations using the ReaxFF HDpoten-
tial tightly matches the corresponding data in the literature. So,
the ReaxFF HDpotential may properly produce the structure of the
tri-atomic molecular ions.
TABLE IV . Average percentage of mono-, di-, tri-, and tetra-atomic species produced
after 1 ns simulation in X D= 0.2, 0.4, 0.5, 0.6, and 0.8 concentrations at 150 K.
Mono-atomic Bi-atomic Tri-atomic Tetra-atomic
4.59±1.83 12.05 ±3.18 72.84 ±1.29 10.52 ±1.32
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FIG. 8 . Schematic representation of the
tri-atomic molecules produced in all of
the simulations performed in this work. In
this picture, letters aandbrepresent the
average bond length, and αrepresents
the average bond angle. For two struc-
tures, the alternative experimental data
are inserted.
IV. CONCLUSIONS
The present study introduces a reactive force field (ReaxFF HD)
for the hydrogen/deuterium plasma, which was developed to allow
an accurate description of chemical species such as H 3+, H 2D+,
D2H+, and D 3+molecular ions in a discharge media. The ReaxFF HD
potential was parameterized against an appropriate training set
based on quantum mechanical data, describing bond dissociation
curves, valence angle distortion, and chemical exchange reactions.
To demonstrate the application of the ReaxFF HDpotential for sim-
ulation of the hydrogen/deuterium plasma, we performed a series
of reactive molecular dynamics simulations for different compo-
sitions of H/D mixtures. In these simulations, we observed some
intermediate molecules (H 2, D 2, and HD) that undergo secondary
reactions to form the tri-atomic molecular ions, such as H 3+, H 2D+,
D2H+, and D 3+, as the most likely products in the hydrogen plasma.
The simulations show that the ReaxFF HDpotential can success-
fully simulate all possible isotope-exchange reactions for the bi- and
tri-atomic ion species within the plasma media. Furthermore, the
ReaxFF HDpotential has a good transferability to reactions taking
place in this media. In particular, apart from the reactions in which
the ReaxFF HDpotential was parameterized, a variety of other chem-
ical reactions involved in the hydrogen plasma are observed during
the simulations. Accordingly, the chemical exchange reactions take
place in such a way as to produce tri-atomic molecular ions, some
of which are richer in deuterium rather than hydrogen, showing the
D-enhancement property at the core of the ReaxFF HDforce field.
Specifically, we find that D 3+and D 2H+molecular ions are produced
more readily than H 3+and H 2D+ions, because of the stronger iso-
tope effects within the ones richer in deuterium. We also found that
at all concentrations of the H/D mixtures, the deuterium fraction
within the tri-atomic molecular ions is higher than the hydrogen
fraction, which is related to the D-enhancement property of the force
field.
This study shows that the framework of the ReaxFF potential
allows simulation and prediction of the isotope effects within the
chemical isotope-exchange reactions. Therefore, it is expected that
the ReaxFF potential could be developed for other isotope-exchange
reactions of reactive and elusive species.
SUPPLEMENTARY MATERIAL
In this supplementary material, the time evolution of bi- and
tri-atomic molecules at other concentrations, and the time evolu-
tion of temperature and total energy are shown in Figs. S1 and S2,respectively. Also, more details regarding the procedure of the
ReaxFF HDparameterization have been reported.
DATA AVAILABILITY
The data that support the findings of this study are available
within the article and its supplementary material.
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1.1723210.pdf | Domain Observations on Iron Whiskers
R. W. DeBlois and C. D. Graham Jr.
Citation: Journal of Applied Physics 29, 528 (1958); doi: 10.1063/1.1723210
View online: http://dx.doi.org/10.1063/1.1723210
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IP: 132.174.255.116 On: Tue, 02 Dec 2014 15:11:57JOURNAL OF APPLIED PHYSICS VOLUME 29, NUMBER 3 MARCH, 1958
Domain Observations on Iron Whiskers
R. W. DEBLOIS AND C. D. GRAHAM, JR.
General Electric Research Laboratory, Schenectady, New York
We have confirmed the appearance in iron whiskers of domain walls which apparently violate the restric
tion that the normal component of magnetization must remain constant across the wall. These walls some
times have a dotted or serrated appearance. We have found such walls in a closure domain structure which
permits the calculation of an approximate value for the surface energy of the serrated wall; this energy is
approximately ten times the energy of a normal 90° wall or five times the energy of a normal 180° wall. We
conclude that the serrated wall contains an inner structure which largely eliminates any free poles.
SCOTT and Coleman, working with iron whiskers,
haye observed domain walls across which the nor
mal component of magnetization is apparently not con
stant.' Some of these walls appear to be serrated or
dotted. We have seen similar walls in whiskers under
conditions which permit us to make some estimate of
the surface energy of a serrated wall.
Our whiskers were grown by hydrogen reduction of
ferrous bromide at 700 to 800°C; they generally grew
with a (100) axis and {IOO} faces. The whiskers re
quired no surface preparation; domain walls were ob
served by the standard colloidal magnetite technique.2
Figure 1 shows the observable structure of a serrated
domain wall. This domain arrangement was obtained
by bending the whisker elastically about an axis per
pendicular to the page so that the top edge was under
tension and the bottom edge under compression.
Figure 2 shows a closure domain structure around an
imperfection at the edge of a whisker. Typical serrated
walls appear at the ends of the 90° closure domains.
FIG. 1. Serrated domain wall in elastically bent 45 p. whisker.
FIG. 2. Serrated domain wallsln closure domain
structure. 75 p. whisker. 1
1 R. V. Coleman and G. G. Scott, Phys. Rev. 107, 1276 (1957).
2 Williams, Bozorth, and Shockley, Phys. Rev. 25, 155 (1943). Figure 3(a) shows the closure domain pattern which
would be expected without serrated walls; comparison
with Fig. 3(b) shows that the appearance of the serrated
walls permits a considerable reduction in the total do
main wall area of the structure. If ')'* is the energy per
unit (projected) area of the serrated walls, and ')'0 is the
surface energy of a 90° wall lying in a {lOO} plane, we
can get an upper limit for the ratio ')'*/')'0 by computing
the difference in domain wall energy (area times sur
face energy) between Figs. 3 (a) and (b), and noting that
configuration b must have the lower energy since it is
the stable configuration observed. If the problem is re
garded as two dimensional, which is equivalent to saying
that the surface structure extends unchanged through
the entire thickness of the whisker, we find that ')'*!'Yo
~ 3.45. The surface energy of a 90° wall lying in a {HO}
plane is taken as (3)!')'o, and of a 180° wall in a {100}
plane as 2')'0.3 If the imperfection extends from the sur
face only to some depth d, then we find ')'* !'Yo~ 3.45
+V1t/2d+l/2d where t and 1 are the dimensions indi
cated in Figs. 3 (a) and (b). In the case we have observed,
Fig. 2, the imperfection is probably nearly equiaxed, so
that f?:!:d. Furthermore, ~5t. This gives ')'*!'Yo ~ 6.7.
-
FIG. 3. Alternative closure domain structures
around an imperfection.
3 L. Neel, Cahiers phys. 25, 1 (1944). (0)
(bl
528
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IP: 132.174.255.116 On: Tue, 02 Dec 2014 15:11:57DOMAIN OBSERVATIONS ON IRON WHISKERS 529
However, in Fig. 2 the domain structure is not exactly
that sketched in Fig. 3(b), but tends slightly toward the
configuration of Fig. 3(a). This indicates that 1'* is
somewhat higher than the calculated value. A reason
able conclusion is that 1'* is of the order of 101'0.
If the serrated domain wall actually consisted of a
single plane wall parallel to the magnetization of one
domain and perpendicular to the magnetization of the
adjacent domain, there would be associated with the
wall a demagnetizing field energy at least several orders of magnitude greater than 1'0.4 Since we have found the
surface energy of the serrated wall to be relatively low,
we conclude that the serrated wall has an internal struc
ture which largely eliminates any free poles. We suspect
that the serrated wall is made up of an array of normal
domain walls, each of which meets the condition that
the normal component of magnetization shall be con
stant across the wall.
4 K. H. Stewart, Ferromagnetic Domains (Cambridge University
Press, New York, 1954), p. 70.
JOURNAL OF APPLIED PHYSICS VOLUME 29. NUMBER 3 MARCH. !958
Magnetization Reversal by Rotation
P. R. GILLETTE AND K. OSHIMA *
Stanford Research Institute, Menlo Park, California
The mechanism of magnetization reversal by simple rotation has been discussed by a number of authors,
but a completely general analysis (applicable to a single-domain ferromagnetic body of any type of
anisotropy, with an applied field in an arbitrary direction) has not been given. The purpose of this paper is to
present such an analysis. General differential equations are derived which can be used to describe the rotation
process in any body for which demagnetization and anisotropy factors can be written. These equations are
then applied to a sphere, a cylinder, and a thin sheet. Both uniaxial and cubic anisotropy are considered for
the sphere and cylinder; only uniaxial anisotropy (the kind observed in deposited films) is considered for
the thin sheet. The resulting differential equations can be solved numerically to obtain output voltages,
switching times, and switching constants.
transform Eq. (1) into the Landau-Lifshitz form,
[(1+a2)/I'Y1 ] (dM/dt) IN analyzing the process of magnetization reversal by
rotation it is customary to begin with a general
equation of motion for the magnetization vector. Both
the Landau-Lifshitz equation! and a modification
proposed by Gilbert2 have been used for this purpose.
However, it can be shown that the predicted reversal
time obtained with the Landau-Lifshitz equation is
zero for infinite damping when actually it should be
infinite for infinite damping as well as for zero damping.
The Gilbert equation leads to a reversal time which
does become infinite for both zero and infinite damping. = -(MXH)- (a/M)[MX (MXH)]. (2)
In the mks system of units Gilbert's equation of
motion for the dipole magnetization vector M under
the influence of a total effective magnetic field H is
dM/dt= -11'1 (MXH)+ (a/M)[MX (dM/dt)], (1)
where M is the magnitude of M (assumed constant), a
is a phenomenological damping constant (assumed
positive), and l' is the gyromagnetic ratio or the average
ratio of the dipole magnetic moment to the angular
momentum associated with the spin of a ferromagnetic
electron (i.e., 11'1 =J.'vglel/2m). It is convenient to
* Now at Ampex Corp., Redwood City, Calif.
1 L. Landau and E. Lifshitz, Physik. Z. Sowjetunion 8, 153
(1935).
2 T. L. Gilbert and ]. M. Kelly, Proceedings of the Pittsburgh
Conference on Magnetism and Magnetic Materials, June 14-16,
1955 (American Institute of Electrical Engineers, New York,
1955), p. 253. If Eq. (2) is divided by M2 and the substitution
r= [I l' I /(1 +a2)]Mt
is made, the following equation is obtained, (3)
If contributions of conduction and displacement
currents and magnetostriction to the effective internal
magnetic field are negligible, H can be written as the
sum of the demagnetization field Hd, the effective
anisotropy field Hk, and the external field Ha. Let
M/M =im • .+jml/+km., (5)
Hd/M+Hk/M
=i( -d",-k",)m",+j ( -dll-kll)mll
+k( -dz-kz)m z,
=i( -n",)m",+j (-nll)mll+k( -nz)m., (6)
(where d"" dll, and d. are demagnetization factors and
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IP: 132.174.255.116 On: Tue, 02 Dec 2014 15:11:57 |
1.2409734.pdf | Emergence and evolution of tripole vortices from net-circulation
initial conditions
L. A. Barba
Department of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom
A. Leonard
Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, California 91125
/H20849Received 13 April 2006; accepted 16 November 2006; published online 3 January 2007 /H20850
The emergence of coherent vortical structures is a hallmark of the evolution of two-dimensional
turbulence. Two fundamental processes of this evolution have been identified in vortex merging andvortex axisymmetrization. The question of whether axisymmetrization is a universal process hasrecently been answered in the negative. In the linear approximation, vortices indeed becomeaxisymmetric, due to shear-enhanced diffusion. In the case of nonlinear interactions, other outcomesare possible; in the present work, we discuss a situation in which the flow reorganizes into a tripolarvortex. By performing an extensive numerical study, spanning the parameter space, we pursue thequestions of what dictates if the flow will become axisymmetric or will develop into a quasisteadytripolar vortex, and what are the stages and the time scales of the flow evolution. The initialcondition in this study consists of a Gaussian monopole with a quadrupolar perturbation. Theamplitude of the perturbation and the Reynolds number determine the evolution. A tripole emergesfor sufficiently large amplitude of the perturbation, and we seek to find a critical amplitude thatvaries with Reynolds number. We make several physical observations derived from visualizing andpostprocessing numerous flow simulations: looking at the decay of the perturbation with respect toviscous or shear diffusion time scales; applying mixing theory; obtaining the first few azimuthalmodes of the vorticity field; and describing the long-time evolution. © 2007 American Institute of
Physics ./H20851DOI: 10.1063/1.2409734 /H20852
I. INTRODUCTION
The emergence, behavior, and persistence of vortices in
two-dimensional flows has for long been a subject of fasci-nation to fluid dynamicists. Particularly in the last two de-cades, since the observation of vorticity concentrations thatarise spontaneously from random initial conditions,
1a huge
interest in these “coherent structures” of turbulence has de-veloped. The behavior of coherent vortices is characterizedby their tendency to assume axisymmetric shapes, their sta-bility, and their evolution in a cascade to larger scales.
The relaxation towards axisymmetry has been discussed
intensely. It was first detected in the evolution of cascadingtwo-dimensional /H208492D/H20850turbulence,
1occurring systematically
after vortex interactions, and then it was argued to be a ge-neric feature of two-dimensional vortices.
2That is, it was
implied that alllocalized and smooth concentrations of vor-
ticity will naturally approach axisymmetry.
The calculations of Melander et al.2use smooth elliptical
vortices as initial conditions, and establish that these behavevery differently from their sharp counterpart, the Kirchhoffvortex. The latter is an exact solution of the Euler equations,consisting in an elliptical patch of constant vorticity whichrotates steadily without change of shape /H20849Ref. 3, Art. 159 /H20850.I t
might seem plausible that the smoothed vortices should be-have nearly in the same way, perhaps slowly becoming axi-symmetric on a diffusion time scale. The results of Melanderet al.
2show, however, that axisymmetrization occurs quite
quickly and that it is an inviscid process. The kinematics ofthe process was then comprehensively explained by and at-
tributed to the formation and evolution of filaments from the
tips of the elliptical compact vortices. Filamentation, it wasargued, brings upon a deviation from the elliptical symmetry,which in turn activates a so-called “axisymmetrization prin-ciple,” relating changes in aspect ratio to the relative orien-tations of vorticity contours and streamlines.
This result, together with demonstrations of stability for
axisymmetric flows under certain conditions,
4led to sugges-
tions that axisymmetric vortices may be attracting states in2D flows at large Reynolds number.
5By means of the WKB
approximation, Bernoff and Lingevitch5determined that
rapid variations of vorticity decay on a fast time scale in theorder of Re
1/3, the shear-diffusion time scale6of passive sca-
lars. The assumption of rapid variation is justified by thecreation of small scales due to spiral wind-up of the pertur-bation in the base vortex. Hence, they show that an axisym-metric, planar vortex returns to axisymmetry after being per-turbed, and it does so at the shear-diffusion time scale /H20849i.e.,
faster than diffusion /H20850. Lundgren
7had previously arrived at
this result by different means, developing asymptotic 2D spi-ral solutions. In the viscous case, and under the assumptionsof large time and small viscosity
/H9263, Lundgren shows that the
characteristic decay time for all the higher harmonics is pro-portional to
/H9263−1/3. Only the axially symmetric component of
the flow decays on the slower time scale proportional to /H9263−1.
It may seem suspect to claim that the shear-diffusion
mechanism which homogenizes a passive scalar in a vortexPHYSICS OF FLUIDS 19, 017101 /H208492007 /H20850
1070-6631/2007/19 /H208491/H20850/017101/16/$23.00 © 2007 American Institute of Physics 19, 017101-1acts in an identical fashion to wane vorticity perturbations.
After all, vorticity is dynamically coupled to the velocity/streamfunction. Lundgren argues that as nonaxisymmetricperturbations become rapidly varying radially, through theeffect of differential rotation, the coupling of streamfunctionand vorticity cancels at leading order. Bernoff andLingevitch
5also rely on the assumption of rapid variation,
and show that far from the core of the vortex the perturbationstreamfunction becomes exponentially small. They then ad-dress the assumption by numerically observing this rapidwinding up of a perturbation in a Lamb vortex. The numeri-cal validation, however, relies on the linearized disturbance
equations.
In brief, a perturbed smooth vortex relaxes and becomes
axisymmetric in the fast shear-diffusion time scale, withinthe context of linear theory. Indeed, the linear approximationresults in an equation for the perturbation vorticity which isexactly the advection-diffusion equation except for a termthat becomes exponentially small far from the core of thebase vortex. Close to the core, on the other hand, the pertur-bation becomes rapidly varying due to the shearing caused
by differential rotation. The perturbation thus becomesstretched to very small scales, where diffusion rapidly acts tohomogenize it. As this occurs, the perturbation is said to be“expelled” to larger radii, where the streamfunction couplingis negligible, and the vorticity perturbation behaves like apassive scalar /H20849see Bernoff and Lingevitch
5/H20850.
The relaxation to axisymmetry in the inviscid case was
analyzed by Bassom and Gilbert,8by means of both asymp-
totics and numerical studies. In the absence of viscosity, thevorticity perturbations continue to cascade to small scalesdue to wind-up, and are never completely diffused. The pro-cess of axisymmetrization is understood in a coarse-grainedsense in this context; i.e., the nonaxisymmetric perturbationtends to zero on average. These studies were limited to suf-ficiently smooth vortices, in the linear approximation.
Full nonlinear simulations of the same flow as used by
Bernoff and Lingevitch,
5consisting in a Lamb vortex with
anm=2 perturbation, produced conspicuously different re-
sults than in the linear case.9It was found that, if the ampli-
tude of the perturbation is large enough, the flow does notrelax to axisymmetry. Instead, one observes the appearanceof a quasisteady, rotating tripole which seems to evolve inthe viscous time scale. Rossi et al.
9suggest that there is a
threshold amplitude below which the flow evolves towardaxisymmetry. Above the threshold, they argue, the negativeportion of the m=2 perturbation does not become mixed, due
to the creation of a separatrix of the streamlines in a framerotating with the structure. In contrast, the test flows ofMelander et al. ,
2leading to the suggestion of universality of
axisymmetry, involved purely positive vorticity distributions.Without negative vorticity, the shear-diffusion effects werenot disrupted. Clearly, if the perturbation vorticity has a con-siderable effect in the streamfunction, then the shear-diffusion mechanism can fail.
Other investigations have also suggested that axisymme-
trization is not a universal process. Simulations using con-tour dynamics methods with contour surgery /H20849see next sec-
tion /H20850by Dritschel
10show an elliptical vortex which evolvesto a nonaxisymmetric structure consisting of an elliptical
core surrounded by weak filaments. Some experimental re-sults in magnetically confined plasmas have exhibited similarnonaxisymmetric asymptotic states.
11Koumoutsakos12stud-
ied numerically, with a high-resolution vortex method, theinviscid evolution of several vorticity profiles. This studysuggested that axisymmetrization occurs for smooth vortices,whereas nonaxisymmetric configurations result from sharpvortices. Further studies using a generic class of vorticeswith sharp edges
13indicated that the steepness of the vortex
determines the degree of its axisymmetrization. Since invis-cid vortices tend to have sharp edges due to “stripping,”
14,15
the implication is that, in the absence of viscosity, vortices in
general do nottend towards axisymmetry. Viscosity would
thus be a contributing factor to axisymmetrization, because ithas the effect of smoothing sharp edges.
In the present paper, we continue the discussion of axi-
symmetrization /H20849or lack thereof /H20850of planar vortices. Using the
Lamb-Oseen vortex with a superposed m=2 perturbation of
moderate amplitude, a numerical study is performed span-ning the parameter space given by
/H9254and Re, the amplitude of
perturbation and the Reynolds number, respectively. Rossi et
al.9gave results for /H9254=0.25 and Re=103,5/H11003103,1 04,
which we have reproduced. In addition, we present calcula-tions with varying perturbation amplitudes, and attempt todetermine whether there is in fact a threshold amplitude thatseparates two asymptotic states /H20849axisymmetric and tripolar /H20850.
The possible relationship between this threshold amplitudeand Reynolds number is sought, and several observationsare made regarding the nonlinear evolution of the perturbedvortex.
II. NUMERICAL METHOD
In the numerical study of two dimensional vortical
flows, the preferred methods of computation seem to be con-tour dynamics methods and pseudospectral methods. Con-tour dynamics, introduced by Zabusky et al. ,
16consists in a
Lagrangian scheme that calculates the inviscid evolution of
vorticity jumps in flows consisting of piecewise constant dis-tributions of vorticity. They suffer the problem of productionof finer and finer structures, due to filamentation, but this hasbeen addressed by the variant called contour surgery.
10,17
For applications in viscous vortex interactions and tur-
bulence, many workers use the pseudospectral methods. Thisis the case in the studies of Melander et al.
2that put forth the
axisymmetrization principle. Most pseudospectral calcula-tions use hyperviscosity; it has been suggested, however, thatthe hyperviscosity operator may introduce spuriousdynamics.
18In studies of vortex stripping with combined
strain and diffusion effects,19it was found to introduce un-
physical numerical artifacts. On the other hand, the standardspectral method calculates a periodic domain and often alarge computational box is needed for simulation of an un-bounded vortex.
Presently, we use a two-dimensional viscous vortex blob
method.
20An inviscid vortex method was used in the work
of Koumoutsakos12on elliptical vortices, whereas Rossi
et al.9used a viscous method based on core spreading, de-017101-2 L. A. Barba and A. Leonard Phys. Fluids 19, 017101 /H208492007 /H20850scribed by Rossi.21The method used herein also utilizes the
core spreading approach to viscosity, but applies a new con-cept of meshless spatial adaption; the method was fullytested and presented by Barba
22and Barba et al.23
The vortex method was implemented in parallel, using
the PETSc library.24The calculations presented here were
carried out in 8–12 processors of a Beowulf cluster, and thenumber of vortex particles was in the order of 10
4. The cur-
rent limitation is mainly one of memory, as a square matrixof rank equal to the number of particles is built on initializa-tion and upon each process of spatial adaption. As describedin detail by Barba
22and Barba et al. ,23the method utilizes
radial basis function interpolation to find the circulationstrength of the vortex particles on initialization and after spa-tial adaption. Radial basis function interpolation is a highlyaccurate method for interpolation of scattered data,
25–27but it
requires the solution of a global influence problem. Precon-ditioned iterative methods are thus necessary,
28and presently
these were provided within the PETSc library. In the future,the efficiency of the implementation can be improved byusing a matrix-free approach, which is possible in PETScusing the “matrix shell” object. PETSc is an advanced libraryfor large-scale scientific computation, and provides all thefunctionality needed for the parallel vector and matrix as-sembly and most operations, doing most of the messagepassing so that the codes have a minimal amount of directcalls to message passing interface functions.
The numerical parameters used in these computations
were: an initial spacing of the particles h=0.15, and some-
times 0.12 or 0.096 /H20849the particles are placed on nodes of a
triangular lattice /H20850; an initial overlap of particles h/
/H9268=0.8,
where /H9268stands for the core radius of the vortex blobs. The
core basis function is a Gaussian: /H208492/H9266/H92682/H20850−1exp /H20849−r2/2/H92682/H20850.
The time step is /H9004t=0.5 and integration is performed via
fourth-order Runge-Kutta; frequency of spatial adaption isevery ten time steps /H20849resulting in a maximum core size due to
core spreading of 0.1901 for Re=10
4, and 0.2125 for Re
=103, for the coarsest case where h=0.15 and /H9268=0.1875 /H20850.
The above parameters relate to a base vortex that has totalcirculation equal to 1.0, and a Gaussian radius equal to 2.0.For details of accuracy studies with the present vortexmethod, see Refs. 22and29.III. EMERGENCE OF A TRIPOLE FROM PERTURBED
VORTICES
The flow under study consists of a Gaussian vortex with
a superposed m=2 /H20849quadrupolar /H20850perturbation that can be of
order 1. The initial condition is given by /H9275=/H9275o+/H9275/H11032, with
/H9275o/H20849x/H20850=1
4/H9266exp/H20873−/H20841x/H208412
4/H20874, /H208491/H20850
/H9275/H11032/H20849x/H20850=/H9254
4/H9266/H20841x/H208412exp/H20873−/H20841x/H208412
4/H20874cosm/H9258, /H208492/H20850
where /H9275ostands for the base vorticity, /H9275/H11032for the perturba-
tion, and /H9258=arg /H20849x/H20850. The main case discussed in Rossi et al.9
corresponds to /H9254=0.25 and Re=104, with Re= /H9003//H9263/H20849total cir-
culation divided by the viscosity /H20850. Figure 1shows the evolu-
tion of the total vorticity for this case. /H20849See Barba et al.23for
a comparison of results with our method to the results ofRossi et al. ,
9using similar line-contour plots of total vorticity
and perturbation vorticity. /H20850
The appearance of the tripole in the simulation depicted
in Fig. 1is significant for several reasons. The tripolar vortex
is much less common than the monopole and the dipole innumerical and experimental observations. A notable experi-ment in which a tripole emerged from an unstable vortex inrotating flow is that of van Heijst and Kloosterziel.
30At the
time, tripoles had not been observed in geophysical flows,but these authors offered that “it is not unlikely that oceanicexamples will be found in the near future.” Shortly after, anobservation was made in images taken by the NOAA 11satellite of what appears to be a tripolar structure. The obser-vation corresponds to a vortex of about 50–70 km in diam-eter, which lived for almost a year in the Bay of Biscay.
31
This oceanic vortex, named F90, exhibited tripolar structurefor only a few days, however. Therefore, it is not a definitiveobservation of an oceanic tripolar vortex.
Numerical evidence of a vortex tripole was first seen by
Legras et al. ,
32who observed its spontaneous emergence in
two-dimensional, forced, homogeneous turbulence. More de-tailed simulations showing flows evolving to tripoles wereperformed by Carton et al. ,
33who used spectral methods to
study perturbed, linearly unstable axisymmetric vortices.
FIG. 1. Initial condition of Eqs. /H208491/H20850
and /H208492/H20850relaxing to a quasisteady tri-
pole; Re=104,/H9254=0.25; 14 equally
spaced contours of vorticity normal-ized by
/H9275max/H208490/H20850.017101-3 Emergence and evolution of tripole vortices Phys. Fluids 19, 017101 /H208492007 /H20850They proposed as the mechanism for generation of tripoles
the growth and saturation of instabilities. In turn, more de-tailed experiments were presented by van Heijst et al. ;
34
these show the generation of compact tripoles from stirring-
induced vortices, and the photographs provided in their paperare quite striking. Laboratory tripoles are usually formed insituations of rotating flow, with initial conditions consistingof isolated monopoles. An isolated or shielded vortex con-sists of a core of vorticity surrounded by a ring of oppositevorticity, with zero total circulation. These structures mayexperience the growth of a wavenumber 2 perturbation, lead-ing to a drastic change in topology /H20849the formation of the two
disconnected satellites from the outer ring of negative vortic-ity/H20850. The present work, in contrast, does not study shielded
vortices /H20849which are unstable /H20850, but stable Gaussian mono-
poles, with a finite amplitude perturbation /H20849which destabi-
lizes them /H20850.
IV. DETAILED NUMERICAL STUDY
A. General characteristics of the flow evolution
The numerical experiments in the present paper treat
nonlinearly perturbed vortices with a quadrupolar perturba-tion which is often large. As shown in the first frame of Fig.1, the initial condition has regions of negative vorticity,
which occupy two disconnected inclusions on opposite sidesof the core of positive vorticity. One could in fact say that the
initial condition is already a sort of tripole; it is, however, farfrom steady state. The tripole vortex obtained at later times/H20849last frame of Fig. 1/H20850is a quasisteady flow structure. During
the flow evolution, perhaps similar to the case of shieldedmonopoles, there is a change in the flow topology which inthis case can be described by the “pinching” of the contourlevel of zero vorticity, as seen in Fig. 2to happen between
t=500 and t=600.
Figure 2presents a useful way to visualize the flow; it
consists of gray-scale contour levels of the logarithm of vor-ticity. The absolute value of the vorticity is used, to avoidproblems with the logarithm of a negative value, and theeffect is one of emphasizing the region where the vorticitychanges sign. This boundary between positive and negativevorticity winds up around the base vortex, and at some timepinches, forming the isolated satellites of negative vorticitythat characterize a tripole. After this transient, during whichthe vorticity reorganizes into a quasisteady tripole, the struc-ture decays slowly and sometimes exhibits oscillations.
In contrast to the situation depicted in Fig. 2, if the initial
amplitude of the perturbation is /H20849relatively /H20850small, the bound-
ary between positive and negative vorticity lies farther awayfrom the core. Subsequently, as it winds up, it folds flat ontoitself, without pinching to enclose a region of negative vor-ticity. This is the situation shown in Fig. 3, which corre-
FIG. 2. Same case as Fig. 1;p l o to f
the logarithm of /H20841/H9275/H20841, emphasizing the
level zero of vorticity. Black in thegray scale is saturated at level 10
−6
/H20849enhanced online /H20850.
FIG. 3. Plot of the logarithm of
/H20841/H9275/H20841, emphasizing the level zero of vor-
ticity; Re=104,/H9254=0.10 /H20849enhanced
online /H20850.017101-4 L. A. Barba and A. Leonard Phys. Fluids 19, 017101 /H208492007 /H20850sponds to /H9254=0.10, for which the flow is seen to relax towards
axisymmetry.
A further insight into the evolution of the flow for large
and small amplitude initial perturbations is obtained fromvisualizing the perturbation itself. By “perturbation” at timest/H110220, we mean the field obtained from subtracting from the
total vorticity the Lamb-Oseen solution. This is shown inFigs. 4and5for the cases discussed above, i.e.,
/H9254=0.25 and
/H9254=0.10 with Re=104. For the larger amplitude, the perturba-
tion vorticity starts winding up, but only the positive partforms a spiral structure. The negative perturbation remains astwo separate inclusions, which come to form the tripole sat-ellites. The spirals of positive perturbation become homog-enized, and commence to merge towards the core, leavingweak filamentary debris that surrounds the whole structure.For the smaller amplitude initial perturbation, the same com-mences to happen, but the negative inclusions becomesqueezed in between the positive spirals. They becomesheared and stretch around the core, and thus become ho-mogenized. Of course, it is not entirely correct to speak of“perturbation vorticity” as we have, by just subtracting aspreading Gaussian from the total vorticity field, because theaxisymmetric base flow and the nonaxisymmetric componentare coupled; i.e., the flow is nonlinear for moderate ampli-tudes /H20849as will be discussed /H20850. However, it is a useful means of
visualizing the evolution of the flow and of characterizingthe decay of the vortex structure.
A very interesting feature stands out in these plots of the
perturbation. Notice in the last frames of Figs. 4and5that a
darker small region remains at the center of the vortex, indi-
cating that the positive perturbation does not completely mixthere. It has been found that the time scale of the shear-diffusion mechanism is slower at the center of a smooth
vortex,
35where the gradient of angular velocity vanishes
/H20849and thus, there is no differential rotation /H20850. Instead of varying
with Reynolds to the power1
3, the shear-diffusion time scale
is order Re1/2at the center of a vortex. We conjecture that
this defect in shear-diffusion is evidenced in the darker cen-ter of the perturbation plots. In the case with the larger per-turbation /H20849Fig.4/H20850, one can see that the center defect stands in
the way of the final merging of the positive perturbation,which in turn contributes to retain an elliptical core, anothercharacteristic feature of tripolar vortices.
B. Parameter study
We now consider situations in which the perturbation
amplitude lies between the two cases discussed above, with
FIG. 4. Perturbation vorticity normal-
ized by /H9275max/H208490/H20850;R e = 1 04,/H9254=0.25.
Gray is zero, black is negative, whiteis positive /H20849enhanced online /H20850.
FIG. 5. Perturbation vorticity normal-
ized by /H9275max/H208490/H20850;R e = 1 04,/H9254=0.10.
Gray is zero, black is negative, whiteis positive /H20849enhanced online /H20850.017101-5 Emergence and evolution of tripole vortices Phys. Fluids 19, 017101 /H208492007 /H20850the same Reynolds number. In these cases, by means of plot-
ting the logarithm of vorticity as before, we observe that thezero-contour level of vorticity folds and pinches leaving asmall satellite of negative vorticity, which decays in a shorttime. The size of these satellites increases with the value
/H9254,
and thus they live longer for larger initial perturbations.
Figure 6corresponds to a calculation with Re=104and
/H9254=0.125. The frames show the times when the zero-contour
of vorticity pinches, and later when the narrow satellites thatwere formed diffuse away. Here we encounter a stumblingblock: Is this a tripole? Structurally, it could be considered atripole with extremely weak satellites /H20849see Fig. 7/H20850. These
very weak satellites disappear almost immediately, and theflow proceeds to become axisymmetric. The conundrum pre-sents itself because we would like to determine a threshold
amplitude for the emergence of the tripole, and moreoverdetermine its relationship with Reynolds number. One could,for example, consider the case in Figs. 6and7to be just
barely above that threshold, as for larger values of
/H9254all cases
will develop a tripolar structure with larger and largersatellites.
In the first stage of this numerical study, 37 different
simulations were carried out as part of a parametric study.Ten simulations were performed for a Reynolds number of10
4, with different amplitudes of perturbation between 0.1
and 0.25. Eleven simulations were carried out for Re=3/H1100310
3, with /H9254=0.1–0.3, and sixteen for Re=103with
/H9254=0.075–0.35. That there is in fact a “threshold” amplitude
separating two sorts of temporal behaviors is illustrated bythe plot in Fig. 8. Each marker in the plot represents one
simulation, parametrized by the initial amplitude of the per-turbation /H20849
/H9254/H20850and Re. The quantity on the plot’s ordinate is
the ratio of minimum to maximum vorticity in the flow /H20849in
absolute value /H20850. This ratio quantifies the importance of the
negative inclusions with respect to the total structure, andthus it is a measure of the nonaxisymmetric state; the ratiotends to zero over time, indicating that the negative satellites
become mixed and decay. Rossi et al.
9plotted this quantity
with respect to the viscous time, for three simulations with
/H9254=0.25 and different Reynolds numbers /H20849their Fig. 9 /H20850; they
conclude that the tripolar structure decays by diffusion. Onthe abscissa of Fig. 8is the value of
/H9254, and the runs with
same Reynolds number have been joined by a dotted line.The quantity /H20841
/H9275min//H9275max/H20841is shown for a time slice t=600 that
corresponds to approximately four turn-over times, which isgenerally enough time for the initial transient when the flowreorganizes. It can be seen that there is a sharp decay of thenegative inclusions at this time for a given value of
/H9254that is
smaller for larger Reynolds number.
To illustrate how the quantity /H20841/H9275min//H9275max/H20841varies in time,
it is plotted in Fig. 9for ten different simulations with
Re=3 /H11003103, at different times. It can be seen that a sharp
drop occurs in this diagnostic at a given time that varies with
/H9254. This is associated with the mixing of the negative vorticity
inclusions, and thus when a tripole is present the sharp dropdoes not occur.
FIG. 6. Plot of the logarithm of /H20841/H9275/H20841, emphasizing the level zero of vorticity;
Re=104,/H9254=0.125.
FIG. 7. Re=104,/H9254=0.125; vorticity normalized by /H9275max/H208490/H20850, contour levels
at/H208510.8, 0.6, 0.4, 0.2, 0.04,−1 /H1100310−4/H20852.
FIG. 8. Ratio of minimum to maximum vorticity /H20841/H9275min//H9275max/H20841,a tt=600, vs
initial amplitude of the perturbation /H9254for the 37 simulations of the param-
eter study /H20849each marker represents one simulation /H20850.
FIG. 9. Ratio of minimum to maximum vorticity /H20841/H9275min//H9275max/H20841vs initial am-
plitude of the perturbation /H9254for ten simulations with Re=3 /H11003103, at differ-
ent times /H20849the arrow indicates increasing time /H20850.017101-6 L. A. Barba and A. Leonard Phys. Fluids 19, 017101 /H208492007 /H20850The above discussion exposes the possibility of a bifur-
cation between the tripole and monopole states, as assumedby Rossi et al.
9It does not, however, aid us in determining
the threshold value of the initial perturbation separating thetwo asymptotic solutions.
Let us for the moment consider the “pinching” of the
zero contour of vorticity to indicate the emergence of a tri-pole, even though it may be very short lived. With thischoice, we can find a “critical” or threshold amplitude of the/H20849initial /H20850perturbation above which the tripole forms, for each
of the three values of Reynolds number used /H20849so far /H20850. This is
plotted in Fig. 10as the three values joined by a solid bold
line /H20849disregard for the moment the dashed extensions to this
line /H20850. Admittedly, this is a rather arbitrary definition of the
threshold, but in the absence of a more categorical or betterdefinition, the results in Fig. 10seem to indicate an approxi-
mate Re
−1/3relationship for the critical amplitude /H20849in the
present range of Reynolds number /H20850. With a different choice
or definition of what constitutes the threshold, perhaps an-other Reynolds number dependence might be found.
Let us now consider that such weak tripoles for which
the satellites are very short lived should be discarded. Onecould decide, for example, that a tripole must exist for atleast half a turnover time. With this new criterion, we findagain the threshold initial perturbation for each Reynoldsnumber, above which the satellites survive at least half aturn. That is, the time between the pinching of the zero con-tour of vorticity and the disappearance of the satellites ofnegative vorticity must be at least
/H9270/2 with /H9270=4/H9266//H9275max /H20849we
consistently used /H9275maxatt=0/H20850. Plotting the result of this
exercise in Fig. 10as the dash-dotted line just above the
solid line, one finds once again a Re−1/3relationship /H20849ap-
proximately /H20850for the critical amplitude. If, furthermore, one
defines the threshold such that the satellites survive one fullturn-over period, then the upper dash-dotted line is obtained,also displaying a Re
−1/3relationship. /H20849In Fig. 10, the error
bars show how closely two consecutive runs are, in theirrespective values of
/H9254, between which the chosen threshold
lies. /H20850The parameter study presented so far—Fig. 10
in particular—was partially motivated by the study ofLe Dizès,
36in addition to Rossi et al. ,9who proposed that
there should be a critical value of /H9254separating tripoles from
monopoles. Le Dizès utilized asymptotic methods to analyzeslightly nonaxisymmetric vortices at high Reynolds number.In Le Dizès, the perturbation parameter measures the size
/H9280
of the nonaxisymmetric disturbance at the vortex core, oncethe asymptotic solution is established. Thus, it is not linearlyrelated to our amplitude parameter
/H9254. This can be seen in Fig.
10, where we also show the corresponding critical /H9280as a
function of Reynolds number, computed using the methodsof Sec. IV G below. /H20849We calculated
/H9280using the amplitude of
them=2 mode; the asterisks on the plot of Fig. 10show the
value of /H9280multiplied by a constant to fit in the same axis. /H20850As
a result of his analysis, Le Dizès he conjectured that thethreshold amplitude for the existence of nonaxisymmetricstates should decrease with increasing Reynolds number.That is clearly in agreement with the present numericalstudy. However, the results discussed above also seem toindicate a Re
−1/3relation, in the given range of Re, which
was not predicted by the asymptotics. Instead, the work ofLe Dizès predicts a Re
−2/3relationship for the critical ampli-
tude of the nonaxisymmetric component /H20849a −2/3 slope is
drawn in Fig. 10for comparison /H20850. He directly compared his
asymptotic results with the numerical solutions found byRossi et al. , and argued that they correspond to the same
solutions. Indeed, good agreement was shown not only in theco-rotating streamlines but also in the value of the angularfrequency.
Perhaps the reason for the disagreement with the predic-
tion of Le Dizès is that he based his study on an equationwhich is obtained at leading order. That is, higher-order non-linear terms were neglected in the governing equation, rep-resenting a radial analog of the critical layer equation.
Nevertheless, we attempt to characterize the relationship
between critical amplitude of the initial nonaxisymmetriccomponent resulting in a tripole vortex, and Reynolds num-ber. To verify whether the critical amplitude of the initial
FIG. 10. Threshold amplitude of the initial perturbation /H9254vs Reynolds num-
ber: for pinching of zero-contour /H20849solid line /H20850, or for half-turn or full-turn
satellite survival /H20849dash-dotted lines above /H20850.
FIG. 11. Decay of the tripolar structure: ratio of minimum to maximum
vorticity vs viscous time, for different values of Re and /H9254.017101-7 Emergence and evolution of tripole vortices Phys. Fluids 19, 017101 /H208492007 /H20850perturbation follows a power law relationship with Reynolds
number, new series of simulations were performed forRe=500 and Re=3 /H1100310
4. We encounter some computational
difficulty in the large or small Reynolds number range, dueto increase of the required memory and CPU time. When Reis smaller, the vortex method with core spreading requiresmore frequent spatial adaption /H20849to control core sizes /H20850, and
also the vorticity support grows considerably due to viscousspreading. This leads to increased problem size and thusCPU time and memory requirements. On the other hand,when the Reynolds number is very large, the resolution mustbe increased, which also leads to increased problem sizes.For this reason, our capacity to extend the curve in Fig. 10is
limited /H20849more effort can still be made to improve code effi-
ciency, however /H20850. The new series with Re=3 /H1100310
4spanned
/H9254=0.09 to 0.2 /H20849eight simulations /H20850, and the new series with
Re=500 spanned /H9254=0.2 to 0.375 /H20849nine simulations /H20850. The
threshold values for the formation of a closed zero-contourof vorticity were found and are plotted in Fig. 10, corre-
sponding to the extensions with dashed line.It is apparent that one cannot put forward a power-law
relationship between Reynolds number and the threshold am-plitude of the initial perturbation that leads to emergence ofthe tripole. The trend in Fig. 10also indicates that there
might be an inviscid limit, a critical value of
/H9254above which
the flow always organizes into a tripole in the large Reynoldsnumber limit. This is intriguing and deserves furtherresearch.
C. Time scale of relaxation
Next, we consider the time-decay of the vortical struc-
ture. Figure 11shows the evolution of the ratio between
minimum and maximum vorticity, the minimum vorticity in-dicating the strength of the negative inclusions. Three differ-ent values of the initial perturbation amplitude and three dif-ferent Reynolds numbers are shown; the decay is plottedwith respect to t/Re, the viscous time scale.
Rossi et al.
9considered the decay of the tripole structure
with regards to this quantity; in Fig. 9 of their paper, theyshow only the case
/H9254=0.25 for three Reynolds numbers: 103,
FIG. 12. Decay of the tripolar structure: ratio of minimum to maximum
vorticity /H20841/H9275min//H9275max/H20841, vs shear-diffusion time scale, for different values of Re
and/H9254.
FIG. 13. Decay of the tripolar structure: negative and positive perturbation
vorticity amplitude /H20849max-norm /H20850vs shear-diffusion time scale, for different
values of Re and /H9254;to=0.017101-8 L. A. Barba and A. Leonard Phys. Fluids 19, 017101 /H208492007 /H208505/H11003103, and 104. They conclude that the structure is decay-
ing on the viscous time scale, as the evolution at various Reseems to scale well with t/Re. We note, however, that by
plotting the decay in the same manner for the smaller ampli-tude perturbations, the scaling no longer applies well /H20849see
Fig.11/H20850. In fact, even for the larger
/H9254=0.25, our results do
not exhibit a clear scaling with Re−1. As shown graphically
and discussed in Ref. 23, our results disagree with those of
Rossi et al. for the lower Re=103, and we attribute this to the
increased numerical diffusion introduced by the frequentsplitting events required in Rossi’s method. In consequence,our results exhibit a retardation in the decay for Re=10
3,i n
comparison with Ref. 9.
Furthermore, Rossi et al. acknowledge that their “great-
est frustration is that /H20851they /H20852do not observe rapid shear-
diffusion mixing in the tripolar attractor.” They speculate ona lack of numerical resolution hindering the observation ofthe mechanism, and also point out that the lack of an analyticsolution for a tripole makes measurement of the relaxationproblematic. We now address this issue of shear-diffusion inthe evolution of this flow.
First, consider simply plotting the same quantity of Fig.
11, but with respect to the shear-diffusion time scale: t/Re
1/3.
For small amplitude perturbations, one finds that this scalingdoes collapse rather well the evolution for different Reynoldsnumbers /H20851see Fig. 12/H20849a/H20850/H20852. As the amplitude of the perturba-
tion is increased, however, the decay curves for the differentvalues of Re start to grow apart and the scaling no longerworks well /H20851Fig. 12/H20849b/H20850/H20852. This result indicates that shear-
diffusion could be active in the attenuation of the small per-turbations but not the large ones. This should not be surpris-ing. Shear-diffusion is a linear mechanism, and as discussedearlier vorticity perturbations can only be considered passiveif they are very small. When the perturbation is large, non-linear effects are important or even dominant, and there is noreason to expect a Re
1/3time scale /H20849so perhaps the frustration
of Rossi et al. was not merited /H20850. At large /H9254, all that can be
observed is a viscous decay of the tripolar structure, asshown in Fig. 11. The reorganization of the vorticity leading
to emergence of the tripole is not effected by shear-diffusionalone; rather, it is produced by a combination of viscosityand nonlinear effects in an interplay that may possibly in-clude some dynamical instability.
Now, consider the decay of the perturbation vorticity
only. Plotting the maximum /H20849positive /H20850and minimum /H20849nega-tive /H20850perturbation field values as they evolve, it can be seen
again that the Re
1/3time scaling appears to apply well only
for the small perturbations. As shown in Fig. 13, once the
perturbation amplitude /H20849max-norm /H20850is large, the behavior be-
comes more complicated. Indeed, for large Re and large /H9254,
the perturbation visibly oscillates during the self-organization stage.
D. Stability of the tripole structure
To test the robustness of the tripole vortex, we per-
formed the following experiments. The final state of thesimulation to a time t=800 /H20849five turn-over times /H20850of the case
with Re=10
4and/H9254=0.25 is used as initial conditions. Before
initiating a continuation run, the particle positions are per-turbed randomly, as in a random walk with a step
/H9280/2. Using
the perturbed particles, a continuation run is performed uptot=1600. Three continuation runs were carried out, with
/H9280=0.01, 0.05, 0.1 /H20849where the value 0.1 is comparable to h,
the interparticle spacing /H20850. The initial conditions thus obtained
are shown in Fig. 14, which includes plots of equal contour
levels of the perturbation vorticity for the three randomlyperturbed cases and the unperturbed one; i.e., Fig. 14/H20849a/H20850cor-
responds to the same field as that shown in the last frame ofFig.4.
The evolution of the case with
/H9280=0.01 is practically in-
distinguishable from the unperturbed case, whereas bothother cases also eventually relax back to the tripole solution.The perturbation vorticity at two different times is plotted inFig. 15for the case
/H9280=0.1 and for the unperturbed case. It
can be seen that the difference in /H9275/H11032is very slight by
t=1600. The flow self-organizes back to the unperturbed
evolution, which is also evident by plotting the normalizedperturbation max-min field values, shown in Fig. 16.
It is quite clear that the tripole is a very robust structure,
once formed. We consider this to be evidence that the tripoleis indeed an asymptotic solution. There remain importantquestions about what are the mechanisms involved in its for-mation, the understanding of which merit further research.
E. Connection with mixing theory
Similarly to the process of vortex merging, which occurs
in several stages of different characteristic time scales, onecan try to characterize the nonlinear relaxation studied herein terms of stages. In the beginning of the first stage, one can
FIG. 14. Initial conditions for tripole runs with perturbed output case Re=104,/H9254=0.25. Contour levels ± /H208510.04:0.02:0.32 /H20852/H20849higher contours not present due to
decay of perturbation vorticity from t=0/H20850.017101-9 Emergence and evolution of tripole vortices Phys. Fluids 19, 017101 /H208492007 /H20850conjecture that the process of mixing is active. When the
amplitude of the perturbation is small, this is the only rel-evant stage, and shear-diffusion dominates at a Re
1/3time
scale. However, with large perturbations, nonlinear effectsbecome important. In particular, it seems that only the posi-tive perturbation is responding to mixing processes when theamplitudes are large.
To support the claim that mixing is active in the first
stage of the process, we look for evidence of the mixing time ,
that is, the characteristic time for the initiation of decay ofthe perturbations. This matter was discussed in detail byMeunier and Villermaux,
37who performed careful experi-
ments of dye mixing in a Lamb-Oseen vortex. They wereable to develop a nearly exact description for this problem,determining the evolution of the scalar concentration and anestimate of the mixing time. We use their result that themaximal concentration of scalar, observed at the center of aspiral branch, at radial distance rfrom the center of the base
vortex, evolves as /H20851their Eq. /H208493.11 /H20850/H20852
c
M/H20849r,t/H20850=c0erf/H208731
/H20881/H9270/H20874, /H208493/H20850
where cMis the maximal concentration, c0is the initial con-
centration, and
/H9270=D
s02t/H208731+/H90032t2
3/H92662r4/H20874. /H208494/H20850
In these equations, the symbols represent: /H9003, the circulation
of the base vortex; r, the radial location of the drop of scalar
with respect to the center of the vortex; s0, the initial size of
the scalar drop; and D, the molecular diffusion rate. Noting
that the scalar concentration in portions of a spirally de-formed drop is constant until the mixing time, denoted t
s,
they are able to obtain an estimate for this time /H20849the concen-
tration varies as an error function, the argument of which isorder 1 until the mixing time, which gives the result /H20850.W e
reproduce here their Eq. /H208493.12 /H20850:
t
s/H20849r/H20850=r2
/H9003/H208733/H92662
16/H208741/3/H20873s0
r/H208742/3/H20873/H9003
D/H208741/3
. /H208495/H20850
Note that the mixing time depends on Pe1/3, where Pe
=/H9003/Dis the Péclet number. After the mixing time, there
should be a decay of the concentration scaling as t−3/2, the
limiting slope of the error function.For the case with Re=104and/H9254=0.25, for which the
evolution of the perturbation amplitudes is depicted in Fig.21, we plot the expected scalar concentration evolution ac-
cording to Eq. /H208493/H20850. The correct parameters for this case are
D→
/H9263=10−4,/H9003=1.0, and r=2, which is the radial location of
the maxima of the perturbation initially, and s0=2.0, which is
the characteristic size of the initial perturbation peaks. Notethat none of these parameters is adjustable, they are given bythe initial flow conditions only. The initial maximal vorticityperturbation is 0.3673 when using
/H9254=0.25.
As shown in Fig. 17, the evolution of the positive part of
the perturbation seems to capture very well the mixing time/H20851the time at which the scalar concentration according to
Eq. /H208493/H20850starts to drop /H20852. The negative perturbation, it seems,
does not evolve according to mixing principles at any time.When performing the same analysis for the smaller value of
/H9254=0.1, however, the results are more difficult to interpret.
This case is presented in Fig. 18, and one can see that the
maximum of the perturbation /H20849the positive maximum indi-
cated by the diamond markers /H20850starts to drop approximately
at the mixing time. The diagnostic used, as in Fig. 17,i st h e
maximum perturbation value in the whole field, whereas themixing theory of Meunier and Villermaux refers to the con-centration peak at the same radial location with respect to thecenter of the vortex. It can be seen when visualizing the
FIG. 15. Perturbation vorticity at two times for continuation runs in the case Re=104,/H9254=0.25. Contour levels ± /H208510.04:0.02:0.32 /H20852/H20849higher contours not present
due to decay of perturbation vorticity from t=0/H20850.
FIG. 16. Perturbation vorticity amplitude /H20849max-norm /H20850normalized by
/H9275max/H208490/H20850vs viscous time, for Re=104and/H9254=0.25; continuation run up to
t=1600 of perturbed case with /H9280=0.1 on the same plot as unperturbed case
/H20849see text /H20850.017101-10 L. A. Barba and A. Leonard Phys. Fluids 19, 017101 /H208492007 /H20850perturbation vorticity that the peak is on the spiral arms and
is being expelled to larger radii /H20849Fig.5/H20850. In the previous case
of/H9254=0.25 this did not present a problem, because the per-
turbation maximum does remain at approximately the sameradial location /H20849this can be seen in Fig. 4/H20850. If we plot the
maximum /H20849and minimum /H20850perturbation found on a circle of
initial radius equal to 2, which spreads at the rate of a Lamb-Oseen vortex, then the decay is considerably faster. This di-agnostic is shown in Fig. 18corresponding to the open circle
and asterisk markers. The effect shown responds to the factthat the maximum of the perturbation is in fact being ex-pelled to larger radius, as it occurs in the passive scalar case.
In brief, there is some evidence that the mechanism that
is active in the most early stage of the relaxation is mixing,though it is inconclusive. It is quite remarkable, however,that the evolution of the positive part of the perturbationcaptures the mixing time so well. Once again, note that thereare no adjustable parameters in Eq. /H208495/H20850. In any case, it is only
the positive part of the perturbation that is susceptible to thismechanism, whereas the negative part decays more rapidly/H20849earlier than the mixing time /H20850in the initial stage. The visu-
alization of the perturbation vorticity shows that the positivepart is subject to spiral wind-up most intensely. The negativepart, in contrast, appears to resist spiral wind-up, which maybe attributed to nonlinear effects.
F. Evolution stages and long-time evolution
A series of continuation runs were carried out, to observe
the long-time evolution of the tripole. Consider the caseRe=3 /H1100310
3and/H9254=0.25. Figure 19shows the logarithm ofvorticity for the time when the zero contour pinches, the end
of the first simulation /H20849t=800 /H20850, and the time on a continua-
tion run when the satellites are about to disappear. The nega-
tive satellites in this case survive for about six turn-overtimes. With Re=10
4and/H9254=0.25, the zero vorticity contour
pinches at t=580 and the negative inclusions survive for a
very long time indeed. As seen in Fig. 20, the satellites are
slowly getting smaller, but by t=4000 they still survive; this
is about 25 turn-over times. They finally die out at t=4600.
Indeed the tripole seems to be a robust, quasisteady structure.
Studying the case Re=104in the last frame of Fig. 13,
which shows the decay of the perturbation amplitude, it isapparent that the relaxation of the flow consists of severalstages. Consider Fig. 21, which shows the evolution in the
viscous time of the amplitude of positive and negative per-turbations, for a continuation run up to t=3200 of the case
Re=10
4,/H9254=0.25. First, there is a rapid decline of the pertur-
bation, followed by a nonlinear adjustment, and finally avery slow evolution. It has previously been asserted by otherauthors that this slow stage is simply viscous decay. Notethat in the final stage depicted, the amplitude of the pertur-bations is between 0.05 and 0.1 /H20849normalized by
/H9275maxat
t=0/H20850, so it is still not very small.
The intermediate stage in the relaxation process is one
where oscillations of the perturbation suggest an interplay ofdynamical effects. This is evident in particular in the largeamplitude cases. During this stage, the flow reorganizes intothe tripole, which is usually completely formed after betweenthree and four turn-over periods /H20849the time taken for the zero
FIG. 17. Perturbation vorticity magnitude normalized by /H9275max/H208490/H20850vs time on
a log-log plot; Re=104and/H9254=0.25. Evolution of maximal concentration
according to Eq. /H208493/H20850in continuous line, and t−3/2slope in dashed line.
FIG. 18. Perturbation vorticity magnitude normalized by /H9275max/H208490/H20850vs time on
a log-log plot; Re=104and/H9254=0.1. Evolution of maximal concentration
according to Eq. /H208493/H20850in a continuous line, and t−3/2slope in dashed line. The
additional curves /H20849open circles and asterisks /H20850correspond to /H9275/H11032measured on
a spreading circle of initial radius r=2.
FIG. 19. Plot of the logarithm of
/H20841/H9275/H20841, emphasizing the level zero of vor-
ticity; Re=3 /H11003103,/H9254=0.25.017101-11 Emergence and evolution of tripole vortices Phys. Fluids 19, 017101 /H208492007 /H20850contour of vorticity to pinch, enclosing the satellites /H20850. The
physical mechanisms at play in this self-organization processare not well understood.
G. Nonlinear evolution of the tripole
We have mentioned that the reorganization of the tripole,
and the decay of the nonaxisymmetric component, exhibit anoscillatory behavior. The oscillations are particularly appar-ent when looking at the decay of the perturbation for thehigher Reynolds number and larger
/H9254cases, as seen in Fig.
13. To educe the oscillatory behavior, we look at the L2-norm
of the perturbation vorticity as it evolves in time. We havealso performed simulations at larger Reynolds number, spe-cifically Re=3 /H1100310
4,1 05, with /H9254=0.2. As seen in Fig. 22,
the oscillations are damped more strongly with decreasingReynolds number, suggesting viscous effects.
We have also acknowledged that there are limitations in
using the perturbation vorticity, as defined here, to character-ize the evolution, because there may be in fact transfer ofenergy to and from the axisymmetric component of the flow.Hence, to complete the analysis of the problem, we now lookat the first few Fourier modes of the vorticity field.
Using the following definition of the modes, we calcu-
late them for n=0, 2, 4:
/H9021
n=1
2/H9266/H20885
02/H9266
/H9275/H20849r,/H9258,t/H20850ein/H9258d/H9258. /H208496/H20850
We numerically obtain the modes from the vorticity field
values on a sampling grid, and choose to look at these inparticular for the case shown in Fig. 22with Re=10
5and/H9254=0.2, for three different times. The times chosen corre-
spond as closely as possible to the first minimum of the plot,the first maximum after this, and the second minimum. Thesehappen at times t=600, t=1100, and t=1700, respectively, as
can be seen in the plot.
First, compare the radial distribution of the amplitude of
the mode /H9021
0for the three times, with a Gaussian distribu-
tion. This is shown in Fig. 23. Clearly, there are deviations
from the Gaussian distribution. In the inner region, the am-plitude of the zeroth mode is slightly larger than a Gaussian;in the tail, it is lesser than the Gaussian and it features avalley and local maximum. Most interestingly, the plotshows how this tail snakes up and down at different times. Att=600 /H20849continuous line /H20850, the tail exhibits the greatest devia-
tion from the Gaussian. This corresponds to the time inFig. 22/H20849c/H20850when the L
2-norm of the perturbation vorticity
reaches its first minimum. At t=1100 /H20849dashed line /H20850, the dis-
tribution of /H90210is closest to the Gaussian, but one still detects
a small hump on the tail /H20849around r=5/H20850. For the next time,
i.e., t=1700 /H20849dash-dotted line /H20850,/H90210has again deviated from
the Gaussian. This series of /H90210/H20849r/H20850plots displays clearly that
the axisymmetric mode of the vorticity is exchanging some
energy with the nonaxisymmetric component, in an oscilla-tory fashion. We suggest that this is a manifestation of thenonlinear character of the evolution.
Figure 24shows the amplitudes for the three modes /H9021
0,
/H90212,/H90214, at the three times discussed in the previous para-
graph. The plots for /H90212show how this mode grows and
shrinks during one oscillation, and in addition displays a ra-dial rearranging. The /H9021
4mode is mostly confined to the
larger radii /H20849around r=4/H20850, indicating perhaps that it captures
the filaments around the structure.
The three modes were also calculated for the main cases
discussed in the beginning sections of the paper, withRe=10
4and/H9254=0.25 and /H9254=0.1; recall that for the first case,
the flow develops into a quasisteady tripole, whereas for thesecond case it evolves towards axisymmetry. For the larger
/H9254, we can see that the 2-mode decays to about half its am-
plitude between t=100 and t=750, but its radial distribution
changes little, with the peak moving only slightly fromr/H110152t o r/H110152/5 /H20849see Figs. 25and26/H20850. There are several
differences with the case with smaller
/H9254=0.1; first, the am-
plitude of the 2-mode was already much smaller at the earlytime, t=100 /H20849in comparison with the previous case with
larger
/H9254/H20850. Most notably, at the later time, t=750, the mode
had experienced considerable radial rearrangement, with thepeak now close to r=4. This is now a signature of the left-
over spiral arms, as the satellites of negative vorticity have
FIG. 20. Plot of the logarithm of
/H20841/H9275/H20841, emphasizing the level zero of vor-
ticity; Re=104,/H9254=0.25.
FIG. 21. Perturbation vorticity normalized by /H9275max/H208490/H20850vs viscous time, for
Re=104and/H9254=0.25; continuation run up to t=3200.017101-12 L. A. Barba and A. Leonard Phys. Fluids 19, 017101 /H208492007 /H20850completely decayed. The maxima are expelled to larger radii,
as would also be the case for shear-diffusion of passivelysmall perturbations.
V. CONCLUSIONS
A two-dimensional vortical flow consisting of a Lamb-
Oseen vortex with a quadrupolar perturbation of finite am-plitude reorganizes into a quasisteady tripole for largeenough perturbation. This paper presents a systematic nu-merical study of this flow situation, including a parametricstudy /H20849in terms of Reynolds number and amplitude of the
initial perturbation /H20850, and a detailed examination of the long-
time relaxation. The parametric study was partly motivatedby previous numerical results of Rossi et al. ,
9by which itwas argued that a threshold perturbation amplitude might
exist separating the domains of attraction of two asymptoticsolutions: the monopole and the tripole. It was also partlymotivated by asymptotic studies of Le Dizès,
36showing that
slightly nonaxisymmetric vortices could survive without ex-ternal forcing, and that the amplitude of the nonaxisymmetriccorrection should exhibit a −2/3 power law with respect toReynolds number.
Analyzing the results of several simulations with each of
three different Reynolds numbers, it was first observed that acritical or threshold amplitude of the initial perturbationleading to the emergence of the tripole behaved approxi-mately as Re
−1/3. Additional series of simulations with larger
and smaller values of the Reynolds number indicated that thepower law relationship only applies in a limited range of Re.As the Reynolds number is increased, the slope of the curvefor critical amplitude decreases. This, in turn, suggests thatthere might be an inviscid limit, i.e., a critical value of theinitial perturbation amplitude above which the flow alwaysreorganizes into a tripole in the large Reynolds number limit.Thus, we find that the theoretical results of Le Dizès do notproperly apply to the flow that we have studied. This is likelythe result of considerable nonlinear effects which cannot becaptured in the theory based on a leading order equation. Themodel of Le Dizès applies in the limit of small perturbationsand large Reynolds number.
Detailed analysis of the simulation data was used to pro-
vide physical insight on the flow situation. A relaxation in theshear-diffusion time scale is apparent for cases when the am-plitude is relatively small, but only a viscous time scale canbe extracted from cases with large amplitude. This is ex-plained by the fact that shear-diffusion is a linear mecha-nism, for passive scalars, and in the present flow nonlineareffects are dominant when the perturbation has largeamplitude.
The very slow decay of the perturbation vorticity after
the initial reorganization stage suggests the possibility of anonlinear amplification mechanism. In fact, we have foundthat there is transfer of energy from the nonaxisymmetricmodes to and from the axisymmetric mode of the vorticity.
The tripole proves to be a very robust structure, as
FIG. 22. Decay in time of the L2-norm of the perturbation vorticity, for three
simulations at large Reynolds number.
FIG. 23. Amplitude of the mode-0, /H20881/H20841/H90210/H208412at three different times, for the
case Re=105and/H9254=0.2.017101-13 Emergence and evolution of tripole vortices Phys. Fluids 19, 017101 /H208492007 /H20850shown by experiments where a random perturbation was in-
troduced to the vortex particles at an intermediate time. Forperturbations of the particle locations as large as to be in theorder of the interparticle spacing in the vortex method, theflow self-organizes back to the tripole solution. This result isconsidered evidence that the tripole is indeed an asymptotic
solution of the Navier-Stokes equations. The mechanisms in-volved in the tripole’s formation, however, are not well un-derstood, and warrant further investigation.
In summary, this work connects with the previous
FIG. 24. Amplitude of three modes,
i.e.,/H90210,/H20881/H20841/H90212/H208412,/H20881/H20841/H90214/H208412as a function of
radius; same case and times as Fig. 23.
FIG. 25. For the three modes, i.e., /H90210,
/H90212,/H90214, the solid line corresponds to
the real part, and the dotted line corre-sponds to the imaginary part; caseRe=10
4,/H9254=0.25, at the indicated
times. /H20849Note that the scale of the yaxis
varies from plot to plot. /H20850017101-14 L. A. Barba and A. Leonard Phys. Fluids 19, 017101 /H208492007 /H20850literature9,36by seeking to determine a threshold amplitude
of the nonaxisymmetric component of the flow such that atripolar vortex can survive. It differs from previous workregarding the tripolar vortex, as obtained in laboratory ex-periments, in the fact that the laboratory tripole is alwaysobtained from the growth and saturation of an instability inan isolated /H20849zero total circulation /H20850vortex. Here, the flow has
total circulation equal to 1.0, and the amplitude of the initialnonaxisymmetric component determines the strength of thesatellites relative to the vortex core. Thus, we can obtain afamily of nonshielded tripoles, with satellites of varying size.We have determined a possible threshold perturbation ampli-tude /H20849acknowledging a degree of arbitrariness in the defini-
tion of what constitutes the threshold /H20850, for different Reynolds
numbers. The critical amplitude is found not to exhibit apower law relationship with Reynolds number, and thus con-tradicts theoretical results by Ref. 36. We think one possible
reason for the disagreement is the considerable nonlinear ef-fects present in the flow, and neglected in the critical layerapproach of Ref. 36. Perhaps the nonaxisymmetric vortices
of Ref. 36are entirely different solutions from the tripolar
vortices studied here. We have explored the signature of non-linear effects by calculating the first few azimuthal modes ofthe vorticity field, at different times in the evolution. Thisreveals that there is a constant exchange of energy to andfrom the axisymmetric component, mostly evident for thelarge perturbations. For the smaller perturbations, one seesalso the 2-mode being expelled to larger radii with time, in asimilar way to shear-diffusion effects. With regard to thedifferent time scales of the evolution of the flow under study,we find that for small perturbations, the decay scales reason-ably well with shear-diffusion time scale, whereas for largeramplitudes only a viscous time scale is observed. Finally, wetested the robustness of this new tripole with nonzero totalcirculation, and find that it is a very stable structure, recov-ering from random perturbations, and surviving for manyturn-over periods.ACKNOWLEDGMENTS
Computing time provided by the Laboratory for Ad-
vanced Computation in the Mathematical Sciences/H20849LACMS /H20850of the University of Bristol /H20849http://
lacms.maths.bris.ac.uk/ /H20850. Thanks to the PETSc team for
prompt and always helpful tech support. L.A.B. thanks S. LeDizès and E. Villermaux for discussions and correspondence.L.A.B.’s travel was possible thanks to a Nuffield FoundationAward.
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1.4862963.pdf | Switching current density reduction in perpendicular magnetic anisotropy spin transfer
torque magnetic tunneling junctions
Chun-Yeol You
Citation: Journal of Applied Physics 115, 043914 (2014); doi: 10.1063/1.4862963
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Appl. Phys. Lett. 93, 232506 (2008); 10.1063/1.3046729
Spin transfer switching in Tb Co Fe Co Fe B Mg O Co Fe B Tb Co Fe magnetic tunnel junctions with
perpendicular magnetic anisotropy
J. Appl. Phys. 103, 07A710 (2008); 10.1063/1.2838335
Switching characteristics of submicrometer magnetic tunnel junction devices with perpendicular anisotropy
J. Appl. Phys. 97, 10C919 (2005); 10.1063/1.1854282
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129.12.30.104 On: Wed, 26 Mar 2014 15:18:32Switching current density reduction in perpendicular magnetic anisotropy
spin transfer torque magnetic tunneling junctions
Chun-Y eol Y ou
Department of Physics, Inha University, Incheon 402-751, South Korea
(Received 28 September 2013; accepted 9 January 2014; published online 27 January 2014)
We investigate the switching current density reduction of perpendicular magnetic anisotropy spin
transfer torque magnetic tunneling junctions using micromagnetic simulations. We find that the
switching current density can be reduced with elongated lateral shapes of the magnetic tunnel
junctions, and additional reduction can be achieved by using a noncollinear polarizer layer. Thereduction is closely related to the details of spin configurations during switching processes with the
additional in-plane anisotropy.
VC2014 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4862963 ]
I. INTRODUCTION
In studies of spin transfer torque magnetic random
access memory (STT-MRAM), the perpendicular magnetic
anisotropy (PMA) free layer is a key issue for the realization
of non-volatile memory devices.1In order to overcome ther-
mal stability problems and achieve a smaller switching cur-
rent density, adapting a PMA free layer is the best solution.2
It is well known that the switching current density, Jc, of the
PMA free layer can be smaller than that of the in-plane free
layer because of the opposite role of demagnetization energy
in the determination of the switching current density. In thein-plane free layer, the inevitable demagnetization energy
acts as an energy barrier in the switching process. However,
the demagnetization energy helps the switching of the PMAfree layer (it causes a lower value of J
c, which has led
to active studies of the PMA STT-MRAM).3,4However, the
value of Jcis still higher than the practical circuit require-
ment; therefore, lowering the value of Jcin the PMA free
layer is still a challenging issue.
Recently, we reported on a series of micromagnetic simu-
lation results for the in-plane free layer of a STT-MRAM. We
found that Jccan be noticeably reduced by introducing a
“noncollinear” polarizer,5,6and strong dependences of Jcon
the exchange stiffness Aexand on the lateral shapes.7,8
Furthermore, if we introduce an artificial symmetry breaking
structure (a small cutting of the one edge of the ellipse), a
much lower value of Jcis found due to the more coherent spin
switching.9Such approaches cannot be explained with a sim-
ple macro-spin model.10–12Only micromagnetic simulations
can reveal such detailed spin dynamics during the switching
process. Therefore, micromagnetic simulations are essentialin order to obtain a deep insight into the switching processes.
In the present study, we applied the aforementioned
approaches to the PMA free layer case. We varied the shape
of the magnetic tunneling junction (MTJ) from a circle to an
ellipse by elongating one axis length. We found that theelongation of one axis resulted in a very different switching
mode compared with the circular MTJ switching mode, and
the value of J
cwas reduced for larger aspect ratio ellipses. In
addition, when we introduced the noncollinear polarizer,13,14
a noticeable reduction in Jcwas observed. We will nowdescribe the detailed spin dynamics that occurred during
the switching processes, and the physical reasons for thereduction in J
cfor various cases. We used micromagnetic
simulations based on the Object-Oriented MicroMagnetic
Framework (OOMMF)15software with the public STT
extension module.16
II. MICROMAGNETIC SIMULATIONS
Figure 1shows a typical PMA MTJ structure. In the fig-
ure, the “F Free,” “Insulator,” and “F Polarizer ” layers represent
the free, insulator, and polarizer layers, respectively. The
thicknesses of the F Free, Insulator, and F Polarizer layers are 2,
1, and 1 nm, respectively. The saturation magnetizations ofF
Freeand F Polarizer are 1.3 /C2106A/m. The interface perpen-
dicular magnetic anisotropy energy Ksof the free and polar-
izer layers is 0.625 and 2.6 /C210/C03J/m2, respectively,
without volume anisotropy energy. We note that the effec-
tive PMA energy, Kef f¼Ks
tF/C01
2l0M2
s, depends on the ferro-
magnetic layer thickness tF. The thinner polarizer layer is
harder than the thicker free layer. The exchange stiffnessenergy ( A
ex¼3.0/C210/C011J/m) and the Gilbert damping con-
stant ( a¼0:02) were fixed in this study, even though the
exchange stiffness energy affected the value of Jc.7A unit
cell size of 1 /C21/C21n m3was used. The positive current is
defined from the polarizer to the free layers. The positive
current prefers anti-parallel (AP) configurations, as shown inFig.1. We applied a specific current density Jfor 10 ns from
the stable spin configurations, and determined whether or not
the switching occurred in order to determine J
c. More details
of the micromagnetic simulations can be found in our previ-
ous report.16For simplicity, we ignored the field-like term,
and all simulations were done at 0 K, thermal activation spinmotion was not considered.
III. DEPENDENCES OF JcON THE LONG-AXIS
LENGTH OF THE ELLIPSE
Our typical experiments for the PMA MTJ employed a
circular shape,1because the general belief is that in-plane an-
isotropy is not important in PMA free layer switching based
on the simple macro-spin model. Figures 2(a)–2(d) show
selected spin configuration snapshots of 40 /C240 nm circular
0021-8979/2014/115(4)/043914/6/$30.00 VC2014 AIP Publishing LLC 115, 043914-1JOURNAL OF APPLIED PHYSICS 115, 043914 (2014)
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129.12.30.104 On: Wed, 26 Mar 2014 15:18:32free layer switching processes. Figure 2(b) shows that the
in-plane components of spin form a counterclockwise rota-
tion at 9.0 ns. Such a circular spin configuration persists for a
while due to the circular rotational symmetry of the system.After 9.5 ns, the rotation symmetry broke down, as shown in
Figs. 2(c) and2(d). After breaking the circular symmetric
spin motion, the switching occurred. We plotted the time-dependent local M
z/Msin Fig. 3in order to better understand
at the five specific points shown in Fig. 3(see A–E in
Fig. 2(a)). The figure clearly shows that points A and B (on
the left side) oscillated together, and points D and E (on the
right side) show out-of-phase oscillations with respect to
A and B in the red box (9–10 ns). The motions of point C(center part of the free layer) are distinguishable from the
others. Such anti-symmetry behaviors were discussed in our
previous study for the in-plane symmetric system.
6,9We
pointed out that the breaking of such asymmetric spin
dynamics (out-of-phase spin motion) can introduce more
coherent spin motion, and can reduce Jc.
In order to break the circular symmetry of the system,
we first elongated one axis length from 40 to 80 nm. Thus,
the shapes of the MTJs are ellipses with different long axislengths a. We depict the micromagnetic simulations results
in Fig. 4for AP-to-P and P-to-AP switching current density.
We find that the elongated ellipse have smaller J
cfor either
cases. It implies that the in-plane anisotropy play an impor-
tant role in the determination of Jc, despite irrelevant contri-
butions of it in the macro-spin model. Here, the in-planeanisotropy occurred as a result of different demagnetization
coefficients of the x-andy-axes. Even though the ellipticity
is not large and the in-plane anisotropy is small, they weresufficient to change the detailed spin dynamics. When the
spins precess, they preferred to tilt toward the x-axis, and the
orbit of spin precession became an ellipse due to the in-planeanisotropy. We note that such reduction seems quite similar
to the in-plane results.
8However, the underlying physics are
totally different. For the in-plane cases, the dependence ofjunction size on the current density is unpredictable and
complicated
8because it is closely related to the domain wall
width and the junction size. In our previous study for the lat-eral shape dependent study for the in-plane cases, we found
that the spin configurations are far from the single domain
during the reversal processes. Excited spin waves withvarious wave vector form complex domain configurations
FIG. 1. Schematics of the magnetic tunneling junction structure of the per-
pendicular magnetic anisotropy free layer. The coordinate system, the cur-
rent direction, and short and long axes of the ellipse are shown.
FIG. 2. Snapshots of spin configurations of the free layer with a circular
shape ( a¼b¼40 nm) at a specific time with a switching current density of
Jc¼2.87/C21011A/m2. (a) t¼8.0, (b) t¼9.0, (c) t¼9.5, and (d) t¼9.75 ns.
In (a), points A–E indicate specific locations in the free layer.
FIG. 3. Time-dependent Mz/Msat points A–E of the circular shape free
layer. Noticeable switching occurs during 9.0–10.0 ns within the red box.
The spin dynamics of points A and B are the opposite of points D and E, and
the center point C is different than the other points.043914-2 Chun-Y eol Y ou J. Appl. Phys. 115, 043914 (2014)
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129.12.30.104 On: Wed, 26 Mar 2014 15:18:32with domain wall, and the excitation are governed by the
exchange stiffness, effective anisotropy energies. Here, the
effective anisotropy included the shape anisotropy, whichstrongly depends on the lateral shape of the MTJ. Since the
typical exchange length of the free layer magnetic material
is comparable with the lateral junction size, the dependenceis not a simple function. Therefore, there are no simple
explanations. However, the dependence of ain PMA is very
predictable, as shown in Fig. 4.
Figures 5(a)–5(d) show selected snapshots for 80
/C240 nm
2cases at various times for the Jc.F i g u r e 5(b) shows
that the in-plane components of the left and right edge areopposite, which is quite different than the circular case
(Fig. 2(b)). After 9.0 ns, the asymmetric spin motion became
more clear (Figs. 5(c) and 5(d)), and the additional shape
in-plane anisotropy plays an important role. Eventually,
switching occurs in this case. However, the switching processdetails are not the same as in the circular case because of the
additional in-plane anisotropy. The time-dependent local
M
z/Msfor the 80 /C240 nm2case is shown in Fig. 6. The figure
shows that the time-dependent Mz/Msat points A and G (the
edges of the ellipse) is almost identical, whereas the motions
of the in-plane component are the opposite, as shown inFigs. 5(b) and 5(c). And, they are already switched after
9.0 ns. Points B and F also showed the same time dependent
and switched about 9.5 ns. Points C and E are similar, butswitching occurs later. The important point is the behavior of
center point D. This point was not switched until 11 ns, despite
the switching was already occurred in the other parts of thefree layer, and that the spin polarized current was already
turned off at 10 ns. However, when the switching occurred at
point D, it switched more abruptly. Similar behavior occurred
FIG. 4. Dependence of Jcon the length of the free layer long axis afor
P-to-AP and AP-to-P switching.
FIG. 5. Snapshots of spin configurations of the free layer with an ellipticalshape ( a¼80,b¼40 nm) at a specific time with a switching current density
ofJ
c¼2.25/C21011A/m2. (a) t¼7.5, (b) t¼8.0, (c) t¼9.0, and (d)
t¼10.5 ns. In (a), points A–G indicate specific locations in the free layer.
FIG. 6. Time-dependent Mz/Msat points A–G of the ellipse shape free layer
(a¼80,b¼40 nm). The spin dynamics of points A and G, E and F, and C
and E are almost identical, whereas the spin motion of center point D ismore abrupt and late.
FIG. 7. Dependence of Jcon the tilt angle of the noncollinear polarizer
layer, with various lengths of free layer long axis afor P-to-AP and AP-to-P
switching043914-3 Chun-Y eol Y ou J. Appl. Phys. 115, 043914 (2014)
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129.12.30.104 On: Wed, 26 Mar 2014 15:18:32for the in-plane free layer case with the elliptical shapes in our
previous studies.6Based on our observations, we conclude
that the additional in-plane anisotropy also plays an important
role in the switching processes, which is in contrast to the sim-ple macro-spin model.
We note that the total current ( J
c/C2area) did not
decrease, while the value of Jcdecreased for the elongated
a-axis case. Therefore, there is no benefit when we consider
real MTJ circuit design. However, we can appreciate the
importance of small in-plane anisotropy in the switching pro-cess. In Sec. IV, we consider a noncollinear polarizer that
leads to additional unidirectional in-plane anisotropy.
IV.JcWITH THE NONCOLLINEAR POLARIZER LAYER
FOR A PMA FREE LAYER
Including our previous studies,5,6it is known that a
noncollinear polarizer17,18leads to reductions in Jcfor anin-plane MTJ. For the PMA case, a noncollinear polarizer
will reduce the value of Jcas shown in Fig. 7. The directions
of the easy axis of the polarizer were varied from 0 to 20/C14
with respect to the film normal in order to introduce the
noncollinear polarizer. We emphasize that tilting of the easy
axis was easily implemented in the micromagnetic simulations;
however, it is not simple in the real experiments. For example,there is a report on a controllable tilting angle by using an
exchange spring system.
19Nguyen et al.19suggested that the
combination of a PMA hard layer with an in-plane soft layergives a continuous tilting angle a s a function of the soft layer
thickness. Since this is general phenomena, any combination of
hard PMA and in-plane soft layers will work.
With a finite tilting angle, the J
cis reduced for the circu-
lar and elliptical shape free layers, and they increase again
after the tilt angle exceeds 5/C14. Such tendencies were discussed
in our previous work5,6for the in-plane anisotropy MTJ. In
our previous studies, we revealed that the non-collinear polar-
izer layer enhanced the coherent spin rotation during theswitching process. With the collinear polarizer layer, the
switching process always involve complex spin configuration
with incoherent spin dynamics for the in-plane case, which issimilar to Figs. 2and5. However, the noncollinear polarizer
breaks the symmetry of the system, and it enhanced the
FIG. 8. Snapshots of spin configurations of the free layer with the circular
shape ( a¼b¼40 nm) at a specific time. The switching current density
(Jc¼1.875/C21011A/m2) is applied, and the tilt angle of the noncollinear po-
larizer layer is 5/C14. (a) t¼8.0, (b) t¼8.5, (c) t¼9.5, and (d) t¼10.0 ns. In
(a), points A–E indicate specific locations in the free layer.
FIG. 9. Time-dependent Mz/Msat
points A–E of the circular shape free
layer. (a) The asymmetric spin dynam-
ics of the left- and right-side edges dis-
appear. (b) Before starting the large
spin motion, the out-of-phase motions
of both edges are still observed.
FIG. 10. Snapshots of spin configurations of the free layer with an ellipticalshape ( a¼80,b¼40 nm) at a specific time. The switching current density
(J
c¼1.81/C21011A/m2) is applied, and the tilt angle of the noncollinear po-
larizer layer is 5/C14. (a) t¼7.0, (b) t¼7.4, (c) t¼8.0, and (d) t¼9.5 ns. In (a),
points A–H indicate specific locations in the free layer.043914-4 Chun-Y eol Y ou J. Appl. Phys. 115, 043914 (2014)
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129.12.30.104 On: Wed, 26 Mar 2014 15:18:32coherent spin motions. It must be noted that our approaches
differ from the work done by Beaujour et al. ;20they investi-
gated the effect of the perpendicular polarizer layer with
in-plane switching layer with additional in-plane reference
layer in order to obtain Tunneling Magnetoresistance (TMR)signal. In this study, we introduce the noncollinear (tilted)
PMA polarizer with PMA free layer.
With a noncollinear polarizer, the additional in-plane
anisotropies have unidirectional natures, while the elliptical
shape free layers have additional uniaxial in-plane anisot-
ropy. Such additional unidirectional in-plane anisotropy willlead to different spin dynamics over all free layer areas.
Figures 8(a)–8(d) show very different spin motions com-
pared with those shown in Figs. 2(a)–2(d). The spin motions
shown in Fig. 2hold circular symmetry. However, coherent
spin motions were observed with the noncollinear polarizerlayer shown in Fig. 8. The differences are clear when
Figs. 2(b) and8(b) are compared.
To obtain a better understanding, the time-dependent
M
z/Msvalues at points A and E are shown in Figs. 9(a) and
9(b). In Fig. 9(a), the spin motions of points A–E are not
strongly correlated after 9.5 ns, while they are strongly corre-lated before 8.5 ns, as shown in Fig. 9(b). The out-of-plane
components, M
z/Ms, for the left and right edges show the
opposite motion as in the collinear polarizer case shown inFig. 6for a small spin motion (Fig. 9(b)). However, when
the spin motion is increasing, additional unidirectional
in-plane anisotropy plays a more important role, and suchcorrelations are broken after 9.2 ns (Fig. 9(a)).
Next, we adapt the noncollinear polarizer for the ellipti-
cal free layers. Snapshots of the spin configuration of80/C240 nm
2are shown in Figs. 10(a) –10(d) . At 7.0 and
7.4 ns (Figs. 10(a) and10(b) ), it appears that the spins are
coherently rotated over the whole areas because the noncol-linear polarizer breaks the symmetry of the system.
However, when we consider the time-dependent M
z/Ms
shown in Fig. 11(b) , we note that the out-of-plane compo-
nents of the spin motion are not very coherent. The center
part (C, D, E) shows strong motion, and the right-hand edge
(H) shows mostly weak spin motions before 8.0 ns.However, after 8.0 ns, the switching occurs from A to H
(from the left edge to the right edge) because of the addi-
tional unidirectional in-plane anisotropy, as shown inFig. 11(a) . Points F, G, and H were switched after the cur-
rent pulse was turned off at 10.0 ns.We observed that the noncollinear polarizer layer
reduced the TMR values. However, with a tilting angle of5
/C14, the TMR value was 99.6% of the collinear TMR.
Therefore, the TMR reduction is not a serious disadvantage
for the noncollinear polarizer system.
V. CONCLUSION
We varied the aspect ratio of the free layers and intro-
duced noncollinear polarizer layers for PMA MTJ structures.We found that an elongated one-axis length introduced addi-
tional uniaxial anisotropy, and the additional anisotropy
broke the circular symmetry of the spin motion during
switching processes. While the spin configuration of the cir-
cular shape PMA free layer holds circular symmetry duringthe switching processes (Fig. 2), the additional in-plane ani-
sotropy promotes a laterally asymmetric spin motion (Fig. 5)
as in the in-plane cases.
6With the additional in-plane anisot-
ropy, the switching current density decreased as shown in
Fig.4. A further reduction can be achieved by employing the
noncollinear polarizer layers. With a tilt angle of 5/C14, noticea-
ble reductions can be established as shown in Fig. 7. The
noncollinear polarizer layer causes unidirectional in-plane
anisotropy, and also breaks the symmetry with respect to thespin dynamics (Figs. 8–11). It is clear that the breaking of
the symmetric spin dynamics helps to reduce the switching
current density, as we already described for the in-planecases.
9Based on our micromagnetic simulations, we suggest
that the switching current densities of PMA free layers
depend on the lateral aspect ratio and the polarizer tiltingangle. In other words, they depend on additional in-plane
anisotropy. Therefore, a more detailed consideration of the
additional in-plane anisotropy should be addressed in futureexaminations of the PMA STT-MRAM.
ACKNOWLEDGMENTS
This work was supported by the Korean Research
Foundation (NRF) (Grant Nos. 2013R1A1A2011936 and
2012M2A2A6004261).
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129.12.30.104 On: Wed, 26 Mar 2014 15:18:32 |
1.1557340.pdf | Increase of magnetic damping in thin polycrystalline Fe films induced by Cu/Fe
overlayers
P. Lubitz, Shu Fan Cheng, and F. J. Rachford
Citation: Journal of Applied Physics 93, 8283 (2003); doi: 10.1063/1.1557340
View online: http://dx.doi.org/10.1063/1.1557340
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/93/10?ver=pdfcov
Published by the AIP Publishing
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128.83.63.20 On: Thu, 04 Dec 2014 14:38:44Increase of magnetic damping in thin polycrystalline Fe films induced
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~Presented on 15 November 2002 !
The ferromagnetic resonance properties of thin polycrystalline layers in the sequence Cu/Fe/Cu/Fe/
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INTRODUCTION
Spin transport in thin film structures consisting of ferro-
magnetic conductors separated by nonmagnetic metal layershas been shown to produce dramatic magnetoresistance phe-nomena, as manifested in so-called giant magnetoresistanceGMR.
1Recent theoretical2,3and experimental studies have
shown that such spin transport can produce torques and othermanifestations of the nonequilibrium spin distribution: cur-rents driven through these structures can induce reversal ofthe static magnetic moment
4and the excitation of nonequi-
librium magnetic precession such as spin wave excitationand amplification.
5Arelated effect recently demonstrated by
Urbanet al.6is a dramatic increase in magnetic relaxation in
a component of a GMR structure, as seen in the increasedferromagnetic resonance ~FMR !linewidth of an Fe layer.
While pure Fe is not usually a component of GMR struc-tures, it was, in fact, a component of one of the first reportedspin-valve type structures,
7indicating moderate spin polar-
ization of an induced current in Fe. Furthermore, Fe has thenarrowest FMR linewidth and therefore the smallest relax-ation rate for magnetic precession of any of the 3 dtransition
metal magnetic materials, as also demonstrated by Urbanet al.
6in epitaxial single crystals and in polycrystalline
films.8,9The reported intrinsic relaxation rate G0’1.26
3108/sec.Another related effect recently predicted3and pos-
sibly observed10is additional damping in single ferromag-
netic layers caused by spin transport out of the layer; thiseffect appears also to be relevant to our measurements.
The effects documented by Urban et al.
6are for epitaxial
films of Fe/Au/Fe where the outer Fe layer is relatively thick.The linewidth of the thinner layer, which can be distin-guished by its higher resonance field ~at which the thicker
layer is not at resonance !, is additionally broadened by an
amount inversely proportional to its thickness—one of thepredictions implied by Berger’s model—but the linewidth ofan isolated layer did not depend on its thickness in the rangestudied. To confirm that this linewidth represented true ‘‘Gil-bert’’ relaxation or viscous damping, they measured thewidth over a wide range of frequencies. For their films, es-sentially the entire linewidth scaled linearly with frequency,as predicted for Gilbert damping. Some studies using only asingle frequency can only be said to give an upper bound onthe relaxation rate;
10yet others in which the contribution of
inhomogeneities, which may be reduced in high fields, isdominant, may underestimate the rate.
11
We have studied a similar system, Cu/Fe/Cu/Fe/Cu,
which was made under less sophisticated laboratory condi-tions but provides comparably narrow FMR widths, consis-tent with Ref. 6, and which verified some of their results.Furthermore, we explored such additional variables as theintermediate Cu layer thickness and the temperature depen-dence of the linewidth of both single Fe films and pairs.
EXPERIMENT
We prepared Cu/Fe multilayer structures using magne-
tron sputtering onto nearly atomically flat Si wafers having anative SiO amorphous surface, as in Ref. 8. In all cases a Culayer was deposited directly onto the wafer before depositionof the first Fe layer. This was found necessary to avoid par-tial oxidation of the interface Fe, which we had found earlierto produce broadening and additional surface anisotropy forFe~Ref. 9 !and Permalloy.
12Subsequent to the first, thinner
Fe layer, we deposited a second or ‘‘spacer’’Cu layer, then a20 nm thick Fe layer, and finally a thick Cu cap, again to
a!Electrnic mail: lubitz@anvil.nrl.navy.milJOURNAL OF APPLIED PHYSICS VOLUME 93, NUMBER 10 15 MAY 2003
8283
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128.83.63.20 On: Thu, 04 Dec 2014 14:38:44avoid oxidation.We made structures with the first Fe layer in
a range of thicknesses from 2 to 5 nm and we exploredintermediate Cu layer thicknesses from 2 to 10 nm.
The static magnetic properties of the films at 300 K were
determined from vibrating sample magnetometer ~VSM !
data. FMR measurements were made with conventionalspectrometers at 300 K and 9.46 and 34.7 GHz and the line-widths reported are the separation of the extrema of the fieldderivative of absorbed power. Some FMR data were alsotaken at temperatures down to 77 K at both frequencies.
RESULTS
VSM data indicate that both Fe layers are nearly isotro-
pic magnetically, with the thinner layer having a coercivefieldH
c, of about 4 Oe and the thicker layer having Hcof
about 10 Oe. The loops generally have remanence of about95%, but the thinner Fe layer loops are sheared slightly whilethe thicker loops are more nearly square.
The relevant VSM result is that, for the composite struc-
ture, the thinner Fe layer behaves very much like an isolatedlayer of comparable thickness for Cu spacer thicknesses of 5nm and above. At 3 and 4 nm Cu, there is still a hint ofseparate switching, although some coupling is evident;roughness ~Ne´el!or possibly exchange may be responsible.
Clearly, the static interaction is insignificant, except perhapsfor the thinnest Cu spacers, especially compared to the addi-tional width and shift of the FMR line position as describedbelow.
In contrast to Urban et al.
6we find that single Fe layers
contain a linewidth term that is roughly proportional to 1/ d
and that scales linearly with frequency, i.e., apparently a vis-couslike damping term ~Fig. 1 !. Note that the linewidth is
closely proportional to frequency, although some data show asmall finite intercept. For this paper we have only used themost widely spaced frequencies conveniently available to de-termine the slope; our earlier work
8and that of others6,11
show negligible departures from linearity within this andeven over a much extended range. Fo ra3n mF e layer, the
1/dterm is comparable to the intrinsic bulk term,
11and to
that reported6for all thicknesses of epitaxialAu/Fe/Au. Since
we are looking for the additional damping induced by anearby Fe layer, we find the damping term implied by Fig. 1to beG5G
01G133/d~nm!and consider this to be thebaseline for the damping of our single thin Fe layers, with
G0’G1’1.2560.103108/sec. Furthermore, the linewidth
of both single layers and thin layers of the pair structure inthis thickness range increases by roughly 10% on cooling to77 K ~again based on widely separated frequency points !,
whereas the intrinsic rate of Fe is widely reported to be in-dependent of Tor even slightly decreasing in this range.
13G1
andG0thus can be inferred to increase by about 10% in this
range. In general, the broader resonances lie higher in fieldby an amount .twice their linewidth, as found in Ref. 8.
For structures with two Fe layers, we primarily set out to
determine the effect of different thickness Cu spacers, so wefixed the Fe layer thicknesses as 4 and 20 nm, as describedabove for the VSM studies. ~We also took limited data for
different Fe thicknesses, generally supporting a 1/ d
dependence
6of the additional linewidth. !Some data are pre-
sented in Fig. 2. This general shape was seen over five dif-ferent series of depositions, although some details changed.Even though the additional layers are deposited entirely sub-sequent to the 4 nm Fe layer, its FMR spectrum is drasticallyaltered compared to single Fe films. The linewidths at bothfrequencies are increased about a factor of 5 for 3 nm of Cu~only one strong FMR line was visible for 2 nm Cu, imply-
ing very strong coupling !and the linewidths for thicker Cu
spacer layers are also considerably larger than for a single Fe4 nm layer, although decreasing rapidly over the range ob-served.
Also striking is the upward shift in resonance field ac-
companying the broadening, as also seen in thinner singlelayers described above, by about twice the linewidth. Urbanet al.
6accounted for an upward shift of the thin Fe FMR with
respect to bulk using a combination of surface anisotropyand a slightly reduced moment. While these are both likelyto be present in some degree, our data require more explana-tion, since our shift increases rapidly with reduced Cu spacerthickness. Comparing the intensities of the FMR lines, asobtained by double integration or line shape fitting, we findtheMof the thin Fe layers is not significantly reduced from
the 20 nm Fe moment. Moments as deduced from VSM alsoseem to be nearly bulklike, and the character of the loops ishardly affected by the presence of the thicker layer, exceptwhen the intervening Cu is very thin.
FIG. 1. Linewidths vs frequency for Cu/Fe/Cu structures for Fe thicknesses
of 2, 3, 5, and 20 nm, showing an increase proportional to 1/ d.
FIG. 2. Linewidth of the 4 nm thick Fe film at 9.46 and 34.7 GHz for the
structures Cu ~2n m !/Fe~4n m !/Cu~x!/Fe~20 nm !/Cu~10 nm !.8284 J. Appl. Phys., Vol. 93, No. 10, Part s2&3 ,1 5M a y 2003
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128.83.63.20 On: Thu, 04 Dec 2014 14:38:44Finally, we observed a few of these structures at 77 K,
and found that the linewidth roughly increased by 10%; sinceonly a small part of the linewidth of these Fe layers is intrin-sic~proportional to G
0), we can infer that the nonintrinsic
but Gilbert-like term increases by about 10%, i.e., compa-rable to but additional to the term G
1required for single Fe
films.
DISCUSSION AND CONCLUSIONS
The broadening and shifts seen for thin Fe single layers
are consistent with some predictions.3Such effects were not
observed in Cu/Py/Cu,10ostensibly because the 5 nm Cu
used was much less than the spin-flip scattering length.3This
suggests that our Cu has a much shorter scattering lengththan usual; alternatively, the spin-flip scattering at the Cu/Feinterfaces may be different from that of Cu/Py. Both circum-stances are not unlikely, since growth of Cu on already finegrained Fe and vice versa in our system requires renucleationand hence grain sizes of only a few nanometers are expected,i.e., an order of magnitude less than for thick Cu or Cugrown on Py. The prediction
3that thegvalue may change,
shifting the resonant field, is not consistent with the analysisof our data at two frequencies, which are satisfied by a singlegand effective M. Furthermore, the shifts cannot be pro-
duced by a static in-plane anisotropy,
6since our data are
nearly isotropic in the film plane. A fixed anisotropy fieldcannot simulate the data for the two frequencies used, unlessit has planar symmetry, in which case it contributes to theeffective Mby a huge amount. Reference 6 invokes such an
anisotropy, but assumes that it is intrinsic to the thinner film,whereas in our case it is a strong function of Cu spacerthickness.
We found that the amount of additional broadening re-
lated to the presence of the second, thicker, Fe layer de-creases rather rapidly with the thickness of increasing inter-vening Cu layer, roughly consistent with an exponentialattenuation length of 3.5 nm, in contrast to the finding of nostrong dependence on spacer thickness for single crystalAu.
6
For our Fe/Cu/Fe system, the observation of a strong de-crease of these effects with increasing Cu thickness is con-sistent with the observations above. Assuming the Bergermodel is correct, the attenuation length for spin polarizationof the current in our Cu is about 3.5 nm; hence the reasongiven
3for the failure to see broadening in Cu/Py/Cu would
not apply to our Cu overlayer, which is thicker than thislength, 10 nm.The origin of shifts seen in our in-plane FMR fields re-
mains enigmatic. Attribution to a change in g value or de-crease in Mis inconsistent with our data. It is tempting to
ascribe the changes to a surface anisotropy, and indeed largesurface anisotropies favoring out-of-plane magnetization arereported in similar systems. However, it seems unlikely thatsuch an anisotropy as large as ;2
pMcould be induced by
subsequent deposition of a second Fe layer tens of latticespacings removed from the surface in question. It seemsmore likely that the origin is a dynamic torque, especiallysince the shift is roughly proportional to the line broadening.Slonczewski’s treatment
14of spin-current effects, in fact,
predicts that in-plane torques can act on M, i.e., act like
in-plane anisotropies. These would have the right magnitudeand symmetry to produce the observed effects and may ex-plain the observed frequency dependence.
Finally, we note that the additional relaxation rates and
shifts increase by about 10% on cooling to 77 K. The GMRalso increases somewhat on cooling, apparently consistentwith Berger’s description of the additional relaxation as be-ing induced by spin transport out of the layer undergoingresonance; this suggests that the shifts may have a similarorigin.
1B. Dieny, V. S. Speriosu, S. S. P. Parkin, B.A. Gurney, D. R. Wilhoit, and
D. Mauri, Phys. Rev. B 43, 1297 ~1991!; S. Colis, M. Guth, J.Arabski,A.
Dinia, and D. Muller, J. Appl. Phys. 91, 2172 ~2002!.
2L. Berger, Phys. Rev. B 54, 9353 ~1996!.
3Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88,
117601 ~2002!.
4F. J. Albert, J. A. Katine, R. A. Buhrman, and D. C. Ralph, Appl. Phys.
Lett.77, 3809 ~2000!.
5M. Tsoi,A. G. M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, and P.
Wyder, Phys. Rev. Lett. 80, 4281 ~1998!.
6R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87, 217204
~2001!.
7A. Chaiken, P. Lubitz, J. J. Krebs, G. A. Prinz, and M. Z. Harford, Appl.
Phys. Lett. 59, 240 ~1991!.
8P. Lubitz, S. F. Cheng, F. J. Rachford, M. M. Miller, and V. G. Harris, J.
Appl. Phys. 91, 7783 ~2002!.
9P. Lubitz, M. Rubinstein, D. B. Chrisey, J. S. Horwitz, and P. R. Brousard,
J. Appl. Phys. 75, 5595 ~1994!.
10S. Mizukami, Y.Ando, and T. Miyazaki, J.Appl. Phys. 226, 1640 ~2001!.
11F. Schreiber, J. Pflaum, Z. Frait, Th. Muhge, and I. Pelzl, Solid State
Commun. 93, 965 ~1995!.
12P. Lubitz, J. J. Krebs, M. M. Miller, and Shufan Cheng, J.Appl. Phys. 83,
6819 ~1998!.
13S. M. Bhagat and P. Lubitz, Phys. Rev. B 10, 179 ~1974!.
14J. C. Slonczewski, J. Magn. Magn. Mater. 247, 324 ~2002!.8285 J. Appl. Phys., Vol. 93, No. 10, Part s2&3 ,1 5M a y 2003
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
128.83.63.20 On: Thu, 04 Dec 2014 14:38:44 |
5.0039771.pdf | J. Chem. Phys. 154, 084304 (2021); https://doi.org/10.1063/5.0039771 154, 084304
© 2021 Author(s).Potential energy surfaces for high-energy N
+ O2 collisions
Cite as: J. Chem. Phys. 154, 084304 (2021); https://doi.org/10.1063/5.0039771
Submitted: 06 December 2020 . Accepted: 20 January 2021 . Published Online: 23 February 2021
Zoltan Varga ,
Yang Liu ,
Jun Li , Yuliya Paukku ,
Hua Guo , and
Donald G. Truhlar
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Potential energy surfaces for high-energy
N + O 2collisions
Cite as: J. Chem. Phys. 154, 084304 (2021); doi: 10.1063/5.0039771
Submitted: 6 December 2020 •Accepted: 20 January 2021 •
Published Online: 23 February 2021
Zoltan Varga,1
Yang Liu,2
Jun Li,2
Yuliya Paukku,1Hua Guo,3
and Donald G. Truhlar1,a)
AFFILIATIONS
1Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota,
Minneapolis, Minnesota 55455-0431, USA
2School of Chemistry and Chemical Engineering & Chongqing Key Laboratory of Theoretical and Computational Chemistry,
Chongqing University, Chongqing 401331, China
3Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico 87131, USA
a)Author to whom correspondence should be addressed: truhlar@umn.edu
ABSTRACT
Potential energy surfaces for high-energy collisions between an oxygen molecule and a nitrogen atom are useful for modeling chemical
dynamics in shock waves. In the present work, we present doublet, quartet, and sextet potential energy surfaces that are suitable for studying
collisions of O 2(3Σ−
g) with N(4S) in the electronically adiabatic approximation. Two sets of surfaces are developed, one using neural networks
(NNs) with permutationally invariant polynomials (PIPs) and one with the least-squares many-body (MB) method, where a two-body part is
an accurate diatomic potential and the three-body part is expressed with connected PIPs in mixed-exponential-Gaussian bond order variables
(MEGs). We find, using the same dataset for both fits, that the fitting performance of the PIP-NN method is significantly better than that of the
MB-PIP-MEG method, even though the MB-PIP-MEG fit uses a higher-order PIP than those used in previous MB-PIP-MEG fits of related
systems (such as N 4and N 2O2). However, the evaluation of the PIP-NN fit in trajectory calculations requires about 5 times more computer
time than is required for the MB-PIP-MEG fit.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0039771 .,s
I. INTRODUCTION
The interactions of nitrogen and oxygen species are impor-
tant in atmospheric and planetary chemistry.1–9In addition, these
interactions are especially important for modeling shock-heated
air in aerospace re-entry.9–23We are developing potential energy
surfaces for simulating collisions between nitrogen and oxygen
species—both four-body potentials involving N 2and O 224–27and
three-body potentials involving N and O collisions with N 2and O 2.
Our initial goal, which governs the present work, is to develop the
potentials needed for modeling Born–Oppenheimer collisions (i.e.,
collisions in which the electronic state does not change) of ground-
electronic-state N, O, N 2, and O 2.28,29In this work, we neglect spin–
orbit coupling; therefore, the electronic spin quantum number ( S) is
conserved.
When collisions occur between collision partners with elec-
tronic degeneracy due to spatial symmetry, as in the presentcase, one must consider more than one potential energy sur-
face.30The goal of the current work is to calculate the global
surfaces of the NO 2system that are needed to describe the
collisions of ground-electronic-state O 2(3Σ−
g) with the ground-
state N atom (4S). Since there is no spatial degeneracy in
the ground states (and spin–orbit coupling is neglected) and
if we make the Born–Oppenheimer approximation, collisions
occur in the lowest-energy doublet ( S= 1/2, where Sis the
quantum number of total electronic spin), lowest-energy quartet
(S= 1 1/2), and lowest-energy sextet ( S= 2 1/2) spin state, and hence,
we need three potentials. These are the surfaces in this paper; they all
correspond to the A′irrep for the spatial part of the electronic wave
function.
To avoid misuse of the present potentials, it is important to
specify that although they are sufficient for O 2(3Σ−
g) + N (4S) col-
lisions, they are not sufficient for NO (2Π) + O (3P) collisions.
The reason for this is as follows: If one considers the collision of
J. Chem. Phys. 154, 084304 (2021); doi: 10.1063/5.0039771 154, 084304-1
Published under license by AIP PublishingThe Journal
of Chemical PhysicsARTICLE scitation.org/journal/jcp
ground-electronic-state NO (2Π) with a ground-state O atom (3P)
because of the higher spatial degeneracy of the collision partners,
collisions occur on the six lowest doublet and six lowest quartet sur-
faces. The current surfaces are important for those collisions but are
just a small part of the required surfaces for studying NO + O colli-
sions. In a similar vein, we note that our previously published N 2+
O2four-body potential27is not applicable for the present purposes
because the four-body potential corresponds to conservation of the
four-body electronic spin, but the spins of the three-body subsys-
tems and atomic subsystem can change when only the four-body
spin is constrained. In summary, the previous N 2+ O 2potential
is not applicable for the present purposes, but the potentials pre-
sented here are sufficient (in the Born–Oppenheimer approximation
and with neglect of spin–orbit coupling) for studying O 2(3Σ−
g) + N
(4S) collisions; however, the present surfaces are not sufficient for
simulating experimental results on NO (2Π) + O (3P) collisions.
Similarly, these surfaces are insufficient for a complete study
of 2O + N intermolecular collisions, where the reactants are the
separated atoms. Such three-body collisions, with all atoms in
their ground electronic states, could occur—even if electronically
adiabatic—on any of nine doublet surfaces, nine quartet surfaces,
nine sextet surfaces, and nine octet surfaces.
Nitrogen dioxide is of special interest because of its role in
environmental chemistry31(smog and ozone production) and in
combustion processes.32Therefore, many local or global surfaces
of the doublet NO 2system have been developed in the past three
decades.33–49The quartet potential of the NO 2system was also
fitted by Sayós et al. ,45but as far as we know, no sextet sur-
face has been fitted yet. It is especially important to eliminate this
lacuna in our simulation database because statistically one half of
the collisions of N(4S) + O 2(3Σ−
g) occur in the sextet state with
only one-third in the quartet state and one-sixth in the doublet
state.
For aerospace applications, the simulations of hypersonic flow
require a very high temperature range, up to 20 000 K or even
30 000 K, and our potentials are designed according to this require-
ment. Thus, the precise structure of the low-energy surfaces, which is
very important for studying ambient-temperature processes, is less
important in our consideration, and we accept a larger absolute error
in fitting surfaces over a wide energy range for high-energy processes
than would be desirable when fitting surfaces in a narrow energy
range for low-energy applications.
In this article, two approaches were used to fit the three poten-
tials of NO 2. The first approach is a permutationally invariant poly-
nomial50,51(PIP) fit in bond order variables to the many-body (MB)
potential part of the potential, retaining only connected terms,52
with separate fits to the two-body parts, where the bond order vari-
ables are taken as mixed-exponential-Gaussians (MEGs).24The full
name of this set of three potentials is MB-PIP-MEG, but we will
use the shorthand name MEG in the rest of this article. The second
approach is a permutationally invariant-polynomial neural-network
(NN) fit,53–55without separating the two-body and three-body parts.
The full name of this set of three potentials is PIP-NN, but we will
use the shorthand name NN in the rest of this article. As we know
from past work56and as we reconfirm below for the present case,
these two approaches are complementary in that the MEG fit is
computationally more efficient while the NN fit has smaller fitting
errors.II. METHODS
A. Electronic structure calculations
All electronic structure calculations are performed with the
2012.1 version of the Molpro software package.57,58We used
dynamically weighted59state-averaged60complete-active-space self-
consistent-field61,62(DW-SA-CASSCF) calculations to obtain the
multireference reference state. The active space consists of 17 elec-
trons distributed in the 12 valence orbitals. In these DW-SA-
CASSCF calculations, for each spin, three states with the same Swere
averaged (we do not need the energies of the two higher states for
fitting, but including them in the calculations makes the energy of
the lowest-energy state smoother as a function of geometry). The
dynamical weighting factor was set to the recommended value,59
which is 3 eV. With the lowest-energy state of the DW-SA-CASSCF
calculation for the given spin state serving as the reference state, a
single-state complete active space second-order perturbation theory
(CASPT2) calculation was carried out with the rs2 keyword;63this
corresponds to using a level shift64of 0.3 hartree and to using the g4
version of the modified Fock-operator.65,66The electron correlation
included all the valence electrons.
The minimally augmented correlation-consistent polarized
valence triple zeta basis set, maug-cc-pVTZ,67is used for all calcula-
tions.
Since a three-atom system always has a plane of symmetry, the
three surfaces were constructed using electronic structure calcula-
tions carried out in Cssymmetry, and we found that, for all three
spin states, the spatial symmetry of the lowest-energy state is A′.
B. DSEC method for NO 2systems
The accuracy of calculated energies was improved by using the
dynamically scaled external correlation27(DSEC) method. In the
DSEC method, the parameter Fof the original scaled external cor-
relation (SEC) method68is replaced by a new parameter p, which is
unitless and equals 1/ F. The general equation for the DSEC energy
at a given geometry is
EDSEC=ECASSCF+p(ECASPT2−ECASSCF), (1)
where p(unlike F) depends on geometry. We parameterize psuch
that the DSEC relative energies agree with best estimate relative
energies at key geometries. The best estimate relative energies are
taken from experiment.
LetrNO1,rNO2, and rO1O2 be the three internuclear distances of
the NO 2system. We define
g1,NO1={0 if rNO1<re,NO
rNO1−re,NO otherwise,(2a)
g1,NO2={0 if rNO2<re,NO
rNO2−re,NO otherwise,(2b)
g1,O1O2={0 if rO1O2<re,OO
rO1O2−re,OO otherwise,(2c)
for the diatomic subsystems, and we define
J. Chem. Phys. 154, 084304 (2021); doi: 10.1063/5.0039771 154, 084304-2
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g2,NO1={0 if rNO1<re,NO (NO 2)
rNO1−re,NO (NO 2)otherwise,(3a)
g2,NO2={0 if rNO2<re,NO (NO 2)
rNO2−re,NO (NO 2)otherwise,(3b)
g2,O1O2={0 if rO1O2<re,OO (NO 2)
rO1O2−re,OO (NO 2)otherwise,(3c)
for triatomic NO 2, where re,Xis an equilibrium internuclear distance
given in Table I. Based on Eqs. (2) and (3), geometry-dependent
weights are defined as
w1,NO1=exp[−aNOg2
1,NO1], (4a)
w1,NO2=exp[−aNOg2
1,NO2], (4b)
w1,NO={w1,NO1 ifw1,NO1>w1,NO2
w1,NO2 otherwise,(5a)
w1,O1O2=exp[−aOOg2
1,O1O2], (5b)
w2=exp[−aNO(NO 2)g2
1,NO1−aNO(NO 2)g2
1,NO2−aOO(NO 2)g2
2,O1O2], (5c)
where axare parameters.
For each of the three NO 2surfaces, the dynamically weighted
scaling factor for a given geometry is
p=1 +c(O2+N)w1,O1O2 +c(NO+O )w1,NO +c(NO 2)w2
+c(cross)3√w1,O1O2w1,NOw2, (6)
where c(O2+N)andc(NO+O) are diatomic scaling factors, c(NO 2)is a
triatomic scaling factor, and c(cross) is a cross-term scaling factor.
LetEtarget
x be the best estimate energy of geometry x, as obtained
from experiment,69,70and a best estimate (BE) relative energy is then
defined by
EBE
xy=EBE
x−EBE
y, (7a)
and a DSEC relative energy is given byΔEDSEC
xy=ECASSCF
x −ECASSCF
y +px(ECASPT2
x −ECASSCF
x)
+py(ECASPT2
y −ECASSCF
y). (7b)
The BE relative energies used to obtain the parameters are
collected in Table I.
First, the range parameters were obtained from Morse models,
as indicated in Table II. Then, the scaling parameters were obtained
from the BE energies in Table I. First, the scaling factors c(O2+N),
c(NO+O) , and c(NO 2)were calculated from Eqs. (7b) and (6) for sta-
tionary structure xby taking the ystructure to be the three separated
atoms (where p= 1). Note that these scaling factors are the same
for doublet, quartet, and sextet surfaces because all three potentials
agree in the asymptotic regions. Then, based on these scaling factors,
c(cross) was adjusted to eliminate the effect of scaling factors c(O2+N)
andc(NO+O) in Eq. (6) at the stationary geometry of the doublet NO 2
structure. Although this parameter was obtained for the doublet sur-
face, the same scaling parameters are applied consistently to all three
NO 2surfaces.
The DSEC parameters are collected in Table II. The applica-
tion of the DSEC correction to the electronic structure calculations
is carried out prior to the fitting procedure.
C. Selection of geometries to include
in the fitting datasets
Most of the geometries used for the fits come from two grids
that were used for all three spin states. In grid 1, the three atoms
are placed in the O1–O2–N order. Then, the r(O1–O2) and r(O2–
N) distances and the α(O1–O2–N) angle were varied. In grid 2, the
three atoms are placed in the O1–N–O2 order, and the r(O1–N) and
r(O2–N) distances and the α(O1–N–O2) angle were varied, with the
restriction that r(O1–N)≥r(O2–N). For these two grids, the values
of the distances used are 0.8 Å–1.6 Å with a 0.1 Å increment plus 1.8
Å, 2.0 Å, 2.2 Å, 2.5 Å, 2.7 Å, 3.0 Å, 4.0 Å, and 5.0 Å, and the angles
were varied from 30○to 180○with a 5○increment. The potentials for
longer distances are dominated by the diatomic potentials, which are
fit separately (see below).
We also carried out a multi-dimensional scan to find regions
with poor data coverage. For this purpose, the Cartesian coordinates
in the dataset were converted to internuclear distances. The OO
internuclear distance is unique, and two NO internuclear distances
were arranged in an ascending order. In the multi-dimensional scan,
the OO and the shorter NO distances were varied from 0.7 Å to
TABLE I . Equilibrium distances and best-estimate relative energies (in kcal/mol) and the calculated CASPT2 relative energies
before the DSEC correction (in kcal/mol).
System State re,X Best estimate CASPT2a
NO 2 X2A1 1.204(NO), 2.215(OO)b0.0 0.0
O2+ N3Σ−
g+4S 1.208c106.9d102.7
NO + O2Π+3P 1.1508c74.5d77.5
O + O + N3P +3P +4S ... 227.1d226.4
aThe geometries were optimized by CASPT2.
bFrom geometry optimized by CASPT2.
cExperimental from Ref. 69.
dExperimental from Refs. 69 and 70.
J. Chem. Phys. 154, 084304 (2021); doi: 10.1063/5.0039771 154, 084304-3
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TABLE II . The parameters of the DSEC method.a
Range parameters Scaling factors
aNO 7.5076bc(O2+N) −0.012 325
aOO 7.0225cc(NO+O) 0.002 178
aNO(NO 2) 7.5076bc(NO 2) 0.012 766
aOO(NO 2) 2.65dc(cross) −0.134 681
aThe range parameters are in Å−2, and the scaling factors are unitless.
bThe square of 2.74 Å−1, which is the Morse parameter of the ground-state NO molecule
derived from Ref. 71.
cThe square of 2.65 Å−1, which is the Morse parameter of the ground-state O 2molecule
derived from Ref. 71.
dThe square root of diatomic parameter aOO.
4.0 Å with a 0.1 Å increment, and the longer NO was run from
the actual value of the shorter NO distance to 4.0 Å, with the three
distances forming a valid triangle structure. To represent the dis-
tance between any geometries in the dataset and any geometry in the
multi-dimensional scan, the absolute values of the differences of the
three internuclear distance pairs (OO, shorter NO, and longer NO)
were calculated and added together. If a geometry point in the multi-
dimensional scan lays further than 0.33 Å from the closest point in
the dataset, then that point is considered to be in a vacant geome-
try region. Since the original geometry grid was already very dense,
we only found 150–200 geometries for each three surfaces, and all of
them were added to the set to be calculated.
After that mentioned above, the points on the grids and those
resulting from the scan were augmented by additional geometries
that were different for the different spin states. For example, these
state-specific points are geometries close to stationary points of that
given potential or geometry points used for tests specific to the given
state. In these calculations, r(O1–O2) is fixed at 1.208 Å, which is
the equilibrium bond length of O 2, and the r(O2–N) distance and
theα(O1–O2–N) angle are varied.
In some cases, points with relatively large fitting errors, as well
as their surroundings, were reinvestigated to see if the error comes
from wrong electronic structure calculations, since one must be
extra careful as to whether multiconfigurational self-consistent field
calculations have converged to the desired wave function. For this
purpose, the calculations were repeated with a different set of guesses
for the orbitals to attempt to get a better solution. There are many
ways to generate an initial guess, and finding a good initial guess
is situation-dependent. In general, an initial guess can come from
a HF calculation with the same spin as the state under considera-
tion, from a higher spin state, or from a (SA-)CASSCF calculation in
which the number of states is different from that used in the actual
calculation. The initial guess can be obtained at the same geome-
try as the actual calculation, or at a different geometry, which can
be relatively far from the geometry of the actual point and possibly
changed gradually along a scanning parameter (at each point of the
scan, the guess comes from the previous calculation) to reach the
geometry point. Trading active orbitals with virtual orbitals is also
used to generate initial guesses. In certain cases, the best guess is the
default option, where natural orbitals of a diagonal density matrix
are constructed using atomic orbitals and atomic occupation num-
bers (in the present work, we have not generated initial guesses bychanging the active space, but that is another option sometimes used
in previous work).
To test the surfaces, two series of trajectory calculations were
carried out with the program ANT72on all the NO 2surfaces. In
the first series of tests, only MEG fits were used, and trajectories
were calculated for O 2+ N and NO + O collisions. For each spin
state, we ran 500 trajectories with relative translational energies in
the range 0.3 eV–4.3 eV, impact parameters in the range 0 Å–1.6 Å,
vibrational quantum numbers in the range 1–8, and initial rotational
quantum numbers in the range 3–200. For these calculations, the
initial atom–diatom separation is 8 Å, and a trajectory is terminated
when any of the internuclear distances becomes longer than 9.2 Å or
if the propagation time reaches 800 fs. The Bulirsch–Stoer integrator
with adaptive step size is used. All of these trajectories ran without a
problem, and the geometry of every second step was saved. Then, we
randomly picked ten trajectories for each potential surface (doublet,
quartet, and sextet) to select 150–200 points for each spin state (the
number of geometry points depends on the trajectory) for additional
electronic structure calculations. The calculated and fitted energies
were compared and found to be in good agreement. In particular,
for the points that were calculated, the maximum deviation was less
than 13 kcal/mol and the mean unsigned errors of the test fits were
2.4 kcal/mol, 3.2 kcal/mol, and 3.4 kcal/mol for the doublet, quartet,
and sextet surfaces, respectively (the acceptable error level was not
formulated as a hard-and-fast rule; we required smaller errors at low
energy and near stationary points, but allowed larger errors on high-
energy repulsive walls). Then, these points were added to the dataset
to make the final fit of each NO 2surface.
In the second series of trajectory tests, all NO 2surfaces fit-
ted by both NN and MEG were used. In these tests, the colliding
partners are O 2+ N. For each surface, altogether, 1100 trajectories
were run with 11 different sets of the collision energy (within range
0.2 eV–30 eV), initial vibrational quantum number (1–12) of the
diatom, and initial rotational quantum number (1–24) of the diatom.
The initial atom–diatom separation was set to 8 Å. The other input
parameters were the same as those for the first series. Since the tra-
jectories did not show any odd behaviors, the fitted NN and MEG
surfaces used in these tests are considered the final ones (these tra-
jectory calculations only served to test the usability of the potentials
and the geometry coverage; they are insufficient to generate cross
sections or rate constants).
Based on the above-mentioned procedures, each of the spin
states has a different number of points used to fit the surfaces. In par-
ticular, the datasets of the doublet, quartet, and sextet surfaces have
8434, 7818, and 8386 points, respectively. The lowest-energy point
on any of the potentials is the equilibrium geometry of the doublet
potential, and this will be used as the zero of energy in the rest of
this article. With this zero of energy, all points used for the fits have
energies below 2000 kcal/mol.
III. INTRODUCTION TO FUNCTIONAL FORMS
OF THE FITS
The potential energy Vfor each spin state is expressed as a
global function Vmodified with a local patch function VPF,
V=VG(r1,r2,r3)+VPF(r1,r2,r3), (8a)
J. Chem. Phys. 154, 084304 (2021); doi: 10.1063/5.0039771 154, 084304-4
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where r1denotes the O1–O2 distance, r2denotes the N–O1 distance,
andr3denotes the N–O2 distance. The patch function is explained
in Sec. III D. Here, we explain the global function.
The NN fit uses VGwithout a many-body expansion, but the
MEG fit uses the following expansion:
VG=V0+∑3
i=1VPA,Z(i)(ri)+VMB(r1,r2,r3), (8b)
where V0is a constant, VPA,Zis a pairwise additive potential (i.e.,
a sum of diatomic potentials) with Z(i) = dNO, qNO, or OO, dNO
denotes doublet NO, qNO denotes quartet NO, OO denotes triplet
O2, and VMBis the many-body term (which is a three-body term
in the present application). The pairwise terms have Z(1) equal to
OO for all three potentials (doublet, quartet, and sextet), Z(2) and
Z(3) equal to dNO for the doublet and quartet surfaces, and Z(2)
and Z(3) equal to qNO for the sextet. The constant V0is set to
227.0 kcal/mol, which is the sum of the dissociation energy of O 2
(120.2 kcal/mol) and singlet N 2(228.4 kcal/mol) molecules minus
the dissociation energy of NO 2(121.6 kcal/mol); this is required to
make the functional form used have the zero of energy specified in
the last paragraph of Sec. II C.
A. Functional form of the NN fit
For each electronic state, the following NN function form53–55
with two hidden layers is used:
VG=b(3)
1+∑K
k=1(w(3)
1,kf2(b(2)
k+∑J
j=1(w(2)
k,jf1(b(1)
j
+∑I
i=1w(1)
j,iGi)))) , (9a)
Gi=ˆSN
∏
i<jplij
ij, (9b)
where the permutation invariant polynomials51,73(PIPs) are used
as the input layer of the NN with the Morse like variables pij
= exp( −λrij) (λis a parameter adjusted to 1.0 Å−1) of internuclear
distances between atoms iandj(i,j= 1–3); ˆSis a symmetrization
operator that permutes the two identical oxygen atoms; Iis the num-
ber of the input PIPs; JandKare the numbers of neurons in the two
hidden layers; fi(i= 1, 2) are nonlinear transfer functions for the
two hidden layers; ω(l)
j,iare weights that connect the ith neuron of
the (l-1)th layer and the jth neuron of the lth layer; b(l)
jis the bias of
thejth neuron of the lth layer; and the ωandbvariables are fitting
parameters.
In the present work, the maximum order of the input PIPs is
3, resulting in 12 PIPs. The fitting parameters were optimized by
nonlinear least squares fitting in which the root-mean-square error,
RMSE =⌟roo⟪⟪op
⌟roo⟪mo⟨⌟roo⟪mo⟨⌟roo⟪⟨o⟪Ndata
∑
i= 1(Ei
output −Ei
target)2/Ndata, (10)
was used to measure the performance of the fitting. The “early stop-
ping” algorithm74was used to avoid overfitting with the dataset ran-
domly divided into three parts: the training (90%), validation (5%),
and test (5%) sets for each NN fitting.Several combinations of the numbers of the neurons in NN
architectures with two hidden layers were tested. Our experience has
been that the improvement of fitting with a larger number of neu-
rons is not drastic. Furthermore, the number of fitting parameters
should be relatively small (less than 1/5) as a ratio to the total num-
ber of data points in the training set. As a result, we typically choose a
structure with a moderate number of neurons that has a sufficiently
high fitting fidelity. Based on these considerations, the final numbers
of the neurons in the two hidden layers ( JandK) were both cho-
sen to be 20, resulting in 701 nonlinear fitting parameters for each
potential. For each architecture, 100 NN trainings with different ini-
tial fitting parameters and different training, validation, and test sets
were carried out. To minimize the random error, the final NN poten-
tial was selected as the average of three best fittings according to the
NN ensemble approach.75
The edge points randomly selected in the validation/test sets
could lead to false extrapolation. Consequently, to decrease such
possible errors, the fit was chosen only if all three sets have simi-
lar RMSEs. The maximum deviation is also used as a criterion for
selecting the final NN potential.
Very recently, another paper appeared76(to be denoted VGJ)
applying neural networks to fitting potential energy surfaces for
aerospace applications, in particular for N 4. Here, we contrast the
approach in that work to our approach here for NO 2and in our own
previous work56on N 4. The key differences are the input coordinates
(permutationally invariant polynomials vs fundamental invariants),
the use of separate diatomic potentials (for NO 2, these were not
used in our NN calculations, but they were used in both their and
our NN work on N 4and in all of our work with conventional least-
squares fits), VGJ’s use of a tapering function to eliminate nonphys-
ical behavior at long-range distances (for our NN fit on N 4and for
all of our work with conventional least-squares fits, we removed the
non-connected terms of the permutationally invariant polynomials
to improve the treatment at long range, but this was not carried out
for the NN fits in the case of NO 2), VGJ’s use of relatively small
networks to keep the computational costs low, our practice of aver-
aging three fits (to further eliminate surface errors), whereas VGJ
apparently did not average.
B. Diatomic potentials
Each diatomic function VPA,Z(ri) has two terms that were
originally introduced in Ref. 25,
VPA,Z(ri)=VSR,Z(ri)+VD3(BJ),Z(ri), (11)
where the long-range term for molecule Z,VD3(BJ), Z(ri), is a damped
dispersion term based on Grimme’s D3 dispersion parameters
with the Becke–Johnson damping (BJ) function.77,78For all three
diatomic potentials, the unitless parameters s6ands8are 1.0 and
2.0, respectively, and we also set a1= 0.5299 a.u. and a2= 2.2 a.u.
based on Ref. 79. For the potential energy curve of the ground state
of triplet O 2(Z= O 2), the C6was fixed at 176.37 kcal ⋅Å6/mol ( C8is
obtained from C6); for more details, see Ref. 25. The parameters of
the short-range term, VSR,O 2(r1), are determined by fitting Eq. (12)
to the accurate O 2potential curve of Bytautas et al.80ForVSR,O 2(r1),
we use the even-tempered Gaussian fitting function of Bytautas
et al. ,79given by
J. Chem. Phys. 154, 084304 (2021); doi: 10.1063/5.0039771 154, 084304-5
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VSR,O 2(r1)=∑7
k=0akexp(−αβkr2
1), (12)
where we obtain the coefficients a kby linear regression and the expo-
nent parameters αandβby nonlinear minimization. The VSR,O 2(r1)
parameters are listed in Table III (they are also given in the supple-
mentary materials of Refs. 25, 26, and 29).
In the case of doublet and quartet NO 2surfaces, the poten-
tial energy curve of ground state doublet NO was used ( Z= dNO).
However, for the sextet NO 2potential, the potential energy curve of
the lowest quartet NO was used ( Z= qNO). In both cases, the NO
distances are denoted with i= 2 and 3 as it was mentioned above.
The BE doublet NO curve, VPA,dNO(ri), was taken from our previous
work,27,28and the 230.11 kcal ⋅Å6/mol C6parameter was used in the
VD3(BJ),dNO (ri) term. The other parameters of the VD3(BJ),dNO (ri) term
correspond to the parameters of VD3(BJ),O2(ri). The VSR,dNO (ri) term
was refitted as a difference of terms VPA,dNO (ri) and VD3(BJ),dNO (ri).
For the fit of VSR,dNO(ri), Eq. (13) was used,
VSR,dNO(ri)=BdNO(∑10
k=1ckXk
i,dNO), (13)
Xi,dNO=exp(−(ri−re,dNO)/α1−(ri−re,dNO)2/α2), (14)
where re,dNO is the equilibrium bond length (1.1508 Å) of doublet
NO. The non-linear parameters α1andα2and the BdNO parameter,
as well as the coefficients, ck, were fitted, and they are collected in
Table IV.
In the case of the quartet NO curve, VPA,qNO (ri), CASPT2 cal-
culations (with the g4 option, a level shift of 0.3 hartree, and the
two 1 s orbitals excluded from the electron correlation) were car-
ried out based on SA(3)-CASSCF(8o,13e) reference wave function,
where the spin state was set to quartet, the three states were dynam-
ically weighted, and Cssymmetry was applied. The NO distance was
scanned from 0.6 Å to 10.47 Å, by a 0.01 Å increment. Based on
the dissociation energy of the doublet NO ( De,dNO = 152.6 kcal/mol)
and the T e(109.9 kcal/mol)81energy between the X2Πanda4Π
states, the dissociation energy of a4Πstate was calculated, and this
BE energy was used in the static version of Eq. (7) to get the scaled
external correlation, i.e., the original SEC, where px=py= 1.13. Since
thea4Πstate dissociates to the same limit as the X2Πstate, we
TABLE III . Re-optimized parameters for the short-range term, VSR,O2(r1), for the
diatomic O 2potential.
Parameter (unit) Value
α(Å−2) 9.439 784 362 354 936 ×10−1
β(-) 1.262 242 998 506 810
a0(millihartree) −1.488 979 427 684 798 ×103
a1(millihartree) 1.881 435 846 488 955 ×104
a2(millihartree) −1.053 475 425 838 226 ×105
a3(millihartree) 2.755 135 591 229 064 ×105
a4(millihartree) −4.277 588 997 761 775 ×105
a5(millihartree) 4.404 104 009 614 092 ×105
a6(millihartree) −2.946 204 062 950 765 ×105
a7(millihartree) 1.176 861 219 078 620 ×105TABLE IV . Re-optimized parameters for the short-range term, VSR,dNO (ri), for
diatomic doublet NO potential.
Parameter Value
α1(Å) 0.896 601 839 395 554
α2(Å2) 2.069 542 710 330 37
BdNO(kcal/mol) −149.478 44
c1 −0.138 534 305 380 708
c2 1.889 990 874 379 91
c3 −4.297 653 558 650 66
c4 21.430 539 556 708 7
c5 −44.347 827 069 026 3
c6 41.072 408 288 420 3
c7 −10.909 962 575 954 4
c8 −10.687 260 815 907 1
c9 9.189 735 457 956 48
c10 −2.201 965 266 358 97
assume that the long-range term in Eq. (11) is very similar for these
two states. Therefore, the parameters set for VD3(BJ),dNO (ri) are used
for the term of VD3(BJ),qNO (ri) as well. Then, the VSR,qNO (ri) term was
fitted as a difference of terms VPA,qNO (ri) and VD3(BJ),qNO (ri). For the
fit of VSR,qNO (ri), Eq. (15) was used,
VSR,qNO(ri)=BqNO(∑10
k=1ckXk
i,qNO), (15)
Xi,qNO=exp(−(ri−re,qNO)/α1−(ri−re,qNO)2/α2). (16)
Here, re,qNO is the equilibrium bond length (1.4219 Å)52of a4Π
NO. The parameter BqNOwas fixed at 42.7 kcal/mol. The non-linear
parameters α1andα2, as well as the coefficients, ck, were fitted, and
they are collected in Table V.
TABLE V . Optimized parameters for short-range term, VSR,qNO (ri), for diatomic
quartet NO potential.
Parameter Value
α1(Å) 0.512 278 626 249 86
α2(Å2) 1.122 416 809 737 78
c1 −0.803 736 202 408 199
c2 12.803 364 029 556 2
c3 −38.692 021 976 777 6
c4 69.709 031 444 894 8
c5 −77.248 460 068 776 5
c6 49.276 565 279 387 6
c7 −15.248 280 905 310 6
c8 0.323 472 459 125 756
c9 0.990 405 801 350 144
c10 −0.174 629 129 756 616
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C. Many-body potential of the MEG fit
The many-body term of the potential energy is expressed as
VMB(r1,r2,r3)=l
∑
connected,
n1+n2+n3Dn1n2n3S[Xn1
1Xn2
2Xn3
3], (17)
where S[...] is a permutationally invariant polynomial basis func-
tion obtained by symmetrization of a primitive monomial basis
function, as also used by Xie and Bowman.50,51The restriction to
connected terms was introduced in Ref. 82. For all three potentials,
twelfth-order ( l= 12) many-body functions were used. The bond
order variables, Xi, are mixed-exponential-Gaussian functions,24
Xi=exp[−(ri−re,Z)/aZ−(ri−re,Z)2/bZ], (18)
where aZ,aZ, and re,Z(Zis either O 2or NO) are nonlinear parame-
ters collected in Table VI.
As we already used recently for fitting the N 2O and O 3sur-
faces,28,29the four-body frame of a general A 2B2system was applied.
Considering this general scheme as an O 2N2system, one of the N
atoms was placed far apart from the other three atoms.
To carry out of the fit of the many-body term, the following
error function is minimized:
F=∑n
j=1Wj(V0,PA
j−Vj+∑m
k=1dksjk)2, (19)
with respect to the linear coefficients dk, where mand nare the
number of basis functions and the number of fitted data points,
respectively, V0,PA
j is the sum of the constant and pairwise terms at
geometry point j,Vjis the energy of geometry point j,dkis the kth
Dn1n2n3coefficient, sjkis the kth basis function S[Xn1
1Xn2
2Xn3
3]eval-
uated at geometry point j, and Wjis a weighting function used to
avoid too much emphasis on the high-energy data points,
Wj={1 for Vj≤(Ec+Esh)
[(Ec+Esh)/Vj]pforVj>(Ec+Esh),(20)
where Ecis a parameter of the fitting process that reduces the weights
of very-high-energy data points. Parameter Eshis arbitrarily set equal
to 121.6 kcal/mol, which is the energy difference of stationary points
N2+ O 2and NO 2+ N, i.e., the difference between the reference ener-
gies of the four- and three-body frames. We chose Ecand the power
pto be 227.0 kcal/mol and 1.5, respectively, for all three NO 2fits.
TABLE VI . The nonlinear parameters of the many-body MEG variables for the NO 2
surfaces.
Surface Z r e,Z(Å) aZ(Å) bZ(Å2)
Doublet O 2 1.208 1.350 2.75
NO 1.1508 1.150 2.75
Quartet O 2 1.208 0.97 1.51
NO 1.1508 0.75 1.30
Sextet O 2 1.208 1.25 2.10
NO 1.4219 0.94 3.90The doublet, quartet, and sextet MEG surfaces of NO 2were
fitted by a modified version of our PIPFit program.83
D. Patch functions for the MEG and NN fits
Test fits showed that the functional forms of the terms
described so far are not flexible enough to properly fit the barrier
between the2A1and2B2minima of doublet NO 2.
We first consider the MEG fit. We originally used a 10th-order
MEG fit [ l= 10 in Eq. (17)], and increasing this to a 12th-order fit did
not solve the problem (we have not seen signs of overfitting at the
12th order, but going to higher order could be risky in that regard).
The switching of the2A1and2B2states is only a few kcal/mol higher
in energy than the energy of the minimum energy structure of the
2B2state, and the crossing seam of these states is also very close to the
minimum energy structure of2B2. Since the MEG fitting function
does not have the flexibility to follow the proper shape of these states,
the location of the2B2structure rather appears as a shoulder instead
of a well. To restore the barrier and give a well shape of the surface
around the2B2structure, we added a local patch function, VPF(r1,
r2,r3), to the doublet NO 2potential,
VPF(r1,r2,r3)=a0exp[Y(r1,r2,r3)], (21)
Y(r1,r2,r3)=−(r2−rfp)2/a1−(r3−rfp)2/a1
−[(r2
2+r2
3−r2
1)/(2r2r3)−cosθfp]2/a2. (22)
The parameters of the patch function are collected in Table VII.
The first two terms on the right-hand side of Eq. (22) correspond
to the two N–O distances. By plotting the potential in the geom-
etry region near the crossing seam of the2A1and2B2states close
to the minimum of the2B2state, we found that using the O–N–
O angle ( α1) is more straightforward than using the O–O distance
to describe the shape of the crossing seam. Therefore, the third
term on the right-hand side of Eq. (22) uses the deviation of O–
N–O angle from a fixed angle. Nevertheless, this patch function
was parameterized such that it decreases very quickly as the bond
lengths start deviating from the focus point (defined by rfpandθfp).
Figure 1 shows an example cut with three sets of data: the tar-
get DSEC-CASPT2/maug-cc-pVTZ energies, the fitted MEG surface
without the patch function, and the fitted MEG surface with the
patch function.
When the NN fits were used, we found the same problem.
The NN fit has a better performance than that of the MEG fit, i.e.,
the energies of the NN fit lie closer to the energies in the dataset,
TABLE VII . Parameters of the patch functions of the doublet NO 2surfaces.
Parameter (unit) MEG NN
a0(kcal/mol) 5.0 3.0
a1(Å2) 0.010 0.008
a2(unitless) 0.002 0.0008
rfp(Å) 1.25 1.25
θfp(deg) 107.0 107.0
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FIG. 1 . An example ( r2=r3= 1.25 Å) showing the effect of the patch function on
the MEG fit to the doublet surface.
but the cusp of the state crossing of the2A1and2B2states is
still cut off. Thus, this patch function was also added to the NN
fit, but with a different set of parameters, which are also given in
Table VII.
For the quartet and sextet NO 2surfaces, such patch functions
are not needed, and they were not applied.
IV. RESULTS AND DISCUSSION
The fitting errors of the NN and MEG fits are collected in
Table VIII, in which the mean unsigned errors (MUEs) and root-
mean-square errors (RMSEs) are shown for various energy ranges,
as well as for the entire (2000 kcal/mol) energy range [recall that
all energies are relative to the equilibrium energy of NO 2(2A1)].
Although the order of the many-body part in the MEG fits increased
to 12 from the previously used 9 (for N 4) or 10 (for all other systems),
the MEG fits are still unable to match the performance of the NN fits.
This observation is consistent with our recent comparison of the N 4
potential using different fitting techniques,56and it is attributable to
the ultra-flexibility84–87of the NN functional form.
Table IX compares the current fits for NO 2to some of our pre-
vious fits for similar systems, in particular to the 11A′, 11A′′, 13A′,
13A′′, 15A′, and 15A′′surfaces of O 329and the3A′and3A′′sur-
faces of N 2O.28The error of the MEG fit of2A′of NO 2is about half
that of the MEG fit of the 11A′and 11A′′surfaces of O 3. There
is another approximate halving of the error in going to the NN fit.
For the surface with middle spin state, the improvement of the quar-
tet NO 2surface compared to the triplet O 3and NO 2surfaces is less
than that in the singlet and doublet surfaces; the MUE of the4A′sur-
face of NO 2is about 70% of the average MUE of the two triplet O 3
surfaces and about 90% of the two triplet N 2O surfaces. For this4A′
surface of NO 2, the MUE of the NN fit is 1/3 of that of the MEG fit.
The MUEs of the current and previous MEG fits are similar for the
high-spin-state surfaces. The performance of the NN fit is outstand-
ing for the6A′surface of NO 2, and the MUE of the NN fit is about
1/6th of that of the MEG fit.TABLE VIII . Mean unsigned error (MUE) and root-mean-square error (RMSE) of the
fitted potential energy surfaces with respect to DSEC-CASPT2/maug-cc-pVTZ results
for various energy ranges (in kcal/mol).
NN MEG
Energy range Number of points MUE RMSE MUE RMSE
Doublet
0≤E<100 1664 0.8 1.3 1.3 1.9
100≤E<200 3559 0.8 1.4 1.2 1.8
200≤E<400 2177 0.8 1.8 1.6 2.6
400≤E<1000 880 1.6 2.5 4.2 5.8
1000<E>1850 154 2.3 3.5 8.6 10.8
All data 8434 0.9 1.7 1.8 3.1
Quartet
76<E<100 443 0.3 0.4 0.9 1.3
100≤E<200 3769 0.7 1.2 2.0 2.7
200≤E<400 2277 0.7 1.4 2.0 2.7
400≤E<1000 1107 1.7 2.9 3.8 5.4
1000<E>1914 222 2.2 3.5 10.3 14.3
All data 7818 0.8 1.7 2.4 4.0
Sextet
106≤E<200 2493 0.3 0.4 1.5 2.0
200≤E<400 3800 0.6 1.0 2.6 4.1
400≤E<1000 1728 0.8 1.4 5.5 8.1
1000≤E<1998 365 0.7 1.1 7.3 9.8
All data 8386 0.5 1.0 3.1 5.1
Some of the improvements in the current fits are due to the fact
that the fits of the NO 2surfaces used significantly more points than
were used in the previous O 3and N 2O surfaces. Since the O 3system
has a higher permutation symmetry than the other two systems, one
expects to require lower number of points for that system than for
the others; however, the fits of the NO 2surfaces also used about 3.5
times more points than those used for the N 2O surfaces, although
both have the same permutational symmetry. Not only does having
more points in the fitting datasets improve the quality of the fits for a
given order of polynomial but also it allows the application of higher
order without overfitting; thus, the present fits used l= 12, whereas
we used many-body functions with l= 9 for our N 4fit and l= 10 for
all our other fits prior to the present work.
Table IX also compares the data distribution in the energy bins
for the O 3, N 2O, and NO 2systems. For the O 3system, the general
trend is that the number of points used was a continuously decreas-
ing function of the bin energy. Also, as the spin increased from low
to high, the relative contribution of the higher energy bins increased
since higher spin states are usually more repulsive than the lower
spin states, and the energies of all spin states are given relative to the
global energy minimum, which belongs to the deepest energy well of
the lowest spin state. The middle or high spin states usually do not
have such deep wells as can be seen, for instance, in Figs. 7, 10, and
13 of Ref. 29 for O 3or later in this article for NO 2. In the cases of
NO 2and N 2O, this trend with spin state is the same as that for O 3,
but the number of points does not monotonically decrease as a func-
tion of bin energy. In the article on O 3,29the fitting errors suggested
that a fit becomes easier and more accurate for higher spin states.
J. Chem. Phys. 154, 084304 (2021); doi: 10.1063/5.0039771 154, 084304-8
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TABLE IX . Comparison of the mean unsigned error (MUE), the number of points, and the distribution of points in the different
energy bins (in kcal/mol) of the current fits of NO 2and our previous fits of O 3and N 2O.
Low spin Middle spin High spin
O3a11A′11A′′13A′13A′′15A′15A′′
MUE: MEG ( l= 10) 4.0 3.3 3.6 3.1 3.4 2.9
Number of points 1686 1622 1645 1605 1617 1587
0≤E<100 (%) 56 50 48 50 36 35
100≤E<200 (%) 18 21 21 21 27 26
200≤E<500 (%) 15 13 15 15 17 18
500≤E<1000 (%) 8 13 12 11 15 14
1000<E(%) 3 3 4 3 5 6
N2Ob 3A′ 3A′′
MUE: MEG ( l= 10) 2.9 2.5
Number of points 2298 2280
0≤E<100 (%) 25 26
100≤E<200 (%) 49 46
200≤E<350 (%) 19 20
350≤E<1000 (%) 6 8
1000<E(%) 0.2 0.3
NO 22A′ 4A′ 6A′
MUE: MEG ( l= 12) 1.8 2.4 3.1
MUE: NN 0.9 0.8 0.5
Number of points 8434 7818 8386
0≤E<100 (%) 20 6 ...
100≤E<200 (%) 42 48 30
200≤E<400 (%) 26 29 45
400≤E<1000 (%) 10 14 21
1000<E(%) 2 3 4
aReference 29.
bReference 28.
FIG. 2 . Contour map of2A′potential of
NO 2, where r3=r2. The increment in
the contours is 10 kcal/mol; the energies
are 0 kcal/mol–300 kcal/mol (left—MEG,
right—NN). The energy of the plateau is
added at the upper right corner.
J. Chem. Phys. 154, 084304 (2021); doi: 10.1063/5.0039771 154, 084304-9
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FIG. 3 . Contour map of4A′potential of
NO 2, where r3=r2. The increment in
the contours is 10 kcal/mol; the energies
are 0 kcal/mol–300 kcal/mol (left—MEG,
right—NN). The energy of the plateau is
added at the upper right corner.
This could be rationalized since a higher spin state surface is—in
general—more repulsive than a lower spin state surface and usually
has lower number of minima. This same trend is observed for the
NN fits of the NO 2surfaces, but it is reversed for the MEG fits of the
NO 2surfaces.Figures 2–13 are contour plots of the2A′,4A′, and6A′surfaces,
where the left and right panels are the MEG and the NN fits, respec-
tively. As a reminder, r1is the O–O distance and r2andr3are the two
N–O distances; the O–N–O and N–O–O bond angles are α1andα2,
respectively.
TABLE X . Optimized coordinates and energies of selected structures of the MB-PIP-MEG and PIP-NN fits.a
Structure Fit r1(Å) r2(Å) r3(Å) α1(deg) α2(deg) V(kcal/mol)
D1-min MEG 1.201 1.201 137.3 0.0
(2A1minimum) NN 1.214 1.214 135.4 0.0
D2-min MEG 1.286 1.286 104.1 36.8
(2B2minimum) NN 1.286 1.286 105.0 33.9
D3-ts MEG 1.221 1.221 180.0 38.8
NN 1.212 1.212 180.0 36.9
D4-sts MEG 1.710 1.710 54.9 179.4
NN 1.719 1.719 55.0 175.6
D5-min MEG 1.933 1.148 124.8 70.9
NN 1.985 1.145 124.5 68.7
D6-sts MEG 1.257 1.634 180.0 141.4
NN 1.246 1.616 180.0 139.2
Q1-min MEG 1.291 1.291 126.4 88.0
NN 1.294 1.294 130.6 85.8
Q2-sts MEG 1.361 1.361 180.0 127.1
NN 1.344 1.344 180.0 129.2
Q3-sts MEG 1.695 1.695 54.0 162.3
NN 1.694 1.694 54.4 163.8
S1-min MEG 1.343 1.778 180.0 190.2
NN 1.308 1.773 180.0 186.3
S2-ts MEG 1.548 1.548 180.0 194.8
NN 1.565 1.565 180.0 191.6
S3-(s)ts MEG 1.875 1.875 58.3 223.1
NN 1.828 1.828 57.5 224.0
aThe names are given in format mn-c, where mis the first letter of the multiplicity (D, Q, or S), nis the serial number of the struc-
ture, and cis the character of the stationary point; min—minimum, ts—transition structure, and sts—second-order transition
structure (hilltop). Coordinate r1is the O–O distance, and r2andr3are the two N–O distances; the O–N–O and N–O–O bond
angles are α1andα2, respectively.
J. Chem. Phys. 154, 084304 (2021); doi: 10.1063/5.0039771 154, 084304-10
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FIG. 4 . Contour map of6A′potential of
NO 2, where r3=r2. The increment in
the contours is 10 kcal/mol; the energies
are 0 kcal/mol–300 kcal/mol (left—MEG,
right—NN). The energy of the plateau is
added at the upper right corner.
In Figs. 2(2A′), 3(4A′), and 4(6A′), the α1angle is varied
from 30○to 180○and the two N–O distances ( r2=r3) are var-
ied from 1.0 Å to 3.5 Å. In these three figures, the region of O 2
is well separated from N ( α1<45○and longer N–O distances),
and the three surfaces are nearly identical because the interac-
tion of O 2(3Σ−
g) with the N atom (4S) has not yet split the spin
states. For shorter N–O distances and larger α1angle, the three sur-
faces are very different; next, we consider these regions of stronger
interaction.
The dominant feature in this region of the doublet surface is
the well of ground-state NO 2(2A1). The geometry of the minimum-
energy structure, as optimized by the Polyrate program88with the
fitted surfaces, is given in Table X as structure D1-min. The small
well of the2B2structure (D2-min in Table X) only appears in Fig. 2
as a small distortion of the contours at of α1≈100○, and it makes
the dominant well of the2A1structure somewhat asymmetric. At
α1= 180○, there is a transition structure (D3-ts) that connects two
D1-min structures. The wells in the NO 2region and the region of
O2+ N are separated by a relatively high barrier; in Fig. 2, a
second-order transition structure (D4-sts) is the lowest energy point
of this barrier.One can see a well in the quartet surface (Fig. 3) in the
NO 2region, similar to the doublet well (Fig. 2), but the quar-
tet well is not as deep. Table X has the coordinates of the min-
imum structure Q1-min, and in Fig. 3, the top of the inversion
barrier at α1= 180○is a second-order transition structure (Q2-sts),
unlike the case in Fig. 2. To get an inversion transition structure
between two Q1-min minima, we attempted to keep the bending
imaginary frequency of the linear and symmetric Q2-sts structure
by making the two NO distances different. For small or moder-
ate distortion of the two NO bond lengths, the optimization led
back to Q2-sts, and for significant distortion, the longer NO bond
broke.
As on the doublet surface, the wells in the NO 2region and
the region of O 2+ N are separated by a relatively high bar-
rier on the quartet surface, and the geometry of the second-order
transition structure (Q3-sts) of the quartet surface is very similar to
that of D5-sts of the doublet surface.
In the plots of Fig. 4 (in which the two NO distances are kept
equal), the sextet surface also shows a shallow well in the NO 2
region centered around a linear transition structure (S1-ts), which
leads to an asymmetric linear minimum, S2-min, with different NO
FIG. 5 . Contour map of2A′potential of
NO 2, where r3= 1.151 Å. The incre-
ment in the contours is 10 kcal/mol; the
energies are 0 kcal/mol–300 kcal/mol
(left—MEG, right—NN). The energy of
the plateau is added at the upper right
corner.
J. Chem. Phys. 154, 084304 (2021); doi: 10.1063/5.0039771 154, 084304-11
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FIG. 6 . Contour map of4A′potential of
NO 2, where r3= 1.151 Å. The incre-
ment in the contours is 10 kcal/mol; the
energies are 0 kcal/mol–300 kcal/mol
(left—MEG, right—NN). The energy of
the plateau is added at the upper right
corner.
FIG. 7 . Contour map of6A′potential of
NO 2, where r3= 1.151 Å. The incre-
ment in the contours is 10 kcal/mol; the
energies are 0 kcal/mol–300 kcal/mol
(left—MEG, right—NN). The energy of
the plateau is added at the upper right
corner.
FIG. 8 . Contour map of2A′potential of
NO 2, where r1= 1.208 Å. The incre-
ment in the contours is 10 kcal/mol; the
energies are 0 kcal/mol–300 kcal/mol
(left—MEG, right—NN). The energy of
the plateau is added at the upper right
corner.
J. Chem. Phys. 154, 084304 (2021); doi: 10.1063/5.0039771 154, 084304-12
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FIG. 9 . Contour map of 4A′potential of
NO2, where r1 = 1.208 Å. The incre-
ment in the contours is 10 kcal/mol; the
energies are 0 kcal/mol–300 kcal/mol
(left—MEG, right—NN). The energy of
the plateau is added at the upper right
corner.
FIG. 10 . Contour map of6A′potential
of NO 2, where r1= 1.208 Å. The incre-
ment in the contours is 10 kcal/mol; the
energies are 0 kcal/mol–300 kcal/mol
(left—MEG, right—NN). The energy of
the plateau is added at the upper right
corner.
FIG. 11 . Contour map of2A′potential of
NO 2, where α2= 120○. The increment in
the contours is 10 kcal/mol; the energies
are 0 kcal/mol–300 kcal/mol (left—MEG,
right—NN). The energy of the plateau is
added at the upper right corner.
J. Chem. Phys. 154, 084304 (2021); doi: 10.1063/5.0039771 154, 084304-13
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FIG. 12 . Contour map of4A′potential of
NO 2, where α2= 120○. The increment in
the contours is 10 kcal/mol; the energies
are 0 kcal/mol–300 kcal/mol (left—MEG,
right—NN). The energy of the plateau is
added at the upper right corner.
distances; thus, it is not shown in the plots of Fig. 4. For the sextet
state, the top of the barrier between the NO 2region and the region
of O 2+ N is higher in energy than the barriers for the doublet and
quartet surfaces. This is a transition structure by the MB-PIP-MEG
fit, but it is a second-order transition structure by the PIP-NN fit. For
this reason, this structure is denoted as S3-(s)ts in Table X to reflect
both possibilities.
In Figs. 5(2A′), 6(4A′), and 7(6A′), theα1angle is varied from
30○to 180○and one of the two N–O distances ( r2) is varied from 1.0
to 3.5 Å, while the other N–O distance ( r3) is fixed at 1.151 Å. Just
like Fig. 2, Fig. 5 contains the well of2A1structures (D1-min) since
it is lower in energy than the plateau, which corresponds to NO(2Π)
+ O(3P). Since the well region of the quartet surface in Fig. 3 is higher
in energy than the energy of NO(2Π) + O(3P), the well mainly dis-
appears in Fig. 6. The sextet surface, Fig. 7, is many repulsive, and
the plateau lies much higher in energy than those of the doublet and
quartet surfaces due to the quartet spin state of NO (4Π).
In Figs. 8(2A′), 9(4A′), and 10(6A′), theα2angle is varied from
30○to 180○and one of the two N–O distances ( r2) is varied from1.0 Å to 3.5 Å, while the O–O distance ( r1) is fixed at 1.208 Å.
The doublet surface (Fig. 8) has two small wells in this cut around
α2= 125○andr2= 1.3 Å, as well as α2= 30○andr2= 2.2 Å. Both wells
are part of the well of the same NOO structure (D5-min); the differ-
ence is only the order of the atoms. This cut also has a hill surface
feature around α2= 180○andr2= 1.6 Å, which belongs to structure
D6-sts. The quartet (Fig. 9) and sextet (Fig. 10) surfaces are mainly
repulsive for this cut.
In Figs. 11(2A′), 12(4A′), and 13(6A′), the α2angle is fixed at
120○, and the O–O distance ( r1) and one of the two N–O distances
(r2) are varied from 1.0 Å to 3.5 Å. The doublet and quartet sur-
faces are again very similar since both the O 2(3Σ−
g) + N(4S) and the
NO(2Π) + O(3P) channels appear in both spin states. In the sextet
surface, the channel of O 2(3Σ−
g) + N(4S) is still there, but in the other
channel, O(3P) has to combine with NO(4Π); thus, it has higher
energy than the doublet or quartet surfaces.
The computational cost using the surfaces is an important fac-
tor in dynamics simulations. To compare the compute times of
the NN and MEG fits, we used the data from the second series of
FIG. 13 . Contour map of6A′potential of
NO 2, where α2= 120○. The increment in
the contours is 10 kcal/mol; the energies
are 0 kcal/mol–300 kcal/mol (left—MEG,
right—NN). The energy of the plateau is
added at the upper right corner.
J. Chem. Phys. 154, 084304 (2021); doi: 10.1063/5.0039771 154, 084304-14
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trajectory calculations described in Sec. II C, where 1100 trajecto-
ries were run for each of the six surfaces. Both the NN and the
MEG subroutines have analytical gradients, and those were used
(unlike the case for the N 4system,56for NO 2, the evaluation of
the analytical gradients89of the neural network fit is faster than the
evaluation of numerical gradients). We know from a similar com-
parison made for N 4system56that if the Bulirsch–Stoer integrator
with adaptive step size is used in trajectory propagation, then the
cost ratio of the trajectories is dominated by the cost of calculating
the gradients. Averaging over the whole second series of trajecto-
ries (3300 trajectories for each fitting method), we found that the
average compute time for NN trajectories is 4.9 times larger than
that for MEG trajectories (a ratio of 4.6 for the doublet, 5.0 for
the quartet, and 5.2 for the sextet). The ratio of compute times is
closer to unity for NO 2than for N 4, where the ratio of compute
time for NN to MEG was about 17.56However, in the MEG fits of
NO 2, we use a higher-order polynomial (12th order) than that used
for N 4(9th order); thus, the lower ratio of compute times in the
present work is, at least, partially due to making the MEG fit more
expensive.
V. SUMMARY
In this work, we provide potential energy surfaces for study-
ing high-energy collisions between nitrogen atoms and oxygen
molecules. The doublet, quartet, and sextet A′surfaces presented
here are suitable for collisions of O 2(3Σ−
g) with N(4S). The sur-
faces were fitted two ways, using both the MB-PIP-MEG method
and the PIP-NN method against datasets with DSEC-corrected
CASPT2//DW-SA(3)-CASSCF(12o,17e) calculations. The neural
network fit has superior performance to MB-PIP-MEG, although
its gradient evaluation for trajectory simulations is 5 times more
expensive than the gradient of MB-PIP-MEG.
SUPPLEMENTARY MATERIAL
The supplementary material contains the fitting datasets of the
doublet, quartet, and sextet A′potential energy surfaces of NO 2, the
subroutines of the MB-PIP-MEG and the PIP-NN fits of the sur-
faces (which provide both the energy and the gradients calculated
analytically), and two examples of Molpro input files.
ACKNOWLEDGMENTS
Continuing discussions with Tom Schwartzentruber and
Graham Candler are greatly appreciated. Computational resources
were provided by the Department of Aerospace Engineering and
Mechanics at the University of Minnesota and by the Minnesota
Supercomputing Institute. The work of Y.L. and J.L. was sup-
ported by the National Natural Science Foundation of China (Grant
No. 21973009) and the Chongqing Municipal Natural Science
Foundation (Grant No. cstc2019jcyj-msxmX0087). The work of
H.G. and D.G.T. was supported by the U. S. Department of Energy,
Office of Science, Office of Basic Energy Sciences, under Award No.
DE-SC0015997.DATA AVAILABILITY
The data that support the findings of this study are
available within the article in the supplementary material.
The potential energy surfaces are also available in Potlib :
http://comp.chem.umn.edu/potlib/.
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Published under license by AIP Publishing |
5.0010798.pdf | Appl. Phys. Lett. 117, 062406 (2020); https://doi.org/10.1063/5.0010798 117, 062406
© 2020 Author(s).Magnetic domain wall curvature induced by
wire edge pinning
Cite as: Appl. Phys. Lett. 117, 062406 (2020); https://doi.org/10.1063/5.0010798
Submitted: 14 April 2020 . Accepted: 06 July 2020 . Published Online: 14 August 2020
L. Herrera Diez
, F. Ummelen , V. Jeudy , G. Durin , L. Lopez-Diaz
, R. Diaz-Pardo , A. Casiraghi
, G.
Agnus , D. Bouville , J. Langer
, B. Ocker
, R. Lavrijsen
, H. J. M. Swagten , and D. Ravelosona
Magnetic domain wall curvature induced by wire
edge pinning
Cite as: Appl. Phys. Lett. 117, 062406 (2020); doi: 10.1063/5.0010798
Submitted: 14 April 2020 .Accepted: 6 July 2020 .
Published Online: 14 August 2020
L.Herrera Diez,1,a)
F.Ummelen,2V.Jeudy,3G.Durin,4L.Lopez-Diaz,5
R.Diaz-Pardo,3A.Casiraghi,4
G.Agnus,1D.Bouville,1J.Langer,6
B.Ocker,6
R.Lavrijsen,2
H. J. M. Swagten,2and D. Ravelosona1
AFFILIATIONS
1Centre de Nanosciences et de Nanotechnologies, CNRS, Universit /C19e Paris-Saclay, 91120 Palaiseau, France
2Department of Applied Physics, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
3Laboratoire de Physique des Solides, CNRS, Universit /C19e Paris-Saclay, 91405 Orsay Cedex, France
4Istituto Nazionale di Ricerca Metrologica, Strada delle Cacce 91, 10135 Torino, Italy
5Departamento de F /C19ısica Aplicada, Universidad de Salamanca, Plaza de la Merced s/n., 37008 Salamanca, Spain
6Singulus Technology AG, Hanauer Landstrasse 103, 63796 Kahl am Main, Germany
a)Author to whom correspondence should be addressed: liza.herrera-diez@c2n.upsaclay.fr
ABSTRACT
In this study, we report on the analysis of the magnetic domain wall (DW) curvature due to magnetic field induced motion in Ta/CoFeB/
MgO and Pt/Co/Pt wires with perpendicular magnetic anisotropy. In wires of 20 lm and 25 lm, a large edge pinning potential produces the
anchoring of the DW ends to the wire edges, which is evidenced as a significant curvature of the DW front as it propagates. As the driving
magnetic field is increased, the curvature reduces as a result of the system moving away from the creep regime of DW motion, which impliesa weaker dependence of the DW dynamics on the interaction between the DW and the wire edge defects. A simple model is derived todescribe the dependence of the DW curvature on the driving magnetic field and allows us to extract the parameter r
E, which accounts for the
strength of the edge pinning potential. The model describes well the systems with both weak and strong bulk pinning potentials like
Ta/CoFeB/MgO and Pt/Co/Pt, respectively. This provides a means to quantify the effect of edge pinning induced DW curvature on magnetic
DW dynamics.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0010798
Understanding the behavior of magnetic domain walls (DWs)
when transitioning from full films into patterned structures is of
great importance for developing nanodevices for DW based technol-ogies.
1The analysis of DW dynamics in the so-called creep regime
of motion,2–5where defects play a central role, is a key aspect. In Ta/
CoFeB/MgO/Ta films with perpendicular anisotropy bulk defectdensities, and therefore depinning fields ( H
dep), are relatively low.6–8
In Pt/Co/Pt films, for example, the values of Hdepcan be more than
one order of magnitude higher.2,4,5Due to this low bulk pinning
potential, the DW dynamics can be easily controlled even in fullfilms by artificial pinning imposed through homogeneous materialengineering processes, like light ion irradiation
9–11or pre-patterned
substrates.12
Defects generated through micro/nanostructuring can also have
a great impact on pristine materials. DW velocities even in micrometersize wires have been found to experience a critical decrease below the
creep law dependence at low drive, which scales with the wire width.
13This effect is also accompanied by an increase in the curvature of theDW front and has, therefore, been attributed to edge pinning. A
deeper analysis of the DW curvature in wires is, therefore, needed inview of miniaturization for technological applications.
In this study, we present the analysis of the DW curvature in
as e r i e so f2 0 lm wide Ta/CoFeB/MgO/Ta wires as a function of
the magnetic field. The large edge pinning potential defines curva-tures that at low drive reach the maximum radius R¼w=2, where
wis the wire width. We present a simple model that accounts for
the variations in the DW curvature as a function of the drivingmagnetic field allowing for the extraction of r
E,ap a r a m e t e rt h a t
characterizes the strength of the edge pinning potential. We also
apply the same analysis to Pt/Co/Pt 25 lm wide wires, which pre-
sent a higher intrinsic bulk pinning potential. The model describeswell both systems, which shows that it can be used to assess thestrength of edge pinning and its influence on DW dynamics for
different bulk pinning potentials.
Appl. Phys. Lett. 117, 062406 (2020); doi: 10.1063/5.0010798 117, 062406-1
Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplThe samples investigated are Si/SiO 2/Ta (5 nm)/Co 20Fe60B20
(1 nm)/MgO (2 nm)/Ta (3 nm) films deposited by magnetron sputter-
ing and annealed at 300/C14C, exhibiting perpendicular magnetic anisot-
ropy. A series of 20 lm wires was fabricated by photolithography and
ion beam etching. The 25 lm wide Pt/Co/Pt wires were fabricated by
depositing a stack of Ta (4 nm)/Pt (4 nm)/Co (0.6 nm)/Pt (4 nm) by
magnetron sputtering using e-beam lithography. The final wire struc-
ture was achieved by conducting a lift-off process. Table I presents the
values of saturation magnetization ( Ms), effective perpendicular
anisotropy constant ( Keff), wire width, and depinning field ( Hdep)f o r
the two types of samples. Hdepcorresponds to the experimentally
determined transition between the creep and depinning regimes of
DW motion.4,5,14
Figure 1 shows the Kerr microscopy images of the DW curvature
for (a) the six CoFeB wires investigated, where curved DWs propagate
under a magnetic field of 1.4 mT (expanding domains have a light
gray color). In Fig. 1(b) , the images show the evolution of the DW cur-
vature with the increasing magnetic field in a Pt/Co/Pt wire, where as
the driving field increases, the DW curvature is progressively sup-
pressed. The difference in the applied fields at which large curvatures
are observed for each film scales with the values of Hdep.T h i ss h o w s
that the DW curvature is closely linked to a pinning mechanism.
In order to gain more insight into this behavior, let us first con-
sider a DW propagating in a strip as shown in Fig. 2(a) .I nt h i ss c e -
nario, the propagating DW remains straight and the Zeeman energy
gain ( dES) that the system experiences by letting the DW propagate a
distance Dxis the following:dEs¼/C02Ms/C1H/C1t/C1dAs; (1)
where ti st h et h i c k n e s so ft h em a g n e t i cw i r ea n d dAsis the area swept
by the DW [see Fig. 2(a) ]. Let us now consider again an initial state
where the DW is straight and moves by the same amount ( Dx).
However, the initially straight DW front can now develop a curvature
of radius Ras it propagates, and this straight-to curved DW displace-
ment is depicted in Fig. 2(b) . In this case, the Zeeman energy gain is
expressed as follows:
dEc¼r/C1t/C1dL/C02Ms/C1H/C1t/C1dAc: (2)
The gain in energy is reduced by the appearance of the first term that
accounts for the variation in DW length dLwith respect to the initial
state where the DW is straight. For a straight DW, its length is equal
toL¼w, while for a curved profile, L¼2Rarcsinw
2R/C0/C1.T h i sm e a n s
that the additional energy cost due to the DW curvature is linked to an
increase in the DW length equal to dL¼2R/C1arcsinw
2R/C0/C1/C0w.
The energy gain for a curved wall is also further reduced due to
the smaller area swept by the DW. For a straight DW, the area is
dAS¼w/C1Dx, while for the curved DW, it takes the following form:
dAc¼w/C1ðDx/C0hÞþACS; (3)
where his the sagitta as indicated in Fig. 2(b) andACSis the area of the
circular sector, which can be expressed as follows:
h¼R/C0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2/C0w2
4r
; (4)
ACS¼RL
2/C0w
2/C1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2/C0w2
4r
: (5)
Using these expressions, we can now calculate dEsand compare
it to dEcwhen, for example, the curvature radius is R¼10lma n dTABLE I. Ms,Keff, the wire width ( w), and the depinning fields Hdepfor the Ta/
CoFeB/MgO/Ta and Pt/Co/Pt wires.
Material Width ( lm) M s(A/m) Keff(J/m3) Hdep(mT)
CoFeB 20 8.7 /C21053.4/C21059.7
Pt/Co/Pt 25 1.4 /C21061.3/C2106100.0
FIG. 1. (a) Kerr microscopy images of curved DWs in CoFeB wires. (b) DW curva-
ture in a Pt/Co/Pt wire as a function of the magnetic field.
FIG. 2. Straight DW (a) and DW that transitions between a straight and a curved
profile upon displacement (b).Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 117, 062406 (2020); doi: 10.1063/5.0010798 117, 062406-2
Published under license by AIP Publishingw¼20lm. It is not surprising to find that if no edge pinning is
involved, it is more energetically favorable for the DW to remainstraight for all applied magnetic fields. Figure 3 shows the result of this
calculation considering the CoFeB parameters informed in Table I ;t h e
energy gain of a straight DW (solid blue line) is always more negativethan that of a curved DW (solid green line) in the whole magnetic field
range.
Up to this moment, no edge pinning was considered. As it is
known, the curved DW scenario [ Fig. 2(b) ] is a strategy to overcome
edge pinning. Therefore, in order to accurately compare it to a straightDW scenario [ Fig. 2(a) ], this last configuration should also include an
energy cost associated with keeping the DW straight in the presence of
edge pinning instead of allowing a curvature to appear. In a previousstudy,
13we have proposed an additional energy term to account for
deviations from the creep law that are observed due to a strong edge
pinning potential in CoFeB. This term includes the parameter rEthat
can be used to quantify the strength of the edge pinning and describesthe effect of the pull-back that the DW experiences in a strong edge
pinning potential.
In the present case, the increase in the DW length observed in
curved DWs due to edge pinning is taken into account by the term
r/C1t/C1dL. Let us assume that this energy cost due to edge pinning can
find its counterpart in the hypothetic straight DW scenario as an
increase in the DW energy equal to
r/C1t/C1dL¼dr/C1t/C1w: (6)
In this context, the difference in the DW energy between the initial
and final states of the straight DW after a displacement Dxis proposed
to be given by dr¼2r
E. This increment accounts for the effect of
having two strong anchoring points of the DW at each wire side butno bending; therefore, the expression for dE
stakes the form
dEs¼2rE/C1t/C1w/C02Ms/C1H/C1t/C1dAs: (7)The addition of this term shifts dEstoward higher energies producing
a crossing with dEcas shown in Fig. 3 (dashed blue line). This crossing
is the point where dEs¼dEcand defines a field limit ( HCL)f o rt h ee x i s -
tence of a DW curvature with radius R.A b o v et h i sfi e l dv a l u e ,i ti s
more energetically favorable for the DW to remain straight even in thepresence of edge pinning. Reordering the terms in dE
s¼dEcallows us
to extract the expression for HCL, which has the following form:
HCL¼rE/C1w/C0r/C1dL
2Ms/C1ðw/C1h/C0ACSÞ: (8)
Substituting for dL,h,a n d ACSintroduces Rin the equation and allows
for the evaluation of the dependence of the field limit on a given range
of DW curvatures (1/ R). The experimentally observed DW curvature
dependence on the applied magnetic field (symbols) together with thecalculated H
CL(solid line) is shown in Fig. 4(a) for CoFeB and in (b)
for Pt/Co/Pt. It is worth mentioning once again that this model
describes the field limit under which a given curvature can beobserved, which includes the possibility of one curvature appearing atdifferent magnetic fields below H
CL.
The model presented here is valid up to a maximum curvature of
2/w; however, at low drive, higher curvatures are observed for both
CoFeB and Pt/Co/Pt. This occurs at magnetic fields below the valueneeded to depin the DW from the wire edges, even at the expense of amaximum curvature, but well above that needed to overcome pinning
at the center of the wire. In this case, it may be more energetically
favorable for a DW to increase its length going from the edges to thecenter of the wire to increase the switched area. In this context, theDW front could take a more triangular shape and exhibit at the center
a curvature smaller than w=2. This effect can be visualized in the wires
at the center of Fig. 2(a) . The description of the curvatures above 2/ w
(R<w=2) at low drive is not considered in this model.
The solid curves representing H
CLprovide a dividing line
between curved and straight DWs in a magnetic field map. As men-
tioned, the calculation of these curves takes into account the values ofK
eff,Ms, and the width of the wire, while the value of rEis adjusted to
model the results. The values of rEobtained for CoFeB and Pt/Co/Pt
are 1.01 /C210–2N/m and 6.35 /C210–2N/m, respectively. rEhas been
introduced in a previous study as a means to quantify the strength ofthe edge pinning potential in the context of edge-pinning induceddeviations from the creep law in CoFeB wires.
13In this framework, it
is particularly interesting to compare rEto the DW surface tension
r¼4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAex/C1Keffp,w h e r e Aexis the exchange stiffness constant
(2.3/C210–11J/m for CoFeB and 1.6 /C210–11J/m for Pt/Co/Pt). The
ratio r/rEresults in 1.1 and 0.28 for CoFeB and Pt/Co/Pt wires,
respectively. It has been proposed13that as this ratio decreases, the
edge pinning potential represented by rEprogressively dominates over
the DW surface energy allowing for a DW curvature. However, whencomparing values for different materials, not only the differences in
the intrinsic parameters but also of the bulk pinning potentials need to
be taken into account. The effect of the edge pinning potential dependson its relative strength with respect to the bulk pinning potential, inti-mately related to the value of the depinning field H
dep.
The measured DW curvatures in CoFeB and Pt/Co/Pt cover a
similar range going from those corresponding to R¼w/2 down to low
values corresponding to a large R(higher than R¼w/0.4) for nearly
straight DWs. However, the magnetic field range over which each cur-
vature is observed is significantly different with respect to Hdep,w h i c hFIG. 3. Energy gain dEs(blue line) and dEc(green line) for straight and curved
DWs as a function of the applied magnetic field in CoFeB. The straight DW withoutpinning increases its energy when the term 2 r
E/C1t/C1wis added to account for
edge pinning (dotted blue line).Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 117, 062406 (2020); doi: 10.1063/5.0010798 117, 062406-3
Published under license by AIP Publishingis marked by red lines in Figs. 4(a) and4(b). In CoFeB at Hdep,t h e
maximum DW curvature observed experimentally and also describedby the model is 6 /C210
–4m/C01(R¼w/1.2). The DW curvature is still
relatively pronounced and continues to be well beyond the creep
regime. Only at fields around 30 mT, three times larger than Hdep,d o
the DWs become relatively straight, showing a maximum curvature of2/C210
–4m/C01(R¼w/0.4). In contrast, the Pt/Co/Pt wires at Hdep
already show relatively straight DWs with a maximum curvature
below 2 /C210–4m/C01. This can be related to the large differences of
about one order of magnitude found in the depinning field for the two
materials. This difference also exists in full films and is due to theintrinsic bulk pinning potential that is known to be exceptionally lowin CoFeB materials.
2,6In this context, a relatively small edge pinningpotential can already have a large impact on the dynamics in CoFeB,
while a much larger one is needed to dominate the DW dynamics in
Pt/Co/Pt. Consequently, the edge pinning potential in CoFeB inducesa much stronger effect than in Pt/Co/Pt, reflected here in the strong
and persisting curvature of the DWs even beyond the creep regime. In
contrast, in Pt/Co/Pt, a significant DW curvature induced by edge pin-ning is only observed deep into the creep regime.
This model can also be used to extrapolate the effects of edge pin-
ning to narrower wires. Figure 5(a) shows the calculated H
CLprofiles
using rE¼1.01/C210–2N/m, the intrinsic parameters for CoFeB and
varying the wire width. The HCLvalues increase as the wire becomes
narrower. The plot in the inset of Fig. 5(a) shows the HCLvalues for a
DW radius of w/2 as a function of the wire width; here, a dramatic
increase is observed for narrow wires. This model could, therefore, allow
for the estimation of the effects of edge pinning for a given material and
a particular wire edge structure, for example, linked to the fabricationprocess, as the wire width decreases. The estimation could be conve-niently performed by analyzing the DW dynamics at much larger scales.
For wires widths below 5 lm, it is interesting to compare this
analytical model with micromagnetic simulations. Figure 5(b) shows
the result for CoFeB wires; filled squares and circles represent the cur-
vatures observed for w¼4lma n d w¼2lm with a maximum edge
roughness ( M
ER)o f3 0n m( s e et h e supplementary material ). The DW
curvature follows a similar trend to the analytical model with respect
to magnetic field and w.MERplays a key role; empty squares show the
trend obtained for w¼4lma n d MER¼80 nm, which is reflected in a
shift of the curvatures at all magnetic fields to significantly higher
values. The curvature at 40 mT was also evaluated in a wire with very
rough edges, MER¼300 nm, showing a further increase (crossed
diamond). This is in line with the conclusions made earlier regardingthe effects of the fabrication process on the DW curvature.
The curve obtained using the analytical model for w¼4lmi s
shown in Fig. 5(b) ; the curvatures are significantly larger than those
obtained with micromagnetic simulations even for very large M
ER.I t
is, therefore, important to highlight that the micromagnetic simula-
tions were made using the experimental values presented in Table I
while exploring the effect of variations in the Gilbert damping parame-
tera. The curvatures for w¼4lm in open and full squares and the
FIG. 4. Domain wall curvature as a function of magnetic field in a 20 lm wide
CoFeB wire (a, different colors correspond to the different wires shown in Fig. 1 )
and in a 25 lm wide Pt/Co/Pt wire. The solid lines are the calculation of HCL.
FIG. 5. (a) Calculated HCLprofile for rE¼1.01/C210–2N/m (CoFeB) for different
wire widths. HCLvalues corresponding to R¼w/2 as a function of the wire width
(inset). (b) Comparison with micromagnetic simulations.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 117, 062406 (2020); doi: 10.1063/5.0010798 117, 062406-4
Published under license by AIP Publishingcrossed diamond were calculated using a¼1, all showing relatively
low values. For high MER, changing afrom 1 (crossed diamond) to
0.01 (crossed square) increases significantly the curvature, as seen
at 40 mT. CoFeB is known to have low values of ain the 0.015
range;6therefore, lower avalues not only give curvatures closer to
the analytical model but are also more compatible with experimen-
tal values.
In contrast, for low MER, the effect of varying ais not a domi-
nant feature. This brings attention to the crucial role of disorder inthe DW dynamics in CoFeB. In the present study, perfect wires aresimulated, while studies in the literature show that adding a granu-
lar structure and an anisotropy distribution to simulate disorder
can have a large impact on the DW dynamics. In this context, addi-tional energy dissipation channels appear in the system, whichcould also affect the response to a change in the value of a.
11A
detailed micromagnetic study would be needed to fully characterize
these effects.
Micromagnetic simulations have also been performed for
w¼250 nm and w¼500 nm, where the analytical model finds its limit
of validity since it is based on the one dimensional (1D) model of DWmotion. A dimensionality change may occur from a 2D to a 1D
medium for very narrow wires
15and edge effects can rule over creep
dynamics,16limiting the use of a 1D based model. For w¼500 nm,
MER¼30 nm, and a¼0.015, the internal structure of the DW
presents a large number of Bloch lines and an irregular DW front,showing no curvature for either 10 mT or 20 mT. A similar behavior is
observed for w¼250 nm (see the supplementary material ). This con-
firms the presence of a more complex dynamics at this scale goingbeyond the 1D model and calls for a more careful theoretical analysis.
In conclusion, we present a simple model to describe the depen-
dence of the DW curvature on the applied magnetic fields in wires
with an edge pinning potential. This model allows for the estimationof the magnetic field limit up to which a given DW curvature can beobserved, and it has been applied to two key spintronics materials withlow and high bulk pinning potentials, CoFeB and Pt/Co/Pt. The analy-
sis of the DW curvature also allows for the extraction of the parameter
r
E, which can be used to compare edge pinning potentials in different
devices. The relative strength of the edge pinning potential withrespect to the surface tension of the DW ( r/r
E) and the differences
between the edge and bulk pinning potentials in each material are the
key aspects involved in the description of the DW curvature. It has
also been shown that this model can be used to extrapolate the effectsof edge pinning observed in relatively large wires to smaller dimen-sions and has a good correspondence with the results obtained frommicromagnetic simulations. Therefore, this model can be of consider-
able interest for the understanding and quantification of the effects of
edge pinning in patterned magnetic structures.See the supplementary material for information about micromag-
netic simulations.
We gratefully acknowledge financial support from the
European Union FP7 and H2020 Programs (MSCA ITN Grant Nos.
608031 and 860060), the French National Research Agency (project
ELECSPIN), and Ministerio de Economia y Competitividad of theSpanish Government (Project No. MAT2017-87072-C4-1-P). Theauthors thank G. van der Jagt for useful comments.
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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Published under license by AIP Publishing |
1.4933374.pdf | Resonance Raman spectra of organic molecules absorbed on inorganic
semiconducting surfaces: Contribution from both localized intramolecular excitation
and intermolecular charge transfer excitation
ChuanXiang Ye , Yi Zhao, , and WanZhen Liang,
Citation: The Journal of Chemical Physics 143, 154105 (2015); doi: 10.1063/1.4933374
View online: http://dx.doi.org/10.1063/1.4933374
View Table of Contents: http://aip.scitation.org/toc/jcp/143/15
Published by the American Institute of PhysicsTHE JOURNAL OF CHEMICAL PHYSICS 143, 154105 (2015)
Resonance Raman spectra of organic molecules absorbed on inorganic
semiconducting surfaces: Contribution from both localized intramolecular
excitation and intermolecular charge transfer excitation
ChuanXiang Ye,1Yi Zhao,2,a)and WanZhen Liang1,2,a)
1Department of Chemical Physics, University of Science and Technology of China,
Hefei 230026, People’s Republic of China
2State Key Laboratory of Physical Chemistry of Solid Surfaces, Collaborative Innovation Center of Chemistry
for Energy Materials, and College of Chemistry and Chemical Engineering, Xiamen University,
Xiamen 361005, People’s Republic of China
(Received 25 August 2015; accepted 6 October 2015; published online 20 October 2015)
The time-dependent correlation function approach for the calculations of absorption and resonance
Raman spectra (RRS) of organic molecules absorbed on semiconductor surfaces [Y . Zhao and W.
Z. Liang, J. Chem. Phys. 135, 044108 (2011)] is extended to include the contribution of the inter-
molecular charge transfer (CT) excitation from the absorbers to the semiconducting nanoparticles.
The results demonstrate that the bidirectionally interfacial CT significantly modifies the spectral line
shapes. Although the intermolecular CT excitation makes the absorption spectra red shift slightly, it
essentially changes the relative intensities of mode-specific RRS and causes the oscillation behavior
of surface enhanced Raman spectra with respect to interfacial electronic couplings. Furthermore, the
constructive and destructive interferences of RRS from the localized molecular excitation and CT
excitation are observed with respect to the electronic coupling and the bottom position of conductor
band. The interferences are determined by both excitation pathways and bidirectionally interfacial
CT. C2015 AIP Publishing LLC. [http: //dx.doi.org /10.1063 /1.4933374]
I. INTRODUCTION
Efficient generation and separation of photogenerated
electron-hole pairs are a key requirement for realizing the
photovoltaic cells with high e fficiencies. Extensive investi-
gations have revealed that there are two possible pathways
for photoinduced interfacial electron transfer in dye-sensitized
solar cells. One is from the localized molecular excited state to
the semiconductor conduction band, and the other is generated
by a one-step intermolecular charge transfer (CT) excitation.
In the case of weak coupling between the organic molecule
and the semiconductor, photon absorption creates a localized
excited state of absorber and sequentially the electrons on
the excited state inject into the manifold of electronic states
of semiconductor conduction band,1–5and a so-called photo-
induced electron transfer takes place. In the case of strong
coupling, alternatively, photoexcitation may directly generate
charge separation between the molecule and semiconductor,
named as a direct CT excitation.6–11This CT excitation is
very important especially for the use of sun light because it
can enhance charge separation e fficiency and modify large
band gaps of typical inorganic nanoparticles to a visible photo-
absorption regime. These two possible photoinduced CT path-
ways may be straightforwardly demonstrated by measuring
the optical absorption spectra of the hybrid materials, and
the absorption band similar to that of absorber with a slight
red shift may correspond to photoinduced electron transfer
whereas CT absorption is related to an additional absorption
a)Electronic addresses: yizhao@xmu.edu.cn and liangwz@xmu.edu.cnband at the energies lower than the absorption edge of either
component.10,12
For detailed understanding interfacial CT dynamics, alter-
natively, resonance Raman scattering (RRS) should be a more
powerful and suitable tool. It can provide a good estimation
of vibrational frequencies and reorganization energies,13–16
which are very sensitive to the conformational changes of
systems undergoing photoexcitation. In addition, the analysis
of RRS intensities can obtain excited state dynamics in a
subpicosecond time scale and be particularly useful to describe
the photo-induced ultrafast mode-specific CT processes. In-
deed, several experiments have measured the RRS of dye-
sensitized TiO 2nanoparticles from locally excited states and
CT states.7,17–19We have also theoretically investigated the
RRS of several organic molecules including Duschinsky rota-
tion and Herzberg-Teller e ffects to reveal geometric structure
and excited-state dynamics,20–25and further demonstrated that
the RRS of C343 /TiO 2from the molecular local excitation and
the intermolecular CT excitation have an obviously di fferent
feature.26
The above investigations are essentially based on the
assumption that the RRS are contributed from a single excited
state, either a localized molecular excited state or an inter-
molecular CT state. However, it has been known27,28that a
typical time scale of the interfacial CT is in femtoseconds for
dye-sensitized TiO 2nanoparticles. This ultrafast CT should
strongly a ffect the RRS intensities predicted from a single elec-
tronic state. To incorporate the complicated charge dynamics,
the simplest model should include an electronic ground state,
a localized excited state of the absorber, and a CT state as well
0021-9606/2015/143(15)/154105/9/$30.00 143, 154105-1 ©2015 AIP Publishing LLC
154105-2 Ye, Zhao, and Liang J. Chem. Phys. 143, 154105 (2015)
as their nonadiabatic couplings.17,29–36For the RRS from a
localized excited state, it is generally accepted that the RRS
intensities are suppressed because of losing population on the
excited state by the electrons injecting into the conduction band
of nanoparticles. However, recent several experiments37–39
have found that the RRS of the sole chromophore can be
enhanced by the interactions with inorganic nanoparticles. The
enhancement is commonly explained by the Herzberg-Teller
contribution,40,41which is essentially perturbative and cannot
explain the ultrafast electron injection. Based on Anderson-
Newns type Hamiltonian,42–45which has been widely em-
ployed to explore the interfacial CT dynamics,45–47we have
proposed a time-domain method to calculate RRS from the
local excitation incorporating the interfacial CT dynamics.48
Both the enhancement and de-enhancement of RRS inten-
sities of the chromophore in a hybrid model system have
been predicted, which heavily depend on mode-specific reor-
ganization energies, driving force, and interfacial electronic
coupling.
As the intermolecular CT excitation is incorporated, a
theoretical prediction of RRS becomes more complicated and
is still a challenge task. When the localized molecular excited
state and CT states are not in resonance, RRS calculation is
relatively easy and the Anderson-Newns type model48can
be straightforwardly used by separately considering a locally
excited state and a CT state incorporating ultrafast interfacial
CT dynamics. However, as these states are all in resonance,
one has to simultaneously consider the contributions to RRS
from those states. In this case, some of the new physical
insights related to the CT events and spectra may appear
like the bidirectionally interfacial CT and the interference
of RRS. It should be noticed that the RRS interference has
already been observed experimentally in two coupled excited
states.17,49–52
In this paper, we focus on this challenge problem, i.e.,
we theoretically calculate the RRS of organic molecules ab-
sorbed on inorganic-nanoparticles incorporating the contri-
butions both from the localized molecular excited state and
the intermolecular CT states of hybrid system. Based on
Anderson-Newns type model, we first extend the time-depen-
dent correlation function method proposed in our previous
work48to involve the CT excitation.
In the theoretical model, the continuum conduction
band states of the inorganic-nanoparticle are discretized
into hundreds of states evenly spaced by Legendre poly-
nomials,53,54and the CT states are thus constructed by the
cation state of organic chromophore immersed in a series of
discrete band states. In numerical tests, the vibrational mode
frequencies and the corresponding shifts are taken from the
calculations for the molecule DTB-Pe-(CH) 2-COOH54,55with
respect to the density functional theory (DFT) /time-dependent
DFT (TDDFT). Starting from the investigation of structure
parameter dependence of absorption spectra, we reveal the
contributions from the localized molecular excitation and
the CT excitation to the chemical enhancement in surface
enhanced Raman spectra (SERS)37,56–62and show detailed
interfacial CT dynamics. In addition, the quantum interference
effect of RRS from the locally excited state and the CT states
is further explored. It should be addressed that we neglect theanharmonicity and Duschinsky rotation in the demonstration,
which may importantly a ffect the spectral position and shape,
to dominantly catch the e ffect of CT excitation.
The paper is arranged as follows. In Sec. II, we present
a time-dependent formula of absorption spectra and RRS for
chromophores absorbed on semiconductor surfaces. Sec. III
shows numerical tests, and the concluding remarks are pre-
sented in Sec. IV.
II. THEORETICAL MODEL
Following the similar model used in our previous work,48
in this section, we display the theoretical expressions of
absorption spectra and RRS of organic molecules absorbed
on inorganic semiconductors. Fig. 1 displays the elemen-
tary photo-physical processes involved in spectral measure-
ments. After the photo-excitation, both the localized molecular
excited states |e⟩and the CT states |k⟩of the hybrid system may
be formed starting from the molecular ground state |g⟩. Due to
the interaction between |e⟩and |k⟩, the excited electron transfer
may occur bidirectionally between these two states, which
subsequently a ffects the absorption spectra and RRS. Here,
we have neglected the photoexcitation from the valance band
because the energy band gaps of inorganic semiconductors are
typically much larger than the molecular excitation energy.
The total Hamiltonian for this hybrid system can be written
as follows:
H=Hg+HE, (1)
HE=He+HCT+Ve,k. (2)
Here, Hg=|g⟩Hg(Q)⟨g|andHe=|e⟩He(Q)⟨e|represent the
Hamiltonian of molecular ground and excited states, respec-
tively, and Qdenotes the normal-mode coordinates. HCT
is the Hamiltonian of photoinduced interfacial electron-
transfer states, which includes the Hamiltonian of molecular
cation and the conduction bands of semiconductor, and reads
as42–45
HCT=
k|k⟩εk⟨k|+Hc(Q), (3)
whereεkis conduction band energies and it covers from the
bottom to top of the band energies and Hcrepresents the
molecular cation Hamiltonian. The physical insight of HCT
is understood by that the electrons on the molecular excited
state can transfer into any conduction band energy level. Ve,k
in Eq. (1) denotes the interaction between states |e⟩and |k⟩and
reads as
Ve,k=
k|k⟩Vke⟨e|+H.C., (4)
where Vkeis the electronic couplings between molecular
excited state |e⟩and band state |k⟩, and it is assumed to be
a constant independent of k.
To calculate vibrationally resolved absorption spectra and
RRS, one has to obviously consider molecular vibrational mo-
tions in its ground, excited, and cation states. The molecular
vibrational Hamiltonian is assumed to satisfy an eigenstate
equation154105-3 Ye, Zhao, and Liang J. Chem. Phys. 143, 154105 (2015)
FIG. 1. A schematic diagram of chromophore-semiconductor hybrid sys-
tem including the localized molecular excitation and intermolecular CT
excitation.
Hi(Q)|ni⟩=ϵni|ni⟩, (5)
whereϵniand |ni⟩denote the nth vibrational eigenvalue
and eigenwavefunction in the ith electronic state ( i=g,e,c),
respectively.
During the process of the optical excitation, the transition
dipole moments µek,gis
µek,g=|e⟩µeg⟨g|+
k|k⟩µkg⟨g|. (6)
In our previous work,48we have only considered the transition
from the |g⟩to|e⟩, i.e.,µkgwas set to be 0. If µkgis not zero,
new photophysical phenomena will appear like bidirectionally
interfacial CT and spectral interference. The target of this paper
is to address these phenomena.
With use of Hamiltonian equation (1), vibration-resolved
spectra can be formally calculated. Under the perturbation
approximation for the electric field of light, absorption spectra
can be written as63
αabs(ωI)∝ωI∞
−∞dtei(ωI−ωeg+εg)t−γ|t|Ca(t). (7)
Here,ωIis the incident light frequency, ωegis the energy gap
between state |e⟩and state |g⟩,γis introduced to consider
excited-state life time, and Cais the dipole autocorrelation
function
Ca=
ngPng⟨g,ng|µe−i(He+HCT+Ve,k)tµ|g,ng⟩, (8)
where Pngis the thermal population of the vibrational states in
the electronic ground state. Together with Eq. (6), the correla-
tion function can be written as
Ca(t)=
ngPng[(
n′e⟨ng|n′
e⟩µg,e⟨e,n′
e|
+
k′,nk′
c⟨ng|nk′
c⟩µg,k′⟨k′,nk′
c|)
·exp−i(He+HCT+Ve,CT)t·(
ne|e,ne⟩
×µe,g⟨ne|ng⟩+
k,nkc|k,nk
c⟩µk,g⟨nk
c|ng⟩)]. (9)Obviously, there are four terms in the above equation, in which
two terms come from the interaction, Ve,CT, between states |e⟩
and |k⟩. As these two states are decoupled, i.e., the electronic
coupling term Ve,CTis zero, Ca(t)remains only as two terms,
Ca(t)=
ngPng[
n′e,ne⟨ng|n′
e⟩µg,e⟨e,n′
e|e−iHet|e,ne⟩
×µe,g⟨ne|ng⟩+
k′,k
nk′
c,nkc⟨ng|nk′
c⟩µg,k′⟨k′,nk′
c|
×e−iHCTt|k,nk
c⟩µk,g⟨nk
c|ng⟩]. (10)
Even in this case, the spectral interference will appear because
of the phase interference of these two terms.
For RRS, its di fferential photon scattering cross section is
given by
σ(ωI,ωS)∝4ωIω3
S
9c4S(ωI,ωS). (11)
Here,ωIandωSdenote the frequencies of the incident and
scattering photons, respectively. c is the speed of light. The
resonance Raman line shape S(ωI,ωS)has a form64
S(ωI,ωS)=2π
ng,mgP(ng)Ing,mg(ωI)
×δ(ωS−ωI−εng+εmg), (12)
where Ing,mgcorresponds to the Raman excitation profile for
the transition from the vibrational state |ng⟩to the vibrational
state |mg⟩in electronic ground state |g⟩, and it can be expressed
as the Fourier transform of dipole correlation function Cmn(t)
Ingmg=2π ∞
0dtei(ωI−ωeg+εng)t−γtCmn(t) 2
, (13)
with
Cmn(t)=⟨g,mg|µe−i(He+HCT+Ve,k)tµ|g,ng⟩. (14)
The physical insight of this equation can be explained as fol-
lows. Initially, the hybrid system is in the ground state |g,ng⟩.
After a light-matter interaction, the system begins to propagate
in the mixed molecular excited state and CT states with the
initial condition µ|g,ng⟩. At time t, the wavefunction on this
mixed excited states is projected back to the ground state
|g,mg⟩by dipole moment µ. Compared with the matrix ele-
ments of propagator in Ca(t)of Eq. (8), the only di fference is
that the projected ground state in Cmn(t)is replaced by |g,mg⟩.
Therefore, Cmn(t)has a similar expression to that of Ca(t), and
obviously,
Cmn(t)=(
n′e⟨mg|n′
e⟩µg,e⟨e,n′
e|+
k′,nk′
c⟨mg|nk′
c⟩µg,k′⟨k′,nk′
c|)
·exp−i(He+HCT+Ve,CT)t·(
ne|e,ne⟩
×µe,g⟨ne|ng⟩+
k,nkc|k,nk
c⟩µk,g⟨nk
c|ng⟩), (15)
which still has four terms. Since Ca(t)andCmn(t)include
the same propagate operator, the same numerical technique
can be used in the propagation. It should be addressed that
the calculation of time evolution operator is a di fficult task
because of the coupling between state |e⟩and state |k⟩and154105-4 Ye, Zhao, and Liang J. Chem. Phys. 143, 154105 (2015)
many vibrational basis sets. Here, we do not straightforwardly
calculate the propagator; alternatively, we expand the time-
dependent wavefunction of mixed excited states on diabatic
basis sets as follows:
|ΨE(t)⟩≡exp(−i(He+HCT+Ve,CT)t)µ|g,ng⟩
=
meAme(t)|e,me⟩+
k,mcAk
mc(t)|k,mc⟩.(16)
Then, the coe fficients Ame(t)andAk
mc(t)can be calculated by
solving the time-dependent Schrödinger equation
i∂|ΨE(t)⟩
∂t=(He+HCT+Ve,k)|ΨE(t)⟩ (17)
with the initial condition
Ame(0)=⟨me|µe,g|ng⟩, (18)
Ak
mc(0)=⟨mc|µk,g|ng⟩, (19)
which are determined from Eqs. (9) and (15).
Numerically, one needs to discretize the continuum con-
duction band, which may be directly discretized into the states
|k⟩with a uniform energy di fference. However, it requires
many discretized states for numerical convergence.65To speed
up the convergence, Ak
mc(t)are alternatively transformed into
a set of new coe fficients by48,53,54,66
Ak
mc(t)=N
s=0Cs
mc(t)Us(εk), (20)
where Us(E)is modified Legendre polynomials
Us(E)=
2s+1
EmaxPs(2E
Emax−1), (21)
with the energy Ebeing from ∆EtoEmax+∆E, where ∆E
is the bottom energy of conduction band with respect to the
molecular excited-state energy, and Emaxis the bandwidth.
Assuming that the electronic coupling Vkeis independent of
the band energy εk, and is written as V/√Emax, from Eqs. (16),
(17), and (20), we finally obtain a set of di fferential equations
i∂Ame
∂t=Ameϵme+
nc⟨me|V|nc⟩C0
nc, (22)
i∂C0
mc
∂t=(ϵmc+1
2Emax)C0
mc
+a1C1
mc+
nc⟨nc|V|me⟩Ame, (23)
i∂Cs
mc
∂t=(ϵmc+1
2Emax)Cs
mc+asCs−1
mc
+as+1Cs+1
mc,s=1,2,..., N, (24)
and the initial conditions of Cs
mc(0)become
C0
mc(0)=
EmaxAk
mc(0), (25)
Cs
mc(0)=0,s=1,2,..., N. (26)
The above di fferential equations can be solved by some
standard numerical recipes. Here, a six-order hybrid Gear algo-
rithm coded by us48is employed because it can be easily
applied to solve complex di fferential equations. Once Cs
mc(t)are known, they can be transformed back to get Ak
mc(t). Subse-
quently, the correlation functions, Eqs. (9) and (15), are easily
calculated, and the spectra are calculated by simple Fourier
transforms of these correlation functions.
III. RESULTS AND DISCUSSION
In the calculations of absorption and RRS spectra of the
organic molecules absorbed on inorganic nanoparticles, the
essential structure parameters required are molecular vibra-
tional Hamiltonians on the ground, excited, and cation states,
as well as the conduction band of semiconductors. The molec-
ular vibrational Hamiltonians in normal mode coordinates are
written as
Hi(Q)=
j(P2
j/2+1
2ω2
j(Qj−Qi
j0)2), (27)
where icorresponds to the electronic state of |g⟩,|e⟩, or the
state of molecular cation. For simplicity, we have assumed that
the normal mode coordinates and frequencies in the three elec-
tronic states are same, and the only di fferences between them
are the shift values Qi
j0calculated at the equilibrium geome-
tries of three electronic states. The frequencies and shifts for
eight normal modes with relatively large Huang-Rhys factors
are listed in Table I. Those parameters are applied to mimic
the hybrid system formed by DTB-Pe-CH 2-P(O)(OH) 2(DTB-
Pe) and TiO 2, and they come from a preliminary DFT /TDDFT
calculation with the exchange-correlation functional B3LYP
for the DTB-Pe molecule within Gaussian 09 software.67It
is found that the calculated absorption spectrum of DTB-Pe
based on these eight modes can reach a good agreement with
the experimental data68atωeg=2.72 eV , shown in Fig. 2(a).
The conduction bandwidth Emaxis set to be 2 eV , correspond-
ing to a TiO 2nanoparticle. The number of discretized states
is about 350, which guarantees numerical convergence. It is
noted here that the key point in the simulations is to adopt the
basis sets of the molecular vibrational motions. The molecule
essentially incorporates three di fferent vibrational motions,
corresponding to the electronic ground, excited, and CT states,
respectively. For the molecule having Mnormal modes, the
basis sets are chosen as
|ni⟩=|n1
i,n2
i,..., nM
i⟩, (28)
where nk
idenotes the vibrational state of the kth mode in the
ith electronic state. The basis set number for the di fferent mode
TABLE I. The chosen mode frequencies (in cm−1), shifts (in a.u.), and the
corresponding Huang-Rhys factors (in a.u.) of DTB-Pe.
wi Qi,g Qi,e Qi,CT Se SCT
262 0.0 12.4 14.2 0.09 0.12
388 0.0 14.5 27.3 0.18 0.65
607 0.0 5.2 5.4 0.03 0.04
1052 0.0 0.0 15.5 0.00 0.57
1200 0.0 8.0 3.1 0.17 0.02
1326 0.0 9.0 −12.2 0.24 0.44
1418 0.0 5.7 2.1 0.10 0.01
1674 0.0 8.6 3.0 0.28 0.03154105-5 Ye, Zhao, and Liang J. Chem. Phys. 143, 154105 (2015)
FIG. 2. The experimental and theoretical absorption spectra (a) for the iso-
lated molecule DTB-Pe and (b) for the hybrid system DTB-Pe /TiO 2. The
incident light energy has been set to be the excitation energy of molecular first
singlet excited state. The structure of DTB-Pe is inserted in the left panel.
is taken to be di fferent. The concrete numbers are determined
by the mode frequency and the maximum energy ϵmaxreached
by the incident electromagnetic field, i.e., we use ϵnk
i=ϵmaxto
obtain the maximum value of nk
i.
With the above set of parameters kept unchanged, we
manipulate other parameters like electronic coupling Vke, the
energy di fference, ∆E, between the molecular excited state and
the bottom of the conduction bands, and the transition dipole
moments to show the parameter-dependence of the spectra.
These dependences should be important for better understand-
ing interfacial CT processes and designing dye-sensitized solar
cells.
A. Absorption spectra
With the vibrational-mode parameters listed in Table I,
one can fit the experimental absorption spectrum for a single
molecule quite well,68as shown in Fig. 2(a). To obtain the
agreed absorption spectrum for DTB-Pe-TiO 2hybrid sys-
tem, one needs to know the electronic coupling values and
the energy gap ∆E. According to the experimental measure-
ment68and current calculations, ∆Eis about−1.0 eV . When
we set the electronic coupling Vketo be 0.7 eV and the
transition dipole moment µgkto be zero, the experimental
absorption spectra for the hybrid system can be well re-
produced, and the results are displayed in Fig. 2(b). The
obvious deviation from the absorption spectrum of the isolated
molecule is that the spectrum of the hybrid system has a
slight red shift and the second low-lying absorption peak
becomes higher than the lowest-energy peak. It is noted that
except the value of electronic coupling, the other parame-
ters applied to fit the experimental absorption spectra are
obtained either from experimental measurements or ab initio
calculations. Therefore, the large value of Vke=0.70 eV
may imply the real situation of the electronic coupling between
DTB-PE molecule and TiO 2surface. In addition, it reveals that
the strong interaction between the molecular local excited state
and the photoinduced CT states plays an important role on
the spectral line shape, which a ffects the relative intensities
FIG. 3. The calculated absorption spectra of DTB-Pe /TiO 2hybrid system at
the di fferent electronic coupling values of Vke=0.7 eV (a); Vke=0.5 eV
(d); and Vke=0.9 eV (e). (b) and (c) correspond to the separate contribution
from the localized intramolecular excitation and intermolecular CT excitation
atVke=0.7 eV .
of the absorption peaks compared to the spectrum of isolated
molecule.
In spite of the encouraging results above, one does not
really know how the direct intermolecular CT excitation con-
tributes to the spectra and the structural parameters a ffect
the absorption spectra. We thus open the intermolecular CT
absorption channel and use di fferent electronic coupling values
to simulate the spectra. Fig. 3(a) displays the absorption spectra
with the same parameters used in Fig. 2 except µgk=µge.
The results are nearly consistent with those in Fig. 2. To reveal
the contribution from the CT excitation, we further display the
individual spectral components from the localized molecular
excited state (Fig. 3(b)) and the CT states (Fig. 3(c)). It is
seen that the CT component is quite small, which may be
explained by the high density of states of |k⟩states because the
excited population, roughly similar to that on the molecular
excited state, is distributed in the conduction band, and only
a small component of population in the resonant electronic
state has a contribution to CT absorption. Interestingly, the
experimental spectra can also be reasonably fitted under a
very unphysical situation µge:µgkto be 1:20 with a smaller
electronic coupling 0 .45 eV . Although this ratio of transition
dipole moments is unrealistic, it gives us an uncertainty to
fit the absorption spectra with a set of parameters. We thus
conclude that it is not accurate enough to extract the structural
parameters only from the absorption spectra. Meanwhile, with
different values of electronic coupling as shown in Figs. 3(d)
and 3(e), the simulated spectral line shapes and peak positions
vary. In Sec. III B, we will focus on the RRS to check its
sensitivity to the structural parameters.
B. Resonance Raman spectra: Effect
of intermolecular CT excitation
It is known that RRS can reveal detailed mode-specific
vibrational motions and interfacial CT processes, and we thus
calculate the RRS with inclusion of the contributions from
the localized molecular excitation and the CT excitation. In
order to get better insight into the contribution from the CT154105-6 Ye, Zhao, and Liang J. Chem. Phys. 143, 154105 (2015)
FIG. 4. Resonance Raman spectra (a) for a single molecule, (d) for a hybrid
system, and (b) and (c) for the individual contributions of the localized
intramolecular excitation (LE) and intermolecular CT excitation (CT).
excitation, Fig. 4 displays four types of RRS, in which (a)
corresponds to a single molecule itself, (b) to the molecular
localized excited-state contribution ( µg,e=1.0,µg,k=0) in
the hybrid system, (c) to the CT excitation ( µg,e=0,µg,CT
=1.0), and (d) to the mixed states of the localized molecular
excitation and the CT excitation ( µg,k=1.0,µg,e=1.0). In
the calculations, the electronic coupling and ∆Eare kept to be
0.1 eV and−0.3 eV , respectively.
Fig. 4 clearly shows that the RRS from the localized
molecular excited state is very similar to that of single molecule
itself although the strengths decrease slightly because we use
a weak electronic coupling, leading to an unobvious inter-
facial CT. The relative peak strengths can be approximately
explained by the Savin’s relation69–71Ii/Ij=ω3
iQ2
i0/(ω3
jQ2
j0),
where Iiis the i-th mode strength. For the RRS from the CT
excitation, however, the intensities of all modes are quite weak,
and in addition, the Savin’s relation is broken down.
To reveal the physical insight behind, we display in Fig. 5
the charge population dynamics related with the spectral calcu-
lation. Obviously, the CT dynamics in case of the CT excitation
(Fig. 5(b)) is di fferent from that for the localized molecular
excitation (Fig. 5(a)). In the case of the molecular excita-
tion, the charge smoothly transfers from the molecule to the
conduction bands, and it may explain the weak RRS inten-
sities compared to those of RRS in the isolated molecule.
FIG. 5. Interfacial electron injection dynamics. Initially, (a) the excited elec-
tron is located on the molecule, (b) on the semiconductor, and (c) on both the
molecule and semiconductor.However, in the CT excitation, only a few percent of charge
transfers from the conduction bands to the molecule,
subsequently transfers back immediately, and finally the
charge distribution reaches a microcanonical equilibrium
distribution. The broken-down of Savin’s relation may come
from the dynamic interference caused by those forward and
backward charge transfer in the conduction bands. Although
most charge is still in conduction bands, we think that this
little fraction of charge (about 1 /Nwith N=350) is extremely
important for the determination of RRS intensities because
it may be located at the resonance energy level for the CT
excitation. This phenomenon can also explain why the RRS
from the CT excitation are much weaker than those from the
molecular excited state.
As both the local excitation and the CT excitation have
the contributions to RRS, as shown in Fig. 5(c), the interesting
phenomena appear. The intensities of modes with frequencies
1200 cm−1and 1326 cm−1are exchanged compared with those
for the single molecule. Also, the intensities of modes with
388 cm−1and 1674 cm−1are enhanced. Those CT excitation-
induced enhancements are well known in SERS, and the cor-
responding mechanism is named as chemical enhancement.
Indeed, there exists charge exchange between the two states,
shown in Fig. 5, but the charge population always flows from
the molecule to the conduction band because of the high den-
sity of states in the band, and the charge transfer from the band
back into the molecule is too small to be distinguished.
From the above preliminary results, it is expected that the
CT excitation plays an important role on the RRS intensities.
To further demonstrate the e ffect of the CT excitation, we
change the transition dipole moment µg,kto make di fferent CT
excitations to see how the RRS intensities change. To do so,
we introduce a quantity of P=µge/(µge+µgk)to represent
the component of the localized molecular excitation. In the
calculations, µge+µgkis kept as a constant and the total
population on the molecule and semiconductor is normalized
to be 2.
We first demonstrate the initial excitation population with
respect to Pin Fig. 6. Obviously, the initial population is
not linearly proportional to the ratio of dipole moments. This
nonlinear property is partially caused by the Franck-Condon
factors of molecular vibrational wavefunctions. But, the
FIG. 6. The population on the molecular excited state vs the component of
transition dipole moments P=µge/(µge+µgk).154105-7 Ye, Zhao, and Liang J. Chem. Phys. 143, 154105 (2015)
FIG. 7. The resonance Raman spectra at the di fferent values of P=µge/
(µge+µgk).
population always increases with increasing of P, and the
intensities of RRS also increase with increasing of the popula-
tion component on molecule, shown in Fig. 7. The intensities of
some modes are obviously changed, for instance, the intensity
of mode with the frequency of 1326 cm−1gradually decreases
with increasing of the CT excitation whereas the relative
intensity of the mode with 1052 cm−1has an opposite tendency.
The phenomenon is reasonable because the characters of the
RRS should come from the CT excitation when the Pis zero
whereas they are close to the characters of the molecular local
excitation when Pis 0.9.
Now, we focus on the interfacial CT e ffect on the RRS.
In the hybrid system, the electronic coupling Vkeis one of the
dominant parameters for controlling interfacial CT. We thus
calculate the electronic coupling dependence of RRS intensity
with∆E=−0.3 eV and P=0.2. Fig. 8 shows the calculated
results (a) from the molecular local excitation only and (b)
from both the molecular local excitation and the CT excita-
tion. For a purpose of comparison, we have scaled the RRS
intensities to be one at V=0.1 eV . As seen from Fig. 8(a), the
RRS intensities of most modes decrease with increasing of the
electronic coupling, manifesting that the excitation electron on
molecule already transfers into the conduction band before its
FIG. 8. Electronic coupling dependence of RRS intensities (a) arisen by the
intramolecular excitation and (b) arisen from both the intramolecular local
excitation and intermolecular CT excitation.energy radiates back to the molecular ground state. However,
the mode of 1052 cm−1is very special and its intensity is greatly
enhanced. From the geometric parameters listed in Table I, it is
known that the equilibrium shifts for this mode are the same on
the ground and excited states. Under Condon approximation,
this mode is inactive to the RRS in a single molecule. But a
partial charge in the band may transfer back to the molecule as
the molecule is absorbed on the semiconductor. And it is thus
expected that the returned charge makes this mode active.48
This phenomenon should be a typical example of chemical
enhancement in SERS.
Interestingly, the CT excitation can dramatically change
the RRS intensities. Fig. 8(b) obviously displays that the RRS
intensities first increase in weak electronic coupling regime,
and then decrease with increasing of the electronic couplings.
As the electronic couplings become strong enough, the oscil-
lations of intensities with respect to electronic couplings even
take place. This di fferent dependence may come from the
different interfacial CT induced by the CT excitation. Fig. 9
shows the corresponding interfacial CT dynamics. Clearly,
the excitation population on the molecule damps faster with
a stronger electronic coupling. However, the CT excitation
obviously slows the population damping rates and makes more
charge transfer from the band into the molecule than that from
the molecule into the band during an initial short time. The
oscillation behavior of intensities at strong electronic cou-
plings may also be explained by the dynamic properties during
1000–1500 a.u. in which the charges on the molecule are
roughly similar at three electronic couplings of 0 .5, 0.6, and
0.7 eV .
C. Resonance Raman spectra: Interference
As discussed in Sec. II, the interference in RRS may
appear when the photo-induced intramolecular and intermo-
lecular excitations take place simultaneously. Here, we define
the interference by
∆I=Iall−Im−ICT, (29)
where Iallrepresents the total RRS intensities calculated by
Eq. (15), and ImandICTrepresent the contributions from the
intramolecular excitation and the CT excitation in the hybrid
system, respectively, unraveled from the total RRS. It is noted
FIG. 9. Charge dynamics on the molecular excited state.154105-8 Ye, Zhao, and Liang J. Chem. Phys. 143, 154105 (2015)
FIG. 10. The RRS interference ∆Ivs (a) the electronic coupling Vkeat
∆E=−0.3 eV , and (b) ∆EatVke=0.3 eV .
that∆Idoes not only come from the phase interference of two
excited pathways but also incorporate the contribution from
bidirectionally interfacial CT because there are four terms in
Eq. (15).
Fig. 10 displays the interference ∆Ivaried with two impor-
tant interfacial structure parameters: (a) the electronic coupling
Vkeat∆E=−0.3 eV and (b) ∆EatVke=0.3 eV . In the calcu-
lations, we have set P=0.2 to make ImandICThave similar
magnitudes. From Fig. 10(a), it is found that most modes have
constructive interferences except the mode with a frequency
of 1052 cm−1which shows a destructive interference. The
strengths of constructive interferences initially increase with
increasing of the electronic coupling from 0 .1 eV to 0.3 eV
and then become decreasing. This behavior is consistent with
the electronic coupling dependence of RRS intensity shown
in Fig. 8. It is thus expected that interfacial CT process plays
an important role in the interference.72Interestingly, as ∆Eis
small at a given electronic coupling, i.e., the molecular excited
state is close to the bottom of the semiconducting conduc-
tion band, all the tested modes show destructive interferences.
With the increasing of ∆E, most modes become constructive
interference, and interference strengths do not have obvious
changes for ∆E=−0.3 eV and−0.7 eV . This phenomenon
again confirms the importance of interfacial CT. Otherwise,
the interference should be expected to be independent of ∆E
because the correlation function in Eq. (15) on the individual
locally excited state and the CT state should be independent of
∆E.
To further reveal the interference of RRS with respect
to the electronic coupling, we calculate the Raman excitation
profiles with respect to the incident light frequency ωL, which
is commonly used in experiments to demonstrate the inter-
ference e ffect.49,52Fig. 11 displays the calculated results for
three modes (388 cm−1, 1200 cm−1, and 1674 cm−1) with and
without the electronic couplings. It is found that in both cases
three peaks of each mode always exist with respect to ωL,
which correspond to those in absorption spectra. However, the
interfacial CT dramatically a ffects the relative intensities of the
peaks. The peaks located at the low frequency domain decrease
whereas those at the high frequency domain increase as the
FIG. 11. Raman excitation profiles for several modes with the frequencies of
(a) 338 cm−1, (b) 1200 cm−1, and (c) 1674 cm−1. Red and green lines denote
the RRS with the electronic coupling Vke=0.5 eV and Vke=0.0.
electronic coupling increases from 0 .0 eV to 0.5 eV . Although
the tendency of RRS with respect to electronic couplings is
very similar to that in absorption spectra, the peak intensities
of RRS are more sensitive. The second peak intensity is greatly
enhanced for the modes with the frequencies of 388 cm−1and
1200 cm−1whereas the third peak is enhanced for the mode at
1674 cm−1. These properties should be very useful to clarify
the interfacial CT.
IV. CONCLUDING REMARKS
We have expanded the time-dependent correlation func-
tion method, proposed by us recently, to investigate absorp-
tion spectra and RRS for an organic molecule absorbed on
an inorganic semiconductor surface, with incorporation of the
intermolecular CT excitation. The results have shown that
the intensities of RRS are much more sensitive to interfa-
cial structure parameters, like interfacial electronic couplings,
and the energy di fference between the molecular excited state
and the conduction-band bottom, than absorption spectra. The
CT excitation significantly changes the relative intensities of
mode-specific RRS and causes the oscillation behavior of Ra-
man enhancement with respect to interfacial electronic cou-
plings. These properties are confirmed to be closely related
to detailed interfacial CT dynamics. Furthermore, the interfer-
ence of RRS by the CT excitation and localized molecular exci-
tation has been demonstrated, and it is significantly a ffected by
interfacial CT, not only by the two individual excitation path-
ways. The obtained results may be useful for understanding
the relationship between RRS intensities and interfacial CT
dynamics and tuning SERS signals.
ACKNOWLEDGMENTS
The work is partially supported by National Natural
Science Foundation of China (Grant Nos. 21373163 and
21290193) and National Basic Research Program of China
(Grant No. 2011CB808501). The partial numerical calcu-
lations have been done on the supercomputer system in154105-9 Ye, Zhao, and Liang J. Chem. Phys. 143, 154105 (2015)
the Supercomputer Center of University of Science and
Technology of China.
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5.0013408.pdf | J. Appl. Phys. 128, 033904 (2020); https://doi.org/10.1063/5.0013408 128, 033904
© 2020 Author(s).Spin–orbit torque based physical unclonable
function
Cite as: J. Appl. Phys. 128, 033904 (2020); https://doi.org/10.1063/5.0013408
Submitted: 12 May 2020 . Accepted: 18 June 2020 . Published Online: 15 July 2020
G. Finocchio
, T. Moriyama
, R. De Rose
, G. Siracusano , M. Lanuzza , V. Puliafito
, S. Chiappini , F.
Crupi , Z. Zeng
, T. Ono , and M. Carpentieri
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Submitted: 12 May 2020 · Accepted: 18 June 2020 ·
Published Online: 15 July 2020
G. Finocchio,1,a)
T. Moriyama,2
R. De Rose,3
G. Siracusano,4M. Lanuzza,3V. Puliafito,5
S. Chiappini,6
F. Crupi,3Z. Zeng,7
T. Ono,2and M. Carpentieri8,a)
AFFILIATIONS
1Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences, University of Messina,
Messina 98166, Italy
2Institute for Chemical Research, Kyoto University, Uji, Kyoto 611-0011, Japan
3Department of Computer Engineering, Modeling, Electronics and Systems Engineering, University of Calabria, I-87036 Rende,
Italy
4Department of Electrical, Electronics and Computer Engineering, University of Catania, I-95125 Catania, Italy
5Department of Engineering, University of Messina, Messina 98166, Italy
6Istituto Nazionale di Geofisica e Vulcanologia, Via di Vigna Murata 605, I-00143 Roma, Italy
7Key Laboratory of Multifunctional Nanomaterials and Smart Systems, Suzhou Institute of Nano-Tech and Nano-Bionics, CAS,
Suzhou, Jiangsu 215123, People ’s Republic of China
8Department of Electrical and Information Engineering, Polytechnic of Bari, Bari 70125, Italy
a)Authors to whom correspondence should be addressed: gfinocchio@unime.it andmario.carpentieri@poliba.it
ABSTRACT
This paper introduces the concept of spin –orbit-torque-magnetic random access memory (SOT-MRAM) based physical unclonable function
(PUF). The secret of the PUF is stored into a random state of a matrix of perpendicular SOT-MRAMs. Here, we show experimentally and withmicromagnetic simulations that this random state is driven by the intrinsic nonlinear dynamics of the free layer of the memory excited by the
SOT. In detail, a large enough current drives the magnetization along an in-plane direction. Once the current is removed, the in-plane magnetic
state becomes unstable evolving toward one of the two perpendicular stable configurations randomly. In addition, we propose a hybridCMOS/spintronics model to simulate a PUF realized by an array of 16 × 16 SOT-MRAM cells and evaluate the electrical characteristics.Hardware authentication based on this PUF scheme has several characteristics, such as CMOS-compatibility, non-volatility (no powerconsumption in standby mode), reconfigurability (the secret can be reprogrammed), and scalability, which can move a step forward the design
of spintronic devices for application in security.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0013408
I. INTRODUCTION
In the “Internet of Things ”(IoT) era, security is becoming a
crucial aspect. An important path to enhance the security of a
physical device is hardware authentication. To provide such a
secure authentication procedure, hardware cryptographic opera-tions (such as digital signatures or encryption) having a secret keyin a nonvolatile electrically erasable programmable read-onlymemory or battery backed static random access memory are used.
1
However, those approaches are expensive in terms of both areaoccupation and static power consumption and a challengingsolution should be to have hardware authentication based on a
memory technology integrated within the device itself. The use of
physical unclonable functions (PUFs) is a direction to face this
challenge promising to enhance the security of the device at aminimal additional hardware cost.
2
PUFs are innovative primitives that derive a chip-unique chal-
lenge –response mechanism by typically exploiting the randomness
due to manufacturing process variability. In other words, the chal-
lenge –response mechanism converts the unique physical state of
the PUF into digital input-output data. The two main subtypes ofJournal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 128, 033904 (2020); doi: 10.1063/5.0013408 128, 033904-1
Published under license by AIP Publishing.PUFs are “weak PUFs ”and“strong PUFs ”: the former store the
secret in a potentially vulnerable hardware while the latter have
more complex challenge –response behavior from the physical dis-
order characterizing the PUF itself.3The most popular implementa-
tion of weak PUFs is the static random access memory (SRAM)PUFs, while one of the first implementations of a strong PUF is based
on optical scattering.
2Recently, the major semi conductor foundries
have integrated spintronics within the standard CMOS technology intheir processes, in particular, for the fabrication of STT-MRAMs(spin-transfer-torque magnetic random access memories).
4,5This
opens a large number of opportunities, and, precisely, we focus on
possible implementations of PUF with spintronic technology.6
The commercial PUFs have several problems, which can affect
their performance and reliability, related to environmental and/oroperative variations that have to be addressed in the design and testphases, such as temperature and supply voltage dependent response
and electromagnetic interference. Spintronic technology, thanks to
the development of reliable magnetic tunnel junctions (MTJs), canbe used to fix some of those problems. For example, MRAMs havealready been proposed for PUF development, where the secret canbe derived from magnetic textures, such as domain wall,
7or in
STT-MRAMs by switching probability.8,9
Those spintronic solutions are based on MTJs and then suffer
from long term wearout, device to device variations, and also biasdependence. As shown later in the paper, spin –orbit-torque
(SOT)-MRAMs have the potential to solve at least the bias field
issue and from a theoretical point of view are more robust todevice-to-device variations.
Figure 1(a) shows the device to implement the STT-MRAM
based PUF. First, the switching probability of the free layer (FL) as
a function of the current pulses of a certain duration is evaluated
(the voltage is supplied at the terminals “A”and“B”). Once deter-
mined, the current value and duration at which the switching prob-ability is 50%, this is applied to a matrix of STT-MRAMs, where allthe cells have been set at the same initial state, e.g., FL parallel to
the PL (pinned layer), thus driving a random state that generates
the secret key of the PUF.
10STT-MRAMs can be designed with an
energy barrier larger than 60k BT, making the secret thermally very
stable.11In addition, the reading signal is insensitive to voltage fluc-
tuations and the resistance states exhibit a reduced drift for a wide
range of temperature near room temperature [a typical tunneling
magneto resistive (TMR) ratio is of the order of 100%]12that make
the states intrinsically very distinguishable. On the other hand, thederivation of the secret is a major issue being related to the switch-
ing probability that is very sensitive to temperature changes and
device-to-device geometrical variations.
To address this aspect, the idea developed in this work is to
use the spin –orbit-torque MRAMs (SOT-MRAMs) as a building
block for the PUF. A SOT-MRAM is a three-terminal device,
where read and write operations are separated [see Fig. 1(b) ]. The
information is written with the SOT originated by a currentflowing in a heavy metal (HM) (Pt, Ta, W, Ir) mainly due to thespin-Hall effect (J
SHE), while the information is read via the resis-
tance of the MTJ built on top of the HM. The main difference of
the SOT-MRAM based PUF as compared to its STT-MRAM coun-
terpart is the writing mechanism; hence, all the benefits from thereading remain unchanged.The starting point for the development of SOT-MRAMs is
due to the pioneering paper by Miron et al.
13in 2011, where the
switching of a single perpendicular ferromagnet coupled with a
HM having a large spin –orbit coupling, Pt/Co in that paper, driven
by a current flowing into the HM, is demonstrated. A similar resultwas achieved by Liu et al.
14in a three-terminal MTJ having the FL
at the interface with the HM. Micromagnetic simulations per-
formed to study the magnetization dynamics in those devices show
that for perpendicular ferromagnets, the switching occurs via anucleation process and that a stochastic switching is also possible.
15
Here, we demonstrate that an intermediate uniform in-plane state
can be driven by an applied current and external field, and, when
the bias current is switched off, the intermediate state becomes
FIG. 1. (a) A schematic of a two terminals STT-MRAM device with the indica-
tion of the free layer (FL) and pinned layer (PL). A current flowing between the
terminals A and B manipulates the magnetic state of the FL. For this device, a
random bit can be generated by a current pulse that gives rise to a 50% of theswitching probability. (b) A schematic of a three-terminal SOT-MRAM device.The writing and reading currents are separated, the current flowing from terminal
C into the pillar is used to read the resistance of the MTJ while the current
applied to the heavy metal (J
SHE) manipulates the magnetic state of the FL. In
the SOT-MRAM having a perpendicular FL, the J SHE, which has a spin polariza-
tion along the in-plane direction perpendicular to its flowing direction, can drive
the FL magnetization in the in-plane configuration. Once the J SHE is switched
off, the in-plane magnetization can evolve to both positive and negative out ofplane directions with equal probability. This mechanism can be used to generaterandom bits in SOT-MRAMs, which can be used as a secret key in a
SOT-MRAM based PUF . (c) A schematic illustration of the experimental device
with the indication of the coordinate reference system. A dot composed ofCo
40Fe40B20(FCB) (1 nm)/MgO (1.6 nm)/T a (1 nm) is built on top a tungsten
layer. During the measurements, an external field Hexis applied along the x
axis. A current pulse is applied along the y-direction to the W stripe giving rise
to a spin –orbit-torque at the interface with FCB with spin polarization along the
y-direction. Inset: Magneto-optical Kerr microscope image showing an FCB dot(the darker part indicated by the white circle) on the W stripe (indicated by the
red dotted lines).Journal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 128, 033904 (2020); doi: 10.1063/5.0013408 128, 033904-2
Published under license by AIP Publishing.unstable and the magnetization evolves randomly toward the posi-
tive or negative uniform out-of-plane state. A schematic of this
behavior is shown in Fig. 1(b) . While those intermediate states
have been used to demonstrate clocking driven by current in mag-netic logic,
16,17here we show how those configurations can be used
to generate the secret of an SOT-MRAM based PUF. In other
words, our approach does not exploit the intrinsic manufacturing
process variability of the manufacturing process, but the secret keyis introduced by the random time domain evolution of an unstablemagnetic state toward two equally likely different other statescoding the bit “1”and“0,”respectively. Micromagnetic simulations
are employed to reproduce the experimental data and predict the
behavior of the PUF for different temperatures and sizes. The stat-istical properties are then used to design and simulate a demonstra-tor of the SOT-MRAM based PUF with a hybrid CMOS/spintronics model in the Cadence ™circuit design tool.
18
II. DEVICE AND MEASUREMENTS
Magnetic multilayers W(6)/Co 40Fe40B20(1)/MgO(1.6)/Ta(1),
thickness in nm, are prepared by magnetron sputtering. The multi-layers are then patterned into the device shown in Fig. 1(c) where
the ferromagnetic dot has a circular shape with a diameter of 1 μm
on a 5 μm-wide HM strip of Wby e-beam lithography and ion-
milling technique. The Co
40Fe40B20is magnetized perpendicular to
the sample plane due to the strong perpendicular magnetic anisot-ropy (PMA) originating from the Co
40Fe40B20/MgO interface.12
The perpendicular magnetic anisotropy is larger than the
out-of-plane demagnetizing field in order to set the out-of-plane
direction as the easy axis of the magnetization. Typical magneto-optical Kerr microscope image of the device is shown in the insetofFig. 1(c) . The circled part is the location of the dot and it is
where the contrast changes up on the magnetization switching.
The electric current flowing in the W stripe, applied along they-direction, invokes the spin current injection into the ferromag-
netic dots due to the spin-Hall effect with a spin polarization along
the x-direction. The spin current exerts a torque resulting in amanipulation of the FL magnetization.
III. SINGLE DOT MEASUREMENTS
The first set of measurements is performed on a single dot. A
current pulse of 4 mA (voltage 8 V) is applied for a duration of
15 ns together with an external magnetic field of 30 or 40 mT
applied along the x-direction (same direction of the spin polariza-tion). The field amplitude is larger than the in-plane field liketorque
19and the Oersted field originated by the current flowing in
the HM, estimated around 0.5 mT for 4 mA (Biot –Savart law for
the latter), respectively. The current drives the magnetization out of
the equilibrium. When the current pulse is removed, the magneti-zation goes back to the perpendicular direction, both upward anddownward, as indicated by optical imaging using magneto-opticalKerr effect. The switching probability (transition from in-plane to
out-of-plane states) characterizing the final state, achieved for 500
current pulses is displayed in Figs. 2(a) and 2(b), which describe
the MOKE contrast measured after each current pulse. Those dataare summarized in the histograms of Figs. 2(c) and2(d) and in the
probability phase diagrams of Figs. 2(e) and2(f), with the indica-
tion of the switching down-to-up, up-to-down, down-to-down, and
up-to-up, where up (down) is referred to the upward (downward)state of the magnetization. The switching probability to haveupward or downward final state is very close to 50%.
The fact that the switching probability does not depend on the
field amplitude for values between 30 and 40 mT is a great advan-tage for the writing procedure being difficult to set a uniform fieldin the whole SOT-MRAM array.
Such an experimental result is at the basis of the design of
SOT-MRAM based PUF devices. In particular, this configuration
FIG. 2. (a) and (b) Experimental measurements of the magnetization switching, based on the evaluation of the MOKE contrast, for a dot of 1 μm. The switching probability
has been computed considering 500 current pulses, each one long 15 ns, and an in-plane field of 30 and 40 mT . (c) and (d) Histograms of MOKE contrast, i.e. , probability
function of the two out-of-plane directions (down and up) of the magnetization. (e) and (f) Switching probability of the four possible transitions, “down-to-up, ”“up-to-down, ”
“down-to-down, ”and “up-to-up, ”as a response to a current pulse.Journal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 128, 033904 (2020); doi: 10.1063/5.0013408 128, 033904-3
Published under license by AIP Publishing.can be considered as a proof of concept for the generation of a
secret key in a SOT-MRAM array featuring equally distributed “1”
(up state) and “0”(down state) stored bits.
IV. RESULTS OF MICROMAGNETIC SIMULATIONS OF
THE SINGLE DOT
To qualitatively understand the FL magnetization dynamics
described in Sec. III, we have performed micromagnetic simula-
tions carried out by means of a state-of-the-art parallel micromag-netic solver, which numerically integrates the LLG equationincluding the Slonczewski-like torque due to SHE,
20
dm
dτ¼/C0m/C2hEFFþαGm/C2dm
dτ/C0gμB
2γ0eM2
StCoFeαHm/C2m/C2(^z/C2J),
(1)
where mand hEFFare the normalized magnetization and the
effective field of the ferromagnet, respectively. The effective fieldincludes the standard magnetic field contributions, the Oerstedfield and the thermal field, which is included as an additionalstochastic term.
21The effective field and boundary conditions
expressions are included as in Refs. 22and23.τ=γ0⋅MS⋅tis the
dimensionless time, where γ0is the gyromagnetic ratio and MSis
the saturation magnetization of the ferromagnet. αGis the Gilbert
damping, gis the Landè factor, μBis the Bohr Magneton, eis the
electron charge, tCoFe is the thickness of the free layer, αHis the
spin-Hall angle obtained from the ratio between the spin current
and the electrical current, ^zis the unit vector of the out-of-plane
direction, and Jis the in-plane current density injected via the HM.
The parameters used for the CoFeB are saturation magnetization
MS=8 0 0×1 03Am−1, exchange constant A=2 . 0×1 0−11Jm−1,24
and damping parameter αG= 0.03. The magnetic anisotropy
KU=5 . 1 2×1 05Jm−3has been estimated from the experimental
hard-axis magnetization curve and the spin-Hall angle αH=−0.33.25
Figure 3 shows a comparison between micromagnetic simulations
and experimental data of the switching probability to achieve the dif-
ferent transitions for a field of 40 mT and a current of 3.6 mA(J
SHE=4×1 07Ac m−2). Similar results have been achieved for a field
of 30 mT.
Examples of time domain evolution of the magnetization pat-
terns for the four possible transitions (down-to-up, up-to-down,
down-to-down, and up-to-up) are shown in Figs. 4(a) –4(d).
The current pulse drives the initial state ( Fig. 4 first column)
toward a uniform state of the magnetization along the x-directionthat is the same as the direction of the external field ( Fig. 4 , second
column). Once the current is switched off, there is nucleation of
domains ( Fig. 4 , third column) that first expand ( Fig. 4 , fourth and
fifth columns) and then collapse into one single domain state rep-resenting the out-of-plane positive or negative configurations(Fig. 4 , sixth column).
We wish to stress that the main difference of a SOT-MRAM
based PUF compared with its STT-MRAM counterpart relies onthe writing mechanism; hence, all the benefits from the readingremain the same. The generation of random bits in STT-MRAMs
is characterized by the following steps: (i) Characterize the switch-
ing probability as a function of the field and current amplitudewith a fixed pulse duration t
D. (ii) Find the configuration that gives
rise to the 50% switching probability for a given tD(H50%, and
I50%). (iii) Set the initial state of all STT-MRAM memory cells to
the same value. (iv) Write the random state by applying H 50%and
FIG. 4. (a)–(d) Examples of time domain evolution of the spatial distribution of
the magnetization (the color is related to the out-of-plane component of the
magnetization, i.e., red for positive and blue for negative, and the arrows indi-cate the in-plane component of the magnetization). The rows show the snap-shots for the switching processes for a specific transition: (a) “down-to-up, ”(b)
“up-to-down, ”(c) “down-to-down, ”and (d) “up-to-up. ”The main common steps
of the switching process are reported with also the indication of the times.
FIG. 3. A comparison between experimental results and micromagnetic simula-
tions of the switching probability for the four possible transitions, “down-to-up, ”
“up-to-down, ”“down-to-down, ”and “up-to-up, ”as calculated for a field of 40 mT
and a current of 3.6 mA (J SHE=4×1 07Ac m−2).Journal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 128, 033904 (2020); doi: 10.1063/5.0013408 128, 033904-4
Published under license by AIP Publishing.a current pulse tDlong of amplitude I 50%to all the cells. On the
other hand, the way to generate the random state in an array of
SOT-MRAM needs the following steps: (i) Characterize a criticalcurrent curve as a function of the field that depends on the currentpulse duration. See, for example, J
critinFig. 5 computed for a pulse
duration of 10 ns by means of micromagnetic simulations and a
field of 40 mT applied along the in-plane direction, which is paral-
lel to the spin polarization (all the other parameters are reported inSec.IV); set the initial state of all SOT-MRAM memory cells to the
same value. (ii) Apply a current and field that are above critical lineJ
crit(H). Comparing those two procedures, it can be clearly observed
that the random state writing in a SOT-MRAM is more robust. In
fact, the fundamental difference between the STT-MRAM and theSOT-MRAM is the polarization direction of the spin currentapplied on the free layer magnetization. For the STT-MRAM, thespin current comes from the fixed layer and the direction of the
spin current depends on the direction of the fixed layer.
26
Therefore, optimum field and current become really convoluted to
realize the 50% probability. Moreover, it relies on a “transient ”50%
probability. Therefore, the current pulse needs to be shut exactly atthat timing. On the other hand, for the SOT-MRAM, the direction
of the spin current is fixed in the plane regardless of the applied
field. To realize the 50% probability, in principle, the spin currentjust needs to be strong enough to direct the free layer magnetiza-tion in the in-plane direction from which the magnetization goes
back to either perpendicular direction with 50% probability. In the
present case, the applied field is just used to assist the spin current,while it important to fix a working point for current and fieldabove the critical line J
crit(H). This fact brings robustness in terms
of device-to-device variation and temperature changes; in fact, it is
possible to characterize experimentally or calculate those critical
lines for different device shapes and temperatures in order to fix acurrent and field amplitude in order that the writing processbecomes independent of possible variation of geometry from the
nominal size and/or temperature sample. Micromagnetic simula-
tions performed at a fixed temperature T= 300 K for dot diameters
of 1100 and 900 nm and for a fixed diameter 1000 nm at differenttemperatures 250 and 350 K and a dot diameter show that theswitching probability is still closer to 50% (see Fig. 6 ). The micro-
magnetic simulations at different temperatures take into account,
together with the different amplitudes of the thermal field, also thetemperature dependence of the parameters ( M
S,A, and KU) con-
sidering scaling relationships,27,28in particular, their values, which
are summarized in Table I . On the other hand, SOT-MRAMs have
limited integration as compared with STT-MRAM being three-
terminal devices. Recent progresses are very promising29and nano-
fabrication demonstrating the integration of SOT-MRAM withCMOS technology has already been developed.
30We believe that
the idea of SOT-MRAM based PUF technology presented in this
work can stimulate further research in developing high quality
SOT-MRAM considering also the possible advantages discussedabove.
FIG. 5. Micromagnetic simulations of the critical current curve, which indicates
the region of field and current where it is possible to drive the in-plane configu-
ration of the free layer magnetization and then the random state. These calcula-
tions are performed considering the physical parameters as indicated in themain text, a dot having a diameter of 1000 nm and a current pulse long 10 ns.
FIG. 6. Micromagnetic simulations performed at a fixed diameter 1000 nm for
different temperatures: (a) 250 K, (b) 300 K, and (c) 350 K, and at a fixed tem-
perature T= 300 K for different diameters: (d) 900 nm, (e) 1000 nm, and (f)
1100 nm.Journal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 128, 033904 (2020); doi: 10.1063/5.0013408 128, 033904-5
Published under license by AIP Publishing.V. THREE DOTS EXPERIMENTAL CHARACTERIZATION
To show the possibility to write the random state with a single
pulse in different dots sharing the same HM nanowire, we havebuilt a device composed of three dots within a single nanowire real-
ized geometrically as indicated in Fig. 7(a) . The dots have nomi-
nally the same size (diameter of 1 μm), while the center-to-center
distance is fixed at 5 μm in order to have a negligible effect of mag-
netostatic coupling. A MOKE image is shown in Fig. 7(b) .
We have applied a current pulse of 4 mA (voltage 8 V) for a
duration of 15 ns for 100 iterations, while maintaining a field of
40 mT applied along the x-direction. We have found that theswitching mechanism is the similar for each dot and the samecurrent pulse is able to drive the magnetization of each dot out ofthe equilibrium qualitatively similar to what observed in single dot
measurements. By removing the current pulse, the magnetizationsindependently go back to the perpendicular direction, upward or
downward, with a certain probability. By using the optical imaging
magneto-optical Kerr effect, we have identified the final state of themagnetization after the current pulse is switched off for 100 itera-tions. In order to process the data, we have modeled specificregions of interest (ROIs) having 1 μm of diameter, i.e., the same as
the devices, and centered around the locations of the dots [see the
arrows in Fig. 7(b) ]. We have developed an algorithm that system-
atically calculates the average brightness of the ROIs against thesurrounding background areas and compares it to a probabilisticthreshold. The analysis of corresponding cumulative distribution
functions allows us to determine the switching probability for each
dot with high accuracy. In detail, in those regions, once the currentpulse is switched off and the brightness of the regions is evaluated,if the ROI exhibits a different (higher) brightness value against the
background, the magnetization is considered upward, and this is
linked to the presence of the bit “0.”Conversely, if the region
exhibits the same brightness value as in the background, the mag-netization points downward and this corresponds to the bit “1.”
The results of this analysis are reported in Fig. 7(c) . The exten-
sion of the analysis to multiple dots allows us to confirm the intrin-
sic randomness of the device (ideally, it should be close to 50%),which is a crucial property for the unpredictability of the PUFresponses. In particular, this configuration can be used to improvethe robustness of the random state by performing logic operations
among bits generated in different cells.
31
VI. CMOS INTEGRATION OF THE SOT-MRAM BASED
PUF
In Secs. I–V, we have discussed experimental and simulation
analyses performed for magnetic dots with a diameter of 1 μmo n
top of a W strip with 6 nm in thickness. Here, we evaluate theimplications of the proposed PUF approach at the circuit level interms of performance and energy characteristics considering that
the SOT-MRAM device is built as a point contact geometry with a
diameter of 100 nm, similarly to a device already developed.
32
In this way, the reading current is applied locally in the FL. Tothis aim, we have considered a general circuit architecture[see Fig. 8(a) ] featuring four bitcell (BC) blocks, each including a
16 × 16 BC array and read/write (RW) control circuits, and a
decoding block to generate the appropriate read (R) signals for aspecific challenge represented by a 64-bit input address (ADD).The latter block consists of 16, one for each row of the BC arrays,
4-to-16 decoders. Each decoder receives four bits of the challenge
to produce 16R signals (one for each column of the BC arrays) by aTABLE I. Parameters of the CoFeB dot used for the simulations at different temperatures.
T=0K T= 250 K T= 300 K T= 350 K
MS(kA/m) 928.75 830.81 800 766.51
KU(105J/m3) 7.11 5.56 5.12 4.66
A(10/C011J/m) 2.50 2.12 2.00 1.88
Switching probability Down 53%
Up 47%Down 49%
Up 51%Down 43.5%
Up 56.5%
FIG. 7. (a) Schematic of the three dots device. (b) The optical imaging
magneto-optical Kerr effect. (c) Probability of bit “0”and “1”for each dot during
100 iterations.Journal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 128, 033904 (2020); doi: 10.1063/5.0013408 128, 033904-6
Published under license by AIP Publishing.logical AND operation between the decoder output bits and a read/
write (RW) control signal. The generated 16 × 16R signals are then
fed to the four BC blocks, where each one provides 16 output(OUT) bits (one bit for each row), thus obtaining a final 64-bitoutput word as response to a given challenge.
The scheme of the BC blocks is detailed in Fig. 8(b) . The
16 × 16 BC array relies on 16 strings (one for each row), each one
including 16 magnetic dots (one for each column) whose freelayers are contacted to the same HM strip in series. The top of eachdot is also connected to one NMOS access transistor driven by anR signal to form a BC. All the HM strips are connected on one side
to power supply voltage (V
DD) through one PMOS transistor
driven by a write (W) signal and on the other side they aregrounded. In addition, the BCs within the same row share a readbit-line (RBL). Such a scheme allows managing write and readoperations by properly asserting RW, W, and R signals. The write
(or program) operation occurs concurrently for all the BCs belong-
ing to one 16 × 16 array by activating all V
DD-connected PMOS
transistors (all W signals = “0”), while disabling all the NMOS
access transistors (all R signals = “0”by setting the RW control
signal = “0”in the decoding block). In this way, in each row string,
an adequate write current (I write=ISHE) flows from V DDto the
ground through the HM nanowire, thus enabling the SOT-basedswitching mechanism in the 16 magnetic dots. This writing opera-
tion is performed only one time (or whenever a rewrite of the
stored secret keys is required); hence, it does not influence signifi-cantly the power dissipation of the whole PUF circuit duringnormal running. Conversely, the read operation occurs as often asa response has to be provided for a specific challenge. This opera-
tion is implemented along the RBLs by setting the RW control
signal = “1”in the decoding block and hence activating one NMOS
access transistor per row in the 16 × 16 BC arrays (since, for eachrow, only one R signal coming from the decoding block is equal to“1”on the basis of the given challenge), while disabling all
V
DD-connected PMOS transistors (all W signals = “1”). As shown
inFig. 8(b) , for each row, a conventional voltage sensing scheme is
then used to detect the binary digit stored in the SOT-MRAMdevice of the activated BC. In particular, it consists of generatinga read current (I
read) to be applied to the RBL and then to
produce a read bitline voltage (V data), which is compared with a
reference voltage (V ref) by a voltage sense amplifier (SA).
Obviously, the applied I readhas to be sufficiently low to avoid any
manipulation of the magnetic state of the SOT-MRAM devicesduring the reading operation. As a consequence, depending on
the FL magnetization orientation (i.e., up or down state) in the
magnetic device of the activated BC, the developed V
datais lower
FIG. 8. (a) Block diagram of the CMOS/SOT-MRAM based PUF general architecture along with the description of the signals and (b) details of the circuit for the b itcell
(BC) block.Journal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 128, 033904 (2020); doi: 10.1063/5.0013408 128, 033904-7
Published under license by AIP Publishing.or higher than V ref, thus translating into an OUT bit equal to “1”
or“0,”respectively.
We have simulated the above hybrid CMOS/SOT-MRAM
PUF circuit into Cadence ™environment using the transistor
models provided by a 1.8 V 0.18 μm CMOS technology and a
Verilog-A based compact model33to integrate the behavior of
SOT-MRAM devices. For the latter, we have considered the follow-
ing parameters: diameter = 100 nm, tCoFe= 1 nm, MgO oxide
thickness tOX= 1.6 nm, resistance-area (RA) product = 230 Ωμm2,
TMR ratio = 150%, αG= 0.03, αH=−0.33, MS= 800 × 103Am−1,
WHM= 1.5 μm,LHM= 1.5 μm (i.e., the center-to-center distance of
two adjacent dots on the same HM strip), and HM strip thickness
tHM= 6 nm. We have properly sized V DD-connected PMOS transis-
tors driven by W signals to achieve an adequate Iwritepulse ampli-
tude of about 3.6 mA (J SHE=4×1 07Ac m−2). This allows one to
get the FL magnetization in-plane within 1 ns, thus corresponding
to an average write energy per bit (E write_bit) of about 0.39 pJ. We
have also designed the NMOS access transistors of the BCs and theread control circuits to obtain a much lower pulse amplitude forthe I
read(around 10 μA for a pulse duration of 1 ns) in such a way
so as to ensure a low read disturbance probability during sensing
operations. This has led us to estimate an average read energy per
bit (E read_bit) of about 0.12 pJ, including the contributions of all
peripheral circuitry (i.e., the decoding block, the circuits for theI
readgeneration, and the SAs). When no read and write operations
occur (standby phase), the designed circuit exhibits a leakage
current of 0.73 mA. However, thanks to the non-volatility capabilityoffered by the SOT-MRAM devices, the leakage power can becompletely cut off by switching off the power supply during thestandby phase.
VII. SUMMARY AND CONCLUSIONS
This work proposes the realization of a SOT-MRAM based
PUF, where the generation of the secret key is based on driving aperpendicular magnet into an equilibrium configuration (in-plane
direction) by applying a large enough current along with an exter-
nal field. Once the current is switched off, such an in-plane mag-netic state becomes unstable giving rise to a random magnetizationevolution toward the positive or negative out-of-plane stable con-figurations. Moreover, we evaluated the circuit level implications in
terms of energy requirements by integrating the proposed
SOT-MRAM device with a commercial CMOS technology.
Experimental and simulation results reported and discussed in
this work clearly prove that our PUF approach allows ensuring arandomness close to 50%. In addition, when compared to
state-of-the-art CMOS based PUF implementations, the proposed
solution offers the typical advantages of spintronic technology suchas low-power consumption, the radiation hardness (MRAM tech-nology being intrinsically rad-hard is already seen as a major candi-date for radiation-hard memory devices), non-volatile behavior
(hence zero standby power consumption), and reconfigurability,
which is based on the possibility to implement a refresh mecha-nism for the secret key via the SOT and the possibility to update itschallenge –response table. We wish to highlight that a PUF recon-
figurable architecture allows to meet the requirements for some
practical applications and can improve the reliability and securityof a PUF-based authentication system.
34,35In addition, the field
should be applied only during the writing procedure while for the
electrical reading of the information, it is not necessary; hence,there will not be an impact in terms of power consumption of thedevice running in the normal PUF regime. Thanks to all thesefavorable properties along with technological scalability (down to
nanoscale nodes to achieve low-power consumption while main-
taining high thermal stability) and easy integration with CMOSprocesses, SOT-MRAMs based PUFs could represent a realbreakthrough for security applications based on hardwareauthentication.
ACKNOWLEDGMENTS
This work was supported by the “Diodi spintronici rad-hard
ad elevata sensitività —DIOSPIN ”(Grant No. 2019-1-U.0), the
Italian Space Agency (ASI) within the call “Nuove idee per la com-
ponentistica spaziale del futuro, ”the JSPS KAKENHI (Grant Nos.
19K21972 and 17H04924), and the Future Development FundingProgram of Kyoto University Research Coordination Alliance. Thiswork has also been supported by the PETASPIN association.
DATA AVAILABILITY
The data that support the findings of this study are available
within the article.
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Published under license by AIP Publishing. |
1.3598340.pdf | Frequency-shift vs phase-shift characterization of in-liquid quartz crystal microbalance
applications
Y. J. Montagut, J. V. García, Y. Jiménez, C. March, A. Montoya, and A. Arnau
Citation: Review of Scientific Instruments 82, 064702 (2011); doi: 10.1063/1.3598340
View online: http://dx.doi.org/10.1063/1.3598340
View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/82/6?ver=pdfcov
Published by the AIP Publishing
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128.189.203.83 On: Fri, 12 Dec 2014 06:38:32REVIEW OF SCIENTIFIC INSTRUMENTS 82, 064702 (2011)
Frequency-shift vs phase-shift characterization of in-liquid quartz
crystal microbalance applications
Y . J. Montagut,1J. V. García,1Y . Jiménez,1C. March,2A. Montoya,2and A. Arnau1,a)
1Grupo de Fenómenos Ondulatorios, Departamento de Ingeniería Electrónica, Universitat Politècnica
de València, Spain
2Instituto Interuniversitario de Investigación en Bioingeniería y Tecnología Orientada al Ser Humano,
Universitat Politècnica de València, Spain
(Received 16 March 2011; accepted 16 May 2011; published online 10 June 2011)
The improvement of sensitivity in quartz crystal microbalance (QCM) applications has been ad-
dressed in the last decades by increasing the sensor fundamental frequency, following the increment
of the frequency/mass sensitivity with the square of frequency predicted by Sauerbrey. However, this
sensitivity improvement has not been completely transferred in terms of resolution. The decrease offrequency stability due to the increase of the phase noise, particularly in oscillators, made impossible
to reach the expected resolution. A new concept of sensor characterization at constant frequency has
been recently proposed. The validation of the new concept is presented in this work. An immunosen-
sor application for the detection of a low molecular weight contaminant, the insecticide carbaryl, has
been chosen for the validation. An, in principle, improved version of a balanced-bridge oscillatoris validated for its use in liquids, and applied for the frequency shift characterization of the QCM
immunosensor application. The classical frequency shift characterization is compared with the new
phase-shift characterization concept and system proposed. © 2011 American Institute of Physics .
[doi: 10.1063/1.3598340 ]
I. INTRODUCTION
Acoustic sensing has taken advantage of the progress
made in the last decades in piezoelectric resonators for
radio-frequency (RF) telecommunication technologies. Theso-called gravimetric technique is based on the change in
the resonance frequency experimented by the resonator due
to a mass attached on the sensor surface;
1it has opened
a great deal of applications in bio-chemical sensing in
both gaseous and liquid media.2–10This characteristic al-
lows using the gravimetric techniques based on acousticsensors for a label-free and a quantitative time-dependent
detection.
The classical quartz crystal microbalance (QCM) has
been the most used acoustic device for sensor applications.
However, other acoustic devices such as surface generated
acoustic wave (SGAW) (Ref. 4) and film bulk acoustic res-
onators (FBAR) (Refs. 8and11–14) have been, and are being
used, for the implementation of nano-gravimetric techniquesdue to the feasibility of obtaining much higher resonant
frequency in these devices than in classical QCM resonators.
The absolute frequency/mass sensitivity, given by the ratiobetween the resonant frequency-shift /Delta1fand the surface
mass density shift /Delta1m:S
a=/Delta1f//Delta1m, theoretically increases
with the square of the fundamental frequency;1absolute
sensitivities of a 30 MHz QCM reach 2 Hz cm2ng−1, with
typical mass resolutions around 10 ng cm-2.15A resolution
improvement down to 1 ng cm−2seems to be feasible by
optimizing the characterization electronic interface as well
as the fluidic system. Consequently, much higher sensitivity
a)Author to whom correspondence should be addressed. Electronic mail:
aarnau@eln.upv.es.is expected at higher resonant frequencies. However, the in-
crease in frequency-shift/mass sensitivity has not been pairedwith the expected improvement in terms of limit of detection
(LOD). Effectively, thin film electroacoustic technology has
made possible to fabricate quasi-shear mode thin FBAR,operating with a sufficient electromechanical coupling for
being used in liquid media at 1–2 GHz;
12,16however, the
higher frequency and the smaller size of the resonator result
in that the boundary conditions have a much stronger effect
on the FBAR performance than on the QCM response. Ahigher mass sensitivity is attained, but with an increased noise
level as well, thus moderating the gain in resolution.
13,17
So far only publications of network analyzer based FBAR
sensor measurements have been published in the literature,
which show that the FBAR mass resolution is very similar if
not better than for oscillator based QCM sensors.14,18On the
other hand, the mass sensitivity of Love mode SGAW sensors
has been evaluated.19–21Kalantar and coworkers reported a
sensibility of 95 Hz cm2ng−1for a 100 MHz Love mode
sensor, which is much better than the typical values reported
for low frequency QCM technology.22However, Moll and
coworkers reported a LOD for a Love sensor of 400 ng cm−2,
this reveals once again that an increase in the sensitivity does
not mean, necessarily, an increase in the LOD.23Moreover,
these results have been compared with typical 10 MHz
QCM sensors; recently, an electrodeless QCM biosensor
for 170 MHz fundamental frequency, with a sensitivity of67 Hz cm
−2ng−1, has been reported;24this shows that
the classical QCM technique still remains as a promising
technique. The main challenges remain on the improvementof the sensitivity, but with the aim of getting a higher mass
resolution, multi-analysis, and integration capabilities and
reliability.
0034-6748/2011/82(6)/064702/14/$30.00 © 2011 American Institute of Physics 82, 064702-1
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128.189.203.83 On: Fri, 12 Dec 2014 06:38:32064702-2 Montagut et al. Rev. Sci. Instrum. 82, 064702 (2011)
This analysis makes clear that increasing the resonant fre-
quency of the sensor is not the only aspect to keep in mind
for resolution improvement; the configuration setup has animportant role, including the fluidic system and electronic
characterization interface. Once all the care has been taken
in optimizing the fluidic aspects, the role of the electronic in-terface is of maximum relevance.
In practice, focusing on in-liquid QCM applications,
the sensor characterization techniques provide, among other
relevant parameters, the resonance frequency shift of the
sensor:
25,26network or impedance analysis is used to sweep
the resonance frequency range of the resonator and deter-
mine the maximum conductance frequency,27,28which is al-
most equivalent to the motional series resonance frequencyof the resonator-sensor; impulse excitation and decay method
techniques are used to determine the series-resonance or the
parallel-resonance frequency depending on the measuring set-up;
29,30oscillator techniques are used for a continuous mon-
itoring of a frequency which corresponds to a specific phase
shift of the sensor in the resonance bandwidth,31–35this fre-
quency can be used, in many applications, as reference of the
resonance frequency of the sensor; and the lock-in techniques,
which can be considered as sophisticated oscillators, are de-signed for a continuous monitoring of the motional series res-
onance frequency or the maximum conductance frequency of
the resonator-sensor.
36–42In order to assure that the frequency
shift is the only parameter of interest, a second parameter pro-
viding information of the constancy of the properties of liquidmedium is important, for instance, in piezoelectric biosensors;
this parameter depends on the characterization system being:
the maximum conductance or the conductance bandwidth inimpedance analysis, the dissipation factor in decay methods,
and a voltage associated with the sensor damping in oscillator
techniques.
For high frequency resonators only impedance analysis
provides accurate results, but its high cost and large dimen-
sions prevent its use for sensor applications. Consequently,oscillators are taken as alternative for sensor resonance fre-
quency monitoring; the low cost of their circuitry as well as
the integration capability and continuous monitoring are somefeatures which make the oscillators to be the most common
alternative for high resonance frequency QCM sensors. How-
ever, in spite of the efforts carried out to design oscillator
configurations suitable for in-liquid applications
43–51the poor
stability of high frequency QCM systems based on oscilla-tors has prevented increasing the limit of detection despite the
higher sensitivity reported.
52–56
A higher sensitivity is necessary when a higher resolution
is required. This happens in those applications where very tiny
changes in the sensor resonant frequency, due to the perturba-
tion process to be monitored, are expected. These very smallfrequency shifts, in the order of tens of hertz, and mainly due
to a mass transfer effect over different kind of coatings with
different contacting media, must be monitored under very dif-ferent damping conditions which depend on the physical and
geometrical properties of the coating and contacting media.
For instance, in piezoelectric biosensors the coating can beconsidered, in general, like an acoustically thin layer and the
contacting media is a water-like solution which provides arelatively small damping on the sensor. On the contrary, in
other cases like, for example, in some electrochemical appli-
cations where polymer coatings are involved, these tiny fre-quency shifts must be monitored under much higher damping
conditions. Continuous monitoring of very small frequency
shifts at very high resonance frequencies can be easily per-formed with well-designed oscillators; however, when the
quality factor of the resonator-sensor is relatively low, cir-
cuits able to oscillate under these special conditions have to be
performed.
A phase-shift monitoring at a constant frequency in the
sensor resonance bandwidth has been recently proposed as an
alternative characterization method for high resolution QCM
applications.
57In the present article this alternative method is
validated in a real application, and compared with an, in prin-
ciple, improved version of a balanced-bridge oscillator.25,51
The comparison is made with relatively low frequency sen-
sors (10 MHz), where the performance of the oscillator circuit
can be considered nearly ideal.
II. THEORETICAL ASPECTS
For a great deal of in-liquid QCM applications, as it
is the case of QCM biosensors, Martin’s equation (Eq. (1))
is generally applied,58which combines the additive contri-
bution of the mass effect (Sauerbrey1) and the liquid effect
(Kanazawa):59
/Delta1f=−2f2
o
Zcq(mc+mL). (1)
In the former equation, fois the fundamental resonant
frequency, Zcqis the characteristic acoustic impedance of the
quartz, mcis the surface mass density of the coating and mL
=ρLδL/2 where ρLandδL=(ηL/πfoρL)1/2,ηLbeing the liq-
uid viscosity, are respectively, the liquid density and the wave
penetration depth of the acoustic wave in the liquid: mLis, in
fact, the equivalent surface mass density of the liquid, whichmoves in an exponentially damped sinusoidal profile, due to
the oscillatory movement of the surface of the sensor.
According to Eq. (1), the frequency shift, associated
with a certain mass change, increases directly proportional to
the square of the fundamental resonance frequency. Conse-
quently, the most relevant parameter used up to date for thecharacterization of microbalance sensors has been the sen-
sor resonance frequency-shift. However, the great efforts per-
formed to improve the sensitivity of the sensor are useless if
they are not accompanied with an increase in the limit of de-
tection. As mentioned, the increase of the sensor frequencyhas not carried a parallel improvement in the mass resolu-
tion. Effectively, the sensitivity will not be improved if the
frequency stability is not improved as well. In oscillators, theorigin of the frequency instability is the phase instability,
25,57
and a direct relationship can be obtained between a phase shift
and the corresponding frequency shift, through the definitionof the stability factor S
Fof a crystal resonator operating at its
series resonance frequency fo:
SF=/Delta1ϕ
/Delta1ffo=2Q, (2)
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128.189.203.83 On: Fri, 12 Dec 2014 06:38:32064702-3 Montagut et al. Rev. Sci. Instrum. 82, 064702 (2011)
where /Delta1fis the frequency shift necessary to provide a phase
shift/Delta1ϕin the phase-frequency response of the resonator,
around fo, and Qis the series quality factor of the resonator.
According to Eq. (2)the frequency noise /Delta1fnassociated
with a phase noise in the circuitry /Delta1ϕ nis
/Delta1fn=fo
2Q/Delta1ϕ n. (3)
Consequently, because the quality factor of the unper-
turbed resonator is normally reduced proportionally to 1/ fo,
the frequency instability is increased in relation to the square
of frequency. Moreover, the phase response of the electronic
components of an oscillator gets worse with increasing thefrequency, which increases, even more, the noise. Further-
more, if the limit of the detection is assumed to be three times
the level of noise ( /Delta1ϕ
min=3/Delta1ϕ n), the minimum detectable
surface mass density change of a QCM, according to the
definition of the absolute frequency/mass sensitivity Saand
Eq.(3), will be
/Delta1mmin=fo
2QS a/Delta1ϕ min. (4)
The former equation seems to indicate that for a given
minimum detectable phase of the measuring system, the sur-
face mass limit of detection does not depend on the frequency.Fortunately this is not completely true; the liquid medium has
not been taken into account in the obtaining of the previous
equation.
III. FREQUENCY-SHIFT VS PHASE-SHIFT
Following a similar mathematical development described
elsewhere,57the next generalized approximated equation for
the phase-shift of a signal, of constant frequency very close
to the motional series resonant frequency of the resonator-
sensor, to small changes both in the coating mass and liquid
properties, is found:
/Delta1ϕ(rad)=−/Delta1mc+/Delta1mL
mq+mL. (5)
In the former equation mq=Zcq/4foQoand mL,p r e -
viously defined in Eq. (1), can be written as follows mL
=Zcq/4foQL, where Qo=c66/ωoηqis the series quality factor
of the unperturbed resonator, c 66andηqbeing, respectively,
the shear modulus and the effective viscosity of the quartz
crystal, and ωothe angular resonant frequency; and QLis the
series quality factor of the resonator under liquid loading con-
ditions, which is given by the following equation:
QL=Zcq√π
21√fo1√ρLηL.
In many in-liquid QCM applications mLcan be assumed
to be constant and in most of them mq/lessmuchmL; thus Eq. (5)
reduces to
/Delta1ϕ≈−/Delta1mc
mL, (6)
which was previously obtained by the authors elsewhere.57According to Eq. (6)the limit of mass-change detection
/Delta1mmincorresponding to the phase-shift detection limit of the
system /Delta1ϕ minwill be given by: /Delta1mmin≈− mL/Delta1ϕ min. Con-
sequently, the mass resolution increases ( /Delta1mmindecreases)
with the decrease of mL; therefore, because m Ldecreases pro-
portionally to 1/ f1/2, the resolution in the detection of surface
mass density changes increases with f1/2for a given /Delta1ϕ min.
This is not in contradiction with Eq. (4); simply the ef-
fective reduction of the quality factor of the sensor is propor-
tional to 1/f1/2instead of to 1/fwhen the contacting liquid is
considered. This is not true in air because the approximationm
q/lessmuchmLmade in Eq. (5)to obtain Eq. (6)is not acceptable. In
air, an increase in frequency does not improve the limit of de-
tection unless the stability and the phase detection limit of themeasuring system are improved. Curiously, this also happens
when only changes in the liquid properties occur. Effectively,
when the aim is to monitor changes in the properties of theliquid in contact with the sensor: /Delta1(η
LρL)1/2, the phase shift
related to these changes is, according to Eq. (5), given by
/Delta1ϕ≈−/Delta1mL
mL=−/Delta1√ρLηL√ρLηL. (7)
Therefore, it is not possible to increase the resolution in
the detection of changes in the liquid properties by increasing
the frequency. According to the previous considerations, it isimportant to check the limits for which the approximation m
q
/lessmuchmLis acceptable:
The parameter mqis independent of the frequency, and
a reference value of mq=2.2·10−6kg m−2is obtained for a
real 10 MHz AT-cut quartz crystal sensor with a typical Qo
around 105(see definition below Eq. (5)with Zcq=8.838833
106Nsm−3). The approximation can be considered accept-
able for ratios of mq/mL≤0.1. This ratio is given by
mq
mL=2.2·10−6√
4π√f√ρLηL≤0.1. (8)
The former equation indicates that for a given liquid the
ratio mq/mLonly depends on the frequency; the worst case
occurs for low density-viscosity liquids like, for instance,
water where ηLρL=1. In this case the maximum frequency
for which the approximation mq/lessmuchmLis acceptable is around
165 MHz. Moreover, Eq. (5)also indicates that when mL
decreases to a value much smaller than mq, no further
improvement in the resolution can be obtained by increasing
the frequency; this happens for ratios mq/mL≥0.9 which
are obtained, under the previous conditions, for frequencieshigher than 10 GHz.
The previous analysis allows concluding the following
important remarks: (1) the sensitivity of a QCM always in-creases with increasing the frequency; however, the mass res-
olution, which is the parameter of interest, only increases with
the frequency if the noise is, at least, maintained constant orreduced. Moreover, this increase in the mass resolution is only
valid for in-liquid QCM and not for in-gas QCM; and (2) once
all the cares have been taken into account to reduce the pertur-
bations on the resonator-sensor such as temperature and pres-
sure fluctuations, etc., the mass resolution is only depending
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FIG. 1. (a) Description of the phase-shift characterization versus the
frequency-shift method, (b) implementation block diagram.
on the interface system, its stability, and its phase detection
limit.
Consequently, unlike in oscillators for RF applications,
in oscillators based on QCM sensors, the resonator is not in-
cluded in the circuitry with the aim of stabilizing the oscillator
system, although evidently it does it; just on the contrary, theoscillator circuitry should be as ideal as possible for not in-
fluencing the shifts in the sensor phase due to the monitoring
processes. Unfortunately, the implementation of an ideal os-cillator for high frequency QCM sensors, keeping in mind the
low quality factors reached by these sensors under liquid con-
ditions, and the very low phase noise that it is necessary, isnot an easy task.
By keeping in mind the previous considerations, a dif-
ferent approach was recently proposed:
57taking into account
that the expected frequency shifts in those QCM applica-
tions where a high resolution is necessary, for example, in
biosensors, are very small, it could be possible to interrogate
the sensor with an appropriate constant frequency signal, in
the sensor resonance bandwidth, and then measure the changein the phase response of the sensor, while maintaining the
frequency of the testing signal in the resonance bandwidth;
Fig. 1(a) depicts the frequency-shift versus the phase-shift
characterization methods.
A similar approach has been already applied under dif-
ferent conditions by some authors.
60,61The advantage of this
approach is that the sensor is interrogated with an external
source which can be designed to be very stable and with ex-
tremely low phase and frequency noises, even at very highfrequencies. Moreover, a very simple circuit can be used
for the phase-mass characterization approach as depicted in
Fig. 1(b), where a mixer based phase detector is used. In the
next paragraph two practical systems are described which will
be useful for validating the phase-shift against the frequency-
shift characterization. An improved version of the balanced-
bridge oscillator proposed elsewhere
25is extensively tested
under different damping conditions showing the effects of the
FIG. 2. Schematics of the balanced-bridge oscillator proposed.
non-ideal behaviour of oscillators to monitor the frequency-
shifts under different damping conditions. Nonetheless, for
constant damping conditions and low frequency sensors(10 MHz AT-cut quartz resonators), the oscillator can be used
for resonance frequency shift monitoring and therefore be
used to validate the phase-shift method by comparison of theresults.
IV. DESCRIPTION OF THE SYSTEMS
A. Improved balanced-bridge oscillator
The proposed interface is shown in Fig. 2, and is based on
the balanced-bridge oscillator presented elsewhere,51where
the transistors have been replaced by operational transconduc-
tance amplifiers or diamond transistors (OTA 1-2). The mixercircuit based on the integrated circuit (IC) AD835, with two
differential inputs, allows the implementation of a differen-
tial amplification with automatic gain control (AGC), whichminimizes the nonlinearities of the active devices and pro-
vides information about the sensor damping. This configura-
tion allows, in principle, a parallel capacitance compensation
which ideally provides oscillation at zero-phase loop condi-
tion; the parallel circuit L
C-CCis included to drastically re-
duce the loop-gain for undesired frequencies. For ideal com-
ponents and parallel capacitance compensation the oscillation
frequency should be the motional series resonant frequency ofthe sensor under different damping conditions, and the AGC
voltage would provide information about the resonator mo-
tional resistance.
Effectively, the input voltage u
iis transferred to the
emitters and the emitter currents are non-inverted voltage-
converted to the collectors into u1andu2as follows:
u1=uiYXZC, (9a)
u2=uiYCvZC, (9b)
where YX=jωCo+1/Zmis the admittance of the QCM sensor
formed by the so-called static capacitance Coin parallel with
the so-called motional branch whose impedance Zmis formed
by a Rm,Lm,Cmseries equivalent circuit, being Zm=Rm
+j(Lmω−1/ωCm);YCv=jωCv,ZC=RC+j(LCω−1/ωCC)
andωis the angular frequency.
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The voltages at the collectors are differentially amplified
with one of the high input impedance differential amplifiers
of the AD835, and the output signal is level controlled witha multiplier giving the following output signal u
/primeiwhich is
fed-back to the input:
u/prime
i=ADk/parenleftbigg1
Zm+jω(Co−Cv)/parenrightbigg
ZCui. (10)
Because u/primei=uithe final loop condition results:
ADk/parenleftbigg1
Zm+jω(Co−Cv)/parenrightbigg
ZC=1. (11)
Under ideal conditions, it is assumed that the parallel
circuit ZChas been designed to resonate at the oscillation
frequency, therefore ZC≈RC; the parallel capacitance has
been compensated, Co=CV; and the OTAs and the multi-
plier do not produce phase-shifts, it is to say ADandkare real
numbers. Consequently, the loop-phase condition given by
Eq.(11) indicates that, under the previous conditions, the os-
cillation frequency corresponds to the motional series reso-
nant frequency at which Zm=Rm, and the loop-gain at the
oscillation frequency reduces to
ADkRC
Rm=1. (12)
Therefore the automatic gain control voltage, k
=Rm/ADRC, is proportional to the value of the motional
resistance Rmfor given values of the differential gain ADand
the resistance RC. The objective of the AGC is to maintain
constant the amplitude of the signal u1; with this purpose a dc
signal, associated with the amplitude of the sinusoidal signal
u1, is obtained by low pass filtering the output of a multiplier
whose inputs are connected to u1.
Experimental results will provide the level of fulfilment
of the previous equations regarding the degree of ideal perfor-
mance of the components of the oscillator.
B. Phase-shift characterization interface
A schematic interface for the phase-shift characteriza-
tion method was proposed elsewhere57and it has been im-
plemented now for its validation. The core of the interfaceis the sensor circuit which is depicted in Fig. 3. Two paral-
lel branches form a differential circuit. Because the testing
signal u
thas constant frequency ft, the only element in the
circuit which contributes to a change in the phase shift be-
tween the reference signal u1and the signal u2is the change in
the phase-frequency response due to the sensor perturbation.Therefore, this phase-shift can be continuously monitored by
a phase-detector. The IC AD8302 from Analog Devices has
been used for this purpose; it includes a mixer and the low-pass filter (LPF), connected in series behind the signals u
1and
u2, which act as a phase detector for small phase-shifts around
90obetween the input signals.62Thus, for a proper operation
it is convenient to phase-shift 90othe testing signals in each
branch of the sensor circuit; for this purpose the networks
formed by RiandCiat the inputs of the sensor circuit have
been included. The phase-shifting networks formed by Riand
Cimust be designed coherently with the resonant frequency of
FIG. 3. Schematics of the phase-shift characterization system.
the sensor in order to obtain two signals 90ophase-shifted and
of similar amplitude. The IC-AD8302 additionally includes
a block, formed by logarithmic amplifiers, which provides avoltage proportional to the decibel ratio of the input signals u
1
andu2. Both the phase detector and the logarithmic amplifier
block have responses VPHSandVMAG, centred at 900 mV for
90ophase-shift and 0 dB power ratio between the signals u1
andu2. These transfer functions are re-centred around 0 mV
with additional differential amplifiers with appropriate volt-
ages at the reference inputs Vref1andVref2, obtained from a
very stable and low noise voltage reference; this allows pro-viding an additional amplification of signals V
PHSandVMAG.
Wide bandwidth operational amplifiers OPA1-4 are used
to isolate the sensor and the reference network RC-CCfrom
the rest of the circuit. At motional series resonance frequency
(MSRF) the sensor reduces to a motional resistance Rmin par-
allel with the so-called static capacitance C0; therefore for op-
timum operation it is convenient to select RCandCCsimilar
toRmandC0, respectively. Effectively, under these conditions
and at the MSRF of the sensor, the voltages uϕanduAcor-
responding to the phase-shift and to the decibel ratio of the
input signals u1andu2, respectively, should be, ideally, zero;
this provides a way to calibrate the system.
Additionally, far from resonance the sensor behaves like
the parallel capacitance C0, and the network formed by the
resistance Rtand the sensor reduces to a low-pass filter
Rt-C0of very high cutoff frequency around several megahertz.
Consequently slow phase noises in the input testing signal areequally transferred to both branches and eliminated by the dif-
ferential system, and then improving the stability.
V. MATERIALS AND METHOD
A. Sensors and accessories
10 MHz fundamental frequency AT-cut quartz sensors,
with 13.67 mm blank diameter and 5.11 mm of Cr/Au elec-trode diameter (100 Å of Cr and 1000 Å of Au), were used in
the experiments. Two home-made cells were used, one with
a volume capacity of 200 μl for the experiments done in-
batch with different concentrations of glycerol in water, de-
scribed elsewhere,
42and a different one for the experiments
in flow with 30 μl volume capacity (Fig. 4). Other instru-
ments associated with the experiment were: Impedance An-
alyzer HP4291A, frequency meter HP53181A, multimeter
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FIG. 4. (Color online) Flow cell.
HP34401A, RF signal generator model HP8664A, and data
acquisition system AWSense IP/AQ-1 from AWSensors Inc.
B. Circuit implementation
The circuits for both systems were made follow-
ing recommendations applicable to high frequency layout
design63,64and implemented in four-layer surface-mount
technology. The following ICs for the main parts of the os-cillator were used: OPA860 for the OTAs and AD835 for the
implementation of the automatic gain control and differential
amplification. The value of R
Cmust be appropriately selected
to assure the fulfilment of the loop-gain condition, according
to Eq. (12), for given values of ADandk, and for the expected
range of values of Rm, which mainly depends on the damping
media in contact with the sensor. In this particular case a value
ofRC=1K/Omega1has been selected for an Rmrange between
10 and 1.2 K /Omega1, keeping in mind the reachable values of k
(0/lessmuch1,2) and AD≈1.LC-CCfilters have been selected
matched and for a resonance at 10 MHz when using 10 MHzAT-cut quartz resonators. The implemented design is shown
in Figs. 5(a)and5(b).
For the phase-shift characterization interface, phase-
shifting networks formed by R
iandCiwere designed for a
cut-off frequency ( −3dB) at 10 MHz in order to obtain two
signals 90ophase-shifted and of similar amplitude; 500 MHz
unity-gain bandwidth operational amplifiers based on the IC
OPA656 from Texas Instruments were used for OPA1-4; In-
strumentation amplifiers based on the IC AD623 from Ana-log Devices were used for a further amplification of the re-
centred signals V
PHSandVMAG, obtaining a final phase and
magnitude amplifications of 100 mV/oand 300 mV/dB, re-
spectively. The values for the components of the reference
network RCandCCwere selected similar, but using standard
values, to the motional resistance and parallel capacitance of
the 10 MHz sensor under the liquid load conditions presented
in the immunosensor application, where phosphate buffered
FIG. 5. (Color online) (a) Implemented circuit boards of the oscillator,
(b) the final oscillator.
saline (PBS) was used as main working liquid; these values
were measured with the IA and standard values were selectedforR
C=320/Omega1andCC=10 pF; in the experiments Rtwas
selected equal to RC. The implemented design is shown in
Figs. 6(a)and6(b).
C. Experimental methodology
1. Calibration and tuning of the phase-shift
characterization system
Before the monitoring, a calibration step of the phase-
shift characterization system can be performed easily. In thisstep an appropriate frequency is selected in the RF genera-
tor source (10 MHz, for instance, when using 10 MHz sen-
sors), the sensor is removed and substituted with a R
C-CCref-
erence network, in such a way that both differential branches
in the system are identical, and the signals at the inputs of the
AD8302 should be, ideally, of the same amplitude and phase-shifted 90
o. Under this configuration the voltages Vref1and
Vref2are varied for setting the outputs uϕanduAat zero volts;
these voltages should be near 900 mV . After calibration, thesensor is placed again in its original position and loaded with
the working liquid medium, and then the frequency of the RF
generator is varied to find again zero volts at the output u
ϕ.
From then on the system is ready for continuous monitoring
of the voltages uϕanduA, which for small sensor resonant
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FIG. 6. (Color online) Implemented phase-shift system: (a) upper side,
(b) lower side in the customized box.
frequency changes can be directly related to the sensor phase-
shift and damping, respectively.
2. Performance of the oscillator in liquid media
The compensation of the parallel capacitance of the
resonator is useful under heavy load conditions.42In order
to test the effective compensation of the parallel capacitance,
measurements of the oscillating frequency and the voltage k
associated with the motional resistance Rmin the oscillator
were made, using liquid solutions of different volume
concentrations of glycerol in water (0%, 5%, 15%, 25%,
35%, and 45%); this allows covering a density-viscosity
product ( ρη) ranged from 1 to 5.2 kg Pa s l−1; data for the
density and viscosity of the different solutions were takenfrom Weast and Astle (1980).
65Different compensation
capacitors Cvwere used, with values around the expected
parallel capacitance of the sensor (6.8, 10, and 15 pF), toevaluate the parallel capacitance compensation effect. The
oscillating frequency and the voltage k associated with the
motional resistance R
mwere compared with the maximum
conductance frequency and the reverse of the conductance
peak at resonance obtained from the impedance analyzer with
the different liquids. For a complete analysis of the sensor, theconductance, susceptance, impedance phase and impedance
modulus plots, and the equivalent parameters R
m,Lm,Cm,
andC0, obtained with the impedance analyzer around the res-
onance bandwidth, were registered for the sensors in all the
liquids and in air, this last taken as reference. Frequency shifts
related to the frequency in air were taken with the oscillator
and the impedance analyzer for each liquid: the frequency
shift in the oscillator /Delta1fOSC=fOSC(liquid) −fOSC(air) and the
FIG. 7. (Color online) Graphical representation of the impedance phase of
the sensor from the plot of the impedance magnitude, knowing the oscillation
frequency.
frequency shift obtained from the impedance analyzer /Delta1fIA
=fIA(Gmax−liquid) −fIA(Gmax−air), where G maxindicates
the frequency at maximum conductance. The phase of the
sensor at the oscillating frequency was obtained startingfrom the impedance phase plot, obtained with the impedance
analyzer for the sensor in contact with each liquid, as depicted
in Fig. 7. With the aim of performing a detailed study of
the compensation effect of the differential branch formed by
the OTA2 and the capacitor C
v, all the measurements, which
were repeated three times and averaged, were made as wellunder three different configurations of the oscillator: (a) the
oscillator without compensation branch (Fig. 8), (b) oscillator
with the compensation branch and without the capacitorC
v(the same as Fig. 2without C V), and (c) oscillator with
different values of Cv(Fig. 2). Results are presented and
detailed discussed in a subsequent paragraph.
3. Immunosensor
A comparison between the classical technique based on
frequency shift monitoring and the new one based on phase
shift monitoring, using the systems described, under the same
FIG. 8. Oscillator configuration without the differential branch.
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experimental conditions, was performed by implementing a
real application based on the detection of low molecular
weight pollutants; only with this purpose, a piezoelectric im-munosensor for the detection of the pesticide carbaryl, as
a validation model, was developed. The following protocol
described elsewhere was followed:
3the AT-cut quartz crys-
tals were functionalized by immobilizing BSA-CNH carbaryl
hapten conjugate on the sensor surface through the forma-
tion of a thioctic acid self-assembled monolayer. The crys-
tal was placed in a custom-made flow cell (Fig. 4) and in-
cluded in a flow-through setup, controlled by a peristalticpump (Minipuls 3, Gilson), with the injection loop and solu-
tions at the input of the flow cell exchanged by manual Rheo-
dyne valves (models 5020 and 5011, Supelco), according tothe flow system described elsewhere.
66The whole fluidic sys-
tem and the sensor characterization circuit with the sensor cell
were placed in a custom made thermostatic chamber and allthe experiments were performed at 25
oC±0.1oC. To avoid
unwanted disturbances the chamber was placed on an anti-
vibration table.
The immunoassay developed to determine carbaryl was
an inhibition test based on the conjugate coated format, in
which the hapten-conjugate was immobilized on the sensorsurface. A fixed amount of a specific monoclonal antibody
was mixed with standard solutions of the analyte and pumped
over the sensor surface. Since the analyte inhibits antibody
binding to the immobilized conjugate, increasing concentra-
tions of analyte will reduce the phase shift induced on thepiezoelectric sensor and the corresponding demodulated volt-
age.
Different standard concentrations of carbaryl were pre-
pared by serial dilutions in PBS, from a 1 mM stock so-
lution in dimetylformamide at −20
oC. The standards were
mixed with a fixed concentration of the monoclonal antibodyLIB-CNH45 (from I3BH-UPV (Ref. 67)) in PBS. Analyte-
antibody solutions were incubated for 1 h at ambient tem-
perature, then loaded (250 μl) into the injection loop in the
chamber, and finally, when tempered, injected onto the sensor
surface. The phase-shift was monitored in real-time for each
analyte concentration during 12 min, as the binding betweenfree antibody and the immobilized conjugate took place. Re-
generation of the sensing surface was performed using diluted
hydrochloric acid, 0.1M HCl, to break the antibody-hapten
linkage, at a flow rate of 280 μl/min for 4 min, and then
with the working buffer solution – phosphate buffered saline– 0.005% tween 20 (PBST) – for 2 min at the same flow rate.
Stabilization of the initial signal was achieved again at a flow
rate of 30 μl/min for 2 min. A complete assay cycle took
20 min. This protocol was performed with both characteri-
zation systems and the results are discussed in a subsequent
paragraph.
VI. RESULTS AND DISCUSSION
A. Performance of the oscillator in liquid media
1. Results
A linear correlation was found between the motional re-
sistance Rmvalues, obtained from the impedance analyzer,and the voltage k values, obtained from the oscillator, with a
correlation coefficient higher than 0.998 in all the cases. Aver-
age deviations between the value of Rmobtained from the lin-
eal regression, under the restricted condition of Rmbeing 0 for
k=0, to be coherent with Eq. (12), and those obtained from
the IA, were smaller that 5.5%, reaching the value of 1.1% forthe highest values of R
m; a uniform reduction of the deviation
is observed as the Rmvalues increase. This can be explained
taking into account that the OTA1 operates at higher current
gains for smaller liquid loads, and that the operation of the
amplifier gets farther from the ideal behaviour as the gain in-creases.
Figure 9(a) collects the frequency shifts obtained with
the oscillator ( /Delta1f
OSC=fOSC(liquid) −fOSC(air)) and the
impedance analyzer ( /Delta1fIA=fIA(Gmax−liquid) −fIA(Gmax
−air)) for all the liquids and with the different circuit set-
ups and compensation capacitors Cv. Figure 9(b) shows the
impedance working phase of the sensor, obtained as ex-
plained, under the different liquid loads and for the different
oscillator set-ups and compensation capacitors Cvdepicted in
Fig. 9(a).T h e Rmvalues in Figs. 9(a) and9(b) are only in-
cluded as reference and are near but not necessarily equal to
the corresponding Rmvalues obtained from the impedance an-
alyzer with the corresponding liquids; however these values
forRmwere used in the numerical calculations performed in
the discussion below.
The main objective of using the balanced-bridge os-
cillator is the parallel capacitance compensation under theideal assumptions made in Sec. IV A . The results shown in
Figs. 9(a)and9(b) allow making the following remarks:
(a) If the parallel capacitance was compensated with an ap-
propriate value of compensation capacitance C
v, the os-
cillation frequency shift should be in coincidence, or
at least to be very near to the MSRF shift given by
the impedance analyzer; however the results depicted in
Fig. 9(a) show that none of sets of the oscilla-
tor frequency shifts for the different liquids fits the
MSRF shifts of the analyzer, only the set of frequency
shifts corresponding to the compensation capacitor Cv
=6.8 pF seems to be parallel to the results obtained with
the impedance analyzer which are taken as reference.
The averaged value of the static parallel capacitance C0
obtained from the impedance analyzer for all the liquids
was 7.37 pF, with a maximum deviation of 1.5%, it is to
say very near to 6.8 pF. This could make to think that akind of compensation occurs; however this is not reason-
able by taking into account the following remark.
(b) According to Fig. 9(b), the sensor working phase for C
v
=6.8 pF shows nearly a constant working phase for the
whole range of liquid loads (average of −28owith a de-
viation of 0.4o). This is in contradiction with the tracking
of the MSRF, since different sensor phases correspond to
different MSRFs associated with different liquid loads.
In any case, the behaviour depicted in Figs. 9(a)and9(b)
should be explained in the following aspects:
(a) Why is there a sensor working phase at which the
oscillation frequency shifts in the working liquid
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FIG. 9. (Color online) (a) Frequency shifts measured with the oscillator and
the impedance analyzer between the sensor in contact with liquid and in con-
tact with air under different liquids, configuration setups, and compensationcapacitors. The frequency shifts measured with the IA were taken at conduc-
tance peak; (b) phase of the sensor impedance under the different conditions
depicted in (a).
solutions, with regard to air, have a parallel behaviour
to those associated with the MSRF shifts, obtained from
the impedance analyzer?
(b) Why does exist, in the proposed balanced-bridge oscilla-
tor, a compensation capacitance Cvat which the working
phase of the sensor in the oscillator is nearly constant for
a relative wide range of liquid loads? Moreover, is theresome relation between this value of compensation capac-
itance and the static parallel capacitance of the sensor
C
0, or the very close values are only a coincidence?
2. Discussion
In relation to the first question previously posed, some
authors have already described the fact that there are certain
working phases of the sensor at which the obtained frequency
shifts with regard to the sensor in air have a similar trendto those obtained for the MSRF shifts, depending on the
range of loads considered.
33,35,68,69Figures 10(a) and10(b)
show this statement. In Fig. 10(a) the frequency shifts of
the sensor under different loading conditions, with regard to
the sensor in air, are plotted together with the MSRF shifts
FIG. 10. (Color online) (a) Comparison made by numerical simulation
among the frequency shifts corresponding to a certain phase of the sensorimpedance, with respect to the frequency of the unperturbed sensor ( /Delta1ω
ϕ
=ωϕ−ω0), and the motional series resonant frequency shifts under the
same conditions ( /Delta1ω s=ωs−ω0); (b) Frequency deviation between the fre-
quency of the sensor under a certain phase of the sensor impedance and the
motional series resonance frequency ( /Delta1ωϕs=ωϕ−ωs).
for the same loading conditions; in Fig. 10(b) the difference
between the frequency shifts, at a given phase of the sensor,
and the MSRF shifts in Fig. 10(a) are shown for a better visu-
alization. The plots in Fig. 10(a) have been obtained through
the corresponding mathematic equations which are derived
in the Appendix for non-disturbing the attention of the
reader. The plots have been obtained for different phases of
a sensor with a 10 MHz series resonance frequency in air,with different R
massociated with the different loads and with
the constant motional and static capacitances of 31.5 fF and
7.37 pF, respectively, obtained with the sensor used in thiswork. As can be observed in Fig. 10(a) , and previously
discussed from other authors, different working phases of
the sensor are more appropriate for certain ranges of loadsand, additionally, can extend the operation range of the
oscillator. Thus, for extending the application to heavy loads
a working phase between −30
oand−40ois advisable. It can
be observed as well that for a range of loads between 300 and
600/Omega1ofRm, a working phase of −28oprovides a frequency
shift approximately parallel to the MSRF shift, which is thebehaviour observed in our experiment. The question now
is: why does it occur for a compensation capacitance of C
v
=6.8 pF. In order to clarify this aspect a deeper analysis of
the proposed oscillator shown in Fig. 2is necessary, which is
derived next.
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V oltages u1andu2in the circuit depicted in Fig. 2can be
mathematically expressed in polar form as follows, where the
phase of the input voltage uiis taken as zero reference:
u1=ui|YX||ZRLC|/angbracketleftϕOT A 1+ϕRLC−ϕZX/angbracketright, (13a)
u2=ui|jωCv||ZRLC|/angbracketleftϕOT A 2+ϕRLC+π/2/angbracketright, (13b)
where ϕOTA1is the phase shift provided by the OTA1, ϕOTA2is
the phase shift given by the OTA2, ϕRLCis the phase shift pro-
vided by the RC-LC-CCparallel branch, ϕZXis the phase shift
associated with the sensor impedance ZX,YX=1/ZXis the ad-
mittance of the sensor, ZRLCis the impedance of the RC-LC-CC
parallel branch, and j ωCvis admittance of the compensation
capacitor Cv.
On the other hand, the phase shift associated with the dif-
ferential amplifier in the AD835 should be very small, be-cause the voltage peak is maintained constant thanks to the
AGC, the frequency is always around 10 MHz for all the liq-
uid loads, the differential gain A
Dis 0 dB ( AD=1), and the
differential error phase is very small;70therefore, under the
configuration setup depicted in Fig. 8, where the differential
branch associated with the OTA2 has been removed, the volt-
ageui/primecan be expressed as follows:
u/prime
i=uik|YX||ZRLC|/angbracketleftϕOT A 1+ϕRLC−ϕZX/angbracketright, (14)
where the phase shift associated with the AD835 has been
neglected.
Figure 9(a) shows that the sensor phase in the con-
figuration set-up where the differential branch is removed
(Fig. 8) changes with the liquid load, as follows (phase shift
and the reference Rmvalue are written in couples as: Rm,
ϕZXo): 220 /Omega1,−22o; 300 /Omega1,−19o; 340 /Omega1,−18o; 440 /Omega1,
−16o; 640 /Omega1, –13o; and 750 /Omega1,−10o. The phase shift in-
creases with increasing the gain (decreasing the load) as itwas expected. Because u
i/primeis fed-back to the input, ui/prime=ui
and Eq. (14) reduces as follows:
k|YX||ZRLC|/angbracketleftϕOT A 1+ϕRLC−ϕZX/angbracketright=1. (15)
Consequently, the loop-phase oscillation condition im-
poses that ϕC=ϕZX, where ϕC=ϕOTA1+ϕRLC; that is, the
phase of the sensor compensates the phase shift due to the
rest of the elements of the oscillator. Therefore the different
working phases of the sensor depicted in Fig. 9(b),f o rt h e
configuration setup without differential branch, give us the
phase shift value due to the oscillator circuitry, mainly due
to the OTA1, for the different liquid loads as follows ( Rm,
ϕC): 220 /Omega1,−22o; 300/Omega1,−19o; 340/Omega1,−18o; 440/Omega1,−16o;
640/Omega1,−13o; and 750 /Omega1,−10o.
When the differential branch is added, the phase shift as-
sociated with the OTA2 depends on the value of Cv, since the
gain of OTA2 changes approximately from 0.5 to 1 for valuesofC
vfrom 6.8 pF to 15 pF, respectively; however this phase
shift is much smaller than in the OTA1 because the smaller
change in the gain. Therefore the voltage u2can be expressed
as follows:
u2=ui|jωCv||ZRLC|/angbracketleftϕ/prime
C+π/2/angbracketright, (16)
where ϕ/primeC=ϕOTA2+ϕRLCUnder this configuration the voltage u0can be written as
follows:
u0=u1−u2=ui(|YX||ZRLC|/angbracketleftϕOT A 1+ϕRLC−ϕZX/angbracketright
−|jωCv||ZRLC|/angbracketleftϕ/prime
C+π/2/angbracketright) (17)
or alternatively:
u0=|u0|/angbracketleftϕ/angbracketright, (18)
where | u0|=A2+B2–2AB cos( β−α) and tan ϕ=(Asinα
−Bsinβ)/(Acos α−Bcosβ); A, B, αandβbeing: A
=|ui/bardblYX/bardblZRLC|,B=|ui/bardbljωCv/bardblZRLC|,α=ϕC−ϕZXandβ
=ϕ/primeC+π/2.
Therefore ui/prime=ui=k|u0|<ϕ> and the loop-phase con-
dition establishes that ϕmust be zero for oscillation. Conse-
quently, now it is possible to check if there is a value of Cv
which makes tan ϕ=0 for a relatively constant value of the
sensor phase for all the liquid loads in the working range. For
that the following expression must be fulfilled:
tanϕ=sinα−B
Asinβ
cosα−B
Acosβ=0, (19)
where B/A=ωCv/|YX|.
The modulus of the admittance | YX| and the value of the
reactance Xmdepend on the phase of the sensor and can be
written, as a function of the admittance phase of the sensorϕ
YX=−ϕZXandRmas follows (derived from Eqs. (A1) and
(A2) in the Appendix):
|YX|=Rm/radicalbig
1+tan2ϕYX
R2m+X2m, (20a)
Xm=1
ωC0−/radicalBig/parenleftbig1
ωC0/parenrightbig2+4/parenleftbigRmtanϕYX
ωC0−R2m/parenrightbig
2. (20b)
Now, a numerical calculation can be performed in order
to obtain the values of ϕZX=−ϕYXwhich make ϕ=0i n
Eq.(19), for the different liquid loads and for the different
values of Cv. For that, the following data were taken into
account:
(a) Values for ϕCwere taken from the results obtained with
the configuration set-up without the differential branch(Fig. 8), indicated above, for each reference value of R
m
associated with a certain liquid load.
(b) The value of ϕ/primeCwas assumed to be constant for the dif-
ferent values of CVand equal to the value of ϕCfor a Rm
=750/Omega1,t h i si s ϕ/primeC=−10o; the reason is because the
gains for OTA2 changed approximately from 0.5 to 1 forvalues of C
vfrom 6.8 pF to 15 pF, and the gain for OTA1
with a liquid load corresponding to Rm=750/Omega1was 1.3
(≈RC/Rm); therefore similar phase shifts are expected for
similar gains.
(c) The value for the angular frequency ωwas taking for a
frequency of 10 MHz, and the value for the static capac-
itance was taken from the averaged values obtained with
the impedance analyzer, C0=7.37 pF.
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FIG. 11. (Color online) Results of the numerical simulation derived from
Eq.(19).
The results of this analysis are presented in Fig. 11.A s
it can be observed they follow almost the same trend as the
ones obtained in Fig. 9(b) for the different values of Cv;a
phaseϕZX=−26.8owith a deviation of 0.5o, very close to the
one obtained in our experiment, results for all the values of Rm
FIG. 12. Results obtained with the oscillator and IA, for a compensation
capacitor Cv=6.8 pF, under the same conditions as in Fig. 9(a) but with a
parallel capacitance of the sensor purposely increased to 12 pF by adding a
parallel capacitor.
FIG. 13. (Color online) (a) Evolution of the frequency shift in the oscillator,
at different flow rates, under the appropriate conditions for maximum signal,and (b) evolution of the voltage associated with the phase-shift in the phase-
shift characterization system, at different flow rates, under the appropriate
conditions for maximum signal.
with the compensation capacitor Cv=6.8 pF, while the sensor
phase changes, with the same trend, for a different value of Cv.
In order to confirm the previous results and to demon-
strate that the very close values between the value of Cv
=6.8 pF, at which frequency shifts for a constant phase of
the sensor are nearly parallel to MSRF shifts, and the value
ofC0was only a coincidence, an additional experiment was
performed in which the parallel capacitance of the sensor was
changed to 12 pF by an external parallel capacitor. Fig. 12(a)
shows how the results for the oscillation frequency shifts leavethe parallelism to those from the MSRF shift obtained from
the IA, while the sensor phase is maintained relatively con-
stant for all the liquid loads (Fig. 12(b) ).
These results indicate that the expected compensation of
the parallel capacitance is not provided by the implemented
balanced bridge oscillator; the non-ideal behaviour of the os-cillator elements and mainly of the OTAs, despite their ex-
pected good performance at 10 MHz, prevailed on the ex-
pected ideal operation of the oscillator circuit. However, thedifferential branch with the compensation capacitor can be
used to compensate the non-ideal behaviour of the oscillator
circuit, in such a way that the phase of the sensor can remain
relatively constant in a certain range of liquid loads. The value
of the capacitor C
Vfor this purpose depends on the sensor and
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FIG. 14. (Color online) Real time piezoelectric immunosensor response to different concentrations of analyte: (a) with the balanced-bridge oscill ator, and
(b) with the phase-shift characterization system.
on the behaviour of the oscillator circuit and can be obtained
if the phase responses of the OTAs versus the gain are known.
On the other hand, the results show that the oscillator
designed can maintain the oscillation under relatively highdamping loads, and therefore can be used for frequency shift
monitoring in in-liquid microgravimetric applications like, for
instance, in piezoelectric biosensors, where the characteristicsof the fluid medium remain constant. Therefore the imple-
mented oscillator was used, with C
v=6.8 pF, for continuous
monitoring the resonance frequency shift of the sensor in the
immunosensor application performed to validate the phase-
shift characterization method.
B. Immunosensor
1. Results
Figures 13(a) and13(b) show the real time monitoring of
different experiments made with both systems, with the aimof optimizing the flow rate under maximum signal condition;
i.e., when the sample is a solution of a reference antibody
concentration without antigen.
3Because the flow cell was the
same in both cases, and the optimized flow rate depends on
the cell volume, the same speed of 30 μl/min was obtained for
both systems. As it can be observed a very close response wasfound for both systems keeping in mind the different mag-
nitudes involved. A real time signal of the voltage u
ϕ, asso-
ciated with the phase-shift, showed an exponential decay assoon as the molecular interaction occurred after the sample
injection; a similar behaviour was observed when the reso-
nant frequency shift was monitored.
Figures 14(a) and14(b) show a comparison between the
real-time signals obtained for the piezoelectric immunosen-sor, for the same experiment, with the frequency-shift
(Fig. 14(a) ) and phase-shift (Fig. 14(b) ) monitoring systems.
During the experiments, different concentrations of pesticide
in the sample were tested after cyclic regeneration stages de-scribed. Only a representative part of the signals obtained in
the immunoassay, corresponding to analyte concentrations of
10, 20, 100, and 500 μg/l, are shown. As it can be observed
the resistance R
m(voltage k) measured with the oscillator
technique (Fig. 14(a) ) remained constant as expected in these
applications.
2. Discussion
These results validate the new characterization concept
and the implemented interface. Moreover, a reduction of the
noise in the new system was observed as well. Effectively, the
noise level in the oscillator technique was of 2 Hz for a max-imum signal of 137 Hz, while for the phase-shift interface
was of 1 mV for a maximum signal of 200 mV , this indicates
an improvement of three times the maximum signal to noiseratio. Furthermore, it is important to notice that this improve-
ment has been got even with relative low frequency sensors
(10 MHz), where electronic components and circuits have avery good performance in both, the oscillator and the phase-
detector system. Therefore a much more significant improve-
ment is expected to be found with very high fundamental fre-quency resonators.
VII. CONCLUSIONS AND FUTURE LINES
A new characterization concept, particularly useful for
high resolution QCM applications, based on the phase-shift
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monitoring has been compared with the classical concept
of frequency shift monitoring. A balanced bridge oscillator
has been proposed for in-liquid QCM applications andproved to be valid for working with sensors under relatively
heavy loading conditions. It has been demonstrated that the
non-ideal behaviour of the active components which formpart of the oscillator prevent making any preliminary ideal
theoretical presumption of the expected performance under
real conditions; however, despite of the non-ideal behaviour
of the oscillators they can follow being used for QCM appli-
cations under liquid conditions, and specially for relativelylow frequency resonators. Alternatively, the following ad-
vantages are expected with the new characterization concept
based on the phase-shift monitoring at constant frequency:(a) the sensor is interrogated passively with an external
source, which can be designed with high frequency stability
an very low phase noise, even at very high frequencies, (b)the sensor circuit is very simple with high level of integration
capabilities, (c) the open loop configuration, in contrast to
the typical feedback configuration of the oscillator, allows astraightforward noise analysis and minimization, simplifying
the design and implementation of the electronics, (d) sensors
working at the same fundamental resonance frequency couldbe characterized, in principle, with only one source, opening
the possibility of working with sensor arrays for multianalysis
detection.
Following the results presented here, the next step is to
perform experiments with the new systems using high fun-damental frequency BAW resonators based on inverted mesa
technology.
ACKNOWLEDGMENTS
The authors are grateful to the Spanish Ministry of Sci-
ence and Technology for the financial support to this research
under contract reference AGL2009-13511, and to the com-
pany Advanced Wave Sensors S.L. ( www.awsensors.com )f o r
the help provided in the development of some parts of this
work.
APPENDIX: DERIVATION OF THE FREQUENCY SHIFT
BETWEEN THE FREQUENCY OF THE SENSOR AT A
CERTAIN PHASE AND THE MSRF
The admittance of the sensor can be expresses as follows:
YX=jωC0+1
Rm+jXm=Rm
R2m+X2m
+jωC0/parenleftbig
R2
m+X2
m/parenrightbig
−Xm
R2m+X2m. (A1)
Therefore, the phase of the admittance ϕYxcomplies with
the following expression:
tanϕYX=ωC0/parenleftbig
R2
m+X2
m/parenrightbig
−Xm
Rm. (A2)
The reactance Xmat a certain angular frequency ωϕcor-
responding to a sensor operating phase ϕcan be written asfollows:
Xm=Lmωϕ−1
Cmωϕ=(Lq+/Delta1L)ωϕ−1
Cqωϕ
=/Delta1Lωϕ+1
Cqωϕ/parenleftBigg
ω2
ϕ
ω2
0−1/parenrightBigg
, (A3)
where it has been assumed that the change in the reactance
due to the liquid load is due to an inertial effect associated
with the increase in the motional inductance /Delta1Lω,t h em o -
tional capacitance is assumed to be constant and equal to the
value in the unperturbed state, Cq;Lqis the motional induc-
tance in air; and ω0=1/(LqCq)1/2is the resonance angular
frequency in the unperturbed state (air).
For liquid loads /Delta1Lω≈Rm, assuming the motional resis-
tance in air is small, and the former equation can be approxi-mated as follows:
X
m≈Rm+2
Cqω2
0/Delta1ωϕ, (A4)
where /Delta1ωϕ=ωϕ−ω0.
By using Eq. (A4) in Eq. (A2) , the following relationship
is obtained, which relates /Delta1ωϕ, with the sensor admittance
phase, the motional resistance Rmcorresponding to a liquid
load, the static parallel capacitance C0, the motional capaci-
tance Cq, and the unperturbed resonance frequency ω0:
tanϕYX=4C0
C2qω3
0Rm/Delta1ω2
ϕ+4C0ω0Rm−2
CqRmω2
0/Delta1ωϕ
+2ω0C0Rm−1. (A5)
The frequency at conductance peak, obtained from IA
measurements, corresponds, with negligible error in most of
cases, to the MSRF where Xm=0; therefore, from Eq. (A4) ,
the angular frequency shift corresponding to the MSRF for a
certain liquid load with respect to the unperturbed state /Delta1ω s
=ωs−ω0, where ωsis the angular MSRF, results as follows:
/Delta1ω s=−Cqω2
0Rm
2. (A6)
Consequently, the angular frequency shift between the
MSRF and the angular frequency corresponding to a certain
sensor admittance phase, /Delta1ωϕs=ωϕ−ωs, can be obtained
from the previous equations as /Delta1ωϕs=/Delta1ωϕ−/Delta1ω s(/Delta1fϕs
=/Delta1ωϕs/2π). The previous equations have been used, for
10 MHz resonance frequency, Cq=31.5 fF and C0=7.37
pF corresponding to the sensor used, and for different values
of the sensor admittance phase, in the derivation of the results
shown in Figs. 10(a) and10(b) .
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1.4873583.pdf | Domain wall motion driven by spin Hall effect—Tuning with in-plane magnetic
anisotropy
A. W. Rushforth
Citation: Applied Physics Letters 104, 162408 (2014); doi: 10.1063/1.4873583
View online: http://dx.doi.org/10.1063/1.4873583
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/104/16?ver=pdfcov
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132.75.80.183 On: Mon, 08 Dec 2014 12:02:05Domain wall motion driven by spin Hall effect—Tuning with in-plane
magnetic anisotropy
A. W. Rushfortha)
School of Physics and Astronomy, University of Nottingham, University Park, Nottingham NG7 2RD,
United Kingdom
(Received 1 March 2014; accepted 16 April 2014; published online 25 April 2014)
This letter investigates the effects of in-plane magnetic anisotropy on the current induced motion
of magnetic domain walls in systems with dominant perpendicular magnetic anisotropy, where
accumulated spins from the spin Hall effect in an adjacent heavy metal layer are responsible fordriving the domain wall motion. It is found that that the sign and magnitude of the domain wall
velocity in the uniform flow regime can be tuned significantly by the in-plane magnetic anisotropy.
These effects are sensitive to the ratio of the adiabatic and non-adiabatic spin transfer torqueparameters and are robust in the presence of pinning and thermal fluctuations.
VC2014 Author(s).
All article content, except where otherwise noted, is licensed under a Creative Commons
Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.4873583 ]
The properties of magnetic domain walls continue to
stimulate great interest due to the rich underlying physics
and potential for applications in areas such as information
storage1and processing.2Recently, the modulation of the
mobility of magnetic domain walls by electric fields, in the
field driven and current driven regimes, has been considered
as a low power technique to control domain wall motion. Inone scheme, this can be achieved by modification of the
perpendicular magnetic anisotropy through direct electric
field gating of a ferromagnet/dielectric interface.
3–6An al-
ternative, indirect method is the electric field induced strain
generated in a hybrid ferromagnet/piezoelectric structure. A
uniaxial strain in the film plane modifies the in-plane mag-netic anisotropy component, even for a film in which per-
pendicular magnetic anisotropy dominates. This has
emerged as a versatile method to modify the pinning fieldor current,
7or to modify the transition between uniform and
precessional motion in the field and current driven
regimes.8,9A recent study9considered domain wall motion
in a dilute ferromagnetic semiconductor in which the cur-
rent driven domain wall motion is understood to arise from
the spin transfer torque (STT) mechanism. A drawback ofthis material system is the fact that the achievable Curie
temperatures are still far below room temperature.
10More
promising for device applications are metallic systems withperpendicular magnetic anisotropy. In such systems, effects
arising from spin-orbit coupling at a heavy metal/ferromag-
net interface or an asymmetric heavy metal/ferromagnet/ox-ide layer structure can give rise to additional contributions
to domain wall motion. Spin accumulation at the heavy
metal/ferromagnet interface, generated by the spin Halleffect (SHE) in the heavy metal, creates an anti-damping
(also known as Slonczewski-like) torque on the domain
wall.
11Here, this mechanism is referred to as the SHE-STT
to distinguish it from the conventional STT mechanism aris-
ing from the spin polarised current flowing in theferromagnetic layer. The Rashba effect, arising from an
electric field gradient in an asymmetric layer structure, can
give rise to field-like12and Slonczewski-like torques.13,14
Finally, the Dzyaloshinskii-Moriya interaction (DMI) aris-
ing at the heavy metal/ferromagnet interface stabilises the
N/C19eel domain wall configuration with the same preferred
chirality for up/down and down/up domain wall configura-tions.
15Several recent investigations16–18have concluded
that the combination of SHE-STT and DMI effects domi-
nate the domain wall motion in such systems. In combina-tion, they promote a stable domain wall configuration with
an internal angle between the Bloch and N /C19eel types, giving
rise to larger domain wall velocities and uniform motionover a larger range of current densities than in the pure STT
driven case. In these investigations, field-like torques were
found to be relatively small and a less significant contribu-tion to the domain wall velocity. Motivated by these recent
experiments, this letter investig ates the effects of an in-plane
magnetic anisotropy on the current induced domain wallmotion in a system where the anti-damping torque from the
SHE-STT drives the domain wall with and without the pres-
ence of the DMI. It is found that the effects of the in-planemagnetic anisotropy are qualitatively different to the pure STT
driven case,
8,9giving rise to large tuneability of the domain
wall velocity at relatively low current densities where the do-main wall motion is in the uniform flow regime. The magni-
tude and sign of the tuneable velocity is sensitive to the STT
non-adiabaticity parameter, b, and the effects are predicted to
be robust in the presence of pinning and finite temperature.
The domain wall depicted in Figure 1is modelled using
the rigid one-dimensional model modified to include theeffects of STT, SHE-STT, and DMI.
15,16The model has been
shown to provide a good qualitative description of domain
wall motion along nanowire strips with perpendicular mag-netic anisotropy
19and more recently of the combined effects
of the SHE-STT and DMI.16–18The model describes the do-
main wall behaviour through two coupled Eqs. (1)and(2)
based on the position Xalong the nanowire and the azimuthal
angle ua)Author to whom correspondence should be addressed. Electronic mail:
andrew.rushforth@nottingham.ac.uk.
0003-6951/2014/104(16)/162408/4 VCAuthor(s) 2014
104, 162408-1APPLIED PHYSICS LETTERS 104, 162408 (2014)
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132.75.80.183 On: Mon, 08 Dec 2014 12:02:051þa2 ðÞdX
dt¼acDHZ/C0cDHK
2sin 2 /ðÞþ1þabðÞ bJ
þacDp
2HSHEcos/ðÞ þcDp
2HDMIsin/ðÞ;
(1)
1þa2 ðÞd/
dt¼cHZþacHK
2sin 2 /ðÞþb/C0aðÞbJ
D
þcp
2HSHEcos/ðÞ/C0acp
2HDMIsin/ðÞ;(2)
where HZis the magnetic field applied perpendicular to the
layer plane. The parameters, with values appropriate for the
Pt/Co/Pt trilayer systems studied previously16include the
Gilbert damping term, a¼0:2, the gyromagnetic ratio,
c¼2:21/C2105m=As, the domain wall width, D¼10 nm,
and the STT non-adiabaticity parameter, b¼0/C00:4.
Conventional STT enters through the term bJ¼lBPj=eMS,
where jis the current density which is positive for conven-
tional current, P¼0.5 is the spin polarisation, MS¼1.4
/C2106A/m is the magnetization of the film, lBis the Bohr
magneton and e¼/C01.6/C210/C019C is the charge on the elec-
tron. Following Ref. 16, the effects on the domain wall of
SHE-STT and DMI are modelled as effective fields,H
SHE¼/C22hhSHj=2l0eMSt, and HDMI¼D=l0MSD, in the direc-
tions transverse (y) and parallel (x) to the wire, respectively.
hSH¼0:1 is the spin Hall angle, t¼0.6 nm is the film thick-
ness, and D¼/C00.5 mJ/m2is the DMI parameter. The prefer-
ence for forming a Bloch or N /C19eel type wall is modelled by an
effective field HK. Positive (negative) HKfavours the forma-
tion of a Bloch (N /C19eel) wall. In practice, HKcan be tuned by
fabricating nanowires of different widths20or by inducing a
uniaxial strain.9Here, the effects of an induced in-plane mag-
netic anisotropy are modelled as a variation of the sign and
magnitude of HK. To select an appropriate range of values
over which HKcould be tuned, the case of a hybrid piezoelec-
tric/ferromagnet structure is considered. For ferroelectric
single crystals such as [Pb(Mg 1/3Nb2/3)O3](1-x)-[PbTiO 3]x
(PMN-PT) or [Pb(Zn 1/3Nb2/3)O3](1-x)-[PbTiO 3]x (PZN-PT)
uniaxial strains of order e¼2/C210/C03are achievable experimen-
tally.21Taking values of magnetostriction, kS¼6/C210/C05and
Young’s modulus, Y¼2.1/C21011Pa, for bulk Co gives
DHK¼3kSYe=MS/C25650 mT, corresponding to a change of
the in-plane magnetic anisotropy energy, Dkll/C25635kJm/C03.
The coupled Eqs. (1)and(2)were solved numerically by imple-
menting a fourth order Runge-Kutta method with a time step of
1p s .First, consider the case of a domain wall driven by the
combination of SHE-STT and STT. In the absence of con-
ventional STT, the SHE-STT would drive the domain wall to
a Bloch configuration from which the SHE-STT alone can-not drive domain wall motion because the anti-damping tor-
que vanishes in this configuration.
11The inclusion of the
conventional STT provides a mechanism to drive the domainwall configuration away from the pure Bloch wall and ena-
bles the SHE-STT to provide a torque, s
SHEwhich drives do-
main wall motion in combination with STT. This torque is
sensitive to the internal angle of the domain wall
(sSHE¼cm/C2HSHEandjHSHEj/jm/C2rj/sin/, where r
is the unit vector of the accumulated spins and mis the mag-
netic moment) and so depends upon the steady state angle /
at a given current density (i.e., the value of /which satisfies
d/
dt¼0). The equilibrium angle and the corresponding do-
main wall velocity can be tuned through variation of HK.I n
Fig. 2(a), it is observed that negative HKleads to an
enhanced domain wall velocity at finite current densities,
where-as positive HKsuppresses this feature. This can be
understood as a stabilisation of the Bloch wall configurationcaused by positive H
Kwhich results in a diminishing influ-
ence of the SHE-STT. Negative values of HKmake the N /C19eel
wall configuration more stable, resulting in an equilibriumvalue of /6¼90
/C14(see Fig. 2(b)) and a non-zero influence of
the SHE-STT on the velocity. At higher current densities,
the SHE-STT drives the domain wall to the Bloch configura-tion and the velocity is then determined only by the STT
term. The maximum velocity and the current density at
which it is reached increase with increasing negative H
K.
Similar behaviour is observed for b<a(Figs. 2(a)and2(b))
andb>a(Figs. 2(c) and2(d)), although the behaviour is
more complicated in the latter case where the domain wallvelocity can be observed to change sign with respect to the
direction of the current. This arises because the presence of
the non-adiabatic field-like STT can drive /to take equilib-
rium values greater than 90
/C14.
The inclusion of a finite DMI results in an effective field
favouring the formation of a N /C19eel wall. Typical values16of
the DMI parameter, D¼/C00.5 mJ m/C02give HDMI>HSHEfor
modest current densities, making the N /C19eel wall the stable
configuration. Therefore, at low current densities, theSHE-STT is very efficient at driving domain wall motion
and larger velocities can be achieved than in the STT driven
case (Fig. 2(e)). The sign of the DMI parameter promotes a
particular chirality for the domain wall and results in the do-
main wall velocity having the opposite sign to the STT plus
SHE-STT driven case. H
DMIis independent of the current
density, therefore at high current densities the SHE-STT
drives the equilibrium angle towards the Bloch configuration
(/¼90/C14) and the domain wall velocity gradually decreases.
The inclusion of positive or negative HKmodifies the equi-
librium angle of the domain wall resulting in a respective
decrease or increase of the velocity at a given current den-sity. As reported previously,
16it is found that the STT makes
a limited contribution to the domain wall velocity in this
regime.22
It is worth comparing the effects of an in-plane magnetic
anisotropy in the SHE-STT driven regime with that of
the STT driven regime. In the latter case, the effect of the
FIG. 1. (a) Schematic diagram of the Pt/Co interface. The black arrows rep-
resent the direction of the magnetization forming a domain wall in the Co
layer. jis the current density and ris the unit vector of the accumulated
spins generated by the spin Hall effect in the Pt layer. (b) Plan view of the
in-plane component of the magnetization. /is the azimuthal angle of the
magnetization at the centre of the domain wall.162408-2 A. W. Rushforth Appl. Phys. Lett. 104, 162408 (2014)
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132.75.80.183 On: Mon, 08 Dec 2014 12:02:05in-plane anisotropy was to move the onset of precessional
motion (the Walker limit) to higher or lower current den-
sities.8,9In the uniform flow regime, the domain wall veloc-
ity was not modified by the in-plane magnetic anisotropy. Inthe SHE-STT driven regime, uniform motion is stabilised to
higher current densities. The domain wall velocity depends
on the equilibrium angle of the domain wall, which dependsupon the sign and magnitude of H
K. Therefore, an in-plane
magnetic anisotropy can modify the domain wall velocitysignificantly in the uniform flow regime. In the absence ofthe DMI, the exact form of the domain wall velocity as afunction of the current density is sensitive to the ratio b=a,
and an abrupt change of sign as a function of current is pre-dicted in the regime where b>a.
The analysis presented so far neglects the effects of do-
main wall pinning and finite temperatures. Following Refs.
16and19, the effects of pinning were modelled as a spatially
dependent field component in the z-direction given by
H
Z¼/C01
2l0LYt@V
@X, where LY¼100 nm represents the wire width
and V¼V0sin2pX=nðÞ with V0¼1.65/C210/C020J and
n¼30 nm. Thermal effects were modelled as an uncorrelatedGaussian distributed stochastic fluctuation of the z-axis field
with amplitude A¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2aKBT
cl0ML YtDdtq
. Each calculation was aver-
aged over ten such random distributions. These parameters
have been used previously to make qualitative predictions of
the effects of pinning and thermal fluctuations in CoPtCr
(Ref. 18) and Pt/Co/Pt (Ref. 16) layers. Figure 3reveals that
in the case of domain walls driven by SHE-STT and STT,
the effects of in-plane magnetic anisotropy are still present,
but are diminished by pinning and thermal fluctuations.When DMI is also present the effects of in-plane magnetic
anisotropy remain strong in the presence of pinning and ther-
mal fluctuations up to room temperature. This is encouragingfor the prospects for experimental observations of these
effects and the potential applications in device technologies.
An experimental demonstration of the effects predicted
by our calculations would be aided by the fabrication of
multi-layered film structures in which H
SHEand H DMIcan be
set independently by the layer structure. It has been sug-
gested18that this can be achieved in asymmetric structures
such as Pt(thin)/Co(thick)/Ni/Co(thin)/Pt(thick) in which athicker top Pt layer provides a larger H
SHEthan the bottom
FIG. 2. (a) and (c) The calculated domain wall velocity as a function of the current density in the presence of the STT and SHE-STT. (e) As is (c) with the
addition of the DMI mechanism. (b), (d), and (f) The corresponding equilibrium angles as a function of the current density.
FIG. 3. The calculated domain wall velocity as a function of the current density with and without the inclusion of pinning and finite temperature (T ¼300 K).
Error bars, given by the standard deviation of averaging over ten stochastic realisations, are approximately the size of the points plotted and so are not included
on these graphs.162408-3 A. W. Rushforth Appl. Phys. Lett. 104, 162408 (2014)
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132.75.80.183 On: Mon, 08 Dec 2014 12:02:05Pt layer, and a thicker bottom Co layer induces a larger H DMI
due to a greater proximity induced moment in the Pt layer at
the bottom Pt/Co interface. The fabrication of such structures
should make possible an experimental exploration of the pre-dicted effects over a large parameter range.
To summarise, the physics of current induced domain
wall motion in ferromagnetic films adjacent to heavy metallayers is qualitatively different to ferromagnetic films in
which the conventional STT is the main mechanism to drive
the domain wall. In such systems, the DMI and SHE-STT
are responsible for stabilising the domain wall orientation
and inducing uniform domain wall motion over a large rangeof applied current densities. Our calculations reveal that, in
this regime, an induced in-plane magnetic anisotropy can
tune the sign and magnitude of the domain wall velocity sig-nificantly. It is shown that hybrid piezoelectric/ferromagnet
systems such as PMN-PT/PtCoPt are prospective systems in
which to observe these effects.
Discussions with Jan Zemen and Tomas Jungwirth are
gratefully acknowledged. This work was funded by EPSRCGrant No. EP/H003487/1.
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calculations performed with P ¼0.162408-4 A. W. Rushforth Appl. Phys. Lett. 104, 162408 (2014)
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1.860712.pdf | Particle simulation of nonneutral plasma behavior
H. Ramachandran, G. J. Morales, and V. K. Decyk
Citation: Physics of Fluids B: Plasma Physics (1989-1993) 5, 2733 (1993); doi: 10.1063/1.860712
View online: http://dx.doi.org/10.1063/1.860712
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IP: 130.113.111.210 On: Mon, 22 Dec 2014 00:56:59LETTERS
The purpose of this Letters section is to provide rapid dissemination of important new results in the fields regulariy covered by
Physics of Fluids B. Results of extended research should not be presented as a series of letters in place of comprehensive articles.
Letters cannot exceed three printed pages in length, including space allowed for title, figures, tables, references and an abstract
limited to about 100 words. There is a three-month time limit, from date of receipt to acceptance, for processing Letter
manuscripts. Authors must also submit a brief statement justifying rapid publication in the Letters section.
Particle simulation of non-neutral plasma behavior
H. Ramachandran, G. J. Morales, and V. K. Decyk
Department of Physics, University of California at Los Angeles, Los Angeles, California 90024-1.547
(Received 1 March 1993; accepted 30 April 1993)
It is demonstrated that particle-simulation techniques can explore a broad range of dynamical
behavior exhibited by non-neutral plasmas. New features isolated by this approach include
collective relaxation through generation of core and halo populations; self-organization without
radial transport as the plasma is cooled; spontaneous generation of solitonlike structures upon
axial reflection from the external confining potential.
The properties of non-neutral systems confined in traps
have been elucidated in several elegant experiments’>2 in
recent years. As plasmas, these systems have the virtue of
being confinable for extremely long times because conser-
vation laws3 place strict bounds on particle loss. Most of
the detailed experiments in this area have explored the
quasistatic evolution of a well-confined configuration near
thermal equilibrium. The transient, nonequilibrium phase
during which the plasma achieves its near-equilibrium pro-
files is dominated by strong nonlinear wave-particle inter-
actions. These processes are difficult to probe experimen-
tally and are not easily described by analytical methods.
However, advances in experimental sophistication (e.g.,
new ion machines coming on line) and increasing interest
in this field are naturally leading to the consideration of the
nonlinear collective interactions supported by these sys-
tems. Consequently, it is of interest to develop computa-
tional methods that allow the investigation of the fast time-
scale, nonequilibrium properties.
In this Letter, we report the results of a systematic
application of particle-simulation techniques to explore the
broad range of behavior exhibited by non-neutral plasmas.
It is demonstrated that this computational technique (also
known as the particle-in-cell method4) permits the detailed
study of a broad range of dynamical features, including
maintenance of a Vlasov equilibrium; cross-field transport
by neutral collisions; relaxation by collective interactions
between core and halo populations; development of self-
organization as a confined plasma blob is cooled externally.
It is our objective here to highlight the broad range of
dynamics that can be explored and to stimulate new theo-
retical studies and experiments. Future publications will
address, in depth, each of these topics.
The particle simulation code used in this study is a
generalization of a code previously developed5 to investi-
gate heating of neutral plasmas by externally launched ra- dio frequency waves. It consists of an electrostatic,
bounded, magnetized two-and-one-half-dimensional sys-
tem (follows two spatial coordinates and three velocities
for each particle). The magnetic field is aligned with the z
direction and confines particles along the x direction (the
radial direction in a cylindrical experiment), with the y
coordinate ignorable (x, y, and z are orthogonal Cartesian
coordinates). As is usual in such studies, we normalize
lengths to the initial Debye length, &-,, and velocities to
the initial thermal velocity, K Walls located at x= f Lx
make the plasma bounded in that direction. The potential
may be prescribed as a function of z on those walls, and
diverse experimental situations can be reproduced. We
have explored two such schemes: ( 1) a slab equivalent of a
Penning trap in which the vacuum potential is approxi-
mately of the form r$(x,z) ,&($-x2)/( Lz-- Lz), with &,
a controllable parameter and L, and L, the half-lengths of
the system along x and z, respectively; and (2) localized
potential hills well separated along the z direction, in a
manner analogous to the traps developed* at the University
of California, San Diego. The capability of turning on and
off these confining hills permits the investigation of plasma
expansion and recapture with this code. Typically the nu-
merical studies use more than 16x lo3 finite size particles,
with a minimum of 64 computational grids in the x direc-
tion and 512 along z (of course, larger numbers are used
when needed to resolve the dynamics of a given situation).
For the results reported here, the strength of the confining
magnetic field satisfies f&/6+,&2.5 with fi2, the electron
gyrofrequency and Oar the electron plasma frequency. This
choice provides good radial confinement (below the Bril-
louin limit6) and does not require an excessive number of
computation steps (the detailed electron gyromotion is cal-
culated with a resolution of fi2, At< i, which is to be con-
trasted with the guiding-center simulations’ previously
used to explore these systems).
2733 Phys. Fluids B 5 (8), August 1993 089%8221/93/5(8)/2733/3/$6.00 @ 1993 American Institute of Physics 2733
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-32 0 32
x/A0
FIG. 1. Demonstration of quiet equilibrium of a simulated non-neutral
plasma after a time t=435 a;‘. (a) Density profile across the magnetic
field, The solid curve is without collisions and the dashed curve shows the
effect of cross-field transport due to neutral collisions. (b) Self-consistent
velocity profile demonstrating the equilibrium electric field EYa X. Spatial
scale is Debye length a0 and the velocity scale is thermal velocity E
The persistence of a quiet Vlasov equilibrium is dem-
onstrated in Fig. 1. These results correspond to a pure
electron plasma column infinitely long in the z direction
(no axial potential trap is applied). The self-consistent
plasma density profile achieved is shown in Fig. 1 (a). As is
characteristic’ of these systems, it exhibits a nearly square
profile with a sharp edge on a scale of a Debye length. The
self-consistent drift profile (rotation profile in a cylinder)
is shown in Fig. 1 (b). It shows that U,CCX, and hence the
radial electric field E, a X, as expected from the flat region
in Fig. 1 (a), i.e., this is the slab analog of a rigid rotor
equilibrium in a cylinder. Because the y coordinate is ig-
norable in the code, flute instabilities are absent here, so, in
practice, we achieve a projection to lower dimensionality of
a stably rotating column. These features have been ob-
served to hold for times t > 400 fir’. The ability to cor-
rectly simulate such equilibria permits the exploration of
nonlinear plasma waves, cyclotron resonance,’ and Bril-
louin limit.2
Next we highlight the collective relaxation of a non-
equilibrium initial distribution suddenly placed in a Pen-
ning trap, i.e., the opposite of the quiet start shown in Fig.
1. We have identified that the dominant feature in this
complex evolution is the self-consistent development of a
high-density (core) population that exhibits fluidlike be-
havior, and surrounded by a tenuous population of ener-
getic (halo) particles. The role of the core population is to
cancel the external parabolic well of the Penning trap, so
that a very small axial electric field results. However, the
effectiveness of this cancellation decreases with x, hence at
2734 Phys. Fluids B, Vol. 5, No. 8, August 1993 (bl ------------_---
I
_ 100 0 100
llAO
FIG. 2. Collective relaxation of a nonequilibrium initial configuration in
a Penning trap. (a) Axial phase space at t= 192 Sk, ’ shows a high-density
‘kore” of slow particles interacting with an energetic “halo” population.
(b) Configuration space shows the “core” population attains a football
shape and the energetic “halo” forms outer rails. Dashed lines indicate
location of walls.
larger x, a larger fraction of the particles belong to the halo
population [the partition and radial distribution of these
populations are determined by the ratio of the length in the
z direction to that in the x direction (aspect ratio)]. The
axial phase space (v, ,z) at time t= 192/C& is shown in Fig.
2(a) for an aspect ratio of 3. The core corresponds to the
high-density (darker) region of slow particles while the
halo is the encircling ring. The core executes long-lived
breathing oscillations on the wpe time scale. The role of
these collective interactions is to generate a time-dependent
balance between the internal self-repulsion and the external
potential well. The configuration-space shape of the con-
fined plasma is shown in Fig. 2(b). The core particles
achieve a football-like shape bounded by a rail-like struc-
ture associated with the energetic halo particles.
To investigate the long-time behavior, we introduce an
external drag force (i.e., vmv). This process extracts ki-
netic energy (i.e., it cools) and simultaneously induces
cross-field transport that results in the expansion of the
plasma, as shown by the dashed profile in Fig. 1 (a). The
equivalent expansion process caused by collisions with neu-
tral particles has been studied experimentally’o in consid-
erable detail. The isotropic drag slows down the energetic
halo particles and simultaneously damps the collective os-
cillations of the core. In the time-asymptotic limit, the
elongated edge rails in Fig. 2(b) coalesce, and the core
population fully equilibrates with the external trap poten-
tial.
To explore the development of self-organization as the
kinetic energy of the particles is reduced further, we apply
Letters 2734
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1 tick
L
x
A0
O-
i -32
-50 0 50
z/x,
FIG. 3. Development of self-organization in configuration space, as the
system shown in Fig. 2 is externally cooled by a factor of 2~ 10e7. The
ring structure develops sequentially from the outermost region inward.
a drag force only in the ( XJ) plane (anisotropic friction)
in order to prevent the radial expansion [documented in
Fig. l(a)], which ultimately would cause the loss of the
entire plasma blob to the walls. Figure 3 illustrates the
development of spatially correlated rings as the kinetic en-
ergy is lowered; they grow sequentially from the outside
toward the interior and are formed without the particles
ever crossing the magnetic field. Essentially, axial motion
results in locally trapped particles. A detailed sampling of
the particles found near the center of the blob shows that
they tend to organize in parallel planes perpendicular to
the magnetic field, but the outer ones form rings having the
shape of a football, as required to match the external trap
potential. Analogous structures have been observed in del-
icate laser-cooling experiments,2S” where they have been
interpreted as liquid crystals.
Finally, we show an example of a ubiquitous feature
that we have identified in several different studies, but that
appears most clearly during the free expansion and subse-
quent recapture (in a longer cell) of a non-neutral plasma.
The phenomenon takes the appearance of a solitonlike ob-
ject formed when a group of particles approaches its exter-
nal turning point. Spontaneously, the particles are momen-
tarily compressed between the external potential hill and
another group of slower, trailing particles, moving toward
the end along the z direction. Once the structure is gener-
ated, it moves through the plasma in the form of a poten-
tial jump, which continuously accelerates slow particles
that reflect from it; a behavior related to the double layers
studied” in neutral plasmas. A characteristic pair of such
objects, moving toward each other, is illustrated in Fig. 4.
The instantaneous phase space is shown in Fig. 4(a), while
the on-axis (x-0) density and self-consistent potential are
shown in Fig. 4(b) . Multiple encounters of the localized
structures result in a steady state consisting of a relatively
cold core and an energetic halo, which in this configuration 3
-i- o-
-60
lb) 1
, / \ / ” \ T I \ IO”++
I
’ 5- 2/’ ,
1 \ ,/ ’ 1, -OS
I I
I I 5% .I I
I 0 -1.0
-150 0 150
FIG. 4. Example of the spontaneous formation of solitonlike structures
upon reflection from the external trap potential. (a) Instantaneous phase
space showing the concentration of slow particles and the generation of
reflected particles away from the external potential hill. (b) The corre-
sponding axial density profile at x=0, and self-consistent axial potential.
At t=O, the system had an extent of RN& with temperature T and
density ns.
plasmas. Two interesting new features have been docu-
mented here: the development of self-organization without
cross-field transport, and the spontaneous generation of
solitonlike structures upon reflection from the confining
potential. Detailed exploration of the properties briefly out-
lined here promises to yield new insight into the collective
response of non-neutral plasmas.
ACKNOWLEDGMENT
This work is sponsored by the Office of Naval Re-
search.
‘C. F. Driscoll, J. H. Malmberg, and K. S. Fine, Phys. Rev. Lett. 60,
1290 (1988); A. W. Hyatt, C. F. Driscoll, and J. H. Malmberg, Phys.
Rev. Lett. 59, 2975 (1987).
*D. J. Heinzen, J. J. Bollmger, F. L. Moore, W. M. Itano, and D. J.
Wineland, Phys. Rev. Lett. 66, 2080 (1991); J. J. Bollinger and D. J.
Wineland, Sci. Am. 262(l), 124 (1990).
3T. M. G’Neil, Phys. Fluids 23, 2216 (1980).
4C. K. Birdsall and A. B. Langdon, Plasma Physics via Computer Sim-
ulation (McGraw-Hill, New York, 1985).
%. K. Decyk, J. M. Dawson, and G. 5. Morales, Phys. Fluids 22, 507
(1979); V. K. Decyk, G. J. Morales, and J. M. Dawson, Phys. Fluids
23, 826 (1980).
6R. C. Davidson, The Theory of Nonneutral Plasmas (Addison-Wesley,
Redwood City, CA, 1989), p. 102.
‘D. H. E. Dubin and T. M. G’Neil, Phys. Rev. Lett. 60, 511 (1988).
*S. A. Prasad and T. M. O’Neil, Phys. Fluids 22, 278 (1979).
9R. W. Gould and M. A. LaPointe, Phys. Rev. Lett. 67, 3685 (1991).
is nresent for all x. and not iust at the edge. “J. S. deGrassie and J. H. Malmberg, Phys. Fluids 23, 63 (1980). L .,
In summary, it has been demonstrated by key nontriv- “J J Bollinger, S. L. Gilbert, D. J. Heinzen, W. M. Itano, and D. J. . .
ial examples that particle-simulation techniques can ex- Wineland, Strongly CoupIed PIasma Physics, edited by S. Ichimaru
plore a broad range of properties displayed by non-neutral (Elsevier, Amsterdam, 1990), p. 177.
‘*T Sato and H. Okuda, Phys. Rev. Lett. 44, 740 (1980). .
2735 Phys. Fluids B, Vol. 5, No. 8, August 1993 Letters 2735
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1.369787.pdf | Thermal reversal of magnetization as “random walk” dynamics of a nonlinear oscillator
Vladimir L. Safonov
Citation: Journal of Applied Physics 85, 4370 (1999); doi: 10.1063/1.369787
View online: http://dx.doi.org/10.1063/1.369787
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/85/8?ver=pdfcov
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202.28.191.34 On: Sat, 20 Dec 2014 17:04:18Thermal reversal of magnetization as ‘‘random walk’’ dynamics
of a nonlinear oscillator
Vladimir L. Safonova)
Center for Magnetic Recording Research, University of California—San Diego, 9500 Gilman Drive,
La Jolla, California 92093-0401
Coherentrotationofmagnetizationinagrainisrepresentedasthedynamicsofanonlinearoscillator
excited by random force. A stochastic differential equation is introduced to study the role of thermalfluctuations and the corresponding Fokker–Planck equation is obtained. It is shown thatconventional Landau–Lifshitz–Gilbert damping is consistent with thermodynamics only for smalldeviations of the magnetization from the equilibrium state. An exact formula for the first passagetime from the initial potential well is obtained. © 1999 American Institute of Physics.
@S0021-8979 ~99!36808-0 #
I. INTRODUCTION
Micromagnetic dynamics of fine magnetic grains are
usually analyzed as coherent rotation of magnetization in theeffective magnetic field H
eff52](E/V)/]M. Here
E/V5Kusin2u2MsH0cos~fH2u!, ~1!
whereEis the energy, Vis the grain volume, Kuis the
uniaxial anisotropy constant, uis the angle between the mag-
netization M(uMu5Ms) and the axis of anisotropy ~z!, and
H5(H0sinfH,0,H0cosfH) is the magnetic field. The mo-
tion ofMsupplemented by random Gaussian fields was stud-
ied to solve the problem of coherent reversal of magnetiza-tion under thermal agitation ~see, for example, Refs. 1–8 !.
The rotation of Min an effective field can be represented
by a nonlinear oscillator. It is interesting to consider theproblem of magnetization reversal from this point of view aslong as most of the time the vector Mis near its equilibrium
state and it exhibits small amplitude oscillations. This ap-proach was recently proposed in Ref. 9. In this article weshall consider the problem of magnetization reversal as therandom dynamics of a nonlinear oscillator.
II. MODEL
The state of equilibrium of magnetization is defined by
the condition ]E/]u50, or
HKsin2u52H0@sin~fH2u!#, ~2!
whereHK52Ku/Msis the anisotropy field. If u0is a solu-
tion of Eq. ~2!, then it is convenient to introduce, so-called
‘‘axes of quantization’’ associated with the equilibrium di-rection of magnetization by transformations,
x5x
1cosu01z1sinu0,Sx5Sx1cosu01Sz1sinu0,
z52x1sinu01z1cosu0,Sz52Sx1sinu01Sz1cosu0,
y5y1,Sy5Sy1.
Here a classical spin is defined as S5MV/\g.We shall describe oscillations of magnetization in terms
of complex variables a*,awhich are classical analogs of
creation and annihilation operators and can be introduced bya Holstein–Primakoff transformation
10
Sz15S2a*a,S15aA2S2a*a,S25~S1!*.~3!
The energy ~Hamiltonian !, Eq. ~1!, in the vicinity of the
equilibrium state can be approximated by the quadratic form
E~2!/\g5Aa*a1~B/2!~aa1a*a*!, ~4!
whereA5HK@12(3/2)sin2u0#1H0cos(fH2u0) and B
52(HK/2)sin2u0. It is possible to eliminate nondiagonal
terms from Eq. ~4!by a linear canonical transformation,
a5uc1vc*,a*5uc*1vc,
~5!
u5AA1vc
2vc,v52B
uBuAA2vc
2vc.
Thus we find the generalized coordinates of the normal
modes of the system and get the energy of an harmonic os-cillator,
E
~2!/\g5e01vcN,N[c*c5ucu2. ~6!
Here e0is a constant and
vc5g@HKcos2u01H0cos~fH2u0!#1/2
3@HKcos2u01H0cos~fH2u0!#1/2~7!
is the frequency of small oscillations of the magnetization.
Hamilton’s equation for complex amplitude ccan be
written as
id
dtc5vc,E/\b51
\]E
]c*. ~8!
Here, Planck’s constant \is used as a dimensional constant
in order to make c*andcdimensionless. The analog of the
commutator is
vA,Bb[~]A/]c!~]B/]c*!2~]B/]c!~]A/]c*!.
The energy Eq. ~1!can be written in the forma!Electronic mail: safonov@sdmag4.ucsd.eduJOURNAL OF APPLIED PHYSICS VOLUME 85, NUMBER 8 15 APRIL 1999
4370 0021-8979/99/85(8)/4370/3/$15.00 © 1999 American Institute of Physics
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202.28.191.34 On: Sat, 20 Dec 2014 17:04:18E5KuV~12P2~11Q!sin2u01P~11Q!1/2Qsin2u0
2Q2cos2u0!2\gH0S@P~11Q!1/2sin~fH2u0!
1Qcos~fH2u0!#, ~9!
whereQ512(u21v2)(c*c/S)2uv(c21c*2)/SandP
5(u1v)(c1c*)/2AS.
The equations of motion for complex amplitude can be
supplemented by the phenomenological relaxation hand
written as
iSd
dt1h~c*,c!Dc5v˜c~c*,c!c, ~10!
where
v˜c5~1/\c!]E/]c* ~11!
is the nonlinear frequency of oscillations ( v˜c!vcifucu
!0); the nonlinear frequency v˜c50 at the top of the energy
barrier. Passing this point the magnetization changes its di-rection of rotation.
It should be noted that the absolute value of magnetiza-
tion will not be changed for any continuous function
hbe-
cause the spin S5uM(t)uV/\gis an invariant of transforma-
tions, Eqs. ~3!and~5!.
III. STOCHASTIC DIFFERENTIAL EQUATIONS
In order to study the effect of thermal fluctuations we
add a complex Gaussian random force f(t) to Eq. ~10!. The
following expressions:
^f~t!&5^f~t1!f~t2!&50,^f~t1!f*~t2!&52Dcd~t12t2!,
are valid, where ^...&denotes the mean value, Dcis a diffu-
sion parameter, and d(t) is the Dirac delta function.
Equation ~10!with random forcing can be rewritten as a
stochastic differential equation,
dc52~h1iv˜c!cdt1ADc@dW1~t!1idW2~t!#,
ADcdW1~t!5Rf~t!dt,ADcdW2~t!5Tf~t!dt.~12!
HereW1(t) andW2(t) are independent Wiener processes.
Let us rewrite the complex variables as
c5rexp~2if!5exp~m2if!. ~13!
Using Ito calculus,11one obtains
dm52hdt1ADcexp~2m!dW m, ~14!
df5v˜cdt2ADcexp~2m!dW f, ~15!
where
dW m5cosfdW12sinfdW2,
dW f5sinfdW11cosfdW2.
Employing Ito calculus the stochastic differential equa-
tions for amplitude r5ucu5exp(m) are
dr5~2hr1Dc/2r!dt1ADcdWr, ~16!
df5v˜cdt2~ADc/r!dW f, ~17!wheredWr[dW m. Equation ~16!can depend on the phase
fif the nonlinear damping hdepends on f. Below we shall
consider the case when his a function of ronly. In this case
the motion of amplitude ris one dimensional.
One can obtain the corresponding stochastic differential
equation for N5r2:
dN52~2hN1Dc!dt12ANDcdWN, ~18!
wheredWN[dWr.
IV. FOKKER–PLANCK EQUATION
The stochastic differential Eq. ~18!corresponds to the
Fokker–Planck equation of the form11
]P
]t522]
]N@~2hN1Dc!P#12Dc]2
]N2NP. ~19!
From the stationary form of this equation for N>0, we can
obtain
dP/dN1Ph/Dc50. ~20!
In accordance with statistical mechanics
P5constexp ~2E/kBT!, ~21!
whereEis defined by Eq. ~9!andTis the temperature. Sub-
stituting Eq. ~21!into Eq. ~20!, we obtain
Dc\/kBT5h/v˜c5a, ~22!
where ais a dimensionless damping parameter. Thus the
ratio of nonlinear relaxation to nonlinear frequency must notdepend on N~and
f!. This is possible if v˜cdepends only on
N5c*cand
h5av˜c. ~23!
From Eqs. ~9!,~11!, and ~23!it follows that such a situation
occurs if H0iHK. In this case
v˜c5vc~12N/Ntop!,Ntop5S~11H0/HK!. ~24!
Note that conventional Landau–Lifshitz–Gilbert ~LLG!
damping for the case of H0iHKin the nonlinear oscillator
representation has the form12
h~N!5av˜c~N!~12N/2S!. ~25!
ForN!0 formulas ~23!and~25!are equal. Thus the relax-
ation, Eq. ~25!, is consistent with thermodynamics only for
N!2S. This means that the LLG equation with random
fields can give a correct result for the thermal magnetizationreversal only for the case of high energy barriers where themagnetization vector spends most of its time in the vicinityof the bottom of the well.
V. FIRST PASSAGE TIME
Consider the equation
dj~t!5A@j~t!,t#dt1AB@j~t!,t#dW~t!, ~26!
where j(t) is a stochastic variable in the interval ~a,b!, with
aa reflecting and ban absorbing boundary. The mean first
passage time for the equation can be written as114371 J. Appl. Phys., Vol. 85, No. 8, 15 April 1999 Vladimir L. Safonov
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202.28.191.34 On: Sat, 20 Dec 2014 17:04:18t~x!52E
xbdy
c~y!E
ayc~z!
B~z!dz. ~27!
Herex[j(0) is the starting point and
c~y![expHE
ay
dy8F2A~y8!
B~y8!GJ. ~28!
Let us consider Eq. ~16!. In this case j5r,a50,b
5(Ntop)1/2,A52ht1Dc/2r, andB5Dc. The starting
point is defined by the mean occupation number NT:x
5(NT)1/2. By integrating, one obtains
c~y!5~y/a!exp@F~y!#,
where
F~y![21
DcE
0y2
h~N!dN.
After several transformations expression ~27!can be re-
written in a form convenient for numerical calculations,
t5j
agHK~11h!~12n!E
01
dvE
01
duexp~j~11h!2
3z~v!~12u!@22z~v!~11u!#!, ~29!
where
z~v!5~12n!v1n,j5KuV/kBT,h5H0/HK,
and
n[NT
Ntop5*01dxxexp@2j~11h!2x~22x!#
*01dxexp@2j~11h!2x~22x!#.
In the case of a high energy barrier DE[KuV(1
1H0/HK)2@kBTthe first passage time, Eq. ~29!, can be
represented as the Ne ´el formula:
t.~C1/avc!~kBT/DE!nexp~DE/kBT!. ~30!For example, C1.0.63 and n.0.56 for 30 <DE/kBT<60.
In the limit of DE/kBT!`one hasC15Ap/4 and n50.5
which correspond to Brown’s asymptotic result.1,3,5
Thus we have constructed a thermodynamically consis-
tent approach to describe the magnetization reversal in a mi-cromagnetic grain due to thermal fluctuations. The exact for-mula ~29!for the first passage time
tfor any energy barrier is
obtained. It should be noted that Eq. ~29!can be written in a
general form as avct5F@j(11h)2#for any a,V,HK,
Ms,H0, andT.
ACKNOWLEDGMENTS
The author would like to thank H. Neal Bertram, Harry
Suhl, and Eric Boerner for helpful comments and discus-sions. This work was supported by matching funds from theCenter for Magnetic Recording Research at the University ofCalifornia–San Diego. Preliminary ideas for this article weredeveloped at the Toyota Technological Institute. Support byProfessor Takao Suzuki is greatly appreciated.
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1.3246351.pdf | The ground state van der Waals potentials of the calcium dimer and calcium rare-gas
complexes
D. D. Yang, P. Li, and K. T. Tang
Citation: The Journal of Chemical Physics 131, 154301 (2009); doi: 10.1063/1.3246351
View online: http://dx.doi.org/10.1063/1.3246351
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Ab initio intermolecular potential energy surface, bound states, and microwave spectra for the van der Waals
complex Ne–HCCCN
J. Chem. Phys. 122, 174312 (2005); 10.1063/1.1888567
Accurate intermolecular ground state potential of the Ar-N 2 van der Waals complex
J. Chem. Phys. 121, 10419 (2004); 10.1063/1.1809606
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89.133.231.66 On: Fri, 11 Apr 2014 18:24:08The ground state van der Waals potentials of the calcium dimer
and calcium rare-gas complexes
D. D. Yang,1P . Li,1and K. T . T ang1,2, a/H20850
1Institute of Atomic and Molecular Physics, Sichuan University, Chengdu, 610065 Sichuan,
People’ s Republic of China
2Department of Physics, Pacific Lutheran University, Tacoma, Washington 98447, USA
/H20849Received 7 July 2009; accepted 21 September 2009; published online 16 October 2009 /H20850
The entire potential energy curve of the Ca 2ground state generated by the Tang–Toennies potential
model with its parameters specified by the three theoretical dispersion coefficients and theexperimentally determined equilibrium distance and well depth is in excellent agreement with theaccurate experimental potential of Allard et al. /H20851Phys. Rev. A 66, 042503 /H208492002 /H20850/H20852. The reduced
potential of Ca
2is almost identical with that of Hg 2. This leads to the conjecture that the ground state
van der Waals dimer potentials of group IIA, except Be, and group IIB elements have the sameshape, which is different from that of the rare-gas dimers. The potentials of Ca-RG complexes/H20849RG=He,Ne,Ar,Kr,Xe /H20850are generated by the same potential model with its parameters calculated
from the homonuclear potentials of calcium and rare-gas dimers with combining rules. The
predicted spectroscopic constants are comparable to other theoretical computations. © 2009
American Institute of Physics ./H20851doi:10.1063/1.3246351 /H20852
I. INTRODUCTION
The interaction potentials of calcium dimer have been
the subject of many investigations. Especially in recentyears, with magneto-optical traps,
1,2a number of fields in
ultracold atom physics, such as photoassociation,1,3optical
frequency standards,4,5as well as possible Bose–Einstein
condensates,6,7have been initiated. Interaction potentials are
necessary for these investigations.
The discrete many line spectrum of calcium vapor was
first successfully photographed by Balfour and Whitlock.8
Vibrotational analysis of a total of 3800 lines involving 47bands with ground vibrational level up to
v/H11033=7 enabled them
to construct realistic Rydberg–Klein–Rees potentials of theground state X
1/H9018g+and an excited state thought to be A1/H9018u+.
The well depth of the ground state was determined to beD
e=1075 /H11006150 cm−1. With laser induced fluorescence,
Vidal9extended the spectroscopic data of Ca 2tov/H11033=34 with
5846 lines. Based on the inverted perturbation approach, theground state potential energy curve was determined with awell depth of 1095 /H110060.5 cm
−1. The ground state potential
was further refined in 2002 with Fourier-transform spectros-copy by Allard et al.
10This potential was derived from the
experimentally observed frequencies of transitions between acommon excited state and the ground state energy levels.The long range part of this potential was determined by thetransition frequencies from the asymptotic levels of theground X
1/H9018g+state reaching v/H11033=38 measured with the fil-
tered laser excitation technique.11The measured
B1/H9018u+→X1/H9018g+transitions cover over 99.8% of the ground
state well depth, which was determined to be1102.08 /H110060.09 cm−1. This kind of accuracy exceeds that of
the rare-gas dimers, whose potentials are generally consid-ered to be most accurately known.
12
Significant progress has also been made in theoretical
calculations.13–18The early density functional calculation of
Jones13overestimated the well depth by almost 50%. Recent
symmetry adapted perturbation calculations of Bussery-Honvault et al.
18with relativistic correction gave a well
depth of 1113 cm−1, which is only 1% deeper than the ex-
perimental value.
Inspite of the remarkably accurate knowledge of the po-
tential energy curves of calcium dimer and rare-gas dimers,19
the interaction potentials between calcium and rare-gas at-oms are not known to a high degree of accuracy. Althoughthere are a large number of calcium rare-gas collision experi-ments for a variety of exchange processes,
20–23quantitative
interpretations of these experiments are hampered by the lackof reliable potentials. The problem is the difficulty of pro-ducing sufficient density of calcium vapor in the molecularbeam oven because of its high melting point of 1120 K.Therefore one has to resort to laser vaporization which usu-ally produces molecules in their excited states. So the groundstates are omitted in the characterization. As far as we areaware, only the ground state potential of CaAr molecule,among all calcium rare-gas molecules, is determined experi-mentally, and only in one supersonic beam experiment.
24
Theoretically, the most extensively studied mixed sys-
tems are CaAr and CaHe. Unfortunately, results from variouscomputations differ greatly from each other. For interactionswith heavier rare gases, CaKr and CaXe, the only previousavailable potentials are from the coupled cluster calculationswith single, double, and perturbative triple excitations/H20851CCSD /H20849T/H20850/H20852by Czuchaj et al.
25This is also the only set of
calculations where the potential energy curves of Ca-RGa/H20850Electronic mail: tangka@plu.edu.THE JOURNAL OF CHEMICAL PHYSICS 131, 154301 /H208492009 /H20850
0021-9606/2009/131 /H2084915/H20850/154301/7/$25.00 © 2009 American Institute of Physics 131 , 154301-1
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89.133.231.66 On: Fri, 11 Apr 2014 18:24:08/H20849RG=He,Ne,Ar,Kr,Xe /H20850are obtained on the same footing.
In general, the CCSD /H20849T/H20850method systematically underesti-
mates the binding energies for mixed systems if the basis setis not large enough. This was shown in systems where inter-action potentials are accurately known either from theory orfrom experiment.
26,27
An important development in theory is the ab initio cal-
culations of the van der Waals dispersion coefficients.28–35It
is well known that the interaction between two spherical
symmetric atoms in the asymptotic region is given by
V/H20849R/H20850=−C6
R6−C8
R8−C10
R10, /H208491/H20850
where Ris the internuclear separation and Cnare the disper-
sion coefficients. The values of C6,C8, and C10of alkaline
earth interactions calculated by Porsev and Derevianko34us-
ing relativistic many-body perturbation theory are believedto be accurate to 1%. The recent large basis configurationinteraction calculations of Mitroy and Zhang
35differ from
these coefficients by only 2%–3%. In this reference, the co-
efficients for calcium rare-gas interactions are also given.
It is important that the interaction potential energy prop-
erly approach this asymptotic expression, especially for thecold atom collisions. In the accurate calcium dimer hybridpotential of Allard et al. ,
10an elaborate nonlinear fitting pro -
cedure is used to smoothly join Eq. /H208491/H20850to the potential in the
intermediate potential well region at a judiciously chosenpoint R
out. The intermediate potential is given in the form of
/H20858i=0nai/H20851/H20849R−Rm/H20850//H20849R−bRm/H20850/H20852iwith n=20. This intermediate po-
tential is then joined at another point Rinto the short range
repulsion in the form of A+B/R12.
In this paper, we will show that with the given set of
C6,C8,C10and the experimentally determined equilibrium
distance Reand the well depth De, the Tang–Toennies /H20849TT/H20850
potential36for Ca 2can be constructed without any fitting.
This potential is in excellent agreement with the multiparam-
eter hybrid potential of Allard et al. This enables us to easily
compare the shapes of several types of van der Waals poten-tials. Furthermore, since TT potential model can accuratelydescribe both the homonuclear calcium and rare-gas dimers,we investigate the possibility of predicting the interactionpotentials of Ca-RG complexes with combining rules.
Atomic units will be used in all calculations. For com-
parison to literature values, energy unit cm
−1and length unit
angstrom will also be used /H20849for energy: 1 a.u.=2.1947
/H11003105cm−1; for distance: 1 a0=0.5292 Å /H20850.
II. THE GROUND STATE POTENTIAL OF CALCIUM
DIMER
A. The potential model
The potential model proposed by Tang and Toennies
/H20849TT/H20850in 1984 /H20849Ref. 36/H20850will be used to generate the ground
state interaction potential of the calcium dimer. In thismodel, the short range repulsive Born–Mayer potentialAexp /H20849−bR/H20850is added to the long range attractive potential,
which is given by the damped asymptotic dispersion seriesV/H20849R/H20850=Ae
−bR−/H20858
n=3nmax
f2n/H20849bR/H20850C2n
R2n. /H208492/H20850
Based on the form of the exchange correction to the disper-
sion series, TT were led to the conclusion that the dampingfunction is an incomplete gamma function
f
2n/H20849bR/H20850=1− e−bR/H20858
k=02n/H20849bR/H20850k
k!, /H208493/H20850
where bis the same as the range parameter of the Born–
Mayer repulsion on the ground that both the repulsion anddispersion damping are consequences of the wave functionoverlap. For simple systems, such as H
2,H e 2, and HHe,
damped dispersion can be numerically calculated.37–39These
accurately calculated damping functions are all in very good
agreement with Eq. /H208493/H20850. Since its introduction, the model
potential of Eq. /H208492/H20850was tested successfully for several
chemically different types of van der Waalsinteractions.
36,40–42Moreover, this model has been shown to
have a firm foundation within the generalized Heitler–
London theory.43
For many systems, the first three coefficients C6,C8,C10
are available, the series may be terminated at nmax=5. If
more terms are desired, the recurrence relation44
Cn+4=/H20873Cn+2
Cn/H208743
Cn−2
may be used to estimate the higher coefficients. Generally,
terms beyond C16do not make any contribution to the po-
tential.
Thus, the model potential is determined by five param-
eters A,b,C6,C8, and C10. If the first three dispersion coef-
ficients are available, only two parameters /H20849A,b/H20850need to be
known in order to use the TT model. Furthermore, if the
equilibrium distance Reand the well depth Deof the poten-
tial are known, then Aandbcan be determined from Eq. /H208492/H20850
through a simple procedure.45
For Ca 2,De=5.021 /H1100310−3a.u. /H208491102.08 cm−1/H20850and
Re=8.081 a0/H208494.277 Å /H20850are well established by Allard et al.10
Together with the three dispersion coefficients C6=21.21
/H11003102a.u., C8=22.3/H11003104a.u., and C10=21.32 /H11003106a.u.
given in Ref. 34, the Born–Mayer parameters are determined
with the simple program in the Appendix of Ref. 45to be
A= 28.23 a.u., b= 0.9987 a0−1,nmax = 5, /H208494/H20850
A= 22.85 a.u., b= 0.9646 a0−1,nmax = 8. /H208495/H20850
The ground state potential energy curve of calcium
dimer calculated from the TT model with these parameters isshown in Fig. 1and some values are listed in Table I.I nt h e
long range and in the well region, there is a very little dif-ference between the potentials with nmax=5 and nmax=8.
They are not distinguishable in the scale of Fig. 1. This is
because with D
eand Refixed, the present model is self-
adjusting. The difference in the number of terms of the dis-persion series is compensated by the change in Aandbto154301-2 Yang, Li, and T ang J. Chem. Phys. 131 , 154301 /H208492009 /H20850
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89.133.231.66 On: Fri, 11 Apr 2014 18:24:08produce the correct shape near the bottom of the potential
well. The small differences between the two can be seen inthe numerical values of Table I.
B. Comparison to experimental hybrid potential
The shape of the present potential is in excellent agree-
ment with the experimentally determined potential of Allardet al.
10The numerical values reported in Refs. 10and11are
shown in Fig. 1as black dots and are listed in Table I. The
values of distance Rand experimental potential Vare taken
directly from these references. In the present results, onlyfour digits are listed, since the input parameters of the poten-tial model have only four significant numbers.
As seen in Table I, in the entire attractive part of the
potential, the largest difference between the present TTmodel and the experiment is only 4.3% at 7.50 Å fornmax=5. This difference is reduced to 1.8% for nmax=8.
In the repulsive part of the potential, there are large dif-
ferences in absolute values. At R=3.463 Å, the difference is
23% for nmax=5 and 33% for nmax=8. However, in this
region the potential is rapidly rising as Rdecreases. The net
effect is that the repulsive wall of the present potential isshifted slightly to the left of the experimental wall. As shownin Fig. 1, the difference between the present repulsive wall
/H20849solid line /H20850and the experimental repulsive wall /H20849dashed line /H20850
is barely noticeable. For example, at R=3.463 Å the experi-
mental hybrid potential is equal to 1032.16 cm
−1,
Vexpt/H208493.463 Å /H20850=1032.16 cm−1. The present potential for
nmax=5 is equal to the same value at R=3.434 Å, that is,
VTT/H208493.434 Å /H20850=1032.16 cm−1. This means that the present
potential is shifted to the left by 0.029 Å, which is less than
1% of the Rvalue.
Should a more accurate repulsive wall is required, addi-
tional parameters can be introduced in the short range poten-tial for improvement in such a way that the long range struc-ture of the potential is maintained.
46,47The excellent agreement of the predicted potential curve
is especially gratifying in view of the possibility that thecalcium dimer bond is not purely dispersive because of thenear s-pdegeneracy of the valence shell. Like the rare gases,
the alkaline earth atoms have a closed-shell electron configu-ration. But unlike rare-gas atoms, the alkaline earths arechemically active due to low ionization potentials which al-low them to bind ionically with other atoms. This effect isreflected clearly in the well depths D
eof the interatomic
potentials. The well depth of Ca 2/H208491102 cm−1/H20850is one order
of magnitude larger than that of the nearby rare-gas dimer
Ar2/H2084999.6 cm−1, Ref. 19/H20850. Thus one might question whether
the TT model, which was previously used mostly for therare-gas atoms, will still work in such a case. Our presentstudy shows that with the correct dispersion coefficients themodel works for calcium just as well as it does for the rare-gas dimers.
19Presumably the ionic effect is included through
the use of the experimental well parameters /H20849De,Re/H20850.
Recently it has been shown that the TT model can also
accurately describe the dimer potential of mercury,42which
has a closed outer electronic 6 s2shell. However, the mercury
dimer bond is also not purely dispersive but is strengthenedby a covalent exchange contribution.
48The potential well of
the ground state Hg 2is about twice as deep as the nearby
rare-gas dimers.
These excellent results for the entire potential energy
curves provide additional evidence for the uncanny ability ofthe TT model to mimic the universal behavior of van derWaals potentials.
C. Reduced potentials of Ca2,A r2, and Hg2
It is interesting to compare the shapes of these three
types of van der Waals potentials, since they can all be de-scribed by the same TT model. In Fig. 2, the reduced poten-
tials U/H20849x/H20850/H20851=V/H20849xR
e/H20850/De/H20852of Ca 2,A r 2, and Hg 2are plotted
against x/H20851=R/Re/H20852. It is seen that the reduced potential of Ca 2
has a wider potential bowl than that of Ar 2. What is surpris-
ing is that the reduced potentials of Ca 2and Hg 2are almost
identical. Apparently this is due to the fact that both have aclosed outer electronic s
2shell, even though the well depth
of Ca 2/H208491102 cm−1/H20850is almost three times as large as that of
Hg2/H20849380 cm−1/H20850. It is well known that the reduced potentials
of rare-gas dimers are conformal /H20849that is, the reduced poten-
tials of all rare-gas dimers coincide with each other /H20850. There
are evidence that the reduced potentials of group IIB dimers/H20849Zn
2,Cd 2,Hg 2/H20850and of group IIA alkaline earth dimers, with
the exception of Be 2, are also respectively conformal.49This
leads us to conclude that the ground state van der Waals
dimer potentials of all group IIA, except Be, and group IIBelements have the same shape, and this shape is differentfrom that of the rare-gas dimers.
III. THE GROUND STATE POTENTIALS OF CALCIUM
RARE-GAS COMPLEXES
A. Combining rules for calcium rare-gas complexes
The fact that the interatomic potentials of both calcium
and rare-gas dimers are expressible in terms of the TT po-tential model makes it likely that the interaction potential
FIG. 1. The ground state potential of calcium dimer. The solid line is gen-
erated by the TT potential model with its parameters specified by the threedispersion coefficients, well depth, and equilibrium distance. The dashedline is the experimental hybrid multiparameter potential of Allard et al.
/H20849Refs. 10and11/H20850. The black dots are the numerical values given in these
references.154301-3 Calcium dimer and calcium rare-gas potential J. Chem. Phys. 131 , 154301 /H208492009 /H20850
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89.133.231.66 On: Fri, 11 Apr 2014 18:24:08between a calcium atom and a rare-gas atom can also be
described by the TT model. For Ca-RG dimers, the disper-sion coefficients C
6,C8, and C10were recently calculated by
Mitroy and Zhang.35We have also calculated these coeffi -
cients with well established combining rules for dispersioncoefficients
45,50and found a close agreement. For simplicity,
we list the coefficients from Ref. 35in Table IIand use them
in the TT potential model for the calcium and rare-gas inter-actions.
In the TT model, the short range repulsive potential is
expressed in terms of the Born–Mayer form Aexp /H20849−bR/H20850.
There are several combining rules for Aand b.
51–54
Böhm–Ahlrichs54tested four different sets of combining
rules for Aandb/H20851Eqs. /H2084930/H20850–/H2084933/H20850of Ref. 54/H20852. They found thatall of them are useful but the results are not identical. Since
these rules are based on heuristic arguments, their validitycan only be judged by the results they predict.
To choose the appropriate short range combining rules
for calcium rare-gas interactions, we use the experimentallydetermined ground state energy of CaAr as the criterion,since it is the only available experimental result. The poten-tial energies of CaAr calculated with all four sets of combin-ing rules are compared with the experimental values, the onewith the best agreement is used for the predictions of allmembers of the calcium rare-gas systems. By this process,the following set of combining rules /H20851Eq. /H2084931/H20850of Ref. 54/H20852is
selected:TABLE I. Comparison of the calcium dimer potential between the experiment of Allard et al. and the present
TT model.
R/H20849Å/H20850ExperimentaPresent /H20849nmax=5 /H20850 Present /H20849nmax=8 /H20850
Vexpt/H20849R/H20850
/H20849cm−1/H20850VTT/H20849R/H20850
/H20849cm−1/H20850/H20849 VTT−Vexpt/H20850/VexptVTT/H20849R/H20850
/H20849cm−1/H20850/H20849 VTT−Vexpt/H20850/Vexpt
3.463 960 1032.1575 795.3 /H110020.2294 687.2 /H110020.3342
3.555 705 373.1825 248.3 /H110020.3346 174.8 /H110020.5313
3.647 450 /H1100297.557 /H11002169.6 0.7385 /H11002217.9 1.233
3.739 195 /H11002440.6477 /H11002499.1 0.1324 /H11002528.7 0.1997
3.830 940 /H11002691.4483 /H11002724.7 0.0482 /H11002742.4 0.0737
3.922 685 /H11002868.0599 /H11002884.6 0.0190 /H11002894.2 0.0302
4.106 174 /H110021057.5163 /H110021061. 0.0035 /H110021063.0 0.0051
4.197 920 /H110021093.3715 /H110021094. 0.0004 /H110021094.0 0.0007
4.289 664 /H110021101.884 /H110021102. 0.0000 /H110021102.0 0.0000
4.381 409 /H110021090.1029 /H110021090. 0.0001 /H110021091.0 0.0005
4.500 000 /H110021053.4653 /H110021056. 0.0021 /H110021057.0 0.0035
4.761 905 /H11002926.3289 /H11002930.4 0.0043 /H11002934.8 0.0092
5.023 859 /H11002779.6277 /H11002785.3 0.0073 /H11002792.1 0.0160
5.285 714 /H11002640.6405 /H11002642.5 0.0030 /H11002650.6 0.0156
5.547 619 /H11002519.073 /H11002519.0 /H110020.0001 /H11002527.3 0.0159
5.809 524 /H11002417.1011 /H11002413.0 /H110020.0097 /H11002420.9 0.0091
6.071 429 /H11002333.4624 /H11002328.3 /H110020.0155 /H11002335.3 0.0057
6.333 333 /H11002265.8181 /H11002259.0 /H110020.0257 /H11002265.0 /H110020.0030
6.595 238 /H11002211.5934 /H11002205.2 /H110020.0303 /H11002210.3 /H110020.0063
6.726 191 /H11002188.7677 /H11002182.3 /H110020.0343 /H11002186.9 /H110020.0100
6.988 095 /H11002150.2882 /H11002144.1 /H110020.0412 /H11002147.8 /H110020.0165
7.250 000 /H11002119.8441 /H11002114.8 /H110020.0420 /H11002117.8 /H110020.0173
7.500 000 /H1100296.8103 /H1100292.67 /H110020.0428 /H1100295.03 /H110020.0184
8.000 000 /H1100263.7338 /H1100261.15 /H110020.0405 /H1100262.61 /H110020.0176
8.717 949 /H1100236.0021 /H1100234.68 /H110020.0368 /H1100235.38 /H110020.0173
9.435 897 /H1100221.1638 /H1100220.50 /H110020.0312 /H1100220.84 /H110020.0156
10.303 42 /H1100211.7610 /H1100211.47 /H110020.0246 /H1100211.61 /H110020.0132
11.611 11 /H110025.373 00 /H110025.263 /H110020.0205 /H110025.300 /H110020.0139
aReferences 10and11.
TABLE II. Potential parameters for the calcium rare-gas systems, all in atomic unit.
SystemAb
C6 C8 C10 nmax=5 nmax=8 nmax=5 nmax=8
Ca–He 34.42 30.96 1.4310 1.3956 36.59 2139.0 131 900.0
Ca–Ne 75.05 67.52 1.4203 1.3855 71.98 4368.0 279 800.0Ca–Ar 145.3 130.8 1.3390 1.3080 274.0 18 150.0 1 238 000.0Ca–Kr 153.3 137.9 1.3008 1.2715 404.2 27 910.0 1 981 000.0Ca–Xe 163.9 147.5 1.2530 1.2258 629.6 46 750.0 3 559 000.0154301-4 Yang, Li, and T ang J. Chem. Phys. 131 , 154301 /H208492009 /H20850
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89.133.231.66 On: Fri, 11 Apr 2014 18:24:08Aij=/H20851AiAj/H208521/2,bij=2bibj
bi+bj, /H208496/H20850
where a single index indicates the potential parameter for the
like system. Böhm and Ahlrichs found these rules useful bytesting them against the repulsive interactions obtained fromthe self-consistent field /H20849SCF /H20850calculation, which is without
correlation. In the present application, these rules are usedfor the entire repulsive potential. This may be justified by theobservation that the small additional repulsion due to corre-lation is more or less proportional to the SCF repulsion.
36
The parameters Aand bfor the short range repulsive
potentials of Ca-RG systems are calculated from Eq. /H208496/H20850with
the Born–Mayer parameters of the rare-gas dimers given inRef. 19, and those of the calcium dimer given in either Eq.
/H208494/H20850/H20851nmax=5 /H20852or in Eq. /H208495/H20850/H20851nmax=8 /H20852. Both sets of results
are listed in Table II.
B. Results and comparison to previous determinations
The van der Waals potentials of calcium rare-gas sys-
tems are calculated from the TT model of Eq. /H208492/H20850with the
parameters listed in Table II. The full potential energy curves
obtained by summing the dispersion series up to nmax=5
and using the corresponding Aandbfor all these five sys-
tems are shown in Fig. 3. In this figure, the scales and units
are chosen to be the same as in Fig. 1 of Ref. 25, where the
corresponding potentials obtained from CCSD /H20849T/H20850calcula-
tions are shown. Although there will be a small error in theCCSD /H20849T/H20850results because Ref. 25used a large but not a very
large basis set, these two sets of curves are very similar,except for CaHe. They follow the same trend in that thedepth of the potential well D
efor the Ca-RG molecules in-
creases regularly with the size of the rare-gas atoms goingfrom He to Xe. The corresponding equilibrium position R
e
changes very little from one system to another, except that
theReof CaHe in the CCSD /H20849T/H20850calculation is greater than
that of all other Ca-RG systems, whereas in the present com-bining rule results, the R
eof CaHe is slightly smaller than all
other systems. The corresponding curves calculated withnmax=8 are very similar with only a slightly deeper poten-
tial well /H20849see Table III/H20850.
Since the TT potential of Eq. /H208492/H20850is an analytic function,
the derivatives can be easily calculated. The derivatives at R
e
are related to the spectroscopic constants. Explicit formulas
for these relations are given in Ref. 42. The predicted well
depth De, the equilibrium distance Re, the vibrational fre-
quency /H9275e, and the anharmonicity /H9273e/H9275eof Ca-RG dimers
with nmax=5 are listed in Table III. The numbers in the
parentheses are obtained by summing the dispersion serieswith nmax=8 and using the corresponding values of Aand
bin Table II. The differences between these two sets of num-
bers provide an indication to the extent of the uncertainty ofthe present approach.
Among all previous determinations, only one set of data
is from experiment. The spectroscopic constants of CaArwere determined by Kowalski et al.
24with excitation spectra
in a supersonic jet. The well depth De=62/H1100610 cm−1in
Table IIIis quoted directly from Ref. 24, which also gave a
dissociation energy D0of 53/H1100610 cm−1. This experimentwas reinterpreted by Spiegelman et al.59According to them,
the dissociation energy D0should be 72.5 cm−1with an in-
creased uncertainty. With this D0, the well depth can be es-
timated to be De=D0+1
2/H9275e−1
4/H9273e/H9275e=81 cm−1, which is listed
in the parenthesis following 62 /H1100610 in Table III.I fw ep u t
the error also at /H1100610 cm−1, then the present De
/H2084970.74 cm−1/H20850falls in place where these two estimates
overlap.
Theoretically, the ground state potential of CaAr is also
the most extensively investigated among all calcium rare-gassystems. The well depth /H20849D
e=72.6 cm−1/H20850calculated by Zhu
et al.60using standard valence pseudopotential and the con -
figuration interaction method with multi-configuration self-consistent field nature orbitals /H20849MCSCF-CI /H20850is in excellent
agreement with the present result, although the vibrationalconstant /H20849
/H9275e=12.7 cm−1/H20850obtained in this calculation seems
FIG. 2. Comparison of the reduced potentials of Ca2,A r2, and Hg2. Note
that the reduced potentials of Ca2and Hg2are almost identical, while Ar2
has a different shape.
FIG. 3. The ground state van der Waals potentials of CaHe, CaNe, CaAr,CaKr, and CaXe complexes.154301-5 Calcium dimer and calcium rare-gas potential J. Chem. Phys. 131 , 154301 /H208492009 /H20850
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89.133.231.66 On: Fri, 11 Apr 2014 18:24:08too small. On the other hand, the /H9275eobtained by Spiegelman
et al.59in a two electron pseudopotential /H20849PP2 /H20850calculation
are in very good agreement with the present and experimen-
tal results, but the well depth /H20849De=96 cm−1/H20850seems too large.
The interatomic potentials of CaAr and CaNe were com-
puted by Kirschner58at four theoretical levels. The most rig -
orous method employed was Møller–Plesset second orderperturbation /H20849MP2 /H20850with valence plus core electron correla-
tion using large basis functions. The results of this computa-tion are listed in Table IIIunder MP2; the basis set superpo-
sition error corrected D
eare listed in parentheses. This
correction amounts to 25% in the case of CaAr and increasesto 45% for CaNe.
As mentioned earlier, Czuchaj et al.
25used a single ref -
erence coupled cluster approach /H20851CCSD /H20849T/H20850/H20852to obtain ground
state potentials for all calcium rare-gas complexes. For CaAr,they reported D
efrom two sets of calculations: one from two
valence electron calculation /H20849De=61.9 cm−1/H20850and the other
from ten valence electron calculation /H20849De=79 cm−1/H20850. Nor-
mally the small core pseudopotential method /H20849ten valence
electron /H20850is thought to be more accurate, but the two valence
electron result /H2084961.9 cm−1/H20850agrees with experiment. The au-
thors attributed an incomplete treatment of core-valence cor-
relation with the basis set as the reason for the inaccuracy ofthe ten valence electron result. Therefore for all other cal-cium rare-gas systems, only two valence electron resultswere reported, which are listed in Table IIIunder CCSD /H20849T/H20850.
In view of the reinterpretation of the experimental result,
59
these numbers may be too small.
For CaKr and CaXe, the CCSD /H20849T/H20850results are the only
previous determinations. The well depths obtained fromthese calculations are only smaller than the present predic-
tions by 10%–15%. For CaNe, the present predictions arealso in general agreement with previous calculations with thepresent D
esomewhat larger.
For CaHe, there are large differences between various
determinations. The three CCSD /H20849T/H20850calculations, one by
Czuchaj et al. ,25one by Partridge et al. ,41and the other by
Lovallo and Klobukowski,57yielded much smaller well
depths than the present prediction. On the other hand, the
present results are in very good agreement with the multi-
reference configuration interaction /H20849MRCI /H20850results of Stien-
kemeier et al.56and with the surface integral results of
Kleinekathöfer.55
Although no unequivocal conclusion can be drawn for
Ca-RG potentials because of the differences in the results ofvarious calculations, as seen in Table III, the present results
are within the range of other theoretical determinations. Con-sidering the simplicity of the TT potential, this is a remark-ably good performance. The present method should be use-ful, at least, as a starting point for the interplay betweentheory and experiment.
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SystemDe
/H20849cm−1/H20850Re
/H20849Å/H20850/H9275e
/H20849cm−1/H20850/H9282e/H9275e
/H20849cm−1/H20850
Ca–He Present 11.89 /H2084912.46 /H20850 4.92 /H208494.93 /H20850
Surface integrala10.33 5.10
MRCIb12.0 5.4
CCSD /H20849T/H20850c3.33 5.95
CCSD /H20849T/H20850d3.32 6.02
CCSD /H20849T/H20850e4.2 5.85
Ca–Ne Present 19.78 /H2084920.76 /H20850 5.07 /H208495.08 /H20850 12.57 /H2084912.88 /H20850 2.25 /H208492.24 /H20850
MP2f20.98 /H2084911.54 /H20850 5.21 10.6
CCSD /H20849T/H20850e14.5 5.25 6.15
Ca–Ar Present 70.74 /H2084975.35 /H20850 5.10 /H208495.10 /H20850 18.87 /H2084919.33 /H20850 1.36 /H208491.38 /H20850
Expt.g62/H1100610 /H2084981/H20850 18.0 1.9
PP2h96 5.05 18.3 1.77
MCSCF-CIi72.6 5.9 12.7
MP2f107.7 /H2084980.09 /H20850 4.87 18.9
CCSD /H20849T/H20850e61.9 /H2084979/H20850 5.08 13.35
Ca–Kr Present 102.7 /H20849111.9 /H20850 5.11 /H208495.08 /H20850 18.92 /H2084920.10 /H20850 0.98 /H208490.96 /H20850
CCSD /H20849T/H20850e97.5 5.05 14.79
Ca–Xe Present 152.2 /H20849172.5 /H20850 5.15 /H208495.09 /H20850 21.00 /H2084922.85 /H20850 0.82 /H208490.81 /H20850
CCSD /H20849T/H20850e131.4 5.17 15.96
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1.3611446.pdf | Low current density spin-transfer torque effect assisted by in-plane microwave field
Jianbo Wang, Congpu Mu, Weiwei Wang, Bin Zhang, Haiyan Xia, Qingfang Liu, and Desheng Xue
Citation: Applied Physics Letters 99, 032502 (2011); doi: 10.1063/1.3611446
View online: http://dx.doi.org/10.1063/1.3611446
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/99/3?ver=pdfcov
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129.120.242.61 On: Wed, 26 Nov 2014 00:16:23Low current density spin-transfer torque effect assisted by in-plane
microwave field
Jianbo Wang,a)Congpu Mu, Weiwei Wang, Bin Zhang, Haiyan Xia, Qingfang Liu,
and Desheng Xue
Institute of Applied Magnetics, Key Laboratory for Magnetism and Magnetic Materials of MOE,
Lanzhou University, Lanzhou 730000, People’s Republic of China
(Received 29 April 2011; accepted 25 June 2011; published online 19 July 2011)
A strategy is presented to greatly reduce both the critical spin polarized current density and the
magnetization switching time in elliptical magnetic spin valve. This method is a combination of
microwave field and spin polarized current. The numerical simulation at zero temperature showsthat the critical current density and the magnetization switching time are sensitive to the frequency
and the amplitude of microwave magnetic field. A 75% reduction in critical current density is
observed when the frequency of microwave coincides with the natural ferromagnetic resonancefrequency of free layer.
VC2011 American Institute of Physics . [doi: 10.1063/1.3611446 ]
It is well known that spin polarized current can induce
spin transfer torque (STT)1,2on nanomagnet, which can be
strong enough to switch the magnetization3,4and manipulate
domain wall motion5,6without applied field. Also, it pro-
vides a promising mechanism for writing information in
magnetic random access memory (MRAM) and race trackmemory
7for future high density and low power magnetic
date storage devices.8However, high value of critical current
density is the major difficulty in applications,9and less
switching time of magnetization reversal is also an utmost
significance in high density magnetic recording and informa-
tion processing technology.10They are also intriguing ques-
tions to the modern magnetism11–14and studied by many
groups. The thermal fluctuations of magnetization can reduce
the critical current density, which can be proved by the sig-nature of a thermally assisted switching in the spin valve.
9
But the cooling rate and the optical system are the substantialchallenges in the thermally assisted switching.
5In compari-
son with the conventional design with a single free layer
(FL), a composite FL will reduce critical current density, as
shown both in experiments15and micromagnetic simula-
tions.3The intrinsic critical current density15decreased from
2.4/C2107A/cm2to 8.5/C2106A/cm2. To make STT MRAM
compatible with modern metal-oxide semiconductor technol-ogy, the intrinsic critical current density needs to be
decreased to below 2 /C210
7A/cm2,16while keeping data
retention times above the value of 10 yr. On the other hand,fast magnetization reversal has been actively studied by
many methods, such as the laser-induced spin dynamics
12,17
based on the inverse Faraday effect,18the reversal triggered
by external static or alternating magnetic fields,19or by a
spin transfer torque acting on magnetization.10,20The effect
of a weak microwave field in fast magnetization reversal ofperpendicular spin valve has been studied.
21In one word, it
is a pivotal problem to reduce both the critical current den-
sity and the switching time of magnetization in technologyapplications of STT-MRAM.In this letter, a strategy is presented to overcome the cur-
rent limitation. We make use of microwave field to reduce
both critical spin polarized current density and magnetizationswitching time in elliptical magnetic spin valve. The micro-
magnetic results show that frequency and amplitude of
microwave field can strongly affect both critical current den-sity and magnetization switching time. Meanwhile, the mag-
netization reversal mode is also changed during
magnetization reversal process, due to the presence of micro-wave field.
Figure 1(a)shows the schematic diagram of spin valve:
a free magnetic layer, a nonmagnetic spacer, and a polarizerlayer. The polarizer layer is assumed to be fully fixed along
x-axis direction. The elliptical FL with a cross section of
60/C240 nm
2and a thickness of 4 nm is chosen; the ground
state of FL is a single domain. The thermal stability of the
FL, defined as the ratio of the switching energy barrier to the
thermal energy kBTat room temperature, is estimated from
the shape anisotropy to be 39.5. The xandydirections are
aligned with the long and short axis of the ellipse, respec-
tively. The magnetic parameters used in the simulation aretypical for permalloy (Py): saturation magnetization
M
s¼8.6/C2105A/m and exchange stiffness constant
A¼13/C210/C012J/m. The spin polarization rate is 0.3 and the
Gilbert damping ais 0.015. The mesh size is 1 /C21/C21n m3.
The Oersted field induced by current is taken into account in
FIG. 1. (Color online) (a) Schematic diagram of the magnetic nanopillar.
(b) Frequency dependence of critical current density for the amplitude of 0.5
mT. The red dash line is the critical current density without microwave as-
sistance. The inset is the imaginary part of susceptibility spectrum calculated
for the element.a)Electronic mail: wangjb@lzu.edu.cn.
0003-6951/2011/99(3)/032502/3/$30.00 VC2011 American Institute of Physics 99, 032502-1APPLIED PHYSICS LETTERS 99, 032502 (2011)
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129.120.242.61 On: Wed, 26 Nov 2014 00:16:23simulation. The object oriented micromagnetic framework
(OOMMF)22is used to numerically calculate the dynamics
of FL at zero temperature by solving Laudau-Lifshitz-Gilbert
(LLG) equation with a spin-transfer torque term.2,3The elec-
tric current flows perpendicularly to the FL and has a uni-
form distribution, and the linearly polarized in-plane
microwave field assuming the form of H(t)¼H0sin(2 pft)i s
applied perpendicularly to the long axis of ellipse. A natural
ferromagnetic resonance (NFMR) frequency of FL is shown
in the inset of Fig. 1(b) and is 5.5 GHz. The microwave fre-
quency is tuned to NFMR frequency of FL, where a switch-
ing field of FL is smaller than that in the absence ofmicrowave.
23It is difficult to obtain a large amplitude of
high frequency microwave in experiment. Thus, the maxi-
mum value of microwave is 2.5 mT in this letter.
A red horizontal dashed line in Fig. 1(b) indicates criti-
cal current density (2 /C2107A/cm2) in the absence of micro-
wave field. The critical current density as a function ofmicrowave frequency is shown in Fig. 1(b) for a certain
microwave amplitude level of 0.5 mT. The dependence of
critical current density on the microwave frequency is sym-metric with respect to NFMR frequency of FL (5.5 GHz).
The critical current density is reduced from 2.0 /C210
7A/cm2
to 0.51 /C2107A/cm2, and the minimum is obtained at the
microwave frequency of 5.5 GHz. The results in Fig. 1(b)
demonstrate that there exists an optimal frequency foptimal at
which the critical current density is the lowest.
For application in practical devices, the reduction of
critical current density is significant in establishing the reli-
ability of microwave assisted magnetization switching.Fig.2shows the calculated map of the critical current den-
sity map of microwave with the frequency changing from
2.0 to 9.0 GHz and the amplitude varying from 0.5 to 2.5mT. Critical current density is expressed in Fig. 2, and the
horizontal axis and the vertical axis represent the frequency
and the amplitude of microwave, respectively. For each am-plitude, critical current density decreases first and increases
afterward with the increase of microwave frequency, which
is similar to Fig. 1(b). The critical current density is strongly
reduced for microwave frequency around NFMR of FL. It
also decreases with increasing microwave amplitude at a cer-
tain frequency of microwave. With the adjustment of micro-wave frequency and amplitude, critical current density is
reduced from 2.0 /C210
7to 4.5/C2106A/cm2. The minimum is
obtained at the amplitude of 2.5 mT and frequency of 5 (not
5.5) GHz, which is due to the red-shift24,25of the NFMR
peak as the increase of the amplitude of the microwave.
For ferromagnetic single domain film at zero field, there
is an energy barrier between two states of opposite magnet-ization of FL. Magnetization must overcome the energy bar-
rier for reversal by Zeeman field or STT. The intrinsical
mechanism of microwave field assisted is that the large-
angle magnetization precession is excited and lowers the
energy barrier for magnetization reversal in single domainelements. Thus, when microwave frequency varies in the
range of 2.0–9.0 GHz, critical current density can be obvi-
ously decreased with microwave field assisted. The processof microwave field assisted involves the microwave pump-
ing. A energy pumping rate from the microwave field can be
described by P
pumping¼CPmw, where Ppumping is the rate for
energy pumping, Cis the coupling efficiency, and Pmwis the
microwave power,26which is proportional to the microwave
amplitude. So the energy pumping rate increases with theincrease of the amplitude of microwave. And critical current
density also decreases with the increase of microwave
amplitude.
When microwave frequency is close to f
optimal , the cou-
pling efficiency Cis high.24Therefore, the energy pumping
rate of magnetization is also large. Thus energy barrierbetween the two steady states of FL is lower and the critical
current density is decreased to 0.45 /C210
7A/cm2with a
microwave frequency of 5.0 GHz ( foptimal ) at amplitude of
2.5 mT. The optimal frequency foptimal decreases as the
increase of the amplitude of microwave due to the red-
shift24,25of the NFMR peak. So Cis relational with both
microwave frequency and amplitude.
The magnetization switching mechanism is changed
from nucleation reversal to homogeneous reversal due to theintroduction of microwave field. As shown in Fig. 3(a), the
reverse process was quite complex under the action of spin
FIG. 2. (Color online) Critical current density J cas a function of microwave
amplitude H 0and frequency. J cis indicated by a single bar on the right and
ranges between 0.45 /C2107and 0.68 /C2107A/cm2.
FIG. 3. (Color online) Snapshots of some significant magnetization configu-
rations during the magnetization switching (a) without microwave and (b)
with microwave. The arrows indicate the magnetization direction.032502-2 Wang et al. Appl. Phys. Lett. 99, 032502 (2011)
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129.120.242.61 On: Wed, 26 Nov 2014 00:16:23polarized current.27,28A C-like state prior to vortex nuclea-
tion is formed in the element, then nucleation of two vortices
appear at both the left and the right sides of the ellipsoid film
with their cores gyrating in opposite directions. At last, themagnetization of ellipsoid film is completely reversed as two
vortices annihilating. Fig. 3(b)shows the magnetization con-
figuration during microwave assisted switching process, andthe switching dynamics are generally more spatially uniform.
To obtain ultrafast magnetization reversal, the ideal switch-
ing mechanism is a complete coherent rotation of the mag-netization without forming any domain wall during the
process of magnetization reversal. Since domain wall motion
needs more time to achieve magnetization reversal than thatof coherent rotation.
11
The switching mechanism of magnetization is homoge-
neous reversal as a result of the introduction of microwavefield, so the magnetization switching time can be reduced. A
calculated map of the switching time as a function of the
microwave frequency and the microwave amplitude for thecurrent density of 0.8 /C210
7A/cm2is shown in Fig. 4. How-
ever, the magnetization switching cannot be achieved under
the action of current density of 0.8 /C2107A/cm2. It is shown
that the switching time is strongly affected by the variation
of frequency and amplitude of the microwave. There also
exists an optimal frequency foptimal at which the microwave
pumping is most efficient and the magnetization switching
time is the lowest. The magnetization switching time can
achieve 1.28 ns, when microwave frequency and amplitudeare 5.0 GHz and 2.5 mT, respectively.
In conclusion, micromagnetic simulations indicate that
critical current density and magnetization switching time canbe obviously reduced due to the introduction of microwave
field. Minimums of both critical current density and magnet-ization switching time can be obtained, when microwave fre-
quency comes up to the optimal frequency which is the
NFMR frequency of FL. At one time, the mechanism of
magnetization reversal is changed from nucleation reversalto coherent rotation.
This work is supported by National Science Fund for
Distinguished Young Scholars (50925103), NSFC(11074101), and the Fundamental Research Funds for the
Central Universities (lzujbky-2011-54).
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FIG. 4. (Color online) Magnetization switching time as a function of micro-
wave amplitude H 0and frequency for the current density of 0.8 /C2107
A/cm2. The switching time is expressed by a single bar on the right and
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1.4862215.pdf | The spin Hall angle and spin diffusion length of Pd measured by spin pumping and
microwave photoresistance
X. D. Tao, Z. Feng, B. F. Miao, L. Sun, B. You, D. Wu, J. Du, W. Zhang, and H. F. Ding
Citation: Journal of Applied Physics 115, 17C504 (2014); doi: 10.1063/1.4862215
View online: http://dx.doi.org/10.1063/1.4862215
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/17?ver=pdfcov
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86.30.54.96 On: Thu, 15 May 2014 21:05:54The spin Hall angle and spin diffusion length of Pd measured
by spin pumping and microwave photoresistance
X. D. Tao, Z. Feng, B. F . Miao, L. Sun, B. Y ou, D. Wu, J. Du, W. Zhang, and H. F . Dinga)
Department of Physics, National Laboratory of Solid State Microstructures, Nanjing University,
22 Hankou Road, Nanjing 210093, People’s Republic of China
(Presented 5 November 2013; received 22 September 2013; accepted 17 October 2013; published
online 16 January 2014)
We present the experimental study of the spin Hall angle (SHA) and spin diffusion length of Pd
with the spin pumping and microwave photoresistance effects. The Py/Pd bilayer stripes are
excited with an out-of-plane microwave magnetic field. The pure spin current is thus pumped and
transforms into charge current via the inverse spin Hall effect (ISHE) in Pd layer, yielding an ISHEvoltage. The ISHE voltage can be distinguished from the unwanted signal caused by the
anisotropic magnetoresistance according to their different symmetries. Together with Pd thickness
dependent measurements of in and out-of-plane precessing angles and effective spin mixingconductance, the SHA and spin-diffusion length of Pd are quantified as 0.0056 60:0007 and
7.360:7 nm, respectively.
VC2014 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4862215 ]
Spin and charge conversion is one of the essential ingre-
dients of spintronics.1Recently, the spin Hall effect (SHE)
and its inverse effect, ISHE, have drawn increasing attention
as they can achieve the spin and charge conversion in the ab-
sence of magnetic material and magnetic field.2The SHE
refers to the generation of a spin current transverse to the
charge current in a paramagnetic metal or a dopedsemiconductor.
3–5Vice versa, a spin current can give rise to
a transverse charge current, i.e., the ISHE. ISHE has also
been reported in magnetic material very recently, such asPy.
6The efficiency of the spin-charge conversion can be
quantified by a single material-specific parameter, i.e., the
spin Hall angle (SHA), hSH. It is defined as the ratio of the
spin Hall and charge conductivities.7The SHA can be meas-
ured through the nonlocal magneto-transport measure-
ments8,9or the method based on spin pumping due to
ferromagnetic resonance (FMR).10–12Because of the com-
plexity of the interface effect, it is typically difficult to esti-
mate the exact amplitude of the injected pure spin currentwith the first method. While the second method is of more
advantage as the above difficulty can be overcome with addi-
tional FMR measurements. In real experiments, however, theISHE signal generated from spin pumping is typically mixed
with the unwanted effect related to the anisotropic magneto-
resistance (AMR).
2,10–12Therefore, the separation of the
ISHE signal from the other effect is crucial for the SHA
quantification based on spin pumping. In addition, the meas-
ured ISHE voltage depends on the SHA, the amplitude of theinjected pure spin current as well as the spin diffusion length
k
sd. Thus, the correct measurements of the amplitude of the
injected pure spin current and the spin diffusion length arealso very essential.
In our previous paper, we have developed a method to
quantify the spin Hall angle of Pt from spin pumping andmicrowave photoresistance measurements.
12In this method,
the AMR related effect can be excluded under a designed ge-ometry due to its different symmetries with the ISHE. The
effective spin mixing conductance and precessing angles can
be further determined from enhanced Gilbert damping and
microwave photoresistance measurements, respectively. Up
to now, the SHE and ISHE have been mainly discussed in 5dmetals, such as Pt,
10–13Au,2,14and Ta.15Pd, a 4d transition
metal, which also has strong spin-orbit coupling and large
spin Hall conductivity,16however, is less addressed.
Therefore, it is important to quantify the spin Hall angle and
the spin diffusion length of Pd.
Py/Pd bilayers are deposited on GaAs substrate by dc
magnetron sputtering at room temperature, and patterned
into stripes with lateral dimension of 2.5 mm /C220lm.
The samples are placed in the slots between the signal andground line of a coplanar waveguide (CPW) [Fig. 1(a)]. In
this configuration, the rfmagnetic field is perpendicular to
the sample plane. We fix the thickness of Py at 16 nm, andstudy the ISHE of Pd by varying Pd thickness from 3 to
40 nm.
According to the basic theory of spin pumping,
11,17the
precessing magnetization inside the ferromagnet (Py) pumps
a net dcpure spin current into its adjacent nonmagnetic (Pd)
layer. The magnetic field ( H) dependence of the injected
spin current at the interface can be written as
j0
sðHÞ¼/C22h
2g"#
ef ffa1b1DH2=ððH/C0H0Þ2þDH2Þ, where H0is
the resonance magnetic field, DHis the half-width of the
FMR linewidth, and a1andb1are the maximum amplitudes
of the in- and out-of-plane precessing angles of the magnet-
ization at resonance. g"#
ef fis the effective spin-mixing con-
ductance and it can be determined experimentally. After
being injected into Pd layer, the spin current gives rise to a
transverse charge current flowing within the NM layer (withlength L, width w, and resistance R
N). Thus, a dcvoltage
VSP
ISHE can be measured along the x-direction [Fig. 1(b)].a)Author to whom correspondence should be addressed. Electronic mail:
hfding@nju.edu.cn.
0021-8979/2014/115(17)/17C504/3/$30.00 VC2014 AIP Publishing LLC 115, 17C504-1JOURNAL OF APPLIED PHYSICS 115, 17C504 (2014)
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86.30.54.96 On: Thu, 15 May 2014 21:05:54Assuming, the SHA hSHand spin diffusion length ksdare
constants, the integrated ISHE voltage can be calculated as
VSP
ISHE¼hSHksdtanhtN
2ksd/C18/C19
g"#
ef fRNfewa1b1
/C2sina0DH2
H/C0H0 ðÞ2þDH2; (1)
where a0is the angle between Hand the x-axis, as shown in
Fig.1(a). Meanwhile, the microwave also generates an induc-
tion current I1cosðxtÞand the precessing magnetization results
an oscillating resistance RðtÞ¼R0/C0RAsin2½a0þa1ðxtÞ/C138due
to the AMR effect. The combination of both gives rise to an
additional voltage in the FM stripe, VAMRand it is proportional
to sin2 a0.18,19As discussed above, VSP
ISHEandVAMRhave dif-
ferent angular dependences with repect to a0. This provides an
unique opportunity to disentangle VSP
ISHEfrom the mixed signal.
In our measuements, we choose two specific geometries, i.e.,
a0¼90/C14and 270/C14,w h e r e VAMRvanishes but the pure VSP
ISHE
has its maximum amplitude. Fig. 2(a)shows the typical results
of the measured dcvoltage as a function of Hfora0¼90/C14
(black squre) and a0¼270/C14(red circle) with the same micro-
wave power input. Both curves exhibit almost a perfectLorentzian shape, and have opposite sign with respect to each
other. They, however, have slightly different amplitudes. This
seeming contradiction with the symmetry analysis is due todifferent spin current injection efficiencies for these two con-
figurations even under the same microwave input power.
12As
shown in Eq. (1), the injected pure spin current is proportional
to the product of in- and out-of-plane precessing angles, i.e.,
a1b1. These two precessing angles can be determined through
the microwave photoresistance measurements.18Our meas-
ured results show that the precessing angles are a1¼1:41/C14,
b1¼0:41/C14fora0¼90/C14anda1¼1:32/C14,b1¼0:38/C14fora0¼
270/C14for the sample Py(16 nm)/Pd(15 nm) in 8 GHz. They are
indeed different for these two configurations. Remarkably but
not suprisingly, the normalized voltage VSP
ISHE =a1b1fora0¼
90/C14and 270/C14falls into an identical curve, evidencing its pure
spin pumping origin [Fig. 2(b)]. The inset of Fig. 2(b) shows
the Pd thickness dependence of the precessing angles a1and
b1under a0¼90/C14and 270/C14, respectively. In most cases, the
precessing angles for a0¼90/C14is slightly larger than those of
270/C14. We note that, in our measurements, all the data were
obtained with the same input microwave power. But we canfind that the precessing angles change as function of the Pd
thickness. Generally, they become smaller when the Pd thick-
ness increases. This is expected as the screening effect
increases with the Pd thickness. The variation can be as largeas/C2450% from 3 to 30 nm. Thus, we argue that it is very impor-
tant to measure the precessing angles for each individual sam-
ple since the pumped spin current is proportional to theproduct of in and out-of-plan precessing angles. As we
discussed above, V
AMR signal has the symmetry of
VAMRða0Þ¼VAMRða0þ180/C14Þ, we therefore further redefine a
normalized ISHE: ~VSP
ISHE¼VSP
ISHE =a1b1j90/C14/C0VSP
ISHE =/C0
a1b1j270/C14Þ=2 to minimize the residual AMR effect caused by
the small experimental misalignment. This also provides a bet-ter platform to compare ISHE characteristics of each individ-
ual sample since the values are normalized to real microwave
power acting on the Py stripes.
As shown in Eq. (1), the spin pumping voltage also
depends on the effective spin-mixing conductance
g
"#
ef f¼4pMstPyasp=glB. Where tPyandtNare the thicknesses
of FM and NM layers, Msis the saturated magnetization of
permalloy, aspis the enhanced Gilbert damping factor due to
the loss of spin moment during spin pumping.20The damp-
ing factor can be obtained from the linear fit of the
frequency-dependent FMR half linewidth DHthrough
DH¼DH0þ2paf=c. The obtained g"#
ef fincreases with the
increase of Pd thickness and satuarates at about 12 nm
[Fig. 3(a)], which is in good agreement with the results of
Foros et al.20and Shaw et al.21
As discussed above, in order to obtain the spin Hall
angle hSHof Pd, one needs to perform the Pd thickness de-
pendent measurements as it entangles with the spin diffusion
length ksdin the ISHE voltage. By putting all the parameters
that can be measured experimentally to the left side of Eq.
(1), we can acquire
FIG. 1. (a) Experimental geometry for the ISHE measurements via spin
pumping: the Py/Pd bilayer stripes are integrated into the slots of a CPW
and with an out-of-plane microwave magnetic field. (b) Schematic illustra-
tion of the ISHE induced by spin pumping.
FIG. 2. (a) Field dependence of dcvoltage VSP
ISHE (f¼8 GHz) of
Py(16 nm)/Pd(15 nm) at a0¼90/C14(black squre) and a0¼270/C14(red circle).
(b) Normalized inverse spin Hall voltage induced by spin pumping corre-
sponding to Fig. 2(a). The inset shows the thickness dependence of precess-
ing angles for a0¼90/C14anda0¼270/C14, respectively.17C504-2 Tao et al. J. Appl. Phys. 115, 17C504 (2014)
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86.30.54.96 On: Thu, 15 May 2014 21:05:54~VSP
ISHEðH0Þ
eRNwg"#
ef ff¼hSHksdtanhtN
2ksd/C18/C19
: (2)
Thus, hSHandksdcan be obtained simultaneously through
fitting the Pd thickness-dependent measurements. Fig. 3(b)
shows the experimental results for ~VSP
ISHEðH0Þ=eRNwg"#
ef ff
with different Pd thicknesses. To check the reliability of thedata, we repeated the measurements for 15 nm thickness as
shown in Fig. 3(b). The almost identical values evidence the
excellent reproducibility of the data. The small difference inthe data obtained with 8 GHz and 9 GHz excitation strongly
supports the frequency independence of the SHA, which is
expected since what we measured is essentially the dccom-
penent of ISHE.
22The experimental data show excellent
agreement with the fitting utilizing Eq. (2), suggesting the
assumption of constant spin Hall angle and spin diffusionlength in Eq. (1)is valid within the film thickness range we
studied. The fitting simultaneously yields both the spin
Hall angle and the spin diffuion length to be h
SH
¼0:005660:0007 and ksd¼7:360:7 nm, respectively.
The obtained SHA and spin diffusion length are in good
agreement with Mosendz et al. ,10in which they distangled
VAMRandVSP
ISHEby assuming that VAMRhas only asymmetric
component in their particular geometry.
In summary, in combination with the spin pumping and
microwave photoresistance measurements, we quantified thespin Hall angle and the spin diffusion length of Pd using an
out-of-plane microwave magnetic field excitation. We dem-
onstrate that one can disentangle the ISHE signal from theunwanted AMR effect with the designed geometries.
Through the microwave photoresistance measurements, we
found the precessing angles depend on the Pd thickness andthe detailed geometry even with the same input microwave
power. As the injected spin current is proportional to the
product of the in and out-of-plane precessing angles, it is im-portant to measure the precessing angles for each individual
sample. The combination of Pd thickness dependent meas-
urements of the ISHE voltage, effective spin mixing con-ductance and precessing angles yield h
SH¼0:005660:0007
andksd¼7:360:7 nm for Pd.
This work was supported by the State Key Program for
Basic Research of China (Grant No. 2010CB923401), NSFC
(Grants Nos. 11023002, 11174131, and 11374145).
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ef ffor Py/Pd ( tN), which reaches satura-
tion at about 12 nm. (b) Experimental determined Pd thickness-dependent
~VSP
ISHEðH0Þ=eRNwg"#
ef ffatf¼8 GHz (black square) and f¼9 GHz (red circle).
The line is the fitted curve according to Eq. (2).17C504-3 Tao et al. J. Appl. Phys. 115, 17C504 (2014)
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1.2436471.pdf | Effect of 3 d , 4 d , and 5 d transition metal doping on damping in permalloy thin films
J. O. Rantschler, R. D. McMichael, A. Castillo, A. J. Shapiro, W. F. Egelhoff Jr., B. B. Maranville, D. Pulugurtha,
A. P. Chen, and L. M. Connors
Citation: Journal of Applied Physics 101, 033911 (2007); doi: 10.1063/1.2436471
View online: http://dx.doi.org/10.1063/1.2436471
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/101/3?ver=pdfcov
Published by the AIP Publishing
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129.101.79.200 On: Thu, 11 Sep 2014 11:29:21Effect of 3 d,4d, and 5 dtransition metal doping on damping in permalloy
thin films
J. O. Rantschler
Metallurgy Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8552
and Center for Nanomagnetic Systems, N308 Engineering Building 1, University of Houston, Houston,Texas 77204
R. D. McMichael,a/H20850A. Castillo, A. J. Shapiro, W. F. Egelhoff, Jr., and B. B. Maranville
Metallurgy Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8552
D. Pulugurtha and A. P. Chen
Metallurgy Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8552
and Department of Electrical and Computer Engineering, George Washington University, Washington,District of Columbia 20052
L. M. Connors
Metallurgy Division, National Institute of Standards and Technology, Gaithersburg, Maryland 20899-8552and Physics and Astronomy Department, Mail Stop 61, Rice University, Houston, Texas 77005
/H20849Received 20 September 2006; accepted 15 December 2006; published online 14 February 2007 /H20850
The effect of doping on the magnetic damping parameter of Ni 80Fe20is measured for 21 transition
metal dopants: Ti, V , Cr, Mn, Co, Cu, Zr, Nb, Mo, Ru, Rh, Pd, Ag, Hf, Ta, W, Re, Os, Ir, Pt, and Au.For most of the dopants, the damping parameter increases linearly with dopant concentration. Thestrongest effects are observed for the 5 dtransition metal dopants, with a maximum of 7.7 /H1100310
−3per
atomic percent osmium.
I. INTRODUCTION
The dynamic properties of magnetism in thin metal films
have become increasingly important as operating speeds ofinformation storage devices have progressed into the giga-hertz regime, and controlled switching in particular layers ofmagnetic structures has become necessary. Measurements ofdamping in these films and the role of interfaces will beimportant for design and for modeling of magnetic devices.
1
The magnetic damping is often quantified in terms of the
phenomenological Gilbert parameter /H9251in the Landau-
Lifshitz-Gilbert equations of motion,2–4
dM
dt=− /H20841/H9253/H20841/H92620M/H11003Heff+/H9251
MsM/H11003dM
dt. /H208491/H20850
Here Mis the instantaneous magnetization vector, Msis
the saturation magnetization of the film, Heffis the instanta-
neous effective field, /H9253is the gyromagnetic ratio, and /H92620is
the permeability of free space. The first term in /H208491/H20850describes
the precession of the magnetization, and the second termdescribes damping.
Damping is a consequence of the coupling between the
magnetization and the thermal bath that allows dissipation ofmagnetic energy. For most of the magnetic transition metalsand alloys, the damping is small, so that following switchingor some small perturbation, the magnetization precessesmany times before damping out in a time of a few nanosec-onds, typically. For many applications requiring high speedor high data rates, this underdamped behavior is undesirable.Faster magnetic responses could be accomplished with in-
creased damping. On the other hand, the coupling that allowsdamping to the thermal bath also allows the thermal bath todrive fluctuations of the magnetization. These fluctuationsare observable in small magnetic sensors as magnetizationnoise.
5,6For these applications, the noise limit of the sensor
could be improved with reduced damping.
A variety of physical mechanisms for coupling between
the magnetization and the thermal environment in ferromag-netic metals have been discussed, including the weak cou-pling to lattice vibrations,
7,8coupling to two-level
impurities,9and spin-orbit coupling to conduction electrons
near the Fermi surface.10–13Experimental checks of these
conduction electron models have met with mixed success inPermalloy films. For spin-flip scattering, the damping is ex-pected to scale with the conduction electron relaxation rateand therefore with the resistivity. While the resistivity anddamping parameter have been shown to scale with each otheras a function of thickness in some cases,
14they have also
behaved differently both as a function of temperature and asa function of thickness
15in other cases.
When magnetization lies near equilibrium, a linearized
version of the equations of motion /H208491/H20850predict ferromagnetic
resonance /H20849FMR /H20850, which is a resonance in the transverse
susceptibility of the material that occurs when the free pre-cession frequency, f
0=2/H9266/H9275 0of the magnetization is close to
the frequency fpof a small, transverse, pumping field.
In the special case of an ideal, homogeneous film with
Maligned parallel to the applied field, the damping param-
eter/H9251would be simply related to the field-swept linewidth of
the FMR peak,a/H20850Electronic mail: rmcmichael@nist.govJOURNAL OF APPLIED PHYSICS 101, 033911 /H208492007 /H20850
0021-8979/2007/101 /H208493/H20850/033911/5/$23.00 101, 033911-1
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129.101.79.200 On: Thu, 11 Sep 2014 11:29:21/H9004Hp−p=2/H9251/H92750
/H208813/H9253. /H208492/H20850
Because films are generally not ideal, however, it is neces-
sary to explicitly account for extrinsic line broadening due toinhomogeneity when determining
/H9251from experiment. In
measurements where inhomogeneity has not been taken in toaccount, we refer to the resulting value of
/H9251as an effective
damping parameter, /H9251effand we regard this value as an upper
bound on the damping parameter.
One way to tailor the dynamic properties of magnetic
materials is to alloy a promising material with small amountsof another. A few recent studies describe the effects of im-purity doping on effective damping in Permalloy /H20849Ni
80Fe20/H20850,
which is a commonly used magnetic alloy with soft magnetic
properties. Ingvarsson et al. studied films sputtered from al-
loy targets of Ni, Fe, and one of six transition metals, /H20849Nb,
Ru, Rh, Ta, Os, Pt /H20850and found that Os had the largest effect
on the effective damping.16Fassbender et al. have observed
large line broadening in Permalloy films ion implanted withCr, measuring an effective Gilbert parameter of 0.055 foronly 4% Cr doping, in comparison to an undoped value of0.008.
17
Bailey et al. investigated thin Permalloy films doped
with rare earth elements Gd and Tb.18With only 2% Tb, the
effective damping was increased by an order of magnitude,and much smaller effects were observed with Gd doping. Alater study showed that the effective damping increase in theTb-doped samples was related to increased damping, notinhomogeneity.
19For a series of rare earths, Sm through Ho,
Reidy et al. have reported trends in the impurity damping in
Permalloy.20One puzzling aspect of these studies is that, at
least for Tb dopants, the impurity damping is very sensitiveto deposition conditions. The sensitivity of the damping tothe rare earth dopants was shown to be dependent on thesputtering pressure in Ref. 18and the values in Ref. 20are
modest by comparison and comparable with the values wereport below.
This paper presents a survey of the effects of transition
metal dopants on the magnetic damping of Permalloy, withparticular attention paid to the effects of inhomogeneity onthe measurements. Details of sample preparation, composi-tional characterization, and dynamic measurements are pre-sented in Sec. II, and the results are discussed in Sec. III.
II. EXPERIMENTAL METHOD
Films were deposited using dc magnetron sputtering
with a system base pressure of less than 10−6Pa onto 1
/H1100315 cm2Si substrates having 250 nm of thermal oxide. Per-
malloy /H20849Ni80Fe20/H20850films were codeposited with one of 21
dopants from the transition metal group /H20849Ti, V , Cr, Mn, Co,
Cu, Zr, Nb, Mo, Ru, Rh, Pd, Ag, Hf, Ta, W, Re, Os, Ir, Pt,and Au /H20850. The Permalloy target was centered over the sub-
strate, while the dopant target was situated closer to one end,as shown in the inset of Fig. 1/H20849a/H20850. Films were grown to a
nominal thickness of 25 nm and capped with 5 nm of Al
2O3.
Effects due to interfaces on the measurement of damping1,14
and surface anisotropy are expected to be minimal in thisthickness range.14,21Phase separation and other microstruc-
tural effects have not been ruled out but are unlikely due tolow adatom mobility at room temperature. We expect, there-fore, that equilibrium solubility limits will generally be ex-ceeded in our films.
Because of nonuniform atom flux, the actual thickness
varied over the sample. Samples were split lengthwise, withone half reserved for concentration measurements and theother half cut into 5 mm segments for FMR measurements.This gave us the possibility of measuring thirty samples perdopant element, although not all samples were measured. Anadditional undoped Permalloy sample was deposited in thesame geometry as a control sample.
Chemical composition was determined by means of
scanning electron microscopy and energy dispersive spec-troscopy /H20849EDS /H20850. The x-ray spectra were acquired at 15 keV
for Ti–Cu and Hf–Au dopants and at 20 keV for Zr–Pd dop-ants using 200 s at each point along a line scan of usually 60points per sample strip. Quantitative analysis with pure ele-ment standards was performed using a correction procedurefor thin films on substrates that yielded composition valuesand an estimate of the film thickness. Uncertainty in the dop-ant concentration measurement is typically on the order of 1or 2 at. %. For the Ru, Rh, Ag, W, and Pt doped samples, theEDS spectra of the dopant was complicated by the presenceof peaks from Si. For these elements, concentration measure-ments were obtained from samples on Ti foil substrates usingnominally identical deposition conditions.
The most significant variations in the dopant concentra-
tion occurred closest to the dopant target, while nearer the farend, the concentration was nearly constant. See Fig. 1/H20849a/H20850.
The Ni to Fe ratio was nearly constant to within measure-ment uncertainty, as shown in Fig. 1/H20849b/H20850.
Ferromagnetic resonance was detected as a change in
transmission of a 9.00 GHz signal through a coplanar wave-guide with the sample film suspended just over the surface.Sensitivity was enhanced by field modulation and phase sen-sitive detection. For each sample, approximately 20 reso-nance spectra were recorded over a range of applied fieldangles spanning an in-plane direction and the sample normal.
To separate damping and extrinsic line broadening con-
FIG. 1. Deposition of films used in this study. Figure /H20849a/H20850shows the dopant
content profile for Au and includes a schematic of the deposition geometryfor all films. Figure /H20849b/H20850shows the ratio of Ni to Fe ratio profile of the Au
doped film as measured by EDS.033911-2 Rantschler et al. J. Appl. Phys. 101, 033911 /H208492007 /H20850
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129.101.79.200 On: Thu, 11 Sep 2014 11:29:21tributions to the measured linewidths, we use a two step
process, taking advantage of the fact that damping andbroadening exhibit very different angular dependences as theapplied field is rotated out of plane.
First, the angular dependence of the resonance field was
fitted by corresponding fields determined from a magneticfree energy model,
F=−
/H92620M·H+1
2/H92620MsHu/H20873M·nˆ
Ms/H208742
. /H208493/H20850
The first term is the Zeeman energy, and the second term is
the perpendicular anisotropy energy, with the film normal nˆ.
The perpendicular anisotropy field Hu, is the sum of the de-
magnetization field, a smaller, typically negative contributionfrom the surface anisotropy, and the effects of film stresses.The resonance field fits yielded values of H
uand/H9253. Plots of
Huand a discussion are given in the appendix.
The second step in obtaining /H9251values was fitting the
angular dependence of the linewidth /H9004Hto a combination of
the linewidth due to Gilbert damping and the inhomogeneousbroadening. The most common form of inhomogeneousbroadening we observed in these samples was characteristicof variations in the perpendicular anisotropy, for which weuse the linewidth model,
1,22–24
/H9004Hp−p=dHr
d/H9275/H9251/H9253
/H208813Ms/H20875/H115092F
/H11509/H92582+1
sin2/H9258/H115092F
/H11509/H92782/H20876+/H20879dHr
dHu/H20879/H9004Hu.
/H208494/H20850
The first term in this expression is the linewidth due to
damping and the second is the inhomogeneous broadeningdue to local variations in the out-of-plane anisotropy. Here,
/H9258
and/H9278are the polar and azimuthal angles of Mand the partial
derivatives are evaluated at the equilibrium direction of M.
The derivatives dHr/d/H9275anddHr/dHuin /H208495/H20850are determined
numerically using the Huvalues from the resonance field fit
described above. The parameter /H9004Hurepresents the magni-
tude of the variations in Hu.
Examples of the angular dependences of the damping
linewidth and the linewidth due to anisotropy variations areshown in Fig. 2. The damping linewidth curve has the same
value for in-plane and normally applied fields, with a maxi-mum at intermediate angles. The extrinsic broadening due to/H9004H
uis largest for normal applied fields, with a sharp mini-
mum and a smaller maximum at lower angles.
In a few of the samples we measured, the in-plane line-
width is greater than the perpendicular linewidth. This typeof angular variation cannot be described by /H208494/H20850with positive
/H9004H
u, but it is characteristic of two-magnon scattering.25–28In
these cases, full fits are not practical because of the complex-ity of the two-magnon model, but since the two-magnonlinewidth becomes small for magnetization angles greaterthan 45°, we determine the damping parameter by fitting dataonly in the region near where the field is applied within afew degrees of the surface normal as shown in Fig. 2/H20849b/H20850.
III. RESULTS
Values of /H9251are plotted as a function of dopant concen-
tration in Fig. 3. With a few exceptions such as Cr and Pd,
the damping parameter shows a linear dependence on con-centration.
For each dopant element Z, the measured Gilbert damp-
ing parameter values were fit to linear functions of the dop-ant concentration x,
/H9251Z=/H92510+/H9252Zx. /H208495/H20850
The undoped value /H92510=/H208498.0±0.5 /H20850/H1100310−3obtained from the
control sample was treated as a fixed point.14,29These fits are
reproduced as lines in Fig. 3. The values of /H9252Zthat we obtain
are shown in Fig. 4, a portion of the periodic table of tran-
sition metal impurity damping in Permalloy thin films.
FIG. 2. Extracting damping from rotational FMR studies. Dots are data
points, lines are theoretical. Figure /H20849a/H20850shows local resonance broadening in
a copper doped film, and figure /H20849b/H20850shows two-magnon scattering in a mag-
nesium doped film. A vertical arrow in /H20849b/H20850shows where the magnetization is
inclined 45° out of the film plane.
FIG. 3. Damping parameters for Ni80Fe20thin films as a function of transi-
tion metal dopant concentration. Dopants are grouped by column of theperiodic table. Representative error bars are shown for Cr and Re.033911-3 Rantschler et al. J. Appl. Phys. 101, 033911 /H208492007 /H20850
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129.101.79.200 On: Thu, 11 Sep 2014 11:29:21Figure 4shows two overall trends that offer clues into
the mechanism of the impurity damping. The first trend isthat doping has a stronger effect on damping for the heavierelements in each column. The second trend is clearest withinthe bottom row, where the largest values of
/H9252Zare seen in the
middle of the row.
The trend toward stronger effects in heavier elements
suggests that spin-orbit coupling may play a role. Spin-orbitcoupling is an important aspect of several models of damp-ing via conduction electrons,
11–13where the motion of the
spin component of the magnetization creates changes in theelectronic states via spin-orbit coupling and a “breathing” ofthe Fermi surface. As the energies of some of the electronicstates are raised or lowered through the Fermi level in con-cert with the magnetization motion, these states are repopu-lated via electron relaxation. In the context of these models,the dopants may also affect the electron relaxation time.
Comparison with Hund’s rules for the atomic moments
of transition metals /H20851Fig. 4/H20849b/H20850/H20852shows that the doping effec-
tiveness and the atomic spin follow similar trends. This resultstands in contrast to the effects of rare earth dopants, whichclosely follow the orbital angular momentum Lfor the heavy
rare earths Gd through Ho and the total angular momentumJ=/H20841L−S/H20841for the lighter Sm and Eu. Perhaps more relevantly,
calculated spin moments for 5 dimpurities in ferromagnetic
Fe predict the spin moment of the impurity atom passingthrough zero near Os while the orbital moment simulta-neously reaches its maximum amplitude.
30–32
Lattice expansion may also warrant discussion in a full
explanation of the measured trends. As a function of atomicnumber, the atomic radii of the transition metal elementsshow minima at Ru and Os, coincident with the maximumeffects of doping on damping within the corresponding rowsof the periodic table.The close connection between damping and conduction
electrons
10–13suggests that the dependence of resistivity on
dopant element may be relevant. The residual resistivity of5dtransition metal dopants in Ni shows a maximum at W.
33
While this trend in the resistivity bears some similarity to
trends in the impurity damping above, there are also signifi-cant differences. In particular, both the 4 delements and the
3delements Ti, V , and Cr produce resistivity effects that are
very similar to those of the 5 delements. The strong resistiv-
ity effects contrast with the relatively weak damping effectsof the 3 dand 4 delements.
In this paper, we have described a broad survey of the
effects of transition metal doping on the damping in Permal-loy. We have shown that the strongest effects occur for dop-ants from the 5 drow of the periodic table, reaching a maxi-
mum for osmium. This set of results can serve as a guide fordevelopment of controlled damping in Permalloy and for fur-ther experimental and theoretical studies of impurity damp-ing.
ACKNOWLEDGMENTS
Two of the authors /H20849D.P. and L.M.C. /H20850acknowledge the
support of the NSF-sponsored NIST SURF program. Theauthors thank M. D. Stiles and K. Gilmore for helpful dis-cussions.
APPENDIX
The perpendicular anisotropy field Huis obtained by fit-
ting the angular dependence of the ferromagnetic resonance
FIG. 5. Effective out-of-plane anisotropy fields Hu, as a function of sample
location on the substrate.
FIG. 4. /H20849a/H20850Effect of transition metal doping on Gilbert damping in Ni80Fe20
thin films. The increase in the dimensionless Gilbert damping parameter is
given in units of 10−3per atomic percent of dopant. Error bars reflect the
uncertainty associated with the fitting alone. /H20849b/H20850Hund’s rules for the par-
tially filled dshells of isolated dopant atoms, for comparison.033911-4 Rantschler et al. J. Appl. Phys. 101, 033911 /H208492007 /H20850
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129.101.79.200 On: Thu, 11 Sep 2014 11:29:21field. It is plotted as a function of position along the substrate
in Fig. 5. For many of the dopants, Hugoes through a maxi-
mum near the center of the substrate and decreases near theends. We believe that this effect is an artifact of obliquedeposition at the ends of the sample, which is a consequenceof the sputtering geometry shown in Fig. 1. Due to shadow-
ing effects, oblique deposition is known to produce lessdense film with voids between columnar grains, reducing theaverage moment density of the film and reducing the domi-nant demagnetization field contribution to H
u.34
For many of the dopants, Hudecreases more strongly on
the right end of the plot, i.e., the more heavily doped end ofthe substrate. This trend is expected as the magnetization isdiluted or perhaps more strongly disrupted by the dopantatoms. Depressed values of H
uare also found on the low
dopant end of the samples, but the absence of a correspond-ing increase in
/H9251at low dopant concentrations shows that
this artifact does not strongly affect the damping measure-ments.
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1.4864361.pdf | Dissipative soliton dynamics in a discrete magnetic nano-dot chain
Kyeong-Dong Lee, Chun-Yeol You, Hyon-Seok Song, Byong-Guk Park, and Sung-Chul Shin
Citation: Applied Physics Letters 104, 052416 (2014); doi: 10.1063/1.4864361
View online: http://dx.doi.org/10.1063/1.4864361
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130.113.86.233 On: Mon, 22 Dec 2014 14:06:00Dissipative soliton dynamics in a discrete magnetic nano-dot chain
Kyeong-Dong Lee,1,2Chun-Y eol Y ou,3Hyon-Seok Song,1,4Byong-Guk Park,2
and Sung-Chul Shin1,4,a)
1Department of Physics and Center for Nanospinics of Spintronic Materials, Korea Advanced Institute
of Science and Technology, Daejeon 305-701, South Korea
2Department of Materials Science and Engineering, KI for the Nanocentury, KAIST, Daejeon 305-701,
South Korea
3Department of Physics, Inha University, Incheon 402-751, South Korea
4Department of Emerging Materials Science, DGIST, Daegu 711-873, South Korea
(Received 29 October 2013; accepted 22 January 2014; published online 7 February 2014)
Soliton dynamics is studied in a discrete magnetic nano-dot chain by means of micromagnetic
simulations together with an analytic model equation. A soliton under a dissipative system is
driven by an applied field. The field-driven dissipative soliton enhances its mobility nonlinearly,
as the characteristic frequency and the intrinsic Gilbert damping decrease. During the propagation,the soliton emits spin waves which act as an extrinsic damping channel. The characteristic
frequency, the maximum velocity, and the localization length of the soliton are found to
be proportional to the threshold field, the threshold velocity, and the initial mobility, respectively.
VC2014 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4864361 ]
A soliton carries localized energy density while conserv-
ing its shape and velocity, and it appears in a wide variety of
physical phenomena in several different types.1,2In the real
world, a soliton exists under a dissipative system and moves
in balance with the energy absorption. To initiate its motion,
the soliton has to be supplied with a certain amount ofenergy which is known as the threshold energy. In such a
nonlinear situation far from equilibrium, accurate predictions
of dissipative soliton motion are only possible via numericalsimulations.
3
In a magnetic nanowire, domain walls have been studied
as dissipative solitons.4–6The study of such system reveals
several key issues for practical applications, including good
stability, high speeds, and low thresholds. A reduced thresh-
old field or critical current, related to the intrinsic dampingand effective anisotropy, is desirable for low power con-
sumption.
7The best use of the dynamic feature has been
exploited by the mobility, which becomes nonlinear with ahigh degree of anisotropy,
8,9antiferromagnetic coupling or
Dzyaloshinski K%interaction,10–12additional damping,13–15or
domain wall transformation.16–18
Similar significant issues can be imagined in a discrete
magnetic nano-dot system. In a discrete dipole-field-coupled
chain, as illustrated in Fig. 1(a), simple head-to-head or tail-
to-tail magnetization states are considered as localized dissi-
pative solitons, as shown in Fig. 1(b). This localized state in
a discrete system provides inherent stability and has there-fore been considered as an information unit. As studied in
relation to domain wall motion,
8,14,16,17stability at a high
speed will be enhanced in the presence of extrinsic spinwave damping, such as Cherenkov radiation
8,12or in the
presence of spin wave emission in the wake of soliton propa-
gation,9,13,15as shown in Fig. 1(c). The speed is expected to
be maximized up around 1 km/s and the clock frequency can
exceed 100 GHz with large saturation magnetization.19Byengineering an individual dot shape, the inter-dot distance,
and the dot-to-dot interaction type, the magnetic nano-dot
system is considered as a viable candidate for non-volatileroom-temperature high-speed nano-logic devices.
20–27
However, the dissipative soliton dynamics in this discrete
magnetic nano-dot system has not yet been clearly addressedin relation to its mobility, threshold field, and maximum or
minimum velocity.
Cascaded interaction in the chain structure can reveal
the interaction-sensitive characteristics accurately, which is a
basic property even in a binary dot operation. In this Letter,
dissipative soliton dynamics is studied numerically and ana-lytically in a discrete magnetic dot chain with various inter-
dot distances and degrees of shape anisotropy in an effort to
investigate the clock frequency, the threshold field, and themaximum velocity. Most importantly, the interplay between
the intrinsic and extrinsic damping on mobility could be
FIG. 1. Schematic of the field-driven soliton propagation in a magnetic
nano-dot chain. (a) Head-to-head soliton is moving along the xaxis by the
magnetization reversal of each dot sequentially under the influence of the
external field Bx. (b) The soliton location is represented by the shifting posi-
tion of the in-plane phase angle /. (c) The emission of spin wave in the
wake of soliton propagation. (d) Evolution of Mx=MsatBx¼8 mT with
Ms¼800 emu/cc, a¼0.008, and e¼1.0. The average magnetization of
each dot is shown in time. Color bar represents the amplitude of Mx=Ms.a)Electronic mail: scshin@dgist.ac.kr
0003-6951/2014/104(5)/052416/5/$30.00 VC2014 AIP Publishing LLC 104, 052416-1APPLIED PHYSICS LETTERS 104, 052416 (2014)
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130.113.86.233 On: Mon, 22 Dec 2014 14:06:00determined with numerical simulations analyzed by a model
equation.28–30
The basics of soliton motion along a dot chain should
be discussed at this point. As magnetic soliton propagates,the magnetization of each dot reverses in a sequential man-
ner, as shown in Fig. 1(a). Because dot iexperiences the
opposite dipole field from a rotated dot ( i/C01) which is anti-
parallel to the magnetization direction of dot i, the switching
field of dot iis reduced. With the help of the opposite dipole
field from the left dot, dot ireverses in the dot chain in the
presence of an external magnetic field, B
x. After the reversal
of dot i,d o t( iþ1) experiences the opposite dipole field
from the reversed dot i; therefore, dot ( iþ1) is reversed af-
ter the reversal of dot i,a n ds oo n .A st h es o l i t o ni si nt h e
head-to-head or tail-to-tail magnetization state between
the dots, as shown in Fig. 1(b), the subsequent reversal of
the magnetization causes the soliton to move. This is the ba-
sis of soliton motion. In this process, spin waves can be gen-
erated near the soliton location where the dipole fieldchanges rapidly from the magnetization reversal of the dot,
as shown in Fig. 1(c).
As the dissipative soliton propagates over the sinusoidal
potential surface along a one-dimensional chain, it can be
described by the sine-Gordon (sG) equation.
31,32The
Hamiltonian H¼PHiwith dot-index iis considered, as
follows:5,9,19
Hi¼c2l0
4pa3~Si/C1~Siþ1/C03Sx
iSxiþ1/C16/C17
/C0cBxSx
iþc2l0
2V~N/C1ð~Si/C14~SiÞ:
(1)
Here, cdenotes the gyromagnetic ratio, and /C14represents the
entrywise product, which is also known as the Hadamard orSchur product. The zaxis is aligned along the thickness direc-
tion of the dot. In addition, ais the spatial periodicity, which
is the sum of the inter-dot gap distance gand the dot diameter
d. The first term in the Hamiltonian comes from the
dipolar interaction, the middle term is the Zeeman energy,
and the last term represents the magnetostatic energy fromthe demagnetizing field. The spin angular momentum
vector of the i’th dot is represented by ~S
i¼ðSx
i;Sy
i;Sz
iÞ
¼Sðcoshicos/i;coshisin/i;sinhiÞ, where /iis the rotation
angle of the spin on the xyplane and hiis the angle between
the spin and the xy plane . Additionally, l0denotes the vac-
uum permeability, ~Nis a demagnetizing vector, and Vis the
volume of the dot. To express the model equation simply, we
define the dipolar coupling strength as D¼l0fð3Þ=4pa3,
where the zeta function fð3Þreflects the long-range dipolar
interaction. In addition, we use the relationship of
cS¼m¼MsV, where mandMsare the magnetic moment
and the saturation magnetization of the dot, respectively.With continuum approximation after the introduction of
the Gilbert damping parameter a, the soliton velocity is
determined as
v’c01þx0a=cBx ðÞ2/C16/C17/C01=2
: (2)
Here, c0is the maximum velocity, x0is the characteristic
angular frequency, and ais the damping parameter. For
quick referencing, we summarize the key parameters inTable I.
33
It is interesting to compare the soliton motion with the
domain wall motion. The initial mobility l/C3, which is
defined by dv=dBxjBx¼0, is approximately cL0=a, where the
L0is soliton localization length; L0¼c0=x0¼affiffiffiffiffiffiffiffiffiffiffi
D=Kxp
.
This is analogous to the equation of the domain wall mobil-ity of cD=a, where the Dis the domain wall width. In con-
trast to the viscous domain wall motion,
4,34the velocity
shows a finite value, even when a¼0 irrespective of the
driving field Bx. As a rule, the mobility lis
l¼l/C3ð1þðcBx=x0aÞ2Þ/C03=2: (3)
The mobility decreases from l/C3to zero as the driving field
increases with the scale of the dissipation rate of x0a.A t
high speeds, the spatial extent of the soliton contracts more
via the relationship L0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1/C0ðv=c0Þ2q
. Therefore, the field-
driven soliton motion inherently shows the nonlinearity of its
mobility.
Soliton propagation has been simulated using a single-
domain disk with a diameter of 90 nm and a thickness of
10 nm.35We used the absolute value of the gyromagnetic ra-
tio of cl0¼17:6 MHz/Oe and a cell size of 2 /C22/C210 nm3.
To ensure the moderate strength of the dipolar interaction,
the gap distance of gwas ranged from 6 to 30 nm. To change
the shape anisotropy of the dot, we controlled the ellipticalaspect ratio eof the diameter along the xaxis to the diameter
along the yaxis. The short-axis length of the elliptical dot
was kept at a diameter of 90 nm, and the other axis lengthwas elongated along either the xoryaxis. A chain of 32 dots
was used to reduce the computing time. To form a soliton
state near the first dot on the left side in the chain, an ellipti-cal aspect ratio of 2 was used for the first fixed dot in order
to guarantee a fixed state. The dot identification numbers
were from 0 (fixed dot) to 31 (end dot). To create the initialstate with a single soliton next to a fixed dot, a simulated
hysteresis curve was utilized.
To initiate soliton propagation, the external field B
xis
applied globally along the þxaxis. The field has the logistic
step function shape with a rising time of /C24300 ps. The soli-
ton starts to move at a certain small amplitude of Bx.W eTABLE I. Key parameters for magnetic dissipative soliton in a discrete chain.
Dot periodicity a¼edþg Maximum velocity c0¼cmaffiffiffiffiffiffiffiffiDAzp
Dipolar coupling D¼l0fð3Þ=4pa3 Characteristic frequency x0¼cmffiffiffiffiffiffiffiffiffiffiKxAzp
Demagnetizing factor ( z) Az¼l0Nz=2Vþ3D Localization length L0¼affiffiffiffiffiffiffiffiffiffiffi
D=Kxp
Anisotropy ( x) Kx¼3Dþl0ðNy/C0NxÞ=2V Initial mobility l/C3¼cL0=a052416-2 Lee et al. Appl. Phys. Lett. 104, 052416 (2014)
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130.113.86.233 On: Mon, 22 Dec 2014 14:06:00designate this minimum field strength by the threshold field
Bt. As the amplitude of the external field increases, the soli-
ton moves faster due to the stronger driving field. To inhibit
the reversal of the end dot, dot 31, we increased the ellipticalaspect ratio of the end dot to 1.1. To analyze the soliton
propagation velocity along the chain, the average magnetiza-
tion of each dot M
x=Msis plotted in time as a function of the
dot identification number, as shown in Fig. 1(d). The initial
value of Mx=Ms’/C01. A uniform velocity of the soliton
was obtained by fitting the contour line at Mx=Ms¼þ0:9.
The clock frequency v/a, the soliton velocity v, and the
inter-dot transit time ðv=aÞ/C01as a function of the external
field Bxand the intrinsic damping ainare shown in Fig. 2.A s
shown in Fig. 2(a), the soliton velocity is nonlinearly
enhanced by the strong driving field. Note that there clearlyexists a non-zero threshold field B
tand threshold velocity
vt=a, as marked in Fig. 2(a). The initial mobility or the initial
slope in the curve shows a gradual decrease with ain.A ta
large damping of ain/C240:40,Btis large while vtis small. Bt
is carefully determined with a resolution of 0.1 mT. As
shown in Fig. 2(b), the transit time ðv=aÞ/C01increases almost
linearly when ain/H114070:1. The slope of the transit time becomes
steeper at the smaller field of 3 mT, which reflects the strong
damping effect with a small value of Bx. Figures 2(c) and
2(d) show the effect of the dipolar interaction by varying the
gap distance g, and Figs. 2(e)and2(f)show the effect of the
shape anisotropy by varying the ellipticity e. To analyze
these effect on Bt,vt, and the initial mobility more precisely,
the analytic model equation of Eq. (2)is modified to reflect
Btand vt, after which it is consistently applied to the simula-
tion data.
As a rule, the thresholds Btand vtare obtained at ain
¼0:008 to reduce the computing time, with a value nearly
identical to the case of ain¼0. Considering the threshold as
an offset, the velocity becomes
v/C0vt¼ðc0/C0vtÞð1þðx0a=cðBx/C0BtÞÞ2Þ/C01=2;(4)
wherea¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a2
inþvainaexþa2
exq
: (5)
Here, aexis the extrinsic damping parameter. The effective
damping parameter aof Eq. (5)is described below.
Based on Eq. (4), the simulation data of both the field-
and damping-dependencies in Fig. 2are fitted simultane-
ously. The fitted curves are shown as solid curves. The
ratios of the fitted values of c0andx0to the analytical val-
ues are 1.120 60.006 and 0.84 60.02, respectively. The
well-matched values and small errors overall indicate the
validity of our model equation.
Even in a dissipation-less situation of ain¼0 in Fig.
2(a),vstill shows a well-determined velocity, which does
not diverge nor fluctuate unstably. This implies the existence
of additional extrinsic damping. This type of the extrinsicdamping is reported in the domain wall motion with the
emission of the spin waves.
8,9,13–17
In general, we consider the energy dissipation parame-
ter as a combination of ainandaex. Because a simple linear
combination of ainandaexdoes not reproduce the mobility
well, the function of Eq. (5)is selected to fit the simulation
data adequately. If v¼2, then a¼ainþaex.I fv<2, then
a<ainþaex. This value vrepresents the combined damp-
ing of ainand aex.T h efi t t e d aex¼0:05060:001 and
v¼0.4560.11. The value of aexshows a remarkable ex-
trinsic damping value of /C240.05. The value of vis smaller
than 2, which implies that the single energy source is sharedby the two damping channels of a
inandaex.
Figure 3reveals the key parameters which are closely
related to the initial threshold and the mobility. In Fig. 3(a),
the threshold field Btlinearly increases with x2
0, like the
Peierls-Nabarro potential in the sG equation, where the local-
ization length is smaller than the discrete dot spacing.2In
Fig. 3(b),vtshows a proportional relationship with c0. The
initial mobility l/C3/L0, as shown in Fig. 3(c). Here, the c0,
x0,L0, and l/C3are extracted from the simulation data for a
better correlation to the initial behavior, which is basically
not predicted in the analytic model equation. The c0is
FIG. 2. The clock frequency v/a, the
soliton velocity v, and the inter-dot
transit time ðv=aÞ/C01as a function of
theBxin left panels and the ainin right
panels. The ainin (a), the Bxin (b), the
gap distance gin (c) and (d), and the
elliptical aspect ratio ein (e) and (f)
are varied (Symbol: micromagnetic
simulation data, solid curve: model fit-
ting curve). Fitting error bar has thesimilar size of the data symbol.052416-3 Lee et al. Appl. Phys. Lett. 104, 052416 (2014)
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130.113.86.233 On: Mon, 22 Dec 2014 14:06:00obtained from the maximum velocity, and x0is determined
from the small-angle excitation, where the soliton cannot
move but where the magnetization vector precesses at x0;
@2/=@t2þx2
0/¼0. The value of L0is estimated by c0=x0,
andl/C3is extracted from the initial slope in the vversus Bx
graph.
The sensitive threshold dependence on the shape anisot-
ropy implies the need for crystalline, induced, or interface
anisotropy in magnetic nano-dot devices for practical appli-cations. What is the benefit in these types of anisotropy is no
restriction from patterning resolution, which is now sub-
10 nm in electron beam lithography. In addition, nano-dotwith large aspect ratio might be susceptible to incoherent
rotational motion or imperfect single domain state.
Equivalent effective anisotropy for eof 0.5–2 corresponds to
the range of 0.3 to 30 kJ/m
3, which is available from the
crystalline anisotropy with proper material choice or the
induced anisotropy by stress or directional order.36,37To be
stable from the thermal fluctuation, threshold field needs to
be greater than 3 mT ( Bt>40kBT=MsV),37whereas the Bt
should not be too high for low energy consumption and easy
operation of external field. This could be better achieved by
engineering the gandetogether with crystalline or induced
anisotropy. Note that the minimum actual applied field of B-
Bt>0:1 mT, which is greater than room-temperature fluctu-
ation field of 26 meV, is additionally needed to ensure the
exact positioning of the kink soliton.
To observe the soliton localization visually, Fig. 4(a)
shows the magnetization states of dots 21 to 25 at three dis-
tinct times during the soliton propagation process. The phasedifferences appear to be close to 90
/C14between dots 21 and 22
at 3.00 ns (top row), between dots 22 and 23 at 3.13 ns (mid-
dle row), and between dots 23 and 24 at 3.26 ns (bottomrow). This indicates that the soliton is localized within /C24aof
one dot. As inferred from Fig. 3(c), the estimated L
0for
g¼25 nm is /C240:57a; this is similar to the analytical
L0/C240:58a.
In order to check the spin wave emission in the wake of
soliton propagation, the in-plane magnetization of each dotis represented by its in-plane phase /or//C0180
/C14, as shown
in Fig. 4(b). Here, we can see the ringing wave tail with a
wavelength of /C2412a, which is similar to the calculated spin-
wave wavelength of kw/C2410:8a. This wave changes the po-
larity of its phase in a period of ðv=aÞ/C01/C240.13 ns. This is nu-
merical evidence that the propagating soliton generates spinwaves. Interestingly, the spin wave appears to propagate
only along the left side of the soliton. One of the reasons for
this is that the ringing fluctuation in the wake of solitonpropagation allows the spin wave to be generated easily.
Another possible reason is the different cutoff condition,which favors a left-side sinusoidal wave and a right-side
exponentially damped wave, as shown in Fig. 1(c).
33
When there is no intrinsic damping and a constant bias
field, the extrinsic damping induced by spin wave emission
can be clearly witnessed by the pulse excitation and relaxa-
tion of the local center dots. In Fig. 4(c), a one-shot field
pulse with a small amplitude is applied at the center dots of
15 and 16, after which the precessional relaxation via spin-
wave emission from the center dots is visualized in the time-evolved M
z=Msmap. As in the case of ain¼0, extrinsic
spin-wave damping is obtained from 1 =2pfs, where the fre-
quency fand the relaxation time sare determined from the
fitting of the precessional relaxation of the center dots by the
exponentially decaying sinusoidal function. The obtained
value of aexis 0.03, which is comparable to 0.05 in the curve
fitting of Fig. 2. This calls attention to the non-negligible,
essential role of spin wave damping.
In summary, dissipative soliton dynamics was studied
via micromagnetic simulations and an analytic model equa-
tion in a ferromagnetically coupled discrete nano-dot system.
It was remarkably found that the characteristic frequencyand the intrinsic as well as the extrinsic damping parameter
significantly affect the nonlinearity of the mobility. The
threshold behavior is also investigated with various inter-dotdistances and degrees of shape anisotropy, which reveals the
need for anisotropy engineering in nano-dot devices for prac-
tical applications.
This research was supported by the DGIST R&D Program
of the Ministry of Education, Science and Technology ofFIG. 3. Key parameters for initial threshold and mobility. (a) Threshold field
Btversus characteristic frequency x0=p. (b) Threshold velocity vtversus
maximum velocity c0. (c) Initial mobility l/C3versus localization length L0.
Numbers inside figures show the symbol position of the lowest gandeval-
ues as shown in Fig. 2.
FIG. 4. (a) Magnetization states of dot 21 to 25 at three distinct times during
the soliton propagation with Ms¼800 emu/cc, ain¼0:008, g¼25 nm, and
e¼1. (b) The emitted spin wave is visualized by the in-plane phase /of the
magnetization vector. Phase /greater than 90/C14is plotted by /–180/C14with
the solid symbol, whereas /less than 90/C14of the reversed state is plotted
with the hollow symbol (Vertical black arrow: direction of changing phases
at 3.00 ns, dashed green line: line at Mx=Ms¼60:9, horizontal hollow
arrow: direction of soliton propagation). (c) The Gaussian field pulse with
0.2 mT and 100 ps is applied locally on the dot 15 and 16 along the /C0y
andþydirection, respectively. The spin-wave propagation is visualized in
the time-evolved Mz=Msmap of the dot chain with Ms¼400 emu/cc,
g¼12 nm, and e¼1.00. Color bar represents the amplitude of Mz=Msin ar-
bitrary units.052416-4 Lee et al. Appl. Phys. Lett. 104, 052416 (2014)
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130.113.86.233 On: Mon, 22 Dec 2014 14:06:00Korea (11-IT-01) and by the National Research Foundation of
Korea (NRF) grant funded by the Korea government (MEST)
(2010-0023798, 2010-0022040, 2012R1A1A1041590, and
2013R1A1A2011103).
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1.344643.pdf | Computer simulation of magnetization reversal in fine hexaferrite particles
Yoshinobu Nakatani, Yasutaro Uesaka, and Nobuo Hayashi
Citation: Journal of Applied Physics 67, 5143 (1990); doi: 10.1063/1.344643
View online: http://dx.doi.org/10.1063/1.344643
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142.157.129.8 On: Sat, 13 Dec 2014 14:22:36Particulate and Perpendicular Recording Media Michael P. Sharrock, Chairperson
Computer simulation of magnetization reversal in fine hexaferrite particles
Yoshinobu Nakatani, Yasutaro Uesaka,a) and Nobuo Hayashi
Department o/Computer Science and Information Mathematics, University of Electro-Communications,
Chofu, Tokyo 182, Japan
Magnetization reversal mechanisms in hexaferrite particles were investigated by computer
simulation. The Landau-Lifshitz-Gilbert equation discretized according to the Crank
Nicolson method was solved numerically. The computing region was divided into triangular
prisms of equal size which are accommodated to the contour surfaces of the hexagonal particle.
The demagnetizing field was calculated using a finite-grid method based on the triangular
prisms. The mode of reversal was found to be extremely dependent on the thickness of the
particle. In the case of a thin particle, the reversal started at the particle center and then spread
to the edges of the particle almost coherently at the same radius. In a thick particle, vortexes of
magnetization with opposite polarities of rotation were generated at the top and bottom
surfaces, and the switching was incoherent over the particle thickness. In the case of an
intermediate thickness, the reversal began at the particle center, but it spread to the particle
edges incoherently. The formulation of the calculation and the results of simulation are given.
L INTRODUCTION
Recently, Ba-ferrite particles have aroused great inter
est in their application to high-density magnetic recording, I
Kubo, Ido, and Yokoyama studied magnetization reversal
of a Ba-ferrite particle experimentally2 and explained the
experimental results based on a fanning model with two par
ticles connected to each other. However, no evidence was
shown that most of the particles were connected like twins.
Furthermore, it is not known that the magnetizations rotate
coherently in each single particle. Victora3 recently carried
out computer simulation and found a type of reversal mecha
nism which is quite different from the ones already known.
In this paper we present a numerical scheme, based on the
Landau- Lifshiftz-Gilbert (LLG) equation, to investigate
in detail reversal mechanisms of a hexagonal particle togeth
er with magnetization states derived during the reversal pro
cess,
II. NUMERICAL METHOD
We use a Cartesian frame (x,y,z) with thez axis normal
to the hexagonal base of the particle and the x axis parallel to
one of the edges of the basal plane. Denoting with rand Ms
the gyromagnetic ratio and the saturation magnetization,
respectively, the LLG equation can be arranged as
0) Central Research Laboratory, Hitachi Ltd., Kokubunji, Tokyo 185.
Japan. -(I + a2)M:lr = [MXH'] =FU),
with H' = H + a[MXH]/Ms. where H denotes the effec
tive reversible field, and a denotes the Gilbert's damping
constant. The method of solving the LLG equation is almost
the same as the one developed previously for cubic particles4
except that the Crank-Nicolson method5 is used and that the
spatial derivatives are discretized based on a triangular
mesh.
Figure 1 shows the computing region projected on the
basal plane. Extra cells shown by shaded triangles are added
so that the computing region consists of n y rows of flat trape
zoids each of which contains equal number ( = nx ) oftrian
gles. Cells are attached indices i, j, and k. While i specifies
the location of a cell within a row, j specifies the location of
the row which contains the celt The x-y plane which con
tains the cell is specified by k ( = 1 to n z ).
In the Crank-Nicolson method, I'd is discre
tized straightforwardly as !J.M (i, j,k) I t:.t, with
t:.M = (flMx ,t:.My ,flMz ) denoting the increment in M dur
ing at. The torque F(t) is replaced by [F(t) + F(t + t:.t)]/
2 and is expanded with respect to the components of t:.M to
OUt!): F(t + At) = F(t) + flF(t)/2. The number of in de
pendent variables is reduced to two per computing point
using the relation M·flM = 0 as described in Ref. 4.
The2D Laplacian flu = Uxx + Uyy contained in the spa
tial derivatives of exchange origin in H is discretized as
Au::;-:4(ui+ + ui-+ uj' -3u)/(38b2),
and
5143 J. Appl. Phys. 67 (9), 1 May 1990 0021-8979/90/095i 43-03$03.00 © 1990 American Institute of Physics 5143
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142.157.129.8 On: Sat, 13 Dec 2014 14:22:36FIG. 1. Division of base plane of particle into triangular cells.
for up-and down-pointing center triangles (see Fig. 1), re
spectively. The superscripts i ± and j ± refer to cells
(i ± 1,j,k) and (i,j ± 1,k) neighboring to the cell (i,j,k),
respectively. In each x-y plane, a cell is surrounded by three
nearest neighbors. The distance between the center of a cell
and the center of one of the nearest neighbors is denoted by
8b.
The demagnetizing field, contained in F, is calculated by
collecting the fields due to the charges appearing at the sur
faces of all the prisms in the computing region, 6 as in Ref. 4.
In the Crank-Nicolson scheme, the exchange energy cou
ples the increment f;.MU,j,k) to f;.M in five neighboring
cells. As for the magnetostatic energy, we include couplings
only from the cell (i,j,k) and the five neighboring cells, neg
lecting long-range magnetostatic couplings in the first-order
expansion Il}<', as in Ref. 4. The LLG equation then is re
duced to 2nx nyllz coupled equations of magnetization incre
ments, which are solved uisng the ILUCGS method.7
m. RESULTS
The parameters used arc the following: Ms = 300
emu/cm-\ magnetocrystalIine anisotropy constant
K" = 7.5 X 105 erg/cm3, exchange constant A = 1 X 10-7
erg/cm,8 y= -1.76X107rad/(Oes),a= 1,nx = 19,and
~.,.. -II I I I I h=10oA-II ! I
I~N 200,11. I I I I n I: 300.A
I I 0 I ,
-r I 2 -1 0 1 2 ,3
H(kOe) I I I I I I I: II
I I I I
I : I -1 .,d,.-
FIG. 2. Magnetization ,'urve derived from three different thicknesses.
5144 J. Appl. Phys_, Vol. 67, No.9, 1 May 1990 0·0 (a) h=100A
(b) h=200A
(c) h dOD A
H~ = -35000e
H~ = 1·0 Oe
-1.0 '--+--+~_t--t--'I"--t_+--+-....p~Ia-"""""*,,,"""I-t..;(_n~s) o 0·5 1·0 1-5
FIG. 3. Change of average magnetization M7 with time.
ny = 10. The diameter (length of the longest diagonal) was
fixed at 1000 A, while three kinds of thicknesses h = 100,
200, and 300 A were used. nz was chosen to be 3 for h = 100
and 200 A and 5 for h = 300 A. The magnetic moments were
almost parallel to the z axis in the field-free equilibrium
states, to which a reversing field composed of a z field H;:
and an offset field H:: of 1 Oe was applied.
The magnetization curves derived for the three different
thicknesses are shown in Fig. 2. The switching field Hsw
increases when h is increased. The magnetization, however,
decreases only slightly until H;: reaches Hsw in aU cases.
Figure 3 shows time dependencies of average magneti
zation X(. Transient states during the reversal derived from
H;: of -3500 Oe are shown in Figs. 4-6, where the magnet
ic moment of each ceil is representd by a three-dimensional
top. A time step f;.t of 0.02 ns was used. The case with
h = 100 A. is shown in Fig. 4. The nucleation starts at the
center of each hexagon. The reversed region then spreads to
the edges oftne hexagons almost coherently. The behavior of
magnetic moments is almost the same among the three x-y
planes.
Figure 5 shows the case with h = 300 A. The magnetic
moments in the edge regions of the top and bottom planes
tend to incline first to form vortexes of magnetization with
opposite polarities of rotation. The reversed region spreads
to the centers of x-y planes except in the midplane. In the
midplane the movement of the moments is hindered, at ear-
FIG. 4. Transient states in a lOO-A.-thick particle corresponding to Fig. 3.
The magnetizations in the midplane are shown.
Nakatani, Uesaka, and Hayashi 5144
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
142.157.129.8 On: Sat, 13 Dec 2014 14:22:36FIG. 5. Transient states in a 3OG-A.-thick particle corresponding to Fig. 3.
The magnetizations in the top, mid, and bottom planes are shown.
lier stages, because the moments in neighboring x-y planes
(k = 2 and 4) tend to precess in opposite directions. The
moments in the center of the particle reverse almost coher
ently. The case with h = 200 A. is shown in Fig. 6. The cen
tral part of each hexagon reverses first as in the lOO-A.-thick
particle, but the reversed region spreads to two opposite
edges. As a result, an S-shaped reversed region appears tran
siently.
5145 J. Appl. Phys., Vol. 67, No.9. 1 May 1990 FIG. 6. Transient states in a 200-A.-thick particle corresponding to Fig. 3.
The magnetizations in the midplane are shown.
1'10 CONCl.USION
A formulation to solve the LLG equation for magneti
zation reversal in fine hexagonal particles was outlined. The
simulation results suggest that the reversal takes place
through a variety of incoherent rotations, the mode of which
depends upon the geometric parameters of the particles.
'T. Fujiwara, IEEE Trans. Magn. MAG-23, 3125 (1987), for example.
20. Kubo, T. Ido, and H, Yokoyama, iEEE Trans. Magn. MAG-23, 3140
(1987).
3R, H. Victora,J. App!. Phys. 63, 3423 (1988).
'Y. Nakatani, Y. Uesaka, and N. Hayashi, Jpn. J. App!. Phys. 28, 2485
(1989).
5D. M. Young and R. T. Gregory, A Survey of Numerical Mathematics
(Addison-Wesley, Reading, MA, 1973), Vol. II, for example.
"The surfaces of all the prisms include the surfaces ofthe individual prisms
themselves.
7C. den Heijer, in Proceedings of the International Conference on Simula
tion of Semiconductor Devices and Processes, 1984, p. 267.
"This value is 5 times smaller than the value used in Ref. 3. The effect of the
A value will be considered in detail in a later paper.
Nakatani, Uesaka, and Hayashi 5145
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142.157.129.8 On: Sat, 13 Dec 2014 14:22:36 |
1.4710524.pdf | Measurement and simulation of anisotropic magnetoresistance in single
GaAs/MnAs core/shell nanowires
J. Liang, J. Wang, A. Paul, B. J. Cooley, D. W. Rench et al.
Citation: Appl. Phys. Lett. 100, 182402 (2012); doi: 10.1063/1.4710524
View online: http://dx.doi.org/10.1063/1.4710524
View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v100/i18
Published by the American Institute of Physics.
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Downloaded 07 Oct 2012 to 136.159.235.223. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsMeasurement and simulation of anisotropic magnetoresistance in single
GaAs/MnAs core/shell nanowires
J. Liang,1,2J. Wang,1,2A. Paul,1,3B. J. Cooley,1,2D. W. Rench,1,2N. S. Dellas,1,3
S. E. Mohney,1,3R. Engel-Herbert,1,3and N. Samarth1,2,a)
1Materials Research Institute, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
2Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
3Department of Materials Science and Engineering, The Pennsylvania State University, University Park,
Pennsylvania 16802, USA
(Received 17 January 2012; accepted 17 April 2012; published online 1 May 2012)
We report four probe measurements of the low field magnetoresistance (MR) in single core/shell
GaAs/MnAs nanowires (NWs) synthesized by molecular beam epitaxy, demonstrating clear
signatures of anisotropic magnetoresistance that track the field-dependent magnetization. Acomparison with micromagnetic simulations reveals that the principal characteristics of the
magnetoresistance data can be unambiguously attributed to the nanowire segments with a zinc
blende GaAs core. The direct correlation between magnetoresistance, magnetization, and crystalstructure provides a powerful means of characterizing individual hybrid ferromagnet/semiconductor
nanostructures.
VC2012 American Institute of Physics .[http://dx.doi.org/10.1063/1.4710524 ]
The incorporation of spin-related functionality into
semiconductor nanostructures provides an exciting route for
nanospintronic devices.1Nanodevices derived from MnAs/
GaAs heterostructures present an interesting opportunity inthis context because GaAs is an important semiconductor for
optoelectronics, while MnAs is a ferromagnetic metal with a
Curie temperature above room temperature ( /C24313–350 K,
depending on the strain). Indeed, MnAs/GaAs heterostruc-
tures have excellent compatibility with commonly used
semiconductor devices.
2,3In addition, MnAs is a fundamen-
tally interesting ferromagnet because of the unique compet-
ing interplay between the magnetocrystalline anisotropy and
the shape anisotropy. Recent work has shown that the hetero-
epitaxy of MnAs on GaAs can also be realized in core/shell
nanowires (NWs).5–7Such NWs could allow the study of
magnetization dynamics in restricted nanoscale geometries.
However, probing the magnetization in such individual NWs
is a challenge for conventional magnetometry techniques.Here, we use magnetoresistance (MR) measurements of sin-
gle NW devices in conjunction with micromagnetic simula-
tions to gain insights into the magnetization switchingprocess of core/shell GaAs/MnAs NWs. The methodology
presented here is also applicable to other hybrid core/shell
semiconductor/ferromagnet NWs of current interest.
8–10
The core/shell NW samples studied here were synthe-
sized on GaAs (111)B substrates in an EPI 930 molecular
beam epitaxy chamber. We used a catalyst-free growth tech-nique for the GaAs NWs,
11followed by thin film growth of a
MnAs shell, as detailed in an earlier report.6This growth
technique contrasts with other approaches wherein GaAs/MnAs core/shell NWs are synthesized using a Au catalyst.
5,7
Additionally, we note that the epitaxial orientation relation-
ship between GaAs and MnAs is different from NWs synthe-sized using a Au catalyst due to the different crystal
structures of the GaAs core. This creates a difference in mag-
netocrystalline anisotropy that has a significant impact onthe magnetic domain structure of the MnAs shell and the low
field magnetotransport properties.
Figure 1(a) shows a cross-sectional transmission elec-
tron microscope image of a single core/shell NW with azinc-blende (ZB) GaAs core of /C24200 nm diameter. The
MnAs shell thickness is estimated to be /C2410 nm. The GaAs
core is mostly in the ZB structure with small segments of thewurtzite (WZ) phase; the MnAs shell is crystalline with a
hexagonal NiAs structure. For the segments of the NW with
ZB core the growth direction is along the [111] directionwith six facets belonging to the {110} family. The c-axis
(hard axis) of MnAs lies in plane with the NW facets, at an
angle of /C24653
/C14with respect to the wire axis. The c-axis of
MnAs mirrors itself on adjacent facets. For the WZ part of
the NW, the growth direction is along [001] and the c-axis ofMnAs is along the NW axis.
6
We ultrasonically removed the GaAs/MnAs core/shell
NWs from the substrate and dispersed them onto a Si/Si 3N4
substrate. The sample was then transferred into a dual-beam
focused ion beam (FIB) system (FEI Quanta 200 3D) with in
situscanning electron microscopy (SEM) capabilities. After
oxide layer milling, we deposited four Pt electrodes on single
GaAs/MnAs core/shell NWs for electrical measurements.
We minimized the Gaþion imaging time to reduce contami-
nation and also kept the Gaþion deposition current and
chamber pressure low to minimize the spreading of Pt. Fig-
ure1(b) shows an SEM image of a typical device. Subse-
quent electrical transport measurements were carried out in a
Quantum Design Physical Properties Measurement system
using a standard four-probe AC resistance bridge. We madesure that the contacts are ohmic and used a typical excitation
current of 0.5 lA. We measured the MR over a temperature
range 500 mK to 300 K and in magnetic fields up to 80 kOe.In total, we fabricated and measured four devices. In this let-
ter, we focus on data from only one of these devices; the
other devices show qualitatively similar behavior. In addi-tion, we carried out control measurements using a bare GaAs
NW device (i.e., without any MnAs shell) using the same
a)Electronic mail: nsamarth@psu.edu.
0003-6951/2012/100(18)/182402/5/$30.00 VC2012 American Institute of Physics 100, 182402-1APPLIED PHYSICS LETTERS 100, 182402 (2012)
Downloaded 07 Oct 2012 to 136.159.235.223. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsFIB contacting technique with Pt electrodes. This control
experiment shows that the nominally undoped GaAs core is
highly insulating. Thus, the GaAs core merely serves as a
NW template that supports the metallic MnAs shell whichdominates the transport data discussed in this paper.
Figure 2(a) shows the temperature-dependent resistivity
of a GaAs/MnAs core/shell NW with length /C241.96lm from
room temperature ( /C24298 K) down to /C24500 mK. The resistiv-
ity of the MnAs shell was /C245/C210
/C04X/C1cm at room temper-
ature, which is /C245 times higher than a typical MnAs epilayer
grown on GaAs(001). The temperature dependence of the re-
sistivity shows metallic behavior, similar to that of a MnAs
epilayer between room temperature and 20 K. However, thesomewhat smaller residual resistivity ratio (defined as the ra-
tio of the resistivity at 300 K to that at 4.2 K) indicates that
these MnAs shells are more disordered than epitaxial filmsof similar thicknesses.
12Below T/C2515 K, the resistivity
increases with decreasing temperature (Fig. 2(b)). Analysis
to be reported elsewhere shows a temperature dependence ofthe conductivity r/C24lnT, consistent with the onset of local-
ization in a diffusive two dimensional system. The saturation
of the resistivity at even lower temperatures ( T.1:4 K),
shown in Fig. 2(b), is not yet understood, but could arise ei-
ther from trivial heating effects or from more interesting
dimensional crossover as the relevant length scales (such asthe phase breaking length) increase with lowering
temperature.
Figures 2(c) and2(d) show the MR of a NW device
measured at low temperature ( T¼10 K) in magnetic fieldsapplied parallel and perpendicular to the wire axis, respec-
tively. These measurements were carried out after first satu-
rating the magnetization of the MnAs shell at 80 kOe. We
observe a hysteretic MR at low fields, superimposed upon alinear negative MR that does not saturate even at the highest
fields used in the present measurements (high field data not
shown). The physical origin of this interesting non-saturatingMR will be discussed elsewhere and is likely related to mag-
non contributions as in earlier interpretations of similar
behavior in both ferromagnetic thin films
13and NWs.14We
focus here on the low field MR which we attribute to the
classic anisotropic magnetoresistance (AMR) effect and not
to magnon-related MR as in permalloy NWs.14
For magnetic fields along the wire axis, as we decrease
the magnitude of the field from its maximum value to zero,
the resistance initially increases (linear background) andreaches a maximum after field reversal at /C243.5 kOe (Fig.
2(c)). The MR then abruptly changes, indicating domain
switching. From the epitaxial relationship between the MnAsshell and the GaAs core, we conclude that this MR feature
originates from wire segments with a ZB core. Our reasoning
is as follows: for segments with a WZ core, the MnAs hardaxis is along the wire axis; the strong magnetocrystalline ani-
sotropy energetically disfavors magnetization in that direc-
tion. Sweeping the external magnetic along the hard axiswould cause a coherent rotation of magnetization and the
MR contributions would only show up in the MR back-
ground. Thus, the clear hysteretic behavior we observed inthe parallel field geometry can only be explained with the
FIG. 1. (a) Cross-sectional TEM image
of a GaAs/MnAs core/shell NW. The
GaAs core is in the ZB phase. (b) Single
GaAs/MnAs core/shell NW device con-
tacted by four FIB-assisted Pt electrodes,with L ¼1.96lm. (c) GaAs/MnAs core/
shell structure used in micromagnetic
simulations.182402-2 Liang et al. Appl. Phys. Lett. 100, 182402 (2012)
Downloaded 07 Oct 2012 to 136.159.235.223. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsAMR effect which tracks the magnetization from wire seg-
ments with a ZB core. Figure 2(d) shows the MR with the
magnetic field perpendicular to the wire axis. Here, the MRis more complex: as we decrease the magnitude of the field
from its maximum value to zero, the resistance again initially
increases (linear background); the resistance then dropssharply after field reversal to a minimum at /C244 kOe before
recovering at /C248 kOe. This behavior suggests a two step re-
versal process. Again, contributions from NW segments witha WZ core cannot cause the abrupt changes observed in the
MR data: if this were the case, the magnetization would be
exclusively in the cross section of the WZ wire segmentsfavored by both the external applied field and the magneto-
crystalline anisotropy. Consequently, the angle between the
current and the magnetization would stay close to 90
/C14and
the resulting AMR effect would be orders of magnitude
smaller than the contributions from segments with a ZB
core.
Both hysteretic effects in the measured MR loops persist
up to room temperature. Since AMR is directly connected to
the magnetization of a ferromagnetic sample, we can exploitthe effect as a sensitive probe of the field induced rotation
and switching of the magnetization in these nanostructures.
15
To further gain insight into the magnetization reversal pro-
cess of GaAs/MnAs NWs, we carried out micromagnetic
simulations. The magnetic domain structure of MnAs thin
films has already been the subject of micromagnetic studiesusing finite difference based solvers.
4,16This approach, how-
ever, is not suitable due to the geometry of the core/shell
structure. We therefore employed the open source finite ele-ment code MAGPAR (Ref. 17) to avoid erroneous results
from the staircase approximation.
18The domain configura-
tion was calculated using the Landau-Lifshitz-Gilbert equa-tion with the damping constant a¼0:1. The following
micromagnetic parameters for MnAs were used: exchange
stiffness constant A¼1/C210/C011J/m, uniaxial magnetocrys-
talline anisotropy constant Ku¼/C07:2/C2105J/m3, and satu-
ration magnetization Ms¼8/C2105A/m. These parameters
were employed previously to simulate the magnetic domainstructure of MnAs thin films grown on GaAs(001)
4and on
GaAs(111) substrates.16We varied MnAs shell thickness
and GaAs core diameter from 10 to 20 nm and from 120 to160 nm, respectively. Here, we present the results of a 10 nm
thick MnAs shell on a ZB GaAs core (160 nm in diameter)
being closest to the NW geometry investigated. The core/shell NW geometry caused a very large boundary element
matrix and we therefore limited the wire length to 250 nm.
The finite element mesh was generated using GMESH.
19
The average edge length of the tetrahedral elements around
5.5 nm was chosen close to the micromagnetic exchange
lengths and the field stepping was as small as 25 Oe.
Figure 3(a) shows the simulated hysteresis curve with
the magnetic field applied along the wire axis. The hysteresis
shows a single domain reversal with a coercive field of 4.85kOe in good agreement with the measurements. The energy
barrier separating the two stable domain states is caused by
the magnetocrystalline hard axis being inclined with the wireaxis. We used the simulation results as the input to the stand-
ard heuristic description of AMR:
20
q¼q?þðqk/C0q?Þcos2u; (1)
where qkandq?are the longitudinal and transversal resistiv-
ities with respect to magnetization and uthe angle between
the magnetization and current density. For the NW, the angleis given by the normalized magnetization along the wire
FIG. 2. (a) Resistivity qof the device in Fig. 1(b)
as a function of the temperature with an excitation
current of 0.5 lA. (b) Resistivity vs. temperature
(plotted on a log scale). Experimental MR loops
with a magnetic field applied (c) parallel and (d)perpendicular to wire axis.182402-3 Liang et al. Appl. Phys. Lett. 100, 182402 (2012)
Downloaded 07 Oct 2012 to 136.159.235.223. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsaxis: cos u¼Mk. Note that for MnAs qk<q?.21Figure 3(c)
shows that the AMR effect calculated by Eq. (1)is in very
good agreement with the measured MR loop in Fig. 2(c),
reproducing shape and coercive field well. Note thatalthough there is a reduction in MR with increasing field, it
does not explain the linear background at higher magnetic
fields. The overestimation in coercive field is attributed to anunderestimation of the MnAs shell width. Simulations with
varying NW geometries showed a decrease in coercive field
with increasing shell thickness. The experimental determina-tion of the MnAs thickness from the TEM image was chal-
lenging due to the low contrast between the core and the
shell and is held responsible for this discrepancy.
Figures 3(b) and 3(d) show the simulated hysteresis
curves and AMR loops for magnetic fields applied perpen-
dicular to the wire axis [ x-axis, see Fig. 1(c)]. Two magnet-
ization curves, M
?in the applied field direction and Mk
along the wire axis, are overlayed. The hysteresis reveals
two discontinuous changes at 2.1 kOe and 3.4 kOe: the twofacets aligned within the applied field [upper and lower facet,
cf. Fig. 1(c)] reverse at a lower field, whereas all other facets
that are inclined to the applied field direction switch at thehigher field value at once. The increase in magnetization
along the wire axis M
kbetween the two switching events
originates from the epitaxial relationship, i.e., the inclinationof the hard axis with respect to the wire axis and the alterna-
tion of the angle in adjacent facets. Reducing the magnetic
field from saturation, the magnetization in the facets deviatesfrom the x-direction and aligns in the respective facet planes,
perpendicular to the “local” hard axis, to minimize demag-
netization and magnetocrystalline anisotropy energies. Thiscauses M
kof the upper and lower facet to be antiparallel
with Mkof all other facets, which in turn gives rise to a small
netMkof the wire. Reversing M?is accompanied withreversing Mk. If the upper and lower facets reverse, Mkhas
the same orientation in all facets between the two switching
fields, giving rise to a large Mk. At larger fields, the magnet-
ization of the slanted facets reverses, reducing the net Mk.
The calculated switching fields are smaller than measured. A
possible source of error is the alignment of the NW in the
applied field. Although a sufficiently precise line-up of thenanowire axis was easily achieved, the control over the azi-
muthal orientation is challenging and might cause the dis-
crepancy. Disregarding the linear background, the measuredMR is well reproduced by the AMR effect derived from the
magnetization curve M
k. A two-step reversal process and the
reduction in MR between the two reversal field steps due toa large M
kare correctly predicted.
In conclusion, low field MR measurements of single
GaAs/MnAs core/shell NWs reveal the AMR effect of thewire segments with a ZB GaAs core superimposed on a lin-
ear background. Both MR loops measured for fields perpen-
dicular and parallel to the wire axis are well reproduced bymicromagnetic simulations, even for a relatively complex
geometry. The combination of MR measurements with
micromagnetic simulations thus provides a powerful meansto gain insight into the domain structure and dynamic proper-
ties of functional ferromagnetic nanostructures.
This work is supported by the Penn State Center for
Nanoscale Science under NSF DMR-0820404, and by NSFECCS-0609282, ONR N00014-09-1-0221, and the Penn
State Nanofabrication Facility NSF NNUN ECCS-35765.
1D. D. Awschalom and M. E. Flatte, Nat. Phys. 3, 153 (2007).
2M. Tanaka, Semicond. Sci. Technol. 17, 327 (2002).
3L. Da¨weritz, Rep. Prog. Phys. 69, 2581 (2006).
4R. Engel-Herbert, T. Hesjedal, and D. M. Schaadt, Phys. Rev. B 75,
094430 (2007).
FIG. 3. Simulated magnetic hysteresis curves of
GaAs/MnAs core shell nanowires with a ZB coreand magnetic fields applied (a) along the wire axis
(z) and (b) perpendicular to the wire axis ( x). In
panel (b), we show hysteresis loops for the magnet-
ization component along the applied field direction
xand perpendicular to the wire axis M
?, and along
the wire axis Mk. AMR effect extracted from the
simulated hysteresis curves for magnetic fields (c)parallel and (d) perpendicular to the wire axis.182402-4 Liang et al. Appl. Phys. Lett. 100, 182402 (2012)
Downloaded 07 Oct 2012 to 136.159.235.223. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions5M. Hilse, Y. Takagaki, J. Herfort, M. Ramsteiner, C. Herrmann, S. Breuer,
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1.4944996.pdf | Magnetoelectric domain wall dynamics and its implications for magnetoelectric
memory
K. D. Belashchenko , O. Tchernyshyov , Alexey A. Kovalev , and O. A. Tretiakov
Citation: Appl. Phys. Lett. 108, 132403 (2016); doi: 10.1063/1.4944996
View online: http://dx.doi.org/10.1063/1.4944996
View Table of Contents: http://aip.scitation.org/toc/apl/108/13
Published by the American Institute of Physics
Magnetoelectric domain wall dynamics and its implications
for magnetoelectric memory
K. D. Belashchenko,1O.Tchernyshyov,2Alexey A. Kovalev,1and O. A. Tretiakov3,4
1Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience,
University of Nebraska-Lincoln, Lincoln, Nebraska 68588, USA
2Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA
3Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
4School of Natural Sciences, Far Eastern Federal University, Vladivostok 690950, Russia
(Received 9 January 2016; accepted 18 March 2016; published online 30 March 2016)
Domain wall dynamics in a magnetoelectric antiferromagnet is analyzed, and its implications for
magnetoelectric memory applications are discussed. Cr 2O3is used in the estimates of the materials
parameters. It is found that the domain wall mobility has a maximum as a function of the electric
field due to the gyrotropic coupling induced by it. In Cr 2O3, the maximal mobility of 0.1 m/(s Oe)
is reached at E/C250:06 V/nm. Fields of this order may be too weak to overcome the intrinsic
depinning field, which is estimated for B-doped Cr 2O3. These major drawbacks for device imple-
mentation can be overcome by applying a small in-plane shear strain, which blocks the domainwall precession. Domain wall mobility of about 0.7 m/(s Oe) can then be achieved at E¼0.2 V/nm.
A split-gate scheme is proposed for the domain-wall controlled bit element; its extension to
multiple-gate linear arrays can offer advantages in memory density, programmability, and logicfunctionality.
VC2016 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4944996 ]
Encoding and manipulation of information by the anti-
ferromagnetic (AFM) order parameter have recentlyattracted considerable attention,
1–4and current-induced
switching of a metallic antiferromagnet has been demon-strated.
5Device concepts utilizing a magnetoelectric antifer-
romagnet (MEAF) as the active element are also being
actively pursued for applications in nonvolatile memory andlogic.
6–8The fundamental principle of operation involves the
reversal of the AFM order parameter in the MEAF byapplied voltage in the presence of an external magnetic field,which is accompanied by the reversal of the boundary mag-netization of the MEAF.
7,9,10Little is known, however,
about the fundamental limitations of this approach. Here we
discuss the switching mechanisms, describe the dynamics ofa moving domain wall, estimate the relevant metrics, andpropose a scheme of a memory bit.
We consider the usual case of a collinear MEAF, such
as Cr
2O3, with two macroscopically inequivalent AFM
domains, mapped one onto the other by time reversal. Thedriving force for the switching of such a MEAF is the differ-ence F¼2E^aHin the free energy densities of the two AFM
domains, where ^ais the magnetoelectric tensor.
11Thermally
activated single-domain switching involves a severe tradeoff
between thermal stability and switching time—a long-standing problem in magnetic recording technology.
12
In order to significantly reduce the activation barrier forsingle-domain switching, the applied fields should satisfyaEH/C24K, where Kis the magnetocrystalline anisotropy con-
stant. In Cr
2O3, where a/H1135110/C04(Gaussian units) and K/C252
/C2105erg/cm3,13,14this condition requires EH/C241011Oe
/C2V/cm. Since fields of this magnitude are undesirable for
device applications, we are led to consider inhomogeneousswitching, which involves nucleation of reverse domains anddomain wall motion. The switching time is determined bythe slower of these two mechanisms. Nucleation is arelatively slow thermally activated process, which can be
avoided by device engineering, as discussed below. Theswitching time is then limited by the domain wall motion
driven by the magnetoelectric pressure F.
The magnetic dynamics in an AFM is qualitatively differ-
ent from that in a ferromagnet (FM).
15–17If the magnetostatic
interaction is neglected, a domain wall in an ideal FM with no
damping does not move at all, but rather precesses in theapplied magnetic field. The FM domain wall velocity vin this
case is proportional to the small Gilbert damping parameter a
0.
The magnetostatic interaction lifts the degeneracy of the Bloch
and N /C19eel configurations and blocks the precession, making
v/a/C01
0as long as vdoes not exceed the Walker breakdown
velocity vW.20In contrast, in an AFM the Gilbert damping lim-
its the terminal velocity of the wall. Here we are interested in
the dynamics of a domain wall in a MEAF, such as Cr 2O3,
which is driven by the application of electric and magneticfields. In a finite electric field, a MEAF turns into a nearly
compensated ferrimagnet. As we will see below, the existence
of a small magnetization has im portant consequences for do-
main wall dynamics and has to be taken into account.
We restrict our discussion to the longitudinal magneto-
electric response, in which the magnetization induced bythe electric field is parallel to the AFM order parameter,irrespective of its spatial orientation. This is the case for
the exchange-driven mechanism
21–23of magnetoelectric
response, which dominates in Cr 2O3and many other MEAFs
at temperatures that are not too low. In Cr 2O3, the only non-
zero component of the magnetoelectric tensor in this approx-
imation is ak¼azz, where zlies along the rhombohedral
axis.22,23It is assumed that the electric field is applied across
an epitaxially grown (0001) film.
Adding the Berry-phase and magnetoelectric terms to
the AFM Lagrangian,15–17we can write the Lagrangian den-
sity of a MEAF, valid at low energies, as
0003-6951/2016/108(13)/132403/5/$30.00 VC2016 AIP Publishing LLC 108, 132403-1APPLIED PHYSICS LETTERS 108, 132403 (2016)
L¼ 2/C15JanðÞ/C1_nþ1
2qj_nj2/C0Ajrnj2/C0K abnanb/C16/C17
/C02/C15JcH/C1n; (1)
where nis the unit vector in the direction of the AFM order
parameter (staggered magnetization) L¼ðM1/C0M2Þ=2;M1
andM2are the sublattice magnetizations, J¼ L=ð2cÞis the
angular momentum density on one sublattice, qthe effective
inertia density, Athe exchange stiffness, and Kabthe magne-
tocrystalline anisotropy tensor.18Unless noted otherwise, it
is assumed that the only nonzero component of this tensor is
Kzz¼/C0 K <0. In the first and last terms, /C15¼ðM1/C0M2Þ=
ðM1þM2Þ¼akE=L, and aðnÞis the vector potential of a
magnetic monopole, rn/C2a¼n; this term is the Berry-phase
contribution from the small longitudinal magnetization
M¼ðM1þM2Þ=2 induced by the electric field.17,19The last
term in Eq. (1)is the magnetoelectric energy density;11cis
the gyromagnetic ratio.
The AFM field theory at E¼0 has characteristic scales
of time, length, and pressure
t0¼ffiffiffiffiffiffiffiffiffi
q=Kp
;k0¼ffiffiffiffiffiffiffiffiffi ffi
A=Kp
;/C15 0¼ffiffiffiffiffiffiffi
AKp
; (2)
which have direct physical meaning. /C150is the scale of the do-
main wall energy per unit area. The magnon dispersion xðkÞ
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
x2
0þs2k2p
has a gap x0¼1=t0and velocity s¼k0=t0.
In Cr 2O3x0¼0:68 meV,24hence t0/C251 ps. The magnon ve-
locity is s¼12 km/s.24The length parameter k0¼st0sets
the scale of the domain wall width d.I nC r 2O3we find
k0¼12 nm and d¼pk0/C2538 nm.
The effective Lagrangian for low-energy domain wall
dynamics is obtained by inserting the domain wall profile
coshxðÞ¼tanhx/C0X
k0;/xðÞ¼U; (3)
parameterized by the collective variables XandU, in Eq. (1)
and taking the integral over all space. For the MEAF domain
wall this leads to
L¼1
2M_X2þ1
2I_U2þG_XU/C0VX;UðÞ ; (4)
where M¼2q=k0andI¼2qk0are the mass and moment of
inertia per unit area of the wall, Vis the potential energy of
the wall, which in a uniaxial AFM has no dependence on U,
andG¼4/C15Jis the gyrotropic term coupling the motion of
the wall to its precession, which is proportional to E.
The equations of motion for the collective coordi-
nates17,25are
M€X¼/C0G_U/C0CXX_XþF;
I€U¼G_X/C0CUU_Uþs; (5)
where CXX¼4a0J=k0andCUU¼4a0Jk0are the viscous
drag coefficients proportional to the Gilbert damping param-etera
0, and F¼/C0@V=@X¼2akEzHz¼2/C15LHz. The torque
s¼/C0@V=@Uvanishes in the case of uniaxial anisotropy.
We will first consider the case s¼0 and then address the
role of broken axial symmetry.
AtG¼0 we have a conventional AFM domain wall,
which behaves as a massive particle subject to viscous drag,and whose angular collective variable Uis completely pas-
sive.17However, the gyrotropic coupling Ginduced by the
electric field generates precession of the moving domainwall, which generates additional dissipation. In the steadystate the moving domain wall precesses with the angular fre-quency X¼G_X=C
UU, and the linear velocity of the wall is
v¼F
CXXþG2=CUU: (6)
Thus, the additional dissipation induced by the gyrotropic
coupling reduces the terminal velocity of the domain wall bythe factor 1 þG
2ðCXXCUUÞ/C01.
Substituting the expressions for CXX,CUU, and Gin
Eq.(6), we obtain
v¼2/C15=a0
1þ/C15=a0ðÞ2vmax; (7)
where vmax¼cHzk0=2. The maximum velocity vmaxof the
domain wall is reached at the optimal electric field strengthE
maxcorresponding to /C15¼a0. Interestingly, vmaxdepends
neither on the magnetoelectric coefficient nor on the Gilbertdamping constant.
Using the value c¼1:76/C210
7s/C01/G and a reasonable
field Hz¼100 Oe, we find vmax/C2510:6 m/s. Assuming the
switchable bit size of 50 nm, we estimate the switching timeof about 5 ns. Note that the maximal MEAF domain wallmobility v
max=Hz/C250:1 m/(s Oe) is 2–3 orders of magnitude
smaller in this regime compared to ferromagnets, such aspermalloy.
26
The Gilbert damping constant can be determined from the
relation T¼q=ð2a0JÞ,w h e r e Tis the relaxation time.17To
estimate Tin Cr 2O3, we use the width of the AFM resonance
DH¼900 Oe,14which translates into Dx¼1:6/C21010s/C01
andT¼1=Dx/C2560ps. Using the value K¼2/C2105erg/cm3,14
we find the inertia density q¼2Kt2
0/C254/C210/C019g/cm. The
value of Jis obtained from the local magnetic moment29
2.76lBand volume X/C2550A˚3per formula unit. Putting these
estimates together, we obtain a0/C252/C210/C04.
The relation /C15¼a0then gives Emax/C2560 V/ lmi n
Cr2O3, where we used the peak value ak/C2510/C04reached at
260 K. The magnetoelectric pressure corresponding to E¼
Emaxand Hz¼100 Oe is Fmax¼2a0LHz/C2540 erg/cm3.T o
put this value in perspective, we note that in ferromagneticiron the magnetic field of 100 Oe exerts a pressure of about3/C210
5erg/cm3on the domain walls. The “loss” of four
orders of magnitude in a MEAF is due to the small magni-tude of the magnetic moment induced by the electric field.Alternatively, one can say that a 100 Oe coercivity in anMEAF at E/C24E
maxis equivalent, assuming similar material
quality, to a 10 mOe coercivity in iron. Thus, it is clear thatreasonably fast switching of an MEAF with uniaxial anisot-ropy requires samples of very high quality, unless the tem-perature is close to the N /C19eel point T
Nwhere the domain wall
width diverges and the coercivity becomes small even in
low-quality samples. Indeed, isothermal MEAF switchinghas so far been observed only close to T
N.7
In the presence of lattice imperfections, switching is
possible if the magnetoelectric pressure Fapplied to the132403-2 Belashchenko et al. Appl. Phys. Lett. 108, 132403 (2016)
domain wall exceeds the depinning pressure Fc. Since
TN¼307 K of Cr 2O3is too low for passively cooled com-
puter applications, it needs to be either doped or strained toincrease its T
N. In particular, boron doping on the Cr sublat-
tice has been shown to raise TNsignificantly.30,31Random
substitutional disorder in a doped material leads to an intrin-sic pinning potential and nonzero coercivity. Let us estimatethe effective depinning pressure for this representative case.
For simplicity, we assume that B dopants modify the
exchange interaction locally but do not strongly affect themagnetocrystalline anisotropy. According to Ref. 30, boron
doping enhances the exchange coupling for the Cr atoms thathave a B neighbor by a factor of 2–3. The concentration of Batoms is n¼3x=X, where xis the B-for-O substitution con-
centration. Therefore, we make a crude estimate that theexchange stiffness Ais enhanced by a factor of 2 in regions
of volume 2 X, whose concentration is n.
Leta
/C3be the radius of a sphere with volume 2 X. The
force acting on the domain wall from the vicinity of oneB atom is f/C24ða
/C3=k0Þ3A. The typical pinning force on a
portion of the domain wall of size R2then becomes fpin
/C24ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
nk0R2f2p
. The typical correlation length for the domain
wall bending displacement is the Larkin length Rc,32–35
which is found by equating fpinto the typical elastic force
fel/C24uffiffiffiffiffiffiffi
AKp
produced by the domain wall, where u/C24k0cor-
responds to the situation in which the domain wall deforms
weakly. This gives Rc/C24ffiffiffiffiffiffiffiffi
k0AK
nf2q
. The depinning threshold
can then be estimated as Fc/C24A=R2
c¼nk0Að2X=k3
0Þ2. Using
x¼0.03 and A/C2410/C06erg/cm, we find Fc/C2410 erg/cm3,
which is comparable to the magnetoelectric pressure atH¼100 Oe and E¼E
max, as we have estimated above.
Other imperfections may further increase Fc. Thus, as
expected from the comparison with typical ferromagnets,even weak pinning associated with homogeneous doping canimpede MEAF switching. This sensitivity to lattice disorder,along with the low upper bound on the domain wall mobility,presents serious challenges for the implementation of mag-netoelectric devices.
We will now show that both of these limitations can be
overcome by introducing a relatively small in-plane anisot-ropy component K
yy¼K ?in addition to the axial compo-
nentKzz¼/C0 K . Such in-plane anisotropy can be induced by
applying a small in-plane shear strain to the magnetoelectric
crystal, for example, by using a piezoelectric element, an
anisotropic substrate, or anisotropic thermal expansion in apatterned structure. The physics of domain wall motion atK
?6¼0 is similar to Walker breakdown in ferromagnets,
where the anisotropy with respect to Uappears due to the
magnetostatic interaction.20
In the equations of motion (5)we now have, after integrat-
ing out the domain wall profile (3), a nonzero torque
s¼/C0k0K?sin 2Uper unit area. There is a steady-state solu-
tion with _U¼0a n d v¼F=CXX,a sl o n ga s v<vW,w h e r e
vW=vmax¼2ðK?=FmaxÞ1=2is analogous to the Walker break-
down velocity.20For example, in order to achieve vW
/C241 0 0m / s ,w en e e dt oh a v e K?/H11407900 erg/cm3, which is three
orders of magnitude smaller than K. It is likely that K?of this
order can be achieved with a fairly small in-plane shear strain.Below the Walker breakdown the domain wall velocity
is linear in E:v=vmax¼2E=Emax.A tF>CXXvWthe in-plane
anisotropy can no longer suppress domain wall precession,
so that its velocity becomes oscillatory. The average velocityhas a cusp at F¼C
XXvWand declines with a further increase
inF, as shown in Fig. 1.
In the presence of K?/H11407900 erg/cm3the fields
E/C250:2 V/nm and H/C25100 Oe result in v/C2570 m/s and
F/C25140 erg/cm3. Under these conditions the switching time
of a nanoscale bit can be well below a nanosecond, while themagnetoelectric pressure Fexceeds the intrinsic depinning
field of B-doped Cr
2O3by an order of magnitude. Clearly,
the imposition of in-plane anisotropy offers compellingadvantages for device applications by improving switchabil-
ity and speed.
It is interesting to note that the domain wall mobility
can be changed by orders of magnitude by imposing a non-zeroK
?in the strong-electric-field regime /C15/C29a0. This
peculiar feature of MEAF domain wall dynamics can be
directly checked experimentally.
Devices based on MEAF switching offer a distinct
advantage in terms of energy efficiency. Energy dissipated
when a bit is switched is Edis¼2akEzHzV¼FV, where Vis
the switched volume. This is the energy difference between
the two AFM domain states of the bit. Taking the switching
volume to be a cube with a 50 nm edge, we estimate Edis
/C2410/C014erg for the field magnitudes chosen above. This cor-
responds to an upper limit on the intrinsic power consump-
tion of 1 mW/Gbit, assuming that each bit is switched everynanosecond. Clearly, energy dissipation in a magnetoelectric
memory device would be dominated by losses in the external
circuitry.
As we argued above, fast memory operation should be
based on domain wall-mediated switching. Therefore, it is nec-
essary to design the architecture of a bit in such a way that the
domain wall is not annihilated at the surface as the bit isswitched. One way to achieve this is through the use of a
multiple-gate scheme, as shown in Fig. 2. In this scheme, addi-
tional “set gates” are used to initialize and maintain two differ-ent AFM states at the edges of the active magnetoelectric
layer, which are labeled þand/C0in Fig. 2, thereby trapping
the domain wall inside the device. The set gates need to beactivated only during the write operation, along with the con-
trol gate. Positive or negative v oltage applied to the control
gate selects the AFM domain state in the switched area anddrives the domain wall between the positions shown in the two
panels of Fig. 2. This scheme is somewhat reminiscent of the
spin-transfer torque domain wall device.
27The control gate
FIG. 1. Average domain wall velocity vas a function of E=EmaxatK?¼0
(dotted blue line), 4 Fmax(dashed green), and 16 Fmax(solid).132403-3 Belashchenko et al. Appl. Phys. Lett. 108, 132403 (2016)
can also provide the memory re ad function by employing an
FM layer, coupled via the bou ndary magnetization of the
MEAF to its AFM domain state, and a spin valve or a similarmagnetoresistive element grown on top of it. Alternatively, the
AFM domain state can be detected through the anomalous
Hall effect in a thin non-magnetic control gate.
28
Since the domain wall should fit inside the bit, its width
dsets a limitation for the downward scaling of the length of
the MEAF element. The width of this element, however, can
be significantly smaller. To facilitate downscaling, the do-
main wall width dcan be reduced by increasing the magne-
tocrystalline anisotropy of the MEAF. For example, it isknown that the addition of Al increases Kin Cr
2O3.36
To increase the memory density, the basic element shown
in Fig. 2may be assembled in a linear array, for example, by
using a sequence of gates like þC/C0CþC…, where þand/C0
denote the set gates and C is the control gate. In this way,
each internal set gate protects the domain walls on both sides,
and for a long array the footprint reduces from 3 to 2 gatesper bit. Alternatively, the use of several control gates in
sequence allows for more than two positions for each domain
wall and leads to memory density ðlog
2nÞ=nbits per gate,
where nis the number of control gates in a sequence. The
memory density is lowest for n¼3 but the gain compared to
n¼2o r n¼4 is only about 6%. If all gates are made identi-
cal, a linear array offers an additional possibility for reprog-
ramming, i.e., for designating different gates as þand/C0set
gates; this could be implemented by applying sufficiently longvoltage pulses to the new set gates to allow reliable switching.
Using the bottom electrode, or sections of it, for magnetic
readout could also allow for additional majority-gate function-
ality. Thus, a multiple split-gate architecture could provide
combined memory and logic capabilities.
To conclude, we have described the domain wall dynam-
ics in a magnetoelectric antiferromagnet and discussed its
implications for magnetoelectric memory applications. We
found that the domain wall mobility v/Hin a uniaxial magneto-
electric antiferromagnet reaches a maximum at a certain elec-
tric field E
maxand then declines, which is unfavorable for
device applications. Howev er, the domain wall mobility and
switchability can be greatly improved by imposing a small in-
plane anisotropy, which blocks the domain wall precession,
and using electric fields E/C240:2 V/nm. A split-gate architec-
ture is proposed to trap the domai n wall inside the bit element.
A linear gate array extending this architecture can offer advan-
tages in memory density, programmability, and logic function-ality integrated with nonvolatile memory. While the domain-
wall-driven mechanism allows reliable and fast switching, it
limits the minimum length of the bit to the domain wall width.We thank Christian Binek and Peter Dowben for useful
discussions and the Kavli Institute for Theoretical Physics
for hospitality. K.B. acknowledges support from the
Nanoelectronics Research Corporation (NERC), a wholly-
owned subsidiary of the Semiconductor Research
Corporation (SRC), through the Center for NanoFerroic
Devices (CNFD), an SRC-NRI Nanoelectronics Research
Initiative Center, under Task ID 2398.001. K.B. and A.K.
acknowledge support from the NSF through the Nebraska
Materials Research Science and Engineering Center(MRSEC) (Grant No. DMR-1420645). A.K. acknowledges
support from the DOE Early Career Award No. DE-
SC0014189. O.T. acknowledges support from the DOE
Office of Basic Energy Sciences, Division of Materials
Sciences and Engineering under Award No. DE-FG02-
08ER46544. O.A.T. acknowledges support by the Grants-
in-Aid for Scientific Research (Nos. 25800184, 25247056
and 15H01009) from the MEXT, Japan and SpinNet. This
research was also supported in part by NSF under GrantNo. PHY11-25915.
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|
1.4993433.pdf | Chopping skyrmions from magnetic chiral domains with uniaxial stress in magnetic
nanowire
Yan Liu , Na Lei , Weisheng Zhao , Wenqing Liu , Antonio Ruotolo , Hans-Benjamin Braun , and Yan Zhou
Citation: Appl. Phys. Lett. 111, 022406 (2017); doi: 10.1063/1.4993433
View online: http://dx.doi.org/10.1063/1.4993433
View Table of Contents: http://aip.scitation.org/toc/apl/111/2
Published by the American Institute of PhysicsChopping skyrmions from magnetic chiral domains with uniaxial stress
in magnetic nanowire
Ya nLiu,1NaLei,2,3Weisheng Zhao,2,3Wenqing Liu,4Antonio Ruotolo,5
Hans-Benjamin Braun,6,a)and Y an Zhou7,a)
1College of Sciences, Northeastern University, Shenyang 110819, People’s Republic of China
2Fert Beijing Institute, Beihang University, Beijing 100191, China
3School of Electronic and Information Engineering, Beihang University, Beijing 100191, China
4Department of Electronic Engineering, Royal Holloway University of London, Egham, Surrey TW20 0EX,
United Kingdom
5Department of Physics and Materials Science, City University of Hong Kong, Hong Kong, China
6UCD School of Physics, University College Dublin, Dublin 4, Ireland
7School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen 518172, China
(Received 27 April 2017; accepted 28 June 2017; published online 12 July 2017)
Magnetic skyrmions are envisioned as ideal candidates as information carriers for future spintronic
devices, which have attracted a great deal of attention in recent years. Due to their topologicalprotection, the creation and annihilation of magnetic skyrmions have been a challenging task.
Here, we numerically demonstrate that a magnetic skyrmion can be created by chopping a chiral
stripe domain with a static uniaxial strain/stress pulse. This mechanism not only provides a methodto create skyrmions in magnetic nanostructures but also offers promising routes for designing
tunable skyrmionic-mechanic devices. Published by AIP Publishing.
[http://dx.doi.org/10.1063/1.4993433 ]
Magnetic skyrmions are stable spin textures with the
magnetic moments forming a twist structure, with the key
characteristic that they are topologically protected. After the
first experimental discovery of magnetic skyrmions in MnSi,
1
they have been observed in a variety of magnetic systemsincluding B20-type materials with a bulk Dzyaloshinskii-
Moriya interaction (DMI)
1–7and interfacial symmetry break-
ing multilayers with interfacial DMI.8–12Stability, small size,
and ultralow threshold current density for motion make the
skyrmions very promising for future spintronic device
applications.13–19
As information carriers, the controllable creation of sky-
rmions is prerequisite for real applications. The nucleation of
isolated skyrmions is a challenging task as it necessarily
involves overcoming the topological barrier, e.g., due to lat-
tice discreteness effects.20Several different methods have
been proposed to create individual skyrmions. They can be
created by providing external stimuli to a ferromagneticstate, such as applying a circulating current,
21injecting an
in-plane, out-of-plane polarized current,13,22–24using laser
heating,25or by spin waves.26In addition, skyrmions can be
created by injecting an in-plane current into a notch27and be
converted from a domain wall.28Very recently, Jiang et al.
have shown experimentally that skyrmions can be generated
by pushing chiral stripe domains (CSDs) via a spin current
through a geometrical constriction in heavy metal/ultrathin
ferromagnet/insulator trilayers.29
On the other hand, strain has been used to manipulate
magnetic domains in soft magnetostrictive materials.30–35
Recently, experiments have demonstrated that mechanical
strain or stress is an effective means to control the skyrmionphase.36–38In this letter, we will show that it is possible to
cut a skyrmion from a chiral stripe domain (CSD) by apply-ing an in-plane uniaxial strain. It is well known that the
application of a static uniaxial strain to a magnetostrictive
material can induce an additional uniaxial anisotropy to themagnetic energy.
32The resulting magneto-elastic anisotropy
field can be expressed as
Hstrain¼/C03eiEkðm/C1^nsÞ^ns; (1)
where eiis the uniaxial strain with i¼xandy,^nsis the direc-
tion of the strain, Eis the Young’s modulus, mis the unit
vector of the magnetization, and kis the magnetostriction
constant. This effective field acts on the CSD, modifies themagnetic free energy, and leads to a reorientation of the
magnetization. Figures 1(a)–1(c) schematically show the
mechanism of evolution of a N /C19eel CSD to a skyrmion under
a uniaxial strain along the xaxis. The in-plane component of
the magnetization in the CSD is mainly along the y-axis. In
contrast, the magneto-elastic anisotropy field is along the x
axis [Fig. 1(a)]. Thus, H
strain drags the magnetization to
rotate to the xaxis, which gradually forms a bottleneck area
at the place near that domain end [Fig. 1(b)]. If the strain is
large enough, the strain anisotropy energy is larger than theelastic energy, the CSD will break at the bottleneck area,which results in a circular domain breaking off from theCSD, and finally it evolves to a skyrmion [Fig. 1(c)]. A simi-
lar mechanism can be realized in a Bloch CSD by applying a
uniaxial strain along the ydirection [Figs. 1(d)–1(f) ].
To demonstrate the aforementioned idea, we study the
geometry as shown in Fig. 2for generating skyrmions by an
in-plane uniaxial strain. The sample we considered is a 1.1-
nm-thick ferromagnetic nanostripe with a width of 120 nm
and a length of 1000 nm with a geometrical constriction at
a)Author to whom correspondence should be addressed: beni.braun@ucd.ie
or zhouyan@cuhk.edu.cn
0003-6951/2017/111(2)/022406/5/$30.00 Published by AIP Publishing. 111, 022406-1APPLIED PHYSICS LETTERS 111, 022406 (2017)
the center. The length and width of the constriction are fixed
to be 100 nm and 82 nm, respectively. We assume that theDMI in the ferromagnetic layer is either of bulk or interfacial
type, which will generate Bloch and N /C19eel skyrmions, respec-
tively. The left region of the sample (chiral stripe region) ini-tially supports a CSD, while the ferromagnetic background
is presented in the right region (skyrmion region). The CSD
can be injected into the left hand part by injecting an in-plane current. The constricted region in the center (strain
region) is attached to the commonly used piezoelectric mate-
rial Pb(Zr,Ti)O
3(PZT). A perpendicular magnetic field Hzis
applied to the whole sample. The PZT is polarized along the
x-direction for the interfacial type [Fig. 2(a)] and the y-direc-
tion for the bulk type [Fig. 2(b)]. By applying a voltage to
the PZT, an in-plane uniaxial piezostrain is induced, and
then, this piezostrain is transferred to the ferromagnetic
layer. This strain transfer from a PZT layer to ferromagneticlayers has been experimentally demonstrated in many
works.
30,31,33Here, the in-plane uniaxial piezostrain is
assumed to be uniform in the strain region and vanishing out-side this region. The strain is directed along the x-direction
for the ferromagnet with interfacial DMI and along the y-
direction for the ferromagnet with bulk DMI. In addition tothe strain, an in-plane spin-polarized current is applied to
drive the CSD or the skyrmions to move forward. Thegeometrical constriction in the strain region is set for restrict-
ing the CSD in the chiral stripe region without expanding it
into the skyrmion region.
The study is carried out by means of the public object-
oriented micromagnetic framework (OOMMF) code.
39The
time-dependent magnetization dynamics is governed by the
Landau-Lifshitz-Gilbert (LLG) equation
dm
dt¼/C0 j cjm/C2Heffþam/C2dm
dt/C18/C19
; (2)
where cis the gyromagnetic ratio and ais the Gilbert damp-
ing constant. The effective magnetic field is given byH
eff¼/C01=ðl0MsÞð@W=@mÞ, where l0is the vacuum per-
meability and Wis the magnetic energy of the system con-
sisting of exchange, anisotropy, Dzyaloshinskii-Moriyainteraction (DMI),
40and strain-induced anisotropy energy
terms. The DMI interaction is assumed to be of bulk or inter-
facial origin. The bulk DMI energy density is given byw
DM¼Dm/C1ð r/C2 mÞ, where Dis the DM interaction con-
stant. The interfacial DMI energy density is
wDM¼/C0Dm/C1ð ð^z/C2r Þ/C2 mÞ. The strain-induced anisot-
ropy energy, which originates from the uniaxial strain
applied to the film, can be written as34
Wstrain¼/C03
2eiEkm/C1^ns ðÞ2: (3)
For the simulation of the current in-plane injection to
the nanostripe, the spin-transfer torque (STT) including bothadiabatic and non-adiabatic terms was added to Eq. (2), with
s
STT¼lm/C2@m
@x/C2m/C18/C19
þbl@m
@x/C2m/C18/C19
; (4)
where l¼lBJP=eMs,Msis the saturation magnetization, J
is the current density, Pis the spin polarization, lBis the
Bohr magneton, eis the electron charge, and bis the non-
adiabaticity factor. Because the constriction of the strain
region, the current density is inhomogeneous. The currentdensity we mentioned in this paper is the current density in
the wide region, and the current density in the strain region
isJ
s¼JW=W1, where Jis the current density in the wide
region and WandW1are the width of the wide region and
the strain region, respectively.
Material parameters used for the simulations are set to
represent Co 20Fe60B20. According to the measurement
FIG. 2. Design of the skyrmion gener-
ation device. (a) N /C19eel skyrmion gener-
ation in a ferromagnet/heavy metal
bilayer system, where the interfacial
symmetry breaking introduces the
interfacial DMI. (b) Bloch skyrmion
generation in a ferromagnetic layer
with a bulk DMI. The arrows represent
the in-plane component of the magne-tization, and the colors encode the z
component. The color-scale has been
used throughout this paper.
FIG. 1. Schematic of the cutting effect of a static uniaxial strain on a CSD
and the formation of a skyrmion. (a)–(c) Evolution of a N /C19eel CSD under a
uniaxial strain ex. The blue region indicates a CSD, and the light orange
region indicates the ferromagnetic background. The white arrows show the
in-plane magnetization direction of the CSD. The yellow arrows show thedirection of the effective strain anisotropy field. (d)–(f) Evolution of a Bloch
CSD under a static uniaxial strain e
y.022406-2 Liu et al. Appl. Phys. Lett. 111, 022406 (2017)results in Ref. 29, we use the following parameters: A satura-
tion magnetization of Ms¼6:5/C2105A=m, an exchange
constant of A¼4:8/C210/C012J=m, and a perpendicular anisot-
ropy constant of Ku¼2:3/C2104J=m3. In this work, we set
the DMI constant to D¼1:0/C210/C03J=m2. From Ref. 41,w e
have k¼31 ppm and E¼1:6/C21011J=m3. The sample is
discretized into small cells with a cell size of2/C22/C21:1n m
3, the Gilbert damping constant is a¼0:1,
the non-adiabaticity factor is b¼0:1, and the degree of spin
polarization is P¼0.5.
It should be mentioned that the demagnetization energy
or the magnetic dipole interaction is neglected in our model.
In thin-plate magnets, magnetic skyrmions are often realized
by competition among the magnetostatic energy, theexchange interaction, and the uniaxial magnetic anisot-
ropy.
42–45For our model, comparing the results of with and
without including the demagnetization energy, the influenceof the demagnetization energy approximately reduces to a
perpendicular uniaxial anisotropy with anisotropy constant
K
eff¼0:5l0M2
s. Therefore, the demagnetization energy only
quantitatively affects our results without any qualitative
changes.
Figure 3(a)shows the N /C19eel skyrmion generation process
via the application of a strain pulse to a CSD (see also the
Multimedia view). The CSD in the chiral stripe region is first
driven to the strain region by the in-plane current(t¼0–2.5 ns). At t¼2.5 ns, a uniaxial strain e
xalong the x
direction is applied. Under the effect of the strain, the right
end of the CSD bends upwards, forming a depression in the
bending region ( t¼3.0 ns). The width of the CSD in the
bending region is gradually compressed, and then, a small
domain breaks down from the CSD ( t¼3.5 ns).
Subsequently, the small domain evolves to a skyrmion(t¼4.1 ns). For the demonstration of the application of
strain-induced skyrmion generation in the nanostripe with a
bulk DMI, we use the same parameters to simulate the Bloch
skyrmion generation process. Figure 3(d) shows the Bloch
skyrmion generation process by applying a uniaxial strain
along the ydirection (see also the Multimedia view). The
process is similar to the case of Neel skyrmion generation.
The generation of a skyrmions can also be confirmed by ana-lyzing the topological number of the system. In a planar sys-
tem, the topological number of the system is given by
S¼
1
4pP
x;yqðx;yÞ, where qðx;yÞ¼m/C1ð@xm/C2@ymÞis the
topological density of the cell with a coordinate ( x, y).
Figure 3(b)shows the development of S. Before applying the
strain, the value of Sis 0.5, which is the topological number
of the half skyrmion at the right end of CSD. At t¼3.5 ns, S
suddenly increases to about 1.5. This increment indicates
that one skyrmion is created.
We have confirmed that both N /C19eel and Bloch skyrmions
can be created by cutting CSD with a uniaxial strain.
However, we are interested in the detailed dynamical process
to understand how a skyrmion is created. Figure 4shows the
temporal evolution of the topological density and energy
density together with the magnetization profile within the
dashed square in Fig. 3(a). After the strain is switched on,
like a uniaxial anisotropy along the xdirection acting on the
magnetization, the magnetization rotates towards the xaxis.
This causes the half skyrmion at the right end of the CSD to
bend upward, which induces the formation of a topological
“dip” in the bending region ( t¼3.1 ns). The topological den-
sity of the “dip” is negative and increases gradually. With
the development of the “dip,” a small domain is graduallyseparated from the CSD. At t¼3.47 ns, the small domain
breaks off from the CSD. Interestingly, the small domain has
FIG. 3. The magnetization evolution of
the skyrmion generation is obtained by
applying a strain pulse and an in-plane
current with current density J¼80
MA/cm2,w h e r e Hz¼70 mT. (a)–(c) A
N/C19eel skyrmion is generated by applying
a 1 ns-duration strain pulse exwith a
strength of 0.8% at t¼2.5 ns. (d) and
(e) A Bloch skyrmion is generated by
applying a 1 ns-duration strain pulse ey
with a strength of 0.8% at t¼2.3 ns. (a)
and (d) The snapshots of the magnetiza-
tion configuration at the indicated times.
(b) and (e) The time evolution of the
topological number S. (c) and (f) The in-
plane current and strain pulse applied
during the simulation. (Multimedia
view) [URL: http://dx.doi.org/10.1063/
1.4993433.1 ][ U R L : http://dx.doi.org/
10.1063/1.4993433.2 ]022406-3 Liu et al. Appl. Phys. Lett. 111, 022406 (2017)a peculiar spin structure with a topological number equiva-
lent to zero. One end of it is a half skyrmion, and the other
end is a half anti-skyrmion. The chirality of the half sky-rmion is favored by the DMI, while the half anti-skyrmion isunfavored by the DMI, which increases the energy (indicatedby the green region). Thus, the magnetization tends to be ori-
ented towards the direction favored by the DMI, which
causes the half anti-skyrmion to be compressed into a smallregion that concentrates a large amount of energy[t¼3.47–3.564 ns]. At t¼3.564 ns, the size of the half anti-
skyrmion is compressed to be comparable to a Bloch point ina classical bubble material, and the topological density of
this area reverses under the emission of spin waves
(t¼3.564–3.586 ns), which switches the half anti-skyrmion
to a half skyrmion. The switching increases the topologicalnumber of the small domain to 1 that corresponds to thetopological number of a single skyrmion, in which the spinswrap once around a unit sphere.
20Finally, the small domain
eventually evolves to be a proper skyrmion ( t¼3.6 ns). The
switching from a half anti-skyrmion to a half skyrmion isaccompanied by a change in topological charge, whichincreases by an integer DS¼1. In general, the mechanism
of aDS¼1 topological change entails the formation of a
singularity in the continuum limit. In thick materials, it evolves
the injection and propagation of a Bloch point vertically
through the finite thickness of the sample. In two-dimensionalsamples, a Bloch point cannot exist, an equivalent process hasbeen mentioned,
13,19,46where the singularity is simply formed
by four neighboring moments, and the “injection and prop-
agation” of the singularity then simply amount to the collec-
tive switching of typically three neighboring moments fromplus to minus orientation.
13In our case, the nanostripe is just
1.1 nm thick, and it can be treated effectively as a two-dimensional thin film. So, the switching from a half anti-
skyrmion to a half skyrmion is similar to the process in Ref.13, which evolves the “injection and propagation” of the
singularity.
For this skyrmion generation mechanism, current den-
sity and strain are two important factors. Figure 5shows a
final phase diagram after applying a 3-ns pulse of strain to
the strain region. There exists a threshold current of J
s¼40
MA/cm2and a threshold strain of es¼0.67% for generating
skyrmions from a CSD. Below this threshold current, the
CSD stays in the strain region or is even forced to return tothe chiral stripe region. However, too large a current forcesthe half skyrmion at the end of the CSD to move too quickly
FIG. 4. Detailed dynamics of skyrmion
generation. The top panels display the
snapshot images of the topological
density within the dashed square in
Fig. 2(a) taken at the indicated times
from the perspective views. The mid-dle panels show the corresponding
energy density, and the bottom panels
present the corresponding top view of
the magnetization distribution.
(Multimedia view) [URL: http://
dx.doi.org/10.1063/1.4993433.3 ]
FIG. 5. Effect of the current and strain on the magnetization dynamics. The
left panel shows the final phase diagram after applying a 3-ns pulse of strain
to the strain region, where the current is always turned on, and Hz¼70 mT.
Four different end states are identified (right panel): the CSD is forced to go
back to the chiral stripe region (red square), the CSD stays in the strainregion (hollow circle), the CSD moves to the skyrmion region without being
cut (blue square), and skyrmions form (blue square with red circle).022406-4 Liu et al. Appl. Phys. Lett. 111, 022406 (2017)such that the CSD extends to the skyrmion region without
being cut. When the strain is below the threshold value, the
forces acting on the CSD from the stain are not sufficient to
cut a small domain off the CSD. In contrast, too large strainmay force the CSD to return to the chiral stripe region. Also,
the shape of the strain pulse influences the generation of sky-
rmions. If the strain pulse duration is too short, the cuttingprocess cannot be finished, and so, the skyrmion cannot be
generated. If the strain pulse duration is long enough, the
skyrmion can be created one by one.
Our results have demonstrated that uniaxial strain can
be an effective way to generate skyrmions of both N /C19eel and
Bloch types. This intriguing approach provides a mechanism
to generate skyrmions. In the future, it could be used in
advanced skyrmionic devices, for example, skyrmion race-track memory devices.
47,48
Y.Z. acknowledges support from the National Natural
Science Foundation of China (Project Nos. 1157040329,
11404053, and 11274261) and Shenzhen FundamentalResearch Fund under Grant No. JCYJ20160331164412545.
Na Lei thanks the funding support of National Natural
Science Foundation of China Grant No. 11574018. H. B.Braun is supported by Science Foundation Ireland Grant No.
11/PI/1048.
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1.367742.pdf | High frequency properties of glass-coated microwire
A. N. Antonenko, E. Sorkine, A. Rubshtein, V. S. Larin, and V. Manov
Citation: Journal of Applied Physics 83, 6587 (1998); doi: 10.1063/1.367742
View online: http://dx.doi.org/10.1063/1.367742
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/83/11?ver=pdfcov
Published by the AIP Publishing
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130.113.111.210 On: Sun, 21 Dec 2014 09:31:16High frequency properties of glass-coated microwire
A. N. Antonenko,a)E. Sorkine, A. Rubshtein, V. S. Larin,b)and V. Manov
Advanced Metal Technologies Ltd., 1 Ha’atsmaut St., P.O.B. 2903, Even Yehuda 40500, Israel
Static and high-frequency 1–30 MHz properties of the (Co 1002xMnx)75B15Si10microwires cast
using the modified Taylor’s technique are reported. The hysteresis loop and the frequencydependence of permeability indicate that the longitudinal magnetization process occurs by coherentrotation of the moments. The values of static permeability are in good agreement with thosepredicted by the theory of ferromagnetic resonance where damping is taken into account. Weobserved a close correlation between the static permeability and the high-frequency limit ~the
frequency up to which the real part of permeability is still constant !. For glass-coated CoMn
microwires, this limit is considerably higher than that given by the Snoek equation. © 1998
American Institute of Physics. @S0021-8979 ~98!39611-5 #
INTRODUCTION
Glass-coated amorphous microwires, cast by a modified
Taylor’s technique is a comparatively scarcely studied mag-netic material. Hereafter, we will adhere to the term ‘‘mi-crowire’’ to distinguish this material from amorphous wiresproduced by other methods, for example, by rotating waterquenching. Recently, several papers devoted to the investi-gation of microwires were published.
1,2It was shown that
microwires demonstrate unique magnetic properties. In par-ticular, Fe-based microwires with positive magnetostrictiondemonstrate bistable behavior, with the critical length assmall as 2 mm.
2The natural ferromagnetic resonance in the
2–12 GHz frequency range was observed in Fe-basedmicrowires.
3The permeability of microwires reaches an ex-
tremely high value ~up to 1000 !in this frequency range, that
suggest that microwires may be a promising material forsuper-high-frequency applications.
Co-based microwires with negative magnetostriction are
studied very scarcely. Static magnetic properties of such mi-crowires were reported in Ref. 4. The investigation of theGMI effect indicates that Co-based microwires have a highpermeability in a radio-frequency range of 1–100 MHz,however, there are no relevant publications relating to thedirect measurement of permeability in this frequency range.
The present work reports the measurement of the com-
plex permeability of Co-based microwires in the 1–30 MHzfrequency range. The correlation between static and high-frequency magnetic properties of microwires are studied too.Another aspect of this work is an attempt to provide a theo-retical description of microwire behavior. The dynamic be-havior of a magnetic material such as ferrite or amorphousribbon is rather complicated. In high-frequency magneticfields, both moment rotation and domain wall motion cantake place. Calculations of permeability and of energy losseswith different modes of remagnetization are extremely diffi-cult. From this point of view, the microwire seems to beeasier to model. Unlike other materials, it is possible to use a
simple model for the description of its dynamic properties.
EXPERIMENT
We have investigated microwires of (Co 1002xMnx)75B15
Si10alloys, where xlies in the range from 4 to 8. In this
range of Mn content, the alloy magnetostriction passesthrough zero. The microwires were cast by Taylor’s tech-nique. The diameter of the metal was in the 6–10
mm range,
and the glass coating was 2–4 mm thick.
We have measured static hysteresis loops as well as real
and imaginary parts of the permeability in the 1–30 MHzfrequency range. The static hysteresis loop was plotted usinga conventional hysteresisgraph. The microwire was placed inthe alternating magnetic field of a long solenoid. A signalproportional to the solenoid current was applied to the X
channel of an oscilloscope, while remagnetization signal wasdetected by a small pickup coil and, after integrating, wasapplied to the Ychannel.
The permeability was measured at high frequency using
the LC resonance method. First, the circuit with a corelesscoil was adjusted to resonance at certain frequency by chang-ing the capacitance. The value of the capacitance, C
1and the
quality factor, Q1were measured. The sample consisting of
several microwires was placed into the coil, and then thecircuit was adjusted to resonance again by changing thevalue of the capacitor. The new values, C
2andQ2, were
measured. Real and imaginary parts of permeability werecalculated as follows:
m8511SC1
C221DD2
nd2, ~1!
m95S1
Q221
Q1DD2
nd2, ~2!
whereDis the diameter of coil, dis the diameter of microw-
ire core, and nis the number of microwires.
The static hysteresis loop depends substantially on the
Mn content of the microwire material. Mn-rich microwiresfeature a rectangular hysteresis loop. With decreasing Mna!On leave from «AmoTec» Ltd., bd Dachia 15/78, 2038 Kishinev,
Moldova.
b!«AmoTec» Ltd, bd Dachia 15/78, 2038 Kishinev, Moldova.JOURNAL OF APPLIED PHYSICS VOLUME 83, NUMBER 11 1 JUNE 1998
6587 0021-8979/98/83(11)/6587/3/$15.00 © 1998 American Institute of Physics
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130.113.111.210 On: Sun, 21 Dec 2014 09:31:16content, the value of coercivity decreases and the loop be-
comes gradually unhysteretic. Further decreasing of the Mncontent does not change the shape of the loop, only its slope,which decreases. The anisotropy field of the material thusincreases with the decreasing in the Mn content. The typicalloop of microwires with less then 6% Mn is presented in Fig.1. This loop has an inflection point approximately corre-sponding to the anisotropy field H
k.
The frequency dependences of real and imaginary parts
of the permeability for different values of the anisotropy fieldare presented in Figs. 2 and 3. The real part of permeabilityis nearly constant in a certain frequency range, and it de-creases rapidly as the frequency increases over a criticalvaluef
r. The frequency dependence of the imaginary part
shows a maximum at fr. The static permeability as well as
frdepend on anisotropy field Hk. Table I summarizes the
obtained results.
DISCUSSION
The hysteresis loops we obtained are typical for the case
of an external magnetic field applied perpendicular to theplane of easy magnetization. In our experiments, this plane isperpendicular to the microwire axis. The magnetic anisot-ropy is induced by the stresses quenched in the microwirecore. These stresses arise during the process of microwirefabrication as a result of difference in thermal expansion co-
efficient between glass and metal. As shown in Ref. 5, thevalue of internal stresses
smay reach 1 GPa. If the Mn
content is lower than approximately 6%, the magnetostric-tion constant lbecomes negative. Due to the effect of mag-
netoelastic anisotropy, which is proportional to l
s, the mo-
ments will lie in a plane perpendicular to the microwire axis,and the external magnetic field applied along the microwireaxis will produce a coherent rotation of moments.
At present time, there is not a definite picture of the
magnetic structure in the Co-based microwires. The appear-ance of circumferential anisotropy like in the conventionalamorphous wires seems to be the most probable, however,there is no direct confirmation of this supposition. Neverthe-less, the magnetic properties of microwires may be describedwithout knowing the definite magnetic structure. The furtherconsideration is based on the following assumptions:
~a!The axis of easy magnetization lies in the plane per-
pendicular to the microwire axis. In each point of microwirethe magnetic anisotropy has the same value H
kand magne-
tization vector Mcoincides with the direction of magnetic
anisotropy.
~b!The coherent moments rotation is the only mecha-
nism of remagnetization.
The Gilbert equation for this mechanism is6
dM
dt5nHM3SH2a
nMdM
dtDJ, ~3!
whereMis the magnetization vector, His the magnetic field
vector, nis the gyromagnetic factor, and ais the parameter
which describes losses.
FIG. 1. The typical hysteresis loop of microwire with Mn ,6%.
FIG. 2. Frequency dependence of the real part of the permeability. The
symbols mean experimental data, continuous curves were obtained from theproposed model.
FIG. 3. Frequency dependence of the imaginary part of the permeability.The symbols mean experimental data, continuous curves were obtainedfrom the proposed model.
TABLE I. mstis the static permeability and mr8mr9is the permeability
measured at fr.
Hk(A/m) fr(MHz) mst mr8 mr9
45 5 8000 4000 3000
120 12 4000 2000 1200280 25 2500 1200 400500 30 1000 500 606588 J. Appl. Phys., Vol. 83, No. 11, 1 June 1998 Antonenko
et al.
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130.113.111.210 On: Sun, 21 Dec 2014 09:31:16TheMandHvectors include static ( M0,Hk) and alter-
nating ~m,h!components and we can write:
M5M01m,H5Hk1h. ~4!
Let us divide the microwire on elements which are small
enough to consider that the direction of local anisotropy ineach element is uniform. The assumption ~b!means that the
component m
zhas the same value for each element. So, from
the solution of Eq. ~3!for a small element, it is possible to
obtain the permeability of the whole microwire. We take alocal coordinate system for a small element of the microwiresuch that the Xaxis is directed along the local M
0andHk
vectors, while the Zaxis is directed along the microwire
axis. The effect of neighboring elements may be taken intoaccount by introducing the demagnetization factor, N,s oh
transforms to h2Nm. The components of Nfor the long
cylinder are N
x51/2,Ny51/2,Nz;(l/r)2!0~lis the length
of microwire, ris the radius of microwire !. We confine our-
self to the consideration of dynamic processes in the lowexternal field, such that h!H
kandm!M0. So, we may
neglect terms including products of mh,mm, andhh. With
these assumptions, Eq. ~3!may be rewritten as
dmy
dt5nFmzHk2M0h1M0Nmz1a
ndmz
dtG, ~5!
dmz
dt5nFM0Nmy1Hkmy1a
ndmy
dtG. ~6!
By solving these equations, it is possible to find an ex-
pression for the frequency-dependent complex permeability:
m˜5mstvrvr~11mstN!1iva
vr2~11mstN!2v22v2a212ivvra~11mstN!,
~7!
where vr5nHk,mst5M0/Hk~initial permeability !.
The term ( a/n)(dmz/dt) in Eq. ~5!expresses the de-
magnetization field occurring due to eddy currents. Accord-ing to Ref. 7 the moment rotation in the cylinder of radius r
induces a demagnetization field H
d5(r2/4r)(dM/dt). The
loss factor ais equal to ( nMr2)/4r, where ris the resistivity
of the microwire. The frequency dependences of the real andimaginary parts of the permeability are presented in Figs. 2and 3. These were calculated using the following parameters:
r54310
26m,r51.631026Vm,
M50.8 T, and n52.23105m~As!21.The experimental results obtained are in a good agree-
ment with the above theoretical considerations. The experi-mental values of static permeability and permeability at thecritical frequency, as well as the values of critical frequencydiffer less than 20% from the values predicted by the pro-posed theoretical model. It is worth to note that CoMn mi-
crowires show a lower value of
mr9than that predicted by the
theoretical model especially for the alloys with high anisot-ropy field. It is also interesting that a certain part of the
frequency dependence of
mr8lies above the so-called Snoek
limit line which is described by the equation vm52nM/3
m0.8The Snoek line confines the area in which the depen-
dences m(f) would lie if we take a bulk material with a
single magnetic anisotropy. There are two kinds of consider-ably different magnetic anisotropies in the microwire. Arather low magnetoelastic anisotropy is responsible for thehigh value of static permeability while a strong shape anisot-ropy provides the extension of the frequency range where thereal part of the permeability stays near constant.
CONCLUSION
The Mn content of Co Mn microwires strongly affects
the magnetic properties. While the Mn content is less thanthe critical value of about 6.5% the microwire shows easyaxes in a plane perpendicular to its axis. The remagnetizationof the microwires occurs by moment rotation, which pro-vides a fast response to the applied external magnetic field.The proposed model of the dynamic magnetization process isin a good agreement with the experimental results. Microw-ires exhibit low values of eddy current loss factor and anextended frequency range where the real part of the perme-ability is near constant.
1J. Gonzalez, N. Murillo, V. Larin, J. M. Barandiaran, M. Vazquez, and A.
Hernando, Sens. Actuators A 59,9 7~1997!.
2A. Zhukov, M. Vazquez, J. Velazquez, H. Chiriac, and V. Larin, J. Magn.
Magn. Mater. 151, 132 ~1995!.
3A. N. Antonenko, S. A. Baranov, V. S. Larin, and A. V. Torcunov, Pro-
ceedings of the Ninth International Conference on Rapidly Quenched andMetastable Materials, Bratislava, 1996 ~Supplement !~Elsevier, Amster-
dam, 1997 !, pp. 248–250.
4S. A. Baranov, V. S. Larin, A. V. Torcunov, A. Zhukov, and M. Vazquez,
in Proceedings of the Fourth International Workshop on Non-Crystal. Sol-ids, 1994, edited by M. Vasquez and A. Hernando ~World Scientific,
Singapore, 1995 !, p. 567.
5S. A. Baranov et al., Fiz. Met. Metalloved. 67,7 3~1989!.
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7S. Tikadzumy, Fiz. Ferromagnetizma ~Russo, translated from Jap. !,
‘‘MIR,’’ Moscow, 1987, p. 324 ~1987!.
8W. Gorter, Proc. IRE 43, 245 ~1955!.6589 J. Appl. Phys., Vol. 83, No. 11, 1 June 1998 Antonenko et al.
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130.113.111.210 On: Sun, 21 Dec 2014 09:31:16 |
1.5055741.pdf | Reuse of AIP Publishing content is subject to the terms at: <a href="https://publishing.aip.org/authors/rights-and-permissions">https://publishing.aip.org/authors/rights-
and-permissions</a>. Downloaded to: 129.81.226.78 on 04 December 2018, At: 05:45Size dependence of spin-torque switching in perpendicular magnetic tunnel junctions
Paul Bouquin , Siddharth Rao , Gouri Sankar Kar , and Thibaut Devolder
Citation: Appl. Phys. Lett. 113, 222408 (2018); doi: 10.1063/1.5055741
View online: https://doi.org/10.1063/1.5055741
View Table of Contents: http://aip.scitation.org/toc/apl/113/22
Published by the American Institute of Physics
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Applied Physics Letters 113, 222404 (2018); 10.1063/1.5047418Size dependence of spin-torque switching in perpendicular magnetic tunnel
junctions
Paul Bouquin,1,2,a)Siddharth Rao,2Gouri Sankar Kar,2and Thibaut Devolder1
1Centre de Nanosciences et de Nanotechnologies, CNRS, Universit /C19e Paris-Sud, Universit /C19e Paris-Saclay,
Avenue de la Vauve, 91120 Palaiseau, France
2IMEC, Kapeldreef 75, B-3001 Leuven, Belgium
(Received 10 September 2018; accepted 13 November 2018; published online 29 November 2018)
We simulate the spin torque-induced reversal of the magnetization in thin disks with perpendicular
anisotropy at zero temperature. Disks typically smaller than 20 nm in diameter exhibit coherentreversal. A domain wall is involved in larger disks. We derive the critical diameter of this transi-
tion. Using a proper definition of the critical voltage, a macrospin model can account perfectly for
the reversal dynamics when the reversal is coherent. The same critical voltage appears to matchwith the micromagnetics switching voltage regardless of the switching path. Published by AIP
Publishing. https://doi.org/10.1063/1.5055741
Magnetization reversal in small particles is a long stand-
ing problem
1,2that was recently put in a new context by the
emergence of spin transfer torque (STT) magnetic random
access memories3(MRAM). This technology is based on the
STT-induced manipulation of the magnetization of nano-sized ultrathin disks with perpendicular magnetic anisotropy(PMA). In addition to its fundamental interest, the switching
dynamics is of paramount application importance as it deter-
mines many of the performance metrics of this technology.Unfortunately, experimental investigations are scarce
4–10
probably because of the technical difficulties associated with
the small dimensions and the large frequencies involved in
the switching process. As a result, the switching paths areoften conjectured from reasonable but approximate mod-els,
11,12if not from overly simplified models like the macro-
spin picture13–15whose range of validity is still to establish.
Here, we unravel the size dependence of the switching
dynamics by taking advantage of the accuracy of micromag-netics. We first clarify how to implement STT in a way that
is adequate for magnetic tunnel junctions (MTJs). We then
describe the main switching regimes: coherent for smalldisks versus domain wall (DW) based at larger diameters.We discuss the critical diameter that separates these two
regimes. We then parametrize the macrospin model to
account exactly for the coherent regime. Finally, we describethe size dependence of the switching voltage and provide amodel valid whatever the switching mode. Our results clarify
the predictive capability of the corpus of theories based on
the macrospin model and therefore it has strong implicationsfor magnetic random access memories.
We are interested by the response of PMA disks to the
STT associated with voltage steps applied through a tunnel
junction. A first difficulty arises from the fact that the STT ismost often expressed in units of current densities
18,19while
the applied voltage is a more correct metric in a tunnel junc-
tion context. Indeed, the insulating nature of the tunnel
oxide renders the voltage laterally uniform across the disk,while the current density is not. To implement STT withinmicromagnetics, we start from the Landau-Lifshitz-Gilbert-
Slonczewski equation
_m¼/C0cl
0m/C2Heffþam/C2_mþsSTT; (1)
where mis the normalised magnetization, cis the gyromag-
netic ratio, Heffis the effective field, and ais the damping
constant. We consider a spin polarization along a unit vectorpparallel to the uniaxial anisotropy axis ( z). In this configu-
ration, the field-like STT can be disregarded as it is mathe-matically equivalent to the Zeeman torque of an easy axis
field. We thus reduce the STT to a sole Slonczewski-like tor-
que s
Slonc. Assuming one-dimensional transport along ( z), we
can write a local STT as18
sSlonc¼c/C22h
2el0MstmaggðhÞJðhÞm/C2ðm/C2pÞ: (2)
Here, tmag¼2 nm is the layer thickness, Msis its magnetiza-
tion, and his the local angle between mandp. The STT effi-
ciency20isg¼P
1þP2cosðhÞ, where Pis the spin polarization21
linked to the tunnel magneto-resistance qTMRratio following
P¼ffiffiffiffiffiffiffiffiffiffiffiffi
qTMR
qTMRþ2q
. Note that for simplicity, we disregard the bias
dependence22,23of the MTJ conductance. With that simplify-
ing assumption, the conductance is GðhÞ¼1þP2cosðhÞ
R?where
the median resistance R?¼2RpR0
RpþR0depends on the resistances
of the h¼0 and h¼pstates. It is noticeable that the h
dependences of Jand gcompensate when the STT is
expressed in voltage. Indeed, we can write
sSlonc¼cPV
AR?/C22h
2el0Mstmagm/C2ðm/C2pÞ; (3)
where Ais the disk area. Equation (3)recalls that the STT is
symmetrical with respect to h: in the case of MTJs with nei-
ther applied field nor field-like STT, the voltage-induced h
¼0t opandpto 0 transitions should occur at exactly oppo-
site voltages and follow identical dynamics. We performour micromagnetic simulations using MuMax3.
24In this
software, the implementation of Eq. (3)requires to seta)Electronic mail: paul.bouquin@u-psud.fr
0003-6951/2018/113(22)/222408/4/$30.00 Published by AIP Publishing. 113, 222408-1APPLIED PHYSICS LETTERS 113, 222408 (2018)
/C150¼0;Pmumax ¼P,K¼1, and Jmumax ¼V
AR?(see Sec. III.H
in Ref. 24). For numerical accuracy, the cell size is kept
below 2 /C22n m2, i.e., substantially smaller than the charac-
teristic micromagnetic lengths of our magnetic material
(Table I).
Let us study the size dependence of the switching path
at zero temperature. In order to better evidence the influence
of the diameter, we apply voltages that correct for the slight
dependence of the switching voltage over the junction diam-
eter; the exact procedure will be detailed later. We varied the
applied voltage between 5% and 50% above the normalized
critical switching voltage and found that increasing the volt-
age accelerates the dynamics without altering the nature of
the switching path (not shown). We varied the diameter
between 16 nm and 300 nm. The reversal is coherent below
22 nm, while a DW is involved for disks larger than 26 nmuntil complexity substantially grows for diameters above
50 nm with the appearance of a center domain. The diame-
ters of 20 nm, 40 nm, and 90 nm (Fig. 1) illustrate those three
regimes.
For the 20 nm disk, the magnetization remains uniform
all along the reversal [Fig. 1(a)]. The degree of coherence
during the reversal can be measured by the modulus of the
mean magnetization jj/C22mjj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m
2
xþm2
yþm2
zq
, where miis
the spatial average of the icomponent of m. For 20 nm and
below, there is no perceivable loss in the degree of coherence
[Fig. 1(c), black curve]: the magnetization switches through
a gradual decrease in its zcomponent while the in-plane
components precess in quadrature [Fig. 1(b)]. For a 22 nm
disk, the reversal starts to exhibit a faint transient non-uniformitywhile the reversal remains mostly coherent and precessional
(not shown). A further increase in the diameter leads to a
gradually stronger non-uniformity until a 180/C14DW appears
during the reversal for disks larger than 26 nm. This DWbased reversal is exemplified with the 40 nm disk [Fig. 1(a)]:
the reversal starts by a coherent phase lasting 11 ns duringwhich the magnetization undergoes a growing precessionthat recalls the behavior seen for smaller disks. During thiscoherent initial phase, the m
zcomponent of the magnetiza-
tion is slightly non-uniform in the sense that the precessioncone is more opened at the center than near the disk circum-ference. The larger susceptibility near the disk center can be
understood from the demagnetizing field profile, which is
maximal at the disk center and therefore reduces locally theeffective anisotropy. In the 26 to 60 nm disks, once a largeprecession amplitude is reached, a nucleation occurs whichleads to the creation of a wall near (but not from) an edge ofthe disk as noticed already in Refs. 12and25. Note that the
DW is a genuine 180
/C14wall: it separates regions with h¼0
andh¼p. We emphasize that despite the magnetization
being tilted everywhere before the nucleation, the nucleation
projects the magnetization to either of the two easy direc-
tions, with then no remaining tilt in the domains after theDW creation. The DW then sweeps across the disk in a non-trivial manner until saturation. This DW creation slowsdown the decay of m
z[see the sudden decrease in the slew
rate of mz(t), blue arrow in Fig. 1(c)]. Above diameters of
60 nm, the radial gradient of the precession cone during theinitial coherent phase of the reversal gets even more pro-nounced such that the magnetization at the center dips and areversed domain is formed at the center. The formation ofTABLE I. Material properties used in the micromagnetic simulations, meant to mimic a dual MgO FeCoB-based layer.16,17
MagnetizationDamping
constantAnisotropy
fieldResistance area
product ( h¼0)Tunnel
MagnetoresistanceExchange
stiffnessBloch
lengthExchange
lengthInitial
tilt
MS¼1.2 MA/m a¼0.01 1.566 MA/m 8.55 Xlm2150% 20 pJ/m 8.5 nm 4.7 nm ht¼0¼1 deg.
FIG. 1. Size dependence of the switch-
ing path at zero temperature. (a)
Snapshots of the magnetization during
the switching for different sizes. (b)Typical magnetization trajectory when
the reversal is coherent. (c) Mean value
of the zcomponent of the magnetiza-
tion and modulus of the mean magneti-
zation during the reversal for 3
different sizes.222408-2 Bouquin et al. Appl. Phys. Lett. 113, 222408 (2018)the central domain is a gradual process (no change in topol-
ogy) that does not lead to any specific feature in the mz(t)
curve [red arrow in Fig. 1(c)]. Once created, the central
domain expands till the disk edges; this lasts longer for largerdevices. Once the central domain approaches the disk edge,it wets one or several points of the perimeter of the disk,depending on the disk size. This wetting is sudden and has aclear signature in the m
z(t) curve.
Let us focus on a central result from this study: the criti-
cal diameter dcabove which the reversal involves DWs.
Several expressions were proposed in the literature to predictd
c. In the minimal approach,3,11dcis estimated by comparing
two energies. First is the energy cost of placing a DW alongthe diameter of the disk. In Ref. 3and11, this energy is writ-
ten as 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A
exKdisk
effq
dtmag, where Aexis the exchange stiffness
and Kdisk
effis the effective anisotropy of the disk. These
expressions assume that the disk is larger than the domain
wall, i.e., d>pD, where D¼ffiffiffiffiffiffiffi
Aex
Kfilm
effq
. The second relevant
energy is the one of the systems when the magnetization is
uniformly in-plane:p
4Kdisk
effd2tmagwhich is necessary to over-
come for a coherent reversal. The effective anisotropy of the
disk depends on the demagnetizing factors Nz(d) and Nx(d).
We deduce Nz–Nxwith micromagnetic simulations from the
FMR angular frequency using Hdisk
k;eff¼Hk/C0ðNz/C0NxÞMs
¼xFMR
cl0. Solving self-consistently this minimal approach
would yield a critical diameter of 33.6 nm which is larger
than the DW width but substantially differs from 26 nmobserved in micromagnetics. This difference can result fromone fundamental and two technical deficiencies of the mini-mal approach. Fundamentally, any comparison based on theenergies of static configurations is bound to underestimatethe energy cost of an inherently dynamical process likereversal. However, in STT switching, the energy lost bydamping is supposed to be compensated by STT; therefore,we conjecture that we can overlook this fundamental objec-tion. Besides, the minimal approach confuses the DW energywithin a disk with that within a fictitious infinite film thatwould have the same effective anisotropy as the disk.
Finally, it neglects the dipolar energy gained when breaking
the system into domains. Neglecting the domain-to-domaindipolar coupling is a minute error in the ultrathin limit.
26
Conversely, the imprecision in DW energy can be substan-tial. Indeed, the demagnetizing field in thin films is essen-tially local within a DW,
27such that the wall energy is much
more linked to the effective anisotropy of the film ratherthan the effective anisotropy of the disk and therefore should
be taken as 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A
exKfilm
effq
dtmag.
Altogether, an improved estimate of dcis
dc/C2516
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
AexKfilm
effq
Kdisk
eff¼16
pffiffiffiffiffiffiffiffiffiffiffi
2Aex
l0Mss
/C2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiHk/C0Msp
Hk/C0ðNz/C0NxÞMs;(4)
which also has to be solved self-consistently because ( Nz
–Nx) depends on d. Figure 2compares Eq. (4)with the
micromagnetic dcin a relevant interval of anisotropy field
and exchange stiffness. The matching is satisfactory: as longas the domain wall width exceeds the disk diameter, Eq. (4)
is reliable estimation of dc. Note that for a fast evaluation of
Eq.(4), the demagnetizing factors can be estimated from Eq.
(11) of Ref. 28fortmag/d>0.06 and from Ref. 29otherwise.
The switching dynamics in the macrospin model has
been used extensively in the past to predict the switchingspeeds,
13the error rates,14,30and the stability diagrams.6In
order to assess to predictive ability of these studies, it isimportant to evaluate to what extent the macrospin modelcan be used to mimic the coherent reversal regime. Wedescribe the macropin with its cylindrical coordinates m
zand
/. The LLGS equation of a macrospin reduces14,15to
_mz¼1
sðmzþvÞð1/C0m2
zÞ; (5)
_/¼1
asðmzþa2vÞ; (6)
where s¼1þa2
acl0Hdisk
k;effand v¼V
Vcis the voltage normalized to
the macrospin critical voltage,14i.e., the smallest voltage at
which the h¼pstate gets unstable
Vc¼2aeAR?tmagl0MsHdisk
k;eff
P/C22h: (7)
Two words of caution are needed. First, although here
we restrict to the case of zero applied field, in general, thetime evolution of m
zis triggered by all torques. As a result,
some authors14,31choose to aggregate the external field Hz
and the spin-torque by defining a generalized stimulus and
substitutingV
VcbyV
VcþHz
Hcin Eq. (5). We prefer not to perform
this substitution in Eq. (5)because the same substitution can-
not be applied to Eq. (6): qualitatively, the influence of
Zeeman torque on the precession frequency [Eq. (6)]is way
higher than that of the STT. HzandVcannot be aggregated
when describing the precession. Second, we stress that since
the macrospin is meant to mimic a disk, its Hdisk
k;effmust take
into account the demagnetizing term –( Nz–Nx)Ms.
Figure 3compares the switching dynamics obtained for
the macrospin and for the largest disk showing coherentFIG. 2. Critical diameter above which a 180/C14. DW is observed in the micro-
magnetic simulations and comparison to its analytical estimates. Main panel:
while varying anisotropy field at fixed Aex¼20 pJ/m or while varying
exchange stiffness at fixed Hk¼2.2 MA/m (inset).222408-3 Bouquin et al. Appl. Phys. Lett. 113, 222408 (2018)switching in micromagnetics. Provided that the proper Hdev
k;eff
and the resulting adequate Vcare used, the outcomes of the
two models match for the time evolution of mz[Eq. (5), Fig.
3(a)] and for the instantaneous precession frequency [Eq.
(6), Fig. 3(b)]. For larger disks, the perfect match is main-
tained during the initially coherent phase of the reversal (not
shown), but as expected, the macrospin model fails to
account for the subsequent evolution as soon as a non-uniformity sets in (not shown).
From the previous discussion, we conclude that the mac-
rospin model describes perfectly the coherent reversalregime. Let us now see whether the macrospin critical volt-
age [Eq. (7)]can account for the micromagnetic switching
voltage, including for sizes that lead to non-coherent rever-sal. The voltage that leads to a destabilization of the uni-
formly magnetized state is bound to match the macrospin
critical voltage V
c. However, destabilizing the uniformly
magnetized state is a necessary condition for switching but it
might not be a sufficient condition. Indeed, even if uniform
state is unstable to finite amplitude precession, the precessionamplitude can be limited by non-linearities and not lead to
reversal, as observed in the in-plane magnetized metallic
spin-valve,
32,33in which there is a net difference between
instability and switching. In our case, we find that Vcand the
micromagnetic switching voltage do agree for all investi-
gated disk diameters [Fig. 3(c)]. This indicates that destabi-
lizing the uniformly magnetized state is a necessary and
sufficient condition for switching and that this holds evenwhen the reversal is far from coherent. In short, Eq. (7)is the
switching voltage including for sizes that lead to non-
coherent reversal.
In summary, we have simulated the spin-torque induced
switching of the magnetization of disks with perpendicular
anisotropy. The reversal always starts by the amplification ofa circular precession. For disk diameters below a critical
threshold, the reversal is coherent and can be accounted forby a macrospin model. For larger sizes, a domain wall
appears during the reversal. Energy considerations can pre-dict this critical size. Besides, the macrospin critical voltage
[Eq. (7)] predicts the switching voltage for any sizes, includ-
ing when the reversal is not coherent.
This work was supported by IMEC’s Industrial
Affiliation Program on the STT-MRAM devices.
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1.2993744.pdf | Preliminary mapping of void fractions and
sound speeds in gassy marine sediments from
subbottom profiles
T. G. Leighton
Institute of Sound and Vibration Research, University of Southampton, Highfield, Southampton SO17 1BJ,
United Kingdom
tgl@soton.ac.uk
G. B. N. Robb
National Oceanography Centre, Southampton, European W ay, Southampton, SO14 3ZH, United Kingdom
gbor199@noc.soton.ac.uk
Abstract: Bubbles of gas (usually methane) in marine sediments affect the
load-bearing properties of the seabed and act as a natural reservoir of “green-house” gas. This paper describes a simple method which can be applied tohistorical and future subbottom profiles to infer bubble void fractions andmap the vertical and horizontal distributions of gassy sediments, and the as-sociated sound speed perturbations, even with single-frequency insonifica-tion. It operates by identifying horizontal features in the geology and inter-preting any perceived change of depth in these as a bubble-mediated changein sound speed.
© 2008 Acoustical Society of America
P ACS numbers: 43.30.Ma, 43.20.Hq, 43.30.Pc [AN]
Date Received: June 4, 2008 Date Accepted: September 8, 2008
1. Introduction
The bubbles that occur at many seabed locations1can impact upon the structural integrity and
load-bearing capabilities of the sediment2,3and can be indicative of a range of biological,
chemical, or geophysical processes4(such as global methane budgets). In acoustical terms,5
such bubbles can degrade the operation of a subbottom profiler (Fig. 1) or be inverted to esti-
mate the bubble population. Bubble radii distributions from 10 µm to 20 mm, and void frac-
tions as great as 9%, have been inferred from the inversion of either compressional wavedata,
6–10acoustic backscatter,11–13or two-frequency techniques.14–16While such inversions can
be based on models of bubbles in water, a model incorporating geotechnical properties of the
host sediment surrounding the bubbles (both before and after it is altered by the presence ofbubbles) would be preferable.17
Table 1shows that, in addition to the scattering studies,11–16there have been many
investigations of sound speeds and attenuations of compressional waves in gassy sands andmuds, but only one8of these has even attempted to estimate void fractions (VFs) remotely
(sensitive to VF /H110222%). This report outlines a very basic method by which subbottom profiles
may be rapidly analyzed to estimate the effect of bubbles on the sound speed in the sediment,
and hence map the void fractions, sensitive to VF=0.002% or better. Furthermore, this method
could in principle be applied to single-frequency data. This approach will not replace, but willinstead complement, the large-scale field trials which deploy specialist equipment to monitorgas bubbles in sediment. It provides a method either to exploit archived subbottom profiles, orto survey a large area rapidly with commercial equipment from a small vessel.
This paper illustrates the method on a single historical subbottom profile. Supplemen-
tary data are not available for this trace, and it is fully recognized that lack of supplementarydata on this figure means that the validity of key assumptions (any significant occurrence ofseverely aspherical bubbles,
18,19or bubbles whose resonant frequency is not much less than theT . G. Leighton and G. B. N. Robb: JASA Express Letters /H20851DOI: 10.1121/1.2993744 /H20852 Published Online 15 October 2008
J. Acoust. Soc. Am. 124 /H208495/H20850, November 2008 © 2008 Acoustical Society of America EL313
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 131.211.208.19 On: Sun, 12 Jul 2015 18:35:10insonifying frequency; the horizontal nature of the sediment layering) cannot be tested without
revisiting the site with modern equipment: one purpose of this paper is to show what prelimi-nary estimations can be made from historical data without resorting to such a visit.
2. Method and results
Figure 1shows a chirp subbottom profile containing gassy sediment.20Bubbles cause two
“blanking” zones, at which the majority of the sound is scattered (and multiply reflects), and inwhich no information can be determined on the sediment layering. However, before the geo-logical layering features on either side of the gas pockets become obscured, they appear to dip togreater depths as they approach the “blanking” zones. If a given pair of interfaces in this regionwas to maintain a constant separation, any apparent changes in separation observed in the sub-bottom profile can be directly attributed to a change in sound speed in the medium betweenthem.
33,40
This analysis can be applied to the estimate cs−1, the vertically averaged sound speed
between the seabed and interface 1 in Fig. 1(labeled A1–G1). In a similar way, it can be used to
estimate c1–2, the vertically averaged sound speed between interfaces 1 and 2 (labeled A2–G2).
The two key assumptions, namely that upper and lower interfaces lie parallel to one another, andthat no bubbles exist in the sediment at the location chosen for the calibration (at A), wouldideally be justified by the observation of the two layers appearing horizontal over a more ex-tended region than is visible in Fig. 1: no such data were available with this historical record.
Recognizing this limitation, the estimated sound speeds perturbations are calculated simplyfrom the ratio of the apparent separation to the assumed separation in Fig. 1and are plotted in
Fig.2(a).
Retaining the principle that this is an example of using the simplest assumptions to
gain a preliminary estimation from this historical data, of the many options
21for estimating the
bubble population from Fig. 2(a)this paper will use Wood’ s equation22for an effective two-fluid
medium for illustrative purposes. It is understood that the dynamical response of the bubblepopulation will be influenced by the range of bubble sizes present, and that the bubbles mayindeed depart significantly from sphericity. However, assuming that the bubbles are smallenough to satisfy the quasistatic conditions inherent in Wood’ s equation,
21,22and that the bubbly
sediment can be treated as an effective fluid medium, then the effective compressibility is there-
fore /H208491−/H9252/H20850/H20849/H9267scs2/H20850−1+/H9252/H20849/H9260p0/H20850−1, where /H9252is the void fraction of bubbles, /H9267sand csare the equi-
librium density and sound speed of the bubble-free saturated sediment, p0is the static pressure
at the position of the bubble, and /H9260is the polytropic index17,21of the gas within the bubbles
Fig. 1. A chirp subbottom profile from Strangford Lough, Northern Ireland, which displays sediment layers and gas
“blanking” /H20849where shallow gas prevents the profiler from obtaining information from beneath the gas /H20850. The three
interfaces used in this paper to determine sound speed and void fraction are the seabed /H20849upper dark line /H20850, the
interface denoted by A1 to G1, and the interface denoted by A2 to G2. The chirp signal was a 2–8 kHz linear sweepof 32 ms duration, and the water depth was estimated to be 15.5 m. Reproduced from Ref. 20and annotated by T.
G. Leighton with permission of the National Oceanography Centre, Southampton /H20849J. S. Lenham, J. K. Dix, and J.
Bull /H20850.T . G. Leighton and G. B. N. Robb: JASA Express Letters /H20851DOI: 10.1121/1.2993744 /H20852 Published Online 15 October 2008
EL314 J. Acoust. Soc. Am. 124 /H208495/H20850, November 2008 T . G. Leighton and G. B. N. Robb: Mapping gas in marine sediments
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 131.211.208.19 On: Sun, 12 Jul 2015 18:35:10Table 1. Compilation of measured compressional wave properties of gassy sediments, including frequency f/H20849kHz /H20850,
experimental technique, sediment type, compressional wave velocity ceff/H20849ms−1/H20850, ratio of velocity in gassy sediment
to that in saturated sediment ceff/cs, attenuation coefficient /H9251/H20849dB m−1/H20850, attenuation coefficient of equivalent saturated
sediment /H9251s/H20849dB m−1/H20850, void fraction, and the source reference. The symbol ¯indicates that no available information
is available and the attenuation coefficient of the equivalent sediment was predicted using Hamilton.31Note that
some simplification and interpretation has been required to condense these data onto a single Table, and that datafrom sediment containing both free gas and gas hydrate have been omitted from this compilation, owing to thecomplex four-phase nature of these sediments.
32
f/H20849kHz /H20850 Site Sediment type ceff/H20849m/s /H20850 ceff/cs/H9251/H20849dB/m /H20850/H9251s/H20849dB/m /H20850VF
/H20849%/H20850Reference
no.
¯ Remote Gravel to mud 1210–1650 0.65–0.93 ¯¯¯ 33
¯ In situ Mud 110–304 0.06–0.22 ¯¯ /H110211 6
0–11 In situ Mud 320–1500 0.21–1.02 /H11021230 /H110211.3 /H110216 7
/H110210.4 Remote Sand/silt 1250 0.70 /H110212 8
/H110211 Remote Mud 250–1280 0.17–0.86 ¯¯¯ 34
/H110211 Remote Mud 550–1100 0.38–0.75 ¯¯¯ 35
/H1102112 Remote ¯ 800 0.55 ¯¯¯ 36
/H1102120 Remote ¯ /H110221050 0.60 ¯ 37
0.13–1 In situ Clay 800 0.55 1.4/kHz /H110210.1/kHz 0.065 9
0.2–3.2 Remote Mud and sand 45–122 ¯¯¯¯ 38
0.1–2 Lab Mud/Kaolinite 114–326 0.07–0.21 ¯¯ 0.1–0.4 39
3.5 Remote Sand to mud ¯ 0.77 ¯ 40
0.6–1.2 Remote ¯ 75–170 ¯ 114 ¯¯ 41
1–30 In situ Mud ¯¯ 2–2433 /H110214 ¯ 42
3–20 In situ Mud 852–1526 0.56–1.03 ¯¯¯ 43
3–100 Lab Silty clay 1280 0.88 ¯¯ /H1102120 44
3.5 RemoteSand to sandy
mud 1610–1660 0.89–0.92 ¯¯¯ 45
5–20 In situ Silty clay 1000–1430 0.70–1.00 0–2 10
7.5 In situ Sand to mud 700–1200 0.47–0.81 ¯¯¯ 46
10–100 Lab Sand and soil ¯¯ 2500–7100 /H1102180 ¯ 47
10–1000 Lab Mud 200–500 0.13–0.34 600–4000 /H11021120 0.4–19.8 48
38 In situ Silty clay ¯¯ 40–50 /H110215 0–2 10
40 In situ Mud ¯¯ 13 /H110215 ¯ Cited by 49
40 Lab Sand 305–1706 0.18–1.00 ¯¯¯ 50
40–80 In situ Mud ¯¯ 25–90 /H1102110 ¯ Cited by 49
50 Lab Glass beads 280–1750 /H110220.16 ¯¯ 0.1–100 51
50 Lab Sand 250–1700 /H110220.15 ¯¯ 0.1–100 51
110 In situ Soil 1220–1270 0.84–0.87 105–470 ¯¯ 52
200 Lab Sand 1218–2090 0.58–1.00 ¯¯ 0–100
controlled53and54
300–700 Lab Mud 1400–1700 0.95–1.15 /H11021700 /H1102184 /H110216 7
400 Lab Mud 1522–1523 1.00 ¯¯ 0.1–0.4 39
400 Lab Silty clay 1430 1.00 /H11022500 /H1102148 0–2 10
400 Lab ¯ 736–1372 0.50–0.94 ¯¯¯ 55
400 Lab Mud 1200–1300 0.80–0.87 ¯¯¯ 56
500 Lab Sand to mud 1900–1460 0.62–1.00 ¯¯¯ 57
500–1000 Lab Sand and silt /H110211000 /H110210.56 ¯¯¯ 58
700 Lab Sand 1264–2515 ¯¯¯ 0–100
controlled59
1000 Lab Sand to mud 1300 0.88 ¯¯¯ 46T . G. Leighton and G. B. N. Robb: JASA Express Letters /H20851DOI: 10.1121/1.2993744 /H20852 Published Online 15 October 2008
J. Acoust. Soc. Am. 124 /H208495/H20850, November 2008 T . G. Leighton and G. B. N. Robb: Mapping gas in marine sediments EL315
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 131.211.208.19 On: Sun, 12 Jul 2015 18:35:10(equal to the ratio of the specific heats of the gas, divided by the polytropic coefficient7,10,12),
which is assumed to be 1.3 for nearly adiabatic pulsations of biogenic sedimentary gas.9The
void fraction being small, /H9252/H112701, means that the bubbly sediment can be treated as an effective
fluid medium of density /H9267eff, where /H9267eff−1/H11015/H9267s−1, since if /H9267eff=/H9267s/H208491+/H9004/H9267//H9267s/H20850, then /H9267eff−1/H11015/H9267s−1/H208491
−/H9004/H9267//H9267s+¯/H20850⇒/H9004/H9267//H9267eff/H11015/H9004/H9267//H9267s−/H20849/H9004/H9267//H9267s/H208502+¯if/H9004/H9267//H9267s/H112701. Equating the above effective
compressibility with /H20849/H9267effceff2/H20850−1/H11015/H20849/H9267sceff2/H20850−1(where ceffis the effective sound speed of the
gassy sediment, noting that dissipation is neglected in this definition21), implies /H9252
=/H20849cs2/ceff2−1/H20850/H20851/H9267scs2//H20849/H9260p0/H20850−1/H20852−1. Assuming /H9260p0/H11270/H9267scs2and /H20841cs2/ceff2−1/H20841/H112701, this reduces to
/H9252/H110152/H9260p0
cs2/H9267s/H208731−ceff
cs/H20874. /H208491/H20850
A detailed form of this basic derivation may be found in Ref. 23, and more sophisticated ver-
sions may be used if the above assumptions are violated.17,21Application of Eq. (1)to the data in
Fig.2(a)allows estimation of the vertically averaged void fraction between the top of the seabed
and layer 1 /H20849/H9252s−1/H20850, and the vertically averaged void fraction between layer 1 and layer 2 /H20849/H92521–2/H20850.
At the time the measurements of Fig. 1were taken, techniques for measuring the density and
Fig. 2. /H20849a/H20850The vertically averaged sound speeds between the seabed and interface 1, and between interface 1 and 2;
/H20849b/H20850the vertically averaged void fractions between the seabed and interface 1, and between interface 1 and 2.T . G. Leighton and G. B. N. Robb: JASA Express Letters /H20851DOI: 10.1121/1.2993744 /H20852 Published Online 15 October 2008
EL316 J. Acoust. Soc. Am. 124 /H208495/H20850, November 2008 T . G. Leighton and G. B. N. Robb: Mapping gas in marine sediments
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 131.211.208.19 On: Sun, 12 Jul 2015 18:35:10sound speed in the sediment were not as advanced as they are today. These values are therefore
predicted for the silts and clays typical of this section of Strangford Lough through the use of
empirical regressions,24resulting in a bubble-free sound speed of 1600 m s−1and a bubble-free
density of 2300 kg m−3. Nowadays it is commonplace to examine cores from regions of satu-
rated sediment to determine density from gamma ray attenuation and compressional wave ve-
locity from ultrasonic propagation at 500 kHz (which for these saturated sediments is assumed
to provide a sound speed relevant to the frequencies used in this paper).25
The final parameter required for the calculation is the pressure p0. This was computed
for the central portion of each layer using p0=pA+/H9267wghw+/H9267sgh/2, where pAis the atmospheric
pressure on the survey date /H20849103 kPa /H20850,gis the acceleration due to gravity, /H9267wis the density of
water /H208491000 kg m−3/H20850,hwis the depth of the water /H2084915.5 m /H20850, and his the depth of the sediment
layer below the seabed (5.1 m for layer 1, and 7.4 m for layer 2). This resulted in an average
hydrostatic pressure of 312.5 kPa for layer 1 and 338.5 kPa for layer 2.
Figure 2summarizes the sound speeds and void fractions calculated for each of the
coordinates A to G. The assumptions in the sediment parameters introduce an unknown uncer-tainty.
3. Discussion and conclusion
This report outlines a simple scheme for assessing the sound speed perturbation induced by
bubbles in the seabed, and for estimating the void fraction and extent of the bubble layers (in-cluding changes both in the horizontal and vertical). Although in the present work this tech-nique has only been applied to two sediment layers (three interfaces), the technique can beextended to include more interfaces and even three-dimensional profiles.
26Contrast
enhancement27of the geological layers compared to the bubble scatterers will extend the tech-
nique to higher void fractions, closer to the center of the gas pocket.
The approach provides a quick first-order technique, but the simplicity of its use is
offset by limitations. The conditions may violate key assumptions (quasi-static dynamics ofnoninteracting spherical bubbles), and inhomogeneities may contradict the assumed averaging.Nevertheless, the ease with which first-order environmental data can be gained at little extraeffort using existing technology, and through examination of historical records of subbottomprofiles, offers the possibility of making rapid progress. This is significant given
(i) quantitative sound speeds and void fractions can be estimated from subbottom profiles
which were previously used qualitatively to assess the location of gassy sediments;
(ii) the method shows that the extent of the gas, as indicated by the void fractions shown in
Fig. 2(b), is much greater than the extent of the shadows in Fig. 1; which one might
otherwise consider to represent the location and extent of the gas pocket;
(iii) the method can be implemented without the complex equipment often required to assess
gassy sediments (e.g., difference frequency sonars, CT scanners, etc.);
10,28–30; and
(iv) the method is remote and highly sensitive (compare with Table 1).
This method is not presented as an alternative to the more sophisticated methods under
development, or as competition for the innovative and large-scale field trials specifically de-signed to measure in situ bubble populations in sediments. Instead it is envisaged to be a
complementary technique which would allow the rapid low-cost assessment of a gassy areausing current commercial apparatus, and the new analysis of historical data. Geological exper-tise will be required in each case to assess the likelihood that the interfaces selected are parallel,and that the perceived dipping is solely due to sound speed perturbations
33,37,40(though, be-
cause gassy sediments are dispersive, this can be tested remotely by taking additional profiles ofthe same location at higher frequencies and seeing whether the apparent separations remainconstant). In the model used here, the material parameters of the sediment enter only throughthe sound speed and density of saturated sediment, and there is therefore no reflection of thecomplexity of propagation that can occur in such materials. While this simplification could beovercome through the substitution of improved models for sound speed into this scheme
17(andT . G. Leighton and G. B. N. Robb: JASA Express Letters /H20851DOI: 10.1121/1.2993744 /H20852 Published Online 15 October 2008
J. Acoust. Soc. Am. 124 /H208495/H20850, November 2008 T . G. Leighton and G. B. N. Robb: Mapping gas in marine sediments EL317
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 131.211.208.19 On: Sun, 12 Jul 2015 18:35:10while the assumption of quasi-static dynamics can be replaced using a more sophisticated in-
version routine), the importance of this report lies in expressing such a simple scheme for ob-taining the void fraction and extent (in the vertical and horizontal) of bubble populations inmarine sediment. It can also be applied to tissue, and to domestic products and pharmaceuticalsinto which deliberate target layers could be placed.
Acknowledgments
This work is funded by the Engineering and Physical Sciences Research Council, Grant No.
EP/D000580/1 (Principal Investigator: T.G. Leighton). The authors are grateful to Justin Dix forproviding the background data to Fig. 1(water depth, and the density and sound speed of the
bubble-free sediment) and to Peter Birkin for helpful comment.
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1.4981389.pdf | Magnetization reversal in ferromagnetic wires patterned with antiferromagnetic
gratings
S. R. Sani , F. Liu , and C. A. Ross
Citation: Appl. Phys. Lett. 110, 162403 (2017); doi: 10.1063/1.4981389
View online: http://dx.doi.org/10.1063/1.4981389
View Table of Contents: http://aip.scitation.org/toc/apl/110/16
Published by the American Institute of Physics
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Appl. Phys. Lett. 110, 152401152401 (2017); 10.1063/1.4979840Magnetization reversal in ferromagnetic wires patterned with
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S. R. Sani, F.Liu,and C. A. Rossa)
Department of Materials Science and Engineering, Massac husetts Institute of Technology, 77 Massachusetts Ave.,
Room 13-4005, Cambridge, Massachusetts 02139, USA
(Received 29 January 2017; accepted 6 April 2017; published online 21 April 2017)
The magnetic reversal behavior is examined for exchange-biased ferromagnetic/antiferromagnetic
nanostructures consisting of an array of 10 nm thick Ni 80Fe20stripes with width 200 nm and period-
icity 400 nm, underneath an orthogonal array of 10 nm thick IrMn stripes with width ranging from
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try on the reversal behavior. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4981389 ]
Exchange bias at the interface between a ferromagnet
(FM) and an antiferromagnet (AFM) leads to a shift in the
hysteresis loop of the FM.1–4It is technologically important
for devices such as magnetic random access memory, record-
ing media, and spin valves for hard disk read heads,5–8as
well as providing a vehicle for exploring magnetic exchangephenomena. The exchange bias (EB) has been related to thepresence of AFM domains and uncompensated pinned spins
at the interface.
9,10
The effects of finite feature size on exchange bias have
been explored in a range of thin film and particulate sys-tems,
8,11–19and Malozemoff’s random field model10predicts an
increase in exchange bias as the FM layer becomes thinner.
Experimental investigations of exchange bias in FM/AFM struc-tures with lateral dimensions below 100 nm have primarily been
carried out in thin films where both layers are patterned into the
same dimensions, for example, by etching or liftoff of a FM/
AFM bilayer.
11–14In general, exchange bias and the blocking
temperature decrease with decr easing dimensions, which has
been attributed to formation of spin glass-like regions with
weaker exchange bias at the edge s of the patterned features.17,18
Structures in which the FM and AFM layer are patterned
into different dimensions20–24are much less well studied, even
though FM layers with local regions of exchange bias may be
useful in controlling domain walls in racetrack memory andlogic devices
23a n di nm a g n e t i cb i te n c o d e r s25and mangnonic
crystals.26,27For example, the magnetic properties of
a continuous FM film partly covered with patterned AFM
regions depend on the dimensions of the AFM and the strengthof the exchange bias within the pinned regions, as shown inNiFe/FeMn
24and NiFe/IrMn,28and Co films with O implan-
tation showed local EB due to the formation of CoO in the
implanted regions.21We showed earlier29,30that NiFe films
patterned with IrMn stripes with a width of 100 to 500 nmand a period of 240 nm to 1 lm exhibited either single-step
or two-step reversal depending on the IrMn dimensions. In
this article, we extend this investigation to a structure con-sisting of an array of 200 nm wide NiFe stripes on whichan orthogonal array of sub-500 nm wide IrMn stripes is pat-
terned. This leads to NiFe stripes with unpinned segmentsalternating with exchange-biased segments with lengths
defined by the IrMn spacing and linewidth, respectively.
Experimental results and micromagnetic modeling delineateregimes of stripe width and period in which the pinned and
unpinned segments reverse at different fields.
FM structures partly covered by an AFM layer can be
made by depositing an FM/AFM bilayer and etching the AFMin selected areas, e.g., by ion beam etching,
31but this can alter
the magnetic properties of the FM layer. Ion bombardment
through a mask can be used to locally modulate exchangebias,
21but the spatial resolution of this process is limited.
Alternatively, the FM and AFM layers may be patterned using
two different lithography steps, but the processing compro-
mises the quality of the FM/AFM interface which is critical
for establishing exchange bias. Here, we use an additivemethod
29,30in which a NiFe film is deposited and patterned,
and then, a second lithography step is used to define the
AFM-covered regions. A good quality FM/AFM interface is
obtained by removing about 1 nm of the NiFe film by an ion-
beam etch after the second lithography step and then deposit-ing/C241 nm NiFe followed by an IrMn layer.
Figure 1shows images of the samples and the fabrication
process. Figure 1(a) is a SEM image of the NiFe grating
and Figures 1(b) and1(c) show the double grating pattern.
The gratings were fabricated using interference lithography.
32
A tri-layer resist stack consisting of antireflective coating(ARC), a SiO
2(20 nm) etch stop layer, and Ohka PS4 nega-
tive resist (215 nm) was formed on a silicon wafer, as shown
in the schematic in Figure 1(d). The thickness of the ARC,
which typically ranged from 280 nm to 320 nm, was calcu-
lated based on the thickness of the other layers and the
desired width and periodicity of the grating in order to mini-mize reflections from the Si substrate. The ARC layer was
made by spin-coating and baking at 110
/C14C for 90 s. The SiO 2
layer was deposited by electron beam evaporation, and the
PS4 resist was deposited by spin-coating at 3 krpm followed
by baking at 90/C14C for 90 s, over a hexamethyldisilazane adhe-
sion layer. The resist stack was exposed to the interferencea)caross@mit.edu
0003-6951/2017/110(16)/162403/5/$30.00 Published by AIP Publishing. 110, 162403-1APPLIED PHYSICS LETTERS 110, 162403 (2017)
pattern produced by two beams from a 325 nm wavelength
HeCd laser using a Lloyd’s mirror system, baked and devel-oped, and a two-step reactive ion etch (RIE) process was then
performed to transfer the pattern through the ARC, consisting
of CF
4to etch the SiO 2and O 2to etch the ARC. A Ta seed
layer was deposited using a sputtering system with a base
pressure of 2 /C210/C08Torr and a working pressure of 1 mTorr
Ar, and then, 10 nm of NiFe was deposited with a growth rateof 0.13 nm/s. The residual ARC was removed using a Dow
Microposit 1165 solvent, consisting largely of N-methyl-2-
pyrrolidone (NMP).
To pattern the IrMn grating on top of the NiFe, the same
tri-layer resist stack was deposited. The samples were rotated
90
/C14before being exposed using interference lithography.
The same development and RIE process was used to transfer
the pattern down through the ARC; but before the IrMn
(10 nm) was deposited, the surface of the NiFe was cleanedby ion beam etching for 3 s using 2 /C210
/C04Torr Ar pressure,
beam current 5.5 mA, and voltage 500 V. Based on a control
sample which showed an etch rate of 0.256 nm/s, this processis estimated to remove about 1 nm of the NiFe. NiFe (1 nm)/
IrMn (10 nm) was then sputtered at 1 mTorr Ar pressure at arate of 0.2 nm/s, and the remaining ARC was lifted off. This
results in a double-grating pattern shown in Figure 1(c).
The NiFe stripe width was usually 200 nm with periodic-
ity 400 nm, but the IrMn stripe width varied from 100 nm to
500 nm, and the periodicity from 200 nm to 1 lm. We define
“w” as the width of the IrMn stripes, i.e., the length of thepinned segments of the NiFe wire, and “s” is the length of the
unpinned segments of the NiFe wire. Then, w þs is the period
of the IrMn grating, equal to the length of the repeating struc-ture of the NiFe wire. Unpatterned control samples of NiFe/
IrMn as well as samples with only NiFe gratings (without
IrMn) were also made for comparison. The 5 /C25m m
2sam-
ples made by interference lithography had enough magnetic
moment to be measured using vibrating sample magnetometry
(VSM, ADE model 1660) at room temperature.
For demonstration, the second grating could also be
made using electron beam lithography and a liftoff process.
This replaced the second interference lithography step with a
FIG. 1. (a) SEM image of NiFe grating with period 400 nm produced by the first interference lithography step; (b) SEM image of the final sample with both
gratings present, in which the second grating was made using electron beam lithography. The unpinned regions of NiFe of width s are separated by exchan ge-
coupled regions of width w. (c) SEM image of the final sample with both gratings present, in which the second grating was made using interference lithogr a-
phy. (d) Schematic of the fabrication procedure. In steps 1–3, the tri-layer resist mask is made. In steps 4–5, the Ta/NiFe grating is formed by liftoff . In step 6,
a second grating of 1 nm NiFe/IrMn is formed orthogonal to the first. Prior to deposition of the NiFe/IrMn, the sample undergoes a short ion beam etch to cl ean
the top surface of the NiFe of the first grating.162403-2 Sani, Liu, and Ross Appl. Phys. Lett. 110, 162403 (2017)PMMA resist coating, electron beam exposure and develop-
ment process. Figure 1(b) shows an example in which the
200 nm wide NiFe wires were made as in Fig. 1(a)and subse-
quently were overlaid with a NiFe/IrMn grating made usinge-beam lithography. The highlighted square indicates thecrossing of the NiFe and IrMn wires. However, the area of the
electron-beam-written samples was too small to enable mag-
netometry by VSM. The samples were not capped, but theirmagnetic moment did not degrade during the measurementspresented here.
The exchange bias was set using a field-cooling proce-
dure. Initially, the temperature was raised beyond the block-ing temperature of the IrMn, 520 K, while applying 10 kOe,with the field parallel to the NiFe stripes. Then, the sample
was gradually cooled to room temperature in 30 min. The
samples exhibit training effects,
30,33and therefore, the sam-
ples were field-cooled before each hysteresis measurement.
The magnetic behavior was modeled using the Object-
Oriented Micro-Magnetic Framework (OOMMF),34based
on the Landau-Lifshitz-Gilbert equation35
dM
dt¼/C0cM/C2Heff/C0aM
Ms/C2dM
dt;
where M is the magnetization, cis the electron gyromag-
netic ratio, ais the damping parameter ¼0.5 to ensure rapid
convergence, and H effis the effective field, the sum of the
external magnetic field, anisotropy field, and the demagnet-izing field. The saturation magnetization M
s¼800 emu
cm/C03, and a small randomly oriented anisotropy ener-
gy¼8000 erg/cm3was included. The exchange length of
NiFe is calculated as kex¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
A=2pM2
sq
¼5.7 nm, where A is
the exchange stiffness ¼13/C210/C07erg/cm.36Ac e l ls i z eo f
5n m/C25n m/C25 nm was chosen. An infinite NiFe wire with
periodic regions of exchange bias was modeled using periodicboundary conditions in the wire direction, i.e., the repeatinglength of the model was w þs along the wire axis. We mod-
eled the exchange bias as an external field of 100 Oe applied
only in the region where the IrMn crossed the NiFe wire. Themagnetic ground state of the structure vs. field was deter-mined by randomizing the magnetization direction of each
cell and then allowing the sample to relax to its stable state at
each different applied field value.
The measured hysteresis loops of the double grating
structure for two samples of different dimensions are shown
in Figs. 2(a)and2(b), with both the exchange bias and applied
field parallel to the NiFe stripes. The sample of Fig. 2(a)with
w¼s¼200 nm shows “one-step” switching. Figure 2(a)also
shows the remanence loop of the sample. The remanence loop
overlaps the major loop, showing that there is little reversible
magnetization, which suggests that the shear in the loop origi-nates mainly from the switching field distribution of the mag-netic wires.
In contrast, the sample of Fig. 2(b) with s¼540 nm and
w¼520 nm shows “two-step” switching, defined as the pres-
ence of a kink or a plateau on at least one branch of the loop.
We attribute the two steps to reversal of the pinned and
unpinned regions of the wire. The minor loop in Fig. 2(b)
shows the switching of the unpinned regions of the wires.Generally, the plateau was larger on one branch of the loop
than the other due to the loop shift and change in switchingfield caused by the exchange bias. Magnetic force micros-
copy was used to image a two-step sample at the plateau
where domain walls are expected to be present separating
the regions of opposite magnetization. However, contrast
from domain walls was not visible, which we attribute to the
low moment of the sample and/or perturbation or annihila-
tion of the domain walls caused by the stray field of the mag-netic tip.
A plot showing the occurrence of two-step and one-step
hysteresis loops vs. w and s is shown in Fig. 2(c). For each
pair of dimensions, five samples were measured. A clear
two-step reversal was seen only for the larger dimensions.However, we cannot rule out the possibility in the one-step
samples that the pinned and unpinned regions switch at
slightly different fields, but the plateau is obscured by the
switching field distribution.
In Fig. 2(b), the ratios of s and w suggest that the plateau
on the descending branch would occur at M ¼64 emu cm
/C03.
However, the plateau was observed at 105 emu cm/C03which
suggests that the effective length of the unpinned NiFe seg-
ment exceeds its value deduced from the SEM images by
/C2413 nm on each side of the pinned segment. A similar result
FIG. 2. VSM measurements for (a) one-step switching for a sample with
s¼w¼200 nm, with remanence loop overlaid; (b) two-step switching for a
sample with s ¼540 nm, w ¼520 nm, with a minor loop; (c) For samples
with w /C25s, the regimes of one-step and two-step switching.162403-3 Sani, Liu, and Ross Appl. Phys. Lett. 110, 162403 (2017)was seen in the other samples with two-step switching and
has been reported also for unpatterned NiFe films with IrMn
stripes of dimensions similar to those used here29but not in
larger structures.20,24,28We assume that the exchange bias is
weaker near the edge of the pinned segment, so part of the
nominally pinned NiFe near the edge of the IrMn reverses at
the same field as the unpinned NiFe.
Figure 3shows exchange bias and coercivity as a func-
tion of the segment length for samples with w /C25s (Figure
3(a)) and as a function of the NiFe stripe width (Figure 3(b))
for samples with w ¼s¼200 nm. In the loops with one step,
we assume the pinned and unpinned regions switch together
and the measured loop offset is a volume-weighted averageof the exchange bias of the pinned segments and unpinned
(zero exchange bias) segments. Error bars in (a) represent
the scatter between several measurements. The exchangebias was 30–40 Oe across a range of sample dimensions,
much smaller than that of the unpatterned bilayer test sam-
ple, which had an exchange bias of 80–120 Oe. There waslittle variation in coercivity and exchange bias across the
samples with w /C25s within the range studied. The coercivity
increased gradually with NiFe linewidth whereas theexchange bias was approximately constant.We note that a significant training effect was observed
in the double grating structure caused by the reorientation of
the interface magnetization of the AFM during cycling. The
second and subsequent hysteresis measurement showed littleor no exchange bias, and therefore, data are only reported for
the loop measured directly after field-cooling.
Micromagnetic simulations were performed with the
magnetic field and exchange bias direction parallel to the
NiFe stripe length. The exchange bias of 100 Oe was mod-
eled as a fixed field applied to the pinned regions, Fig. 4(b).
Examples of a half hysteresis loop, i.e., the magnetization
along the stripe after relaxing at different applied fields, are
shown in Fig. 4(a). This shows the evolution from a single-
step loop at w ¼s¼200 nm to a two-step loop for larger
w and s. A similar transition was found previously in contin-
uous NiFe films patterned with IrMn wires.
29
The micromagnetic configuration is uniform within the
NiFe for the single-step loop for w ¼s¼200 nm at all fields
including the switching field, /C050 Oe. In several simulations
with different random initial conditions, the magnetic configu-
ration was oriented either all to the left or all to the right at
/C050 Oe, without any indication that the pinned and unpinned
regions reversed at different fields to yield a non-uniform
FIG. 3. (a) Measured exchange bias
and coercivity versus the IrMn line
width w for samples with w /C25s. (b)
Exchange bias and coercivity versus theNiFe stripe width for w ¼s¼200 nm.
For samples in both (a) and (b) that
show one-step switching, the exchange
bias is extrapolated assuming the mea-
sured exchange bias is the weighted
average of the two regions. The coer-
civity in the two-step loops is the coer-civity of the pinned region.
FIG. 4. (a) Half hysteresis loops for
w¼s¼200 nm to 600 nm based on
OOMMF simulations. (b) Schematic of
a simulation cell showing the exchange
bias region H EB. The applied field H app
is present throughout the structure,
and periodic boundary conditions are
applied at the two ends. (c) Image of the
micromagnetic configuration at the mid-
point of the plateau ( /C050 Oe) for w ¼s
¼300 nm. A pair of domain walls sepa-
rate the reversed (unpinned) segmentand the unreversed (pinned) segment.
(d) Switching behavior phase map
showing regimes of one-step and two-
step reversal. (e) Effect of exchange
bias in the model on the transition
between one-step and two-step reversal,
for w¼s. The error bars are included
because of the limited combinations of
dimensions tested.162403-4 Sani, Liu, and Ross Appl. Phys. Lett. 110, 162403 (2017)magnetization state. In contrast, Fig. 4(c)shows the micromag-
netic configuration for the two-step loop for w ¼s¼300 nm at
/C050 Oe, which is the midpoint of the plateau. The unpinned
region switched but not the pinne d region, leading to a domain
wall at the boundary.
Fig. 4(d) shows a plot delineating the regimes of one-
and two-step switching. The boundary shows one-step switch-ing when w ¼s<300 nm. One-step switching also occurs for
w¼500 nm when s <200 nm or for s ¼450 nm, w <200 nm.
In order for two-step switching to occur, two domain walls
must be present within a distance of w þs. For the NiFe strip
of width 200 nm, the full width at half maximum of thedomain wall is /C24120 nm (Fig. 4(c)). Therefore, the one-step
regime is related to the finite domain wall width and will
occur when the structure is too small to support two stabletransverse domain walls.
Fig.4(e)shows the effect of the magnitude of the model
exchange bias on the boundary between one- and two-stepswitching for w ¼s. As the exchange bias decreases, the
boundary moves to larger dimensions because there is less
exchange energy stabilizing the pinned segment against rever-sal at the same field as the unpinned segment. The experimen-
tal hysteresis loops gave an exchange bias of around 30 Oe
which in the model would lead to a boundary between one-step and two-step switching of /C24370 nm. This is lower than
the experimental boundary, /C24500 nm, but the agreement
seems reasonable considering the simplified treatment ofexchange bias, the use of periodic boundary conditions, and
the lack of thermal effects.
In summary, size-dependent magnetic switching behavior
was studied in 10 nm thick NiFe stripes of typical width
200 nm and periodicity 400 nm, overlaid with orthogonal
10 nm thick IrMn stripes of width ranging from w ¼200 nm
to 500 nm and periodicity ranging from (w þs)¼400 nm to
1lm. The samples therefore consist of NiFe stripes with alter-
nating exchange-biased and unpinned segments of lengthw and s, respectively. The magnetic switching behavior was
mapped out as a function of w and s both experimentally and
through micromagnetic simulations. At low IrMn wire widthsand spacings, the exchange-coupled and unpinned segments
switched at the same field, whereas for larger dimensions,
the unpinned segment switched at a different field from thepinned segment, resulting in a two-step hysteresis loop and
implying that domain walls are formed at the boundary
between the pinned and unpinned segments. The one-step totwo-step transition with increasing dimensions was also found
in NiFe continuous films overlaid with IrMn stripes. This
behavior is relevant to applications such as domain wall mem-ory or logic devices consisting of magnetic wires with pinned
regions,
22in which local exchange bias can control the loca-
tion and movement of domain walls.
The authors gratefully acknowledge the support of the
National Science Foundation and C-SPIN, a STARnet Center
of the Semiconductor Research Corporation supported by
DARPA and MARCO. Facilities of the MIT Center for
Materials Science and Engineering (NSF DMR1419807) andthe NanoStructures Laboratory were used. SRS acknowledges
support from the Swedish Research Council (VR) underContract No. 637-2014-494.
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1.4813449.pdf | Resonant and non-resonant microwave absorption as a probe of the
magnetic dynamics and switching in spin valves
A. A. Timopheev, N. A. Sobolev, Y. G. Pogorelov, A. V. Talalaevskij, J. M. Teixeira et al.
Citation: J. Appl. Phys. 114, 023906 (2013); doi: 10.1063/1.4813449
View online: http://dx.doi.org/10.1063/1.4813449
View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v114/i2
Published by the AIP Publishing LLC.
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Downloaded 15 Jul 2013 to 141.210.2.78. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsResonant and non-resonant microwave absorption as a probe of the
magnetic dynamics and switching in spin valves
A. A. Timopheev,1,a)N. A. Sobolev,1Y . G. Pogorelov,2A. V. Talalaevskij,2J. M. Teixeira,2
S. Cardoso,3P . P . Freitas,3and G. N. Kakazei2,4,b)
1Departamento de F /C19ısica and I3N, Universidade de Aveiro, 3810-193 Aveiro, Portugal
2IFIMUP and IN-Institute of Nanoscience and Nanotechnology, Departamento de F /C19ısica e Astronomia,
Universidade do Porto, 4169-007 Porto, Portugal
3INESC-MN and IN-Institute of Nanoscience and Nanotechnology, 1000-029 Lisbon, Portugal
4Institute of Magnetism, NAS of Ukraine, 03142 Kiev, Ukraine
(Received 13 May 2013; accepted 21 June 2013; published online 11 July 2013)
We use the resonant and non-resonant microwave absorption to probe the dynamic and static
magnetic parameters of weakly coupled spin valves. The sample series include spin valve
structures with varying thickness of the non-magnetic metallic spacer and reference samplescomprised only a free or fixed magnetic layer. Beside the common resonance absorption peaks, the
observed microwave spectra present step-like features with hysteretic behavior. The latter effect is
a direct manifestation of the interlayer coupling between the ferromagnetic layers and provides twostatic magnetic parameters, the switching field and coercivity of the fixed layer. The analysis of the
microwave absorption spectra under in-plane rotation of the applied magnetic field at different
spacer thicknesses permits a deeper insight in the magnetic interactions in this system as comparedto the conventional magnetometry. We combine the standard Smit-Beljers formalism for the
angular dependence of the resonance fields with a Landau-Lifshitz-Gilbert dynamics extended to
describe in detail the intensity of microwave absorption in the spin valves. In this way, we extract aset of parameters for each layer including the effective magnetization and anisotropy, exchange
bias and interlayer coupling, as well as Gilbert damping. The model reproduces well the
experimental findings, both qualitatively and quantitatively, and the estimated parameters are in areasonable agreement with the values known from the literature. The proposed theoretical
treatment can be adopted for other multilayered dynamic systems as, e.g., spin-torque oscillators.
VC2013 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4813449 ]
I. INTRODUCTION
Magnetic multilayers are of great research interest due
to their applications in the fields of data storage and mag-
netic field sensing. Spin valves (SVs)1–3represent one of the
technologically most efficient structures for this purpose,
employing the giant magnetoresistance (GMR) effect4to
detect changes of the relative orientation between magneticmoments of the free and fixed layers.
There are two spatially separated types of interlayer
interactions in SVs: the exchange coupling of the fixed ferro-magnetic (FM) layer to the antiferromagnetic (AFM) one
and the interlayer coupling (IC) between the fixed and
free FM layers. The FM/AFM direct contact leads to theexchange bias effect
5by an exchange bias field, Hex, on the
FM layer. The physics of this effect is rich and still challeng-
ing. Many structural and magnetic aspects of the FM/AFMsystem behavior
6–12are still under intensive study. Among
others, the MnIr-based systems are favored by high Hexval-
ues, high N /C19eel and blocking temperatures (above 700 K and
410 K, respectively),13,14low annealing fields, and a low crit-
ical AF thickness to keep exchange bias stable.15,16Incontrast, the main IC mechanisms in SVs are rather clear:
the Ruderman-Kittel-Kasuya-Yosida (RKKY)17–19indirect
coupling via conduction electrons in the metallic spacer andthe N /C19eel “orange-peel” magnetostatic coupling.
20,21
Typically, increasing surface roughness leads to suppression
of the RKKY interaction and strengthens the N /C19eel “orange-
peel” mechanism. Also a direct exchange between the FM
layers may take place when the surface density of pinholes
in a thin metallic spacer is high enough.22Anyway, all these
mechanisms contribute to the same isotropic Heisenberg
interaction between the two layer magnetization vectors,
defining its effective coupling constant Eic.
Both the IC and Hexparameters in SVs are easily
extracted from magnetization or magnetoresistance measure-
ments. However, usage of the ferromagnetic resonance(FMR) technique can be preferable as permitting, in addition
to those parameters, also quantitative information on the
effective magnetization and internal anisotropy of the FMlayers and their dynamical properties. In some cases, it even
allows extracting static parameters of the FM layers,
23i.e., it
can partially replace magnetostatic measurements.
From the FMR viewpoint, the interlayer coupling has
dynamic and static constituents. The dynamic part always
exists, due to an interaction between the dynamic compo-nents of the magnetization vectors of the free and fixed
layers. The distance between the respective resonance modesa)Author to whom correspondence should be addressed. Electronic mail:
andreyt@ua.pt
b)Now at Department of Electrical & Computer Engineering, National
University of Singapore, Singapore.
0021-8979/2013/114(2)/023906/9/$30.00 VC2013 AIP Publishing LLC 114, 023906-1JOURNAL OF APPLIED PHYSICS 114, 023906 (2013)
Downloaded 15 Jul 2013 to 141.210.2.78. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsincreases as the IC strengthens. The static contribution only
exists in the case of weak coupling when the magnetic
moments precess almost independently. The magneticmoment of each layer is affected by the effective interaction
field produced by the magnetic moment of the other layer
and both resonance modes are shifted in the same directionwhen the IC increases. Generally, the FMR in SVs can ex-
hibit two regimes, those of the strong and weak coupling. In
the first case, the resonance peaks are considered as normalmodes of coupled oscillators. In this regime, the in-plane
angular dependence of the resonance field of each peak
shows simultaneously features of both free and fixed layers,their intensities and linewidths being functions of their rela-
tive positions. In the weak coupling regime, the peaks
behave as two independent resonances stemming from thefree and fixed layers, since the IC dynamic effects are too
weak. A weak interaction field (due to the static contribu-
tion) shifts the resonance field for each peak in the samedirection.
Despite the well-developed formalism of FMR in
SVs,
24–26experimental determination of the IC strength in
weakly coupled SV systems27is a non-trivial task. Several
nanometers thick ferromagnetic layers typically have
reduced saturation magnetization and modified magnetic ani-sotropy parameters and thus the usage of reference book pa-
rameters measured in bulk ferromagnets is not justified.
Moreover, the resonance conditions for FMR usually requiretoo high fields, where the magnetic moments of both fixed
and free layers are already aligned with the external mag-
netic field. In this regime, the above discussed static part ofthe IC appears to be angular independent. Thus, the interac-
tion effect is only reflected in small (of the order of the IC
field on each layer) shifts of the resonance fields. As a result,it is hard to separate the “thin layer” effects from the interac-
tion effects without employing additional experimental
methods and, to the best of our knowledge, there is no publi-cation discussing that.
Here we present a comprehensive analysis of weakly
coupled spin valves based only on microwave absorptionmeasurements. Resonant and also non-resonant features,
such as the switching field and coercivity of the fixed layer,
will be analyzed and modeled on the combined basis of theSmit-Beljers formalism, macrospin approximation, and the
Landau-Lifshitz-Gilbert equation extended here to describe
the non-resonant microwave absorption in spin valves. Weshow that the hysteresis of the microwave absorption,
observed in the field range where the fixed layer is being
unpinned from the antiferromagnet, occurs due to the inter-layer coupling between the FM layers.
II. EXPERIMENTAL DETAILS
Spin valves with the (Glass/Ta(30 A ˚)/Ni 80Fe20(30 A˚)/
Co80Fe20(25 A˚)/Cu( tCu)/Co 80Fe20(25 A˚)/Mn 82Ir18(80 A˚)/
Ta(30 A ˚)) structure were grown by the ion-beam deposition
in a Nordiko3000 system.28The Cu spacer thickness ( tCu)
was varied from 14 to 28 A ˚. Additionally, samples contain-
ing only the free (Glass/Ta/NiFe/CoFe/Cu/Ta) and fixed
(Glass/Ta/Cu/CoFe/MnIr/Ta) layers from these SVs wereprepared at the same conditions, with the same layer compo-
sitions and thicknesses to serve as references for the FMR
studies. A more detailed description of the growth routinecan be found in Ref. 29. As a part of preliminary characteri-
zation, 4-point magnetoresistance measurements were
performed on the SVs. The samples with t
Cu>16 A˚demon-
strated the GMR effect up to 5-6%, while at thinner copper
spacers, an abrupt decrease of the GMR was observed down
to 1% for tCu¼14 A˚.30It also should be noted that rigorous
analysis of FMR spectra requires the in-plane saturation
fields for the isolated free and fixed layers as well as for the
spin valves. Although we did not conduct magnetometry forour samples, we can conclude from the rectangular shape of
magnetoresistance curves
30that both layers in the weakly
coupled spin-valves are easily saturated in several tens ofoersteds.
FMR studies were done at room temperature using a
Bruker ESP 300 E EPR spectrometer at the 9.67 GHz micro-wave frequency. The first derivative of microwave absorp-
tion was registered due to modulation of the applied
magnetic field. For each sample, a series of FMR spectrawere collected for different angles of magnetic field in the
film plane with respect to the internal exchange bias. Each
FMR spectrum was fitted with Lorentzian and/or Gaussianfunctions to obtain the resonance field and linewidth. The
effective spectroscopic splitting factor of the reference com-
posite free layer was extracted from the frequency vs fielddependences measured with a vector network analyzer
(VNA) setup, similar to that described in Ref. 31.
III. EXPERIMENTAL RESULTS AND DISCUSSION
A. Reference free layer
In order to clarify the FMR dynamics in the SVs, the ref-
erence free and fixed layers were tested at the first step of the
study. The FMR line of the reference free layer has a sym-
metric Lorentzian shape with a full width at half maximum(FWHM) of 66 Oe.
The FMR angular dependences were obtained from the
Smit-Beljers formalism
32
@Ufrl
@hm1¼0;@Ufrl
@um1¼0;
x
c/C18/C192
¼1
ðMs1sinhm1Þ2@2Ufrl
@hm12@2Ufrl
@um12/C0@2Ufrl
@hm1@um1 ! !9
>>>=
>>>;;
(1)
where xis the microwave frequency and c¼gl
B//C22his the
gyromagnetic ratio. The Land /C19e spectroscopic factor for this
bilayer was found with VNA setup as g1¼2.10. The mag-
netic energy density (per unit area) for the free layer is writ-ten as
U
frl¼t1ðð2pM2
s1/C0Ku?1Þcos2hm1/C0Kujj1sin2hm1cos2um1
/C0HextMs1cosðuh/C0um1Þsinhm1Þ: (2)
Here, the first term comprises the demagnetization (2 pMs12)
and perpendicular anisotropy ( Ku?1) contributions so that the023906-2 Timopheev et al. J. Appl. Phys. 114, 023906 (2013)
Downloaded 15 Jul 2013 to 141.210.2.78. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionseffective film magnetization is Meff1¼Ms1/C0Ku?1/(2pMs1),
with the bulk saturation magnetization Ms1. The second term
involves the in-plane uniaxial anisotropy ( Kuk1) induced by
annealing the film under applied magnetic field, and the last
one is the Zeeman energy in the external magnetic field Hext
at the angle uhto the in-plane anisotropy axis. The polar
(hm1) and azimuthal ( um1) angles of the free layer’s magnetic
moment M1are defined with respect to the film normal and
to the in-plane anisotropy axis.
The fitting results are shown in Fig. 1by a solid line,
related the effective anisotropy field Hak1¼2Kuk1=Ms1
¼9.2 Oe and the effective film magnetization 4 pMeff1
¼14.2 kG. According to Ref. 33, the average magnetiza-
tion of a FM bilayer made of two ferromagnets with the
volume magnetizations MA
s,MB
sand the corresponding
thicknesses tA,tBis approximated as
4pMav¼4pðtAðMA
sÞ2þtBðMB
sÞ2Þ=ðtAMA
sþtBMB
sÞ:
Using the reference book 4 pMsvalues for Ni 80Fe20
(/C2410 kG) and Co 80Fe20(/C2420 kG) alloys34and respectivethicknesses tA¼30 A˚andtB¼25 A˚(t1¼tAþtB), one gets
4pMav¼16.3 kG, a noticeably higher value than that meas-
ured experimentally. In absence of perpendicular contribu-tion: K
u?1¼0, as repeatedly checked in magnetostatic
studies on such structures, the observed 13% drop of magnet-
ization should be evidently attributed to a reduced magnet-ization near the FM/NM interfaces.
B. Reference fixed layer
In contrast to the free layer, the FMR line of the fixed
reference layer (see Fig. 2(a)) is much broader (FWHM
/C25410 Oe) and has Gaussian shape, indicating magnetic
inhomogeneities near the AF/FM interface. The FMR line
shows two principal features:
(i) A unidirectional anisotropy by the exchange coupling
to the MnIr AFM layer, seen in the 360/C14symmetry of
FMR field angular dependence with the highest valueof/C25950 Oe for H
extantiparallel and the lowest one of
/C2560 Oe for Hextparallel to the exchange bias field.
(ii) Switching of the FMR line position when Hextis
(near) antiparallel to the exchange field. This is due to
switching of the FM layer at Hext/C25420 Oe. A small
jump due to changes in the microwave absorptionconditions is clearly observed in the FMR spectra
(Fig. 2(a)). The jump amplitude and sharpness
decrease as the external field is turned away from theantiparallel orientation to the exchange field. A hys-
teresis of the microwave absorption is observed in
this region for the up- and down-sweeps of the mag-netic field (Fig. 2(b)), denoting that, aside from the
unidirectional anisotropy, the system possesses an in-
plane magnetic anisotropy.
In all the SVs under study, as well as in the reference
fixed layer, the uniaxial anisotropy axis coincides with thatof the exchange bias. This anisotropy is responsible for the
observed hysteresis in the FMR spectra wherefrom two addi-
tional parameters could be extracted: the switching field, H
sw
(the geometrical center of the hysteresis loop), and theFIG. 1. In-plane angular dependence of the FMR for the reference free layer
(circles) and corresponding model fitting (solid line) using Eq. (2)with the
following parameters: Ku?1¼0,Ms1¼1128.5 emu/cm3,Kujj1¼5.2/C2103
erg/cm3.
FIG. 2. FMR of the reference fixed layer: (a) FMR spectra collected at different in-plane orientations of the external magnetic field. (b) Hysteresis of the micro-
wave absorption during the field-up and field-down sweeps. The inset shows the way to extract the switching field and coercivity of the fixed layer.023906-3 Timopheev et al. J. Appl. Phys. 114, 023906 (2013)
Downloaded 15 Jul 2013 to 141.210.2.78. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionscoercive field, Hc(the half distance between crossings of the
field-up and field-down traces with the vertical level of the
loop center), as sketched in the inset of Fig. 2(b).
Empirically, one can differentiate the FMR curves for the
field-up and field-down sweeps to define the positions of the
respective d2P/dH2maxima, then Hswcorresponds to their
half sum, and Hcto their half difference. It should also be
noted that the hysteresis of the microwave absorption,
though caused by that of the static magnetization, does notfollows exactly the latter’s real shape. However, the harder
(more rectangular) the magnetization loop, the sharper is the
step on the microwave absorption curve and the smaller isthe systematic error in determination of the coercivity and
switching field.
At intermediate angles ( u
hfrom 50/C14to 90/C14), there are two
lines in the spectrum (Fig. 2(a)), indicating that switching of
the FMR line position still takes place. The asymmetry of lines
and their overlapping complicate accurate extraction of theresonance fields that is why we did not include these points in
the angular dependence shown by circles in Fig. 3. For the
model fitting, additionally to the terms present in Eq. (1),a
unidirectional anisotropy contribution to the magnetic energy
of the system, E
ex, was included. It has been found from the
out-of-plane FMR measurements that the exchange bias fieldlies in the film plane. As noted above, the uniaxial anisotropy
and exchange bias axes coincide, naturally defining the choice
of azimuthal axis for spherical coordinates, while its polar axisis still aligned with the film normal. Then the magnetic energy
density in the considered system is written as
U
fxl¼t2½ð2pM2
s2/C0Ku?2Þcos2hm2/C0Kujj2sin2hm2cos2um2
/C0HextMs2cosðuh/C0um2Þsinhm2/C138
/C0Eexcosðuu/C0um2Þsinhm2; (3)
where t2is the FM layer thickness, hm2andum2are the polar
and azimuthal angles of its magnetic moment. From the liter-
ature,35the Land /C19e spectroscopic splitting factor for Co 80Fe20
isg2¼2.12.The fitting results are shown on Fig. 3by solid lines. It
was impossible to find appropriate model parameters to fit
accurately the whole angular range of the experiment. Onthe other hand, there is no reason to include additional con-
tributions in the model. As a compromise, two essentially
different sets of parameters were found to provide partial fit-tings over partial angular ranges: (i) the first one is accurate
around the antiparallel orientation but fails outside it and (ii)
the second one is accurate around the parallel and close tothe antiparallel orientation but fails in the intermediate range.
To choose the correct parameter set, one has to discuss this
situation in detail and bring additionally into analysis themeasured values of coercivity and switching field. As can be
seen, the first set only applies in the u
hrange from 100/C14to
260/C14. However, the FM/AFM interlayer coupling
Eex¼5/C210/C02erg/cm2is six times lower than that in Refs.
36and37and twice lower than in Ref. 38. Moreover, the
magnitudes of the coercive field (2 Kuk2/Ms2¼9.5 Oe) and
switching field ( Eex/(Ms2t2)¼189 Oe) extrapolated from the
macrospin approximation would have been much lower than
the experimentally observed, Hc¼50 Oe and Hsw¼420 Oe.
Also the volume magnetization parameter Ms2is unreason-
ably small (only 66% of the expected Ms/C251600 emu/cm3).
In contrast, the second set provides fully adequate Hcand
Hswvalues (54 Oe and 438 Oe, respectively), as well as
Ms2¼1480 emu/cm3, a value that is much closer to the
expected one. It can be physically argued that the both mag-netization and exchange coupling of the fixed layer are
essentially reduced in course of its switching, most probably
due to the aforementioned effects of inhomogeneity and/ordomain structures in the AFM layer.
C. Spin valves
The SV samples were divided into two groups30with
respect to the IC regimes: (i) weak coupling ( tCufrom 17 to
28 A˚) and (ii) strong coupling ( tCufrom 14 to 16 A ˚). The first
group, which is of our interest, showed no direct IC signs in
the FMR angular dependences. The expected unidirectional
contribution from the fixed layer is absent in the angulardependences of the free layer, unlike those observed in the
SVs with t
Cu<17 A˚30or calculated in Ref. 26.
In the weak coupling regime, when the magnetic
moments of free and fixed layers precess almost independ-
ently, a small static exchange field acts on each layer in addi-
tion to the external magnetic field. These interaction fieldsresult from the explicit Heisenberg IC
U
ic¼Eic½coshm1coshm2þsinhm1sinhm2cosðum1/C0um2Þ/C138:
(4)
In our SVs, the existing conditions t1Kujj1/C28t2Kujj2
þEexandMs1ffiMs2indicate that the most prominent inter-
action effect which could appear is a unidirectional anisot-ropy contribution to the angular dependence of the free
layer. However, this expected effect is suppressed due to the
relatively high resonance field ( /C24700 Oe) of the free layer.
As a matter of fact, this resonance field is twice higher of the
exchange bias field, E
ex/(2t2Ms2), and more than an order ofFIG. 3. In-plane FMR angular dependence for the reference fixed layer
(circles) and model fits (solid lines) using two sets of parameters in Eq. (3):
Eex¼0.05 erg/cm2,Kujj2¼5/C1103erg/cm3,Ku?2¼0,Ms2¼1057 emu/cm3to
fit only the range uh/H20678[100/C14,2 6 0/C14], and Eex¼0.162 erg/cm2,Kujj2¼4/C1104
erg/cm3,Ku?2¼0,Ms2¼1480 emu/cm3to fit the whole range of the angles.023906-4 Timopheev et al. J. Appl. Phys. 114, 023906 (2013)
Downloaded 15 Jul 2013 to 141.210.2.78. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsmagnitude higher of the free layer anisotropy field, Kuk1/Ms1.
As a result, the magnetic moments in both layers are practi-cally aligned with the external magnetic field for any in-
plane angle u
h, which causes a constant shift of the FMR
line positions for each layer by the respective interactionfield values, H
ic1¼Eic/(t1Ms1) and Hic2¼Eic/(t2Ms2). Just
this tendency has been observed in our SVs, i.e., a continu-
ous slight decrease of the mean resonance field for both the
free and fixed layers with narrowing tCu. Accordingly, it is
possible to describe the main features of the weakly coupledSVs assuming full independence ( E
ic¼0) of the fixed and
free layers (i.e., using the models by Eqs. (2)and(3)), but
admitting slight alterations of the effective magnetizations inthe layers due to the ignorance of the existing interlayer cou-
pling. Below, a rigorous quantitative analysis describing the
FMR peaks in SVs by normal modes of a system of coupledmagnetic layers will be performed in Secs. IVandV.
The main FMR features will be described on the exam-
ple of the SV with t
Cu¼24 A˚(see Fig. 4). Each spectrum
consists of two resonance lines stemming from the fixed and
free layers and exhibiting 360/C14and 180/C14symmetries, respec-
tively. The free layer FMR in these SVs shows no substantialchanges as compared to the reference free layer. No sizable
unidirectional anisotropy is detected either. The free layer,
being deposited first, preserves its flatness that assures such astable behavior. However, a slight decrease (by /C2420–30 Oe)
of the resonance field is observed, yielding an enhanced
value of calculated M
s1(see the legend to Fig. 4). This fact
points to the above discussed interaction effect and deter-
mines the ferromagnetic character of the interlayer coupling.
In contrast, the fixed layer, deposited atop the free layer
and copper spacer, shows a noticeable decrease of both the
exchange bias and the effective magnetization. It leads to a
substantial decrease of the switching field Hsw. The latter,
determined directly from the hysteresis of the microwave
absorption, drops from 420 Oe (for the reference fixed layer)
to/C24300 Oe in all weakly coupled SVs. The model by Eq. (2)
yields the best fit with the parameters Eex¼0.095 erg/cm2
and Ms¼1155 emu/cm3. The switching field calculated inthe macrospin approximation, Hsw¼Eex=ðt2Ms2Þ¼329 Oe,
agrees well with the value measured from the microwaveabsorption hysteresis. This agreement proves absence of any
measurable perpendicular anisotropy ( K
u?2) in the fixed
layer. While the reduced values of EexandMsdirectly reflect
a relatively high roughness of such an ultra-thin FM layer, it
is evident that any roughness, even on an atomic scale
(/C243–4 A˚), should noticeably influence the magnetic proper-
ties of a 25 A ˚thick FM layer.
Presence of an in-plane uniaxial anisotropy in the fixed
layer with an effective field of /C2435 Oe leads to its distinctive
hysteretic switching (see Fig. 5) with the maximum coerciv-
ity at uh¼180/C14. At intermediate uhangles (between 0/C14and
180/C14), the metastable minima resulting from the uniaxial ani-
sotropy are suppressed by an order of magnitude higher uni-
directional anisotropy, so that the coercivity vanishes. Asimilar effect was also theoretically discussed in Ref. 39.
To prove our statements, we have calculated the angularFIG. 4. In-plane FMR spectra for the spin valve with tCu¼24 A˚(a) and their angular dependences (b). The points represent experimental data and the lines are
the model fits using Eqs. (2)and(3)with the following parameters: Ku?1¼0,Ms1¼1186 emu/cm3,Kujj1¼5.7/C2103erg/cm3,Ku?2¼0,Ms2¼1155 emu/cm3,
Kujj2¼2.0/C2104erg/cm3andEex¼0.095 erg/cm2.
FIG. 5. Hysteresis of the microwave absorption in the SV sample with
tCu¼24 A˚. The inset shows the angular dependences of coercivity obtained
from the experiment (circles) and by the model calculation (solid line) using
Eq.(3)with the parameters indicated in Fig. 4.023906-5 Timopheev et al. J. Appl. Phys. 114, 023906 (2013)
Downloaded 15 Jul 2013 to 141.210.2.78. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsdependence of the fixed layer coercivity (see inset in Fig. 5).
The calculation was done through the energy minimization
of Eq. (3)in the macrospin approximation. Using the param-
eter values extracted from fitting the angular dependence of
the resonance field, we obtained a very similar angular de-
pendence of the coercivity. The value Kuk2¼2.0/C2104erg/
cm3, obtained by the model fitting of this angular depend-
ence, gives the maximum theoretical coercivity of 2 Kuk2/
Ms2¼35 Oe, very close to the measured value. This also
indicates that the magnetization reversal mechanism in the
fixed layer is a coherent rotation and validates the direct
extraction of the switching field and coercivity from theFMR spectra.
IV. THEORETICAL BACKGROUND FOR THE
MICROWAVE ABSORPTION IN SPIN VALVES
In order to theoretically describe both the resonant and
non-resonant microwave absorption in SVs as a function of
the external field Hextand r.f. field h, we have to go beyond
the most common framework of resonance conditions given
by the roots of a secular equation. To this end, we calculate
the internal fields in each FM layer, Hj¼/C0tj/C01rMjUSVþh,using the model energy density USV¼UfrlþUfxlþUic
given by Eqs. (2)–(4), and then solve the non-uniform
coupled Landau-Lifshitz-Gilbert (LLG) equations40,41for
the magnetizations Mjof the free ( j¼1) and fixed ( j¼2)
layers
_Mj¼/C0cjMj/C2HjþajMjðMj/C2HjÞ;j¼1;2;(5)
withcj¼gjlB//C22h. The Gilbert parameters ajdescribe the dissi-
pation of the power absorbed by the sample from the micro-
wave magnetic field h.
To simplify the analysis, we restrict it here to the in-
plane, hh¼p/2, and collinear, uh¼0/C14or 180/C14, geometries
with the z-axis aligned along the exchange anisotropy field,
and choose the microwave magnetic field in the form
h¼exhxcosxt(linearly polarized accordingly to our experi-
mental conditions). In this case, the x- and y-components of
the precessing magnetizations in the two FM layers can be
joined into a 4-spinor w¼(Mx1,My1,Mx2,My2). Then, the
standard solution for the non-uniform matrix equation result-ing from Eq. (5)isw¼^Rf, expressed through the resolvent
matrix ^R
^R¼x
2/C0x11x12þa12x122/C0a1X1x11 c1x21Hic2t2=t1 0
a1X1x12 x2/C0x11x12þa12x1120 c1x22Hic2t2=t1
c2x11Hic2t1=t2 0 x2/C0x21x22þa22x222/C0a2X2x21
0 c2x12Hic2t1=t2 a2X2x22 x2/C0x21x22þa22x2120
BB@1
CCA/C01
;
and the source force spinor f
f¼hxðx11/C0a12x12Þcosxtþa1xsinxt
/C0a1X1cosxtþxsinxt
ðx21/C0a22x22Þcosxtþa2xsinxt
/C0a2X2cosxtþxsinxt0
BB@1
CCA;
where the characteristic frequencies are x11¼c1[Hext62Kuk1/Ms1þS(Hext)Hic1],x21¼c2[Hext62Kuk2/Ms2þHex
þS(Hext)Hic2],xj2¼xj1þ4pMsjcj,Xj¼xj1þxj2, and Hex¼Eex/(Ms2t2). The6sign refers to the field sweep direction (up
or down). The function S(Hext)¼tanh[(– Hextcosuh6Kuk2/Ms2/C0HexþHic2)/(0.2 Kuk2/Ms2)] phenomenologically models the
smooth switching of the fixed layer magnetization in the course of the field sweep, alike standard fits for hysteretic magnetiza-tion curves in bulk ferromagnets.
42As could be seen from the experiment, the hysteresis loop is not absolutely sharp, and the
switching occurs in a certain field range. To consider such a behavior in the model, the function S(Hext) has been designed.
The factor 0.2 Kuk2/Ms2in the denominator scales the field range in which the switching occurs. Then the sought absorption
power (averaged over a precession period) is
P¼x
2pð2p=x
0_USVdt; (6)
and the time dependence appears only due to the dissipation terms in the LLG equations, _USV¼w†^Dw, where the diagonal
dissipation matrix is
^D¼a1c1t1Ms1x122000
0 a1c1t1Ms1x11200
00 a2c2t2Ms2x2220
000 a2c2t2Ms2x2120
BB@1
CCA:023906-6 Timopheev et al. J. Appl. Phys. 114, 023906 (2013)
Downloaded 15 Jul 2013 to 141.210.2.78. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsThe resulting dP/dHcurve (solid line on Fig. 6(a)) dem-
onstrates the FMR absorption peaks fitted to the experimen-
tal data (blue circles) for the particular case of SV witht
Cu¼24 A˚. It is worth noting that no additional “technical
coefficients” were used in this fitting, which undoubtedly
indicates the adequacy of the model and a high quality of the
grown samples. The relative amplitudes of the FMR peaks
are strictly defined by the dissipation matrix ^D. The layer
thicknesses are the same as for the reference layers. Other fit-
ting parameters (listed in the legend to Fig. 6(a)) are almost
the same as those used above to fit the FMR angular depend-ence in this sample within the Smit-Beljers formalism with
no account for IC (Fig. 4(b)). Only a slight reduction of cal-
culated saturation magnetization in both layers is observed,in accordance with our expectations. The reduction is stron-
ger in the fixed layer, as thinner than the free layer so that
the respective interaction field H
ic2is higher. Also, the
decrease of the uniaxial contribution, Kuk2, should be attrib-
uted to the influence of the free layer. Additionally, the
model permits to reasonably estimate the damping parame-ters of the SV system. It is only a small asymmetry of the
stronger peak, hardly seen in the antiparallel and more prom-
inent in the parallel orientation, that remains out of the scopeof this model.
Fig.6(b) shows a low-field part of the simulated micro-
wave absorption spectrum (plotted as P(H) dependence),
where the fixed layer switching takes place. As the interlayer
coupling tends to zero the hysteresis effect becomes hardly
seen on this scale (though it still exists). Thus, the observedjump in the microwave absorption (as well as the hysteresis
of it) in our SVs is a direct manifestation of the interlayer
coupling. The effect is mainly caused by the change of reso-nance field for the free layer when the fixed layer is being
unpinned from the antiferromagnet. Near the antiparallel
magnetic field orientation, the resonance field of the freelayer is much closer to the switching field, H
sw, and its inten-
sity is much higher. When switching of the fixed layer occurs,the IC field on the free layer changes its sign, so that the total
effective field on the free layer increases by 2 Hic1. As a result,
the resonance field decreases by about the same value.
As shown in Sec. III B, a hysteresis of microwave
absorption is clearly seen in the spectra of the reference fixed
layer in absence of the dominating free layer peak. However,
this effect is significantly suppressed in the SVs, due to
reduced exchange bias and magnetic parameters of the fixedlayer. According to our calculations, the microwave absorp-
tion of the fixed layer is negligibly weak near the switching
field so that the observed kink effect stems mostly from thefree layer. It should be also noted that the instrumental deriv-
ative of the absorption spectrum (using a modulation field
and lock-in detection) is not done properly when passingthrough a non-equilibrium magnetic state of the system where
a hysteresis takes place. This explains why there are only
kinks in the experimental spectra shown in Fig. 5, instead of
peaks expected in a true derivative of a step-like curve.
V. THICKNESS DEPENDENCE AND N /C19EEL MODEL
FITTING
A preliminary study on these samples30has shown that
the main interlayer coupling mechanism is N /C19eel’s “orange-
peel” magnetostatic interaction. Here, we evaluate the inter-
layer coupling and extract topological parameters of weakly
coupled spin valves. According to N /C19eel’s paper,20the mag-
netostatic coupling energy of two layers with the saturation
magnetizations Ms1,Ms2, separated by a non-magnetic
spacer of thickness tCu, all of them having the same waviness
of period kand height h, reads
Eic¼p2Ms1Ms2h2
kffiffiffi
2pexp/C02pffiffiffi
2ptCu
k/C18/C19
: (7)
In order to improve the accuracy, the fitting was per-
formed simultaneously on five sets of available experimentalFIG. 6. (a) The solid line shows a simulation of the dP/dH ext¼f(Hext) dependence using the parameters fitted to the FMR data at /h¼180/C14for the SV with
tCu¼24 A˚. Dots are experimental data. The fitting parameters are hx¼0.1673, x/2p¼9.67/C1109rad/s, a1¼0.012, a2¼0.055, t1¼5.0/C110–7cm,
t2¼2.5/C110/C07cm,Ms1¼1165 emu/cm3,Ms2¼1125 emu/cm3,Kuk1¼5.7/C1103erg/cm3,Kuk2¼1.7/C1104erg/cm3,Eic¼0.006 erg/cm2,Eex¼0.092 erg/cm2. (b)
Simulation of the hysteresis for the P¼f(Hext) dependence using the same parameters as for (a) (red curves) and additionally for Eic¼0 (green curve). The fig-
ure shows that the hysteresis of P(Hext) is observable only if the interlayer coupling is non-zero.023906-7 Timopheev et al. J. Appl. Phys. 114, 023906 (2013)
Downloaded 15 Jul 2013 to 141.210.2.78. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsthickness dependences, viz., those of two FMR resonance
fields, each at uh¼0/C14and 180/C14, and that of the switching
field Hsw. The averaged parameters, Ms1¼1155 emu/cm3,
Ms2¼1175 emu/cm3,Eex¼0.094 erg/cm2, were used for the
whole SV series, while the others were taken from the fittingof the dP/dH
ext¼f(Hext) dependences (Fig. 6). Resonance
fields vsIC were found from numerically calculated roots of
the equation det ^R/C01¼0. The switching field was found as
Hsw¼(Eex–Eic)/(t2Ms2). Then the deduced Hres,Hswvalues
as functions of tCuwere fitted by the N /C19eel-type expressions
Eic¼ANexp(/C0BNtCu). Fig. 7shows the fitting results (solid
lines) for the parameters AN¼0.0492 erg/cm2and
BN¼0.0889 A ˚–1. The calculated dependences Hres(tCu) dem-
onstrate decreasing resonance fields for both peaks withincreasing IC. The distance between the resonance modes
increases with increasing coupling as typical of coupled
oscillators. The interlayer coupling parameter E
icdecreases
from 1.1 /C210/C02erg/cm2to 4 /C210/C03erg/cm2with tCu
increasing from 17 to 28 A ˚. As a consequence, the effective
field, Hic1, on the free layer decreases from 17 to 6 Oe. A
similar estimate of Hic2on the fixed layer yields a variation
from 37 to 14 Oe, thus meaning that the switching field
should be reduced by 23 Oe if the parameters Ms2andEex
would be independent of tCu. Actually, as seen from Fig. 7,
these parameters are sensibly dependent on tCuand the
observed scatter of experimental points reflects their smallfluctuations, rather than measurement errors. We were not
able to put error bars on the graph without a deep structural
analysis of each sample. The deduced N /C19eel model parame-
ters, the waviness height of 7.2 A ˚and its periodicity of
100 A˚, look consistent. Direct magnetotransport measure-
ments
29of interlayer coupling done on similar systems have
shown close results.
VI. CONCLUSIONS
We have investigated the weak coupling regime in spin
valves using the ferromagnetic resonance technique. Unlike
the strongly coupled systems, dynamic effects are less pro-nounced in this regime. Thus, precession of magnetic
moment in each layer is practically independent from theother. Only a small static interlayer exchange field exists in
the system and, from the standard FMR viewpoint, it can be
masked by slight changes of the layer’s effective magnetiza-tion. Therefore, identification of interaction effects from
FMR data turns non-trivial and a combination of traditional
FMR analysis with micromagnetics was used for that pur-pose. The static magnetic parameters of the fixed layer were
extracted from the hysteretic effect in microwave absorption
due to switching of the fixed layer.
The study was done on ion-beam deposited NiFe/CoFe/
Cu/CoFe/MnIr spin valves (SVs) with the Cu-spacer thick-
ness t
Cuvarying from 17 to 28 A ˚. The main interlayer cou-
pling mechanism in this series is the N /C19eel’s “orange-peel”
magnetostatic interaction. To clarify the interpretation, we
prepared also reference samples of the fixed (CoFe/MnIr)and free (NiFe/CoFe) layers, respectively. For all the SV
samples with weak interlayer coupling, the resulting FMR is
a superposition of resonance peaks stemming from the freeand fixed layers, with parameters close to those observed in
the reference layers.
The static magnetic properties of the fixed layer, such as
switching field, coercivity and its angular dependence have
been analyzed in the macrospin approximation using the
same expression for the magnetic energy as in the FMRtreatment. The magnetic parameters extracted from the FMR
angular dependences have been used to calculate the static
parameters and to compare them with the experiment.
The developed theoretical approach based on the
Landau-Lifshitz-Gilbert equations for a coupled magnetic
system permitted to quantitatively describe the observeddynamic and static properties of the SVs without additional
adjustment parameters. The damping parameters of the SVs
have been determined. The main IC effects in these SVs arefound in the decrease of resonance fields for both peaks and
hysteretic microwave absorption upon bidirectional magnetic
field sweeps. The first effect is due to a weak, quasi-staticexchange field from one FM layer to another, whiles the sec-
ond effect, being subsequent to the first one, is a shift of the
resonance field of the free layer after switching of the fixedlayer. Finally, the model has been used to extract the inter-
layer coupling strength and topological parameters of the
spin valves. The proposed theoretical treatment can beadapted to describe the interlayer coupling effects in other
multilayered dynamic systems, such as, e.g., spin-torque
oscillators.
43
ACKNOWLEDGMENTS
We would like to say many thanks to Mr. Ivo Mateus
for his professionalism and kind-hearted attitude to our fre-
quent demands for mechanical engineering works.
This work was supported by the Portuguese FCT
through the projects PEst-C/CTM/LA0025/2011, RECI/FIS-
NAN/0183/2012, NanoSciERA/0002/2008, PTDC/EEA-ELC/108555/2008, and PTDC/CTM-NAN/112672/2009,
Grants SFRH/BPD/74086/2010 (A.A.T.) and SFRH/BPD/
72329/2010 (J.M.T.), and the “Ci ^encia 2007” program
(G.N.K.).FIG. 7. N /C19eel model fitting with AN¼0.0492 erg/cm2andBN¼0.0889 A ˚/C01.
The following averaged spin-valve parameters have been used: Ms1¼1155
emu/cm3,Ms2¼1175 emu/cm3,Eex¼0.094 erg/cm2; the other ones are the
same as in Fig. 6.023906-8 Timopheev et al. J. Appl. Phys. 114, 023906 (2013)
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1.354181.pdf | Simulation of magnetization distribution in magnetoresistive film under a
longitudinal bias field
Chiaki Ishikawa, Kaori Suzuki, Naoki Koyama, Kazuetsu Yoshida, Yutaka Sugita, Kiminari Shinagawa,
Yoshinobu Nakatani, and Nobuo Hayashi
Citation: Journal of Applied Physics 74, 5666 (1993); doi: 10.1063/1.354181
View online: http://dx.doi.org/10.1063/1.354181
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/74/9?ver=pdfcov
Published by the AIP Publishing
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130.70.241.163 On: Tue, 23 Dec 2014 14:39:37Simulation of magnetization distribution iin magnetoresistive film under a
longitudinal bias field
Chiaki Ishikawa, Kaori Suzuki, Naoki Koyama, Kazuetsu Yoshida, and Yutaka Sugita
Central Research Laboratory, Hitachi, Ltd., Kokubunji, Tokyo 185, Japan
A. Kiminari Shinagawa
Toho University, Miyama, Funabashi, Chiba 274, Japan
Yoshinobu Nakatani and Nobuo Hayashi
University of Electra-Communications, Chofugaoka, Chofu, Tokyo 182, Japan
(Received 8 February 1993; accepted for publication 2 July 1993)
The effects of longitudinal bias field, used for domain control on the magnetization distribution
in a magnetoresistive (MR) film, have been investigated by computer simulation. The
longitudinal bias field was generated by an exchange-coupled antiferromagnetic or permanent
magnetic film formed on the MR film outside the sensing region. It was assumed that the
magnetization in the part of the MR film on which the bias-generating films were formed was
fixed along the easy axis. The spatial sensitivity of the MR him along the track width was
evaluated by calculating the dependence of the resistance change on the position of a narrow
track recording medium. It was found that the resistance change in the MR film with the
anti-ferromagnetic film was roughly twice as large as the change in the film with the permanent
magnetic tilm. The asymmetric sensitivity profile with respect to reflection about the track width
mid-plane was also obtained. The asymmetry in the track sensitivity proiile was found to be
caused by three factors: asymmetric magnetization distribution about the track width mid-plane
due to the transverse bias field, the difference in angular changes in the magnetization direction
in the left and right regions facing the recording medium, and anisotropic flux propagation in the
MR film.
I. INTRODUCTION
Magnetoresistive (MR) heads have been widely inves-
tigated for use as the reading elements of magnetic heads
because their signal output is higher than inductive-type
read heads. The magnetization distribution in the MR fihn
directly influences the signal response. However, it is dif-
ficult to directly observe the microscopic magnetization
distribution at present because of insufficient resolution of
experimental observation techniques. Thus, micromagnetic
computer simulation’” is a powerful tool for analyzing the
relationships between the head characteristics and the
magnetization distribution of MR film. Heim’ calculated
the spatial sensitivity change along the track width, and
showed the asymmetry in the sensitivity profile with re-
spect to reflection about the track width midplane due to
anisotropic flux propagation in the MR film. However,
Heim did not take into account the effect of the longitudi-
nal bias field applied to the MR tilm for the domain control
of the magnetization distribution.
The authors have previously reported that the longitu-
dinal bias field exaggerates the asymmetry in the sensitivity
profile.6 This article reports the precise origins of the asym-
metry, including results on the spatial sensitivity profile
and the magnetization distribution of the MR film under a
longitudinal bias field used for domain control.
II. CALCULATION METHOD
The longitudinal bias field is calculated using the two
types of bias-generating films, the antiferromagnetic film, and the permanent magnetic film. Figure 1 is a schematic
diagram of an MR film with the bias-generating films
formed on the MR film outside the sensing region and
adjacent to it on both sides. The magnetization distribution
in the MR film without the magnetic shield film is calcu-
lated by considering the uniform transverse bias field and
the stray field from the recording medium. We assume the
following in the calculation:
( 1) The magnetization does not vary in the direction
perpendicular to the film plane (X axis).
(2) There is no magnetic wall in the film.
(3) The magnetization in the part of the MR film on
which bias-generating films are formed is fixed along the
easy (z) axis for the permanent magnet biasing and anti-
ferromagnetic biasing.’
(4) In the case of permanent magnet biasing, the stray
field from the bias-generating film is applied to the MR
sensing region, while no stray field is considered in the case
of antiferromagnetic biasing.
The magnetization distributions in the MR film are deter-
mined by using the Landau-Lifshitz-Gilbert (LLG) equa-
tion:
(I+‘&?= --y[MXhf] -$$MX [MxH]), (1)
where hl is the magnetization, H is the effective field, (Y is
the damping constant, and y is the gyromagnetic ratio. The
effective field consists of four components: demagnetization
field, anisotropy field, exchange field, and externally ap-
plied field. The externally applied field includes the uni-
5666 J. Appl. Phys. 74 (9), 1 November 1993 0021-8979/93/74(9)/5666/6/$6.00 @ 1993 American institute of Physics 5686
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130.70.241.163 On: Tue, 23 Dec 2014 14:39:37Anti-ferromagnetic film or -1.552615 ~.i m
Easy axis MR film permanent ma netic film
I B
-.. - -...- -.,
Medium
FIG. 1. Simulation model of MR film.
form transverse bias field, the stray field from the recording
medium, and the stray field from the permanent magnetic
film used for domain control. The previously developed
backward difference method7 is used to solve Eq. ( 1): We
use the two-dimensional calculation on the bias of assump-
tion (1).
The stray field from the permanent magnetic film is
calculated from the potential generated by the charges on
both of the end surfaces, S (Fig. 2). Figure 2 shows the
change in the z component of the stray field Hz along the
z axis between the permanent magnetic films at the center
of the film height. The field abruptly decreases in the re-
gions of 1 pm from the edges of the permanent magnetic
films.
The stray field from the recording medium is obtained
by the three-dimensional finite element method.’ The MR
film is assumed to be located at the center of the transition
region of the recording medium (Fig. 1). The x compo-
nent of the magnetization MX in the transition region is
approximated by
Mx=Mrtanh 2 ,
( 1 (2)
where Mr is the remanence and a is the transition param-
eter. The z component of the magnetization is assumed to
be uniform. Figure 3 shows the y component of the stray
01 -2 -1 0 1 2
z(rm)
FIG. 2. Stray field from the permanent magnetic film on line A-A’. The
film is 3 pm high and 0.06 pm thick; the remanence Br is 0.7 T. 16
&=a
2 9.6
$ 6.4
x 3.2
I
0
-3.2
= (rm)
(4
12 -
-z a 8- 25 I” 4- -1.5SzS1.5 pm
1 2
Distance from medium y (pm)
03
FIG. 3. Stray field from the recording medium. The medium is 4 ,um wide
and 0.03 /*m thick; the remanence Mr is 6~ lo5 A/m. (a) Field along z
axis at 0.2 pm head-medium spacing. (b) Field along y axis.
field from the recording medium with Mr of 6~ lo5 A/m,
a of 0.21 pm, thickness of 0.03 pm, and track width of 4
pm. The calculated field along the z axis, indicated by the
solid line in Fig. 3(a), is approximated by the dotted
straight lines. Figure 3 (b) shows the profile of fields along
thep axis for - 1.5 pm<;z(1.5 pm, z= If 1.7, =k 1.9, st2.1,
and f 2.3 with respect to the dotted lines in Fig. 3(a).
The resistance is calculated from the magnetization
rotational angle 0i from the easy axis at each spatial mesh
elements according to
, (3)
where p is rcsistivity, Ap/p is the relative variation of p, h
and L are height and width of the mesh element, and t is
the film thickness.
In the calculation, the values used for the saturation
magnetization, anisotropy field, exchange stiffness con-
stant, and gyromagnetic ratio of the MR film were
Ms=8X105 A/m, H/c=400 A/m, A=l.OXlO-” 3/m,
and y=2.2 1 x lo5 m/(A s). The resistivity of the MR film
is set to p=O.25 ,uln m, Ap/p=O.O27. The MR film is 2.6
pm wide, 3.0 pm high, and 0.02 ,um thick; the spacing
between head and medium is 0.1 pm. The spatial mesh size
for solving the LLG calculation is 0.1 ,um. The time step
5667 J. Appl. Phys., Vol. 74, No. 9, 1 November 1993 lshikawa et al. 5667
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130.70.241.163 On: Tue, 23 Dec 2014 14:39:37MR film MR film
+-- A--- -2.6 pm -+:
Transverse bias
field 3.3 kA/m t
FIG. 4. Magnetization configuration in the MR tilm with the antiferro-
magnetic fdm formed outside the sensing region. Magnetization directions
are shown by arrows at intervals of 0.2 pm. The film is 0.02 pm thick.
used is 0.01 ns. The value of bias field Hb is determined
such that the resistivity change due to the magnetic flux
from the recording medium with upward direction equals
the resistivity change due to the magnetic flux from the
recording medium with downward direction.
111. RESULTS AND DISCUSSIONS
A. Magnetization distribution under the transverse
bias field
Figure 4 shows the magnetization distribution in the
MR him that has the antiferromagnetic tihn outside the
sensing region under a uniform transverse bias field Hb of
3.3 kA/m. In the initial state, all the magnetizations are
aligned in the z direction. The magnetization directions of
the convergent state are shown by arrows, and the distri-
bution of the angle 8 between each magnetization direction
and the z axis is shown by color. The magnetization dis-
tribution is asymmetric with respect to the reflection about
the track width midplane. That is, the magnetization an-
gles 8 in the upper left and lower right regions are larger
than those in the upper right and lower left regions. Figure
5 shows the magnetization distribution of the MR film that
has the permanent magnetic film under a transverse bias
field Hb of 6 kA/m. The distribution is similar to Fig. 4.
However, the magnetization rotation angle 8 is overall
lower than in the MR film with the antiferromagnetic lilm.
This is because the stray field from the permanent mag-
netic film increases the effective anisotropy field of the easy
(z) axis.
The above magnetization distributions are caused by
the distribution of the demagnetization field. Figure 6 il-
lustrates the demagnetization field generated by the Transverse bias
field 6.0 kA/m t
FIG. 5. Magnetization configuration in the MR film with the permanent
magnetic film formed outside the sensing region. Magnetization directions
are shown by arrows at intervals of 0.2 pm. The film is 0.02 pm thick.
charges on the boundary of the MR film when the trans-
verse bias field is applied. The charges are indicated by the
plus and minus signs. The demagnetization field Hdl in the
upper left region is the vector sum of the demagnetization
fields Hd(&) and Hd(&), which are mainly generated by
the charges $i on the upper boundary and rJ2 on the left
boundary. Similarly, the demagnetization field Hd, in the
lower left area is mainly affected by the charges r/z and $s.
As shown in this figure, Hd, is larger than Hdl because of
the different charge distribution. Due to the effect of these
inhomogeneous demagnetization field distributions, the
magnetization rotation angle 8 is lower in the lower left
region than in the upper left region. Similarly, 8 is lower in
the upper right region than in lower right region. These
effects result in asymmetric magnetization about the track
width midplane shown in Figs. 4 and 5.
Charge t+bj
~r--c:.+ + + -51
Charge J,LJ~
FIG. 6. Schematic diagram of demagnetization field in MR film under a
transverse bias field.
5668 J. Appl. Phys., Vol. 74, No. 9, 1 November 1993 lshikawa et a/. 5668
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130.70.241.163 On: Tue, 23 Dec 2014 14:39:37Conductor
A
6
FIG. 7. Schematic diagram of left/right asymmetry in the spatial sensi-
tivity profile for the MR film along the track width (see Ref. 9).
B. Spatial sensitivity along the track width
Feng et al. experimentally evaluated the spatial sensi-
tivity along the track width of the MR film by changing the
head sensing position across a narrow recorded track.g The
sensitivity profile was not symmetric with respect to reflec-
tion about the track width midplane (left/right asymme-
try), as shown in Fig. 7.
Using the simulated magnetization distributions men-
tioned in Sec. III A, we analyzed the spatial sensitivity of
the MR iilm in which the magnetization outside the sens-
ing region is fixed. Figure 8 shows the changes in MR
resistance, AR, calculated for various distances, S, between
the center of the head and the center of the narrow track
medium with 0.8 pm track width. AR is the resistance
change due to the opposite signal fields from the recording
medium under the transverse bias field. The left/right
asymmetry in the obtained track sensitivity profile is sim-
ilar to the experimental results.g The maximum of AR for
the MR film is at S= +0.3 pm for the antiferromagnetic
- 2.6pm -----.*
spm
I
i
I i
S (pm)
FIG. 8. Calculated left/right asymmetry in the spatial sensitivity profile
for the MR film under a longitudinal bias.
5669 J. Appl. Phys., Vol. 74, No. 9, 1 November 1993 lshikawa et a/. 5669 Oo IY. 10 * 20 * 30 8 40 3 50 I
A 6’ (degrees)
FIG. 9. Resistivity for different magnetization rotation angles 8 under a
transverse bias field as a function of AO.
lilm and at &= +0.2 pm for the permanent magnetic tilm.
The spatial sensitivity profile is more asymmetrical for the
MR film with the antiferromagnetic film than for the per-
manent magnetic film. The AR of the MR film with the
antiferromagnetic film is roughly twice as large as that
with the permanent magnetic film.
The left/right asymmetries in the sensitivity profiles
are caused by the magnetization distributions under the
transverse bias field, as shown in Figs. 4 and 5. As the stray
field from the recording medium is applied to the lower
region of the MR film, the difference between the magne-
tization rotation angles 8 in the lower right region and in
the lower left region under the transverse bias field causes
the asymmetry in AR. The average 8 values for the lower
right region and the lower left region in Fig. 4 are roughly
40” and 15”. Figure 9 shows normalized resistivity change
Ap as a function of A6 for 8 of 40” and 15”. A6 is the
angular change in the magnetization direction caused by
the stray field from the transition in the medium. Ap for 0
of 40” is found to be roughly twice that for 8 of 15”. This is
the reason for the left/right asymmetry in the track sensi-
tivity profile.
To further investigate the mechanisms of the asymme-
try of the sensitivity profile, the spatial distributions of the
magnetization rotation angle A0 are calculated for S= - 1
pm(a), 0 pm(P), and 1 pm(r) (see Fig. 8). Figure 10
illustrates A6 with the transverse bias field. A6 in the re-
gion facing the recording media is found to be larger in
case (c) than in case (a). The magnetization is more ro-
tated in the right lower region because the demagnetization
field is weaker in that region than in the left lower region
(Fig. 6). The difference in A8 for the two regions also
causes the left/right asymmetry in the spatial sensitivity
profile.
The sensitivity of the MR film is not only determined
by the magnetization in the lower region in the MR film
but also by the magnetization rotation of the other region.
Figures 10(b) and lO( c) shows that the magnetic flux
from the medium propagates through the region where the
demagnetization field is relatively low (Fig. 6), and ob-
liquely toward the upper left region. This anisotropic prop-
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130.70.241.163 On: Tue, 23 Dec 2014 14:39:37MR film r2 >-.--- .-.>
0 Medium (4
(b)
(4
FIG. 10. Distributions of the magnetization angle change A0 caused by
the stray field from the medium. The MR fdm has the antiferromagnetic
tihn under a 3.3 kA/m transverse bias field. The medium is located (a)
under the left side, (b) center, and (c) right side of the MR film.
agation also causes a left/right asymmetry in the sensitivity
profile along the track width because the region where AtI
is high, in case (c), is wider than in case (a), which results
in the different AR for the entire MR film.
Thus several origins for the asymmetric sensitivity pro-
file have been identified. We evaluated their quantitative
contributions to the sensitivity. Table I shows the contri-
bution of the effects on the resistance change AR from the
magnetization change in the region that faces the recording
medium and in which the anisotropic flux is propagated.
AR, is the resistance change in the region rl, where the
stray field from the medium has a direct effect; ARz is for
the other region, r,, where the flux propagation causes the
resistance change shown in Fig. 10. In case (c) (S= 1
pm), the total AR is 1.8 times larger than in case (a)
(S= - 1 pm); the proportion of ARZ in AR is 28% for case
(c) and 8% for case (a). These results show that the effect
from the flux propagation on the total AR is quite different
in cases (a) and (c). Further, AR,, which is the major part
of AR, is about 1.5 times larger for case (c) than for case
(a). This shows that the contribution from the magnetiza- TABLE I. Contributions to resistance change in the MR lihns. Regions r,
and r,, and cases (a) and (c) are shown in Fig. 10.
Case (a) Case (c)
Total resistance
change AR
Resistance change
Art in t,
Resistance change
AR, in r2 2.38(n) 4.21
2.20 3.04
0.18 1.17
tion change in region rl to the asymmetry of the sensitivity
profile is significant.
Figure 11 shows the distribution of A8 in the MR ftlm
with the permanent magnetic film outside the sensing re-
gion when the recording medium is located under the cen-
ter of the MR film (S =0) . In this case, A0 is smaller than
for the MR film with the antiferromagnetic Elm shown in
Fig. 10(b). Furthermore, the flux propagation is not as
oblique as in Fig. 10(b). We believe that the flux propa-
gates towards the upper left region less easily than in the
case of the MR film using the antiferromagnetic film, be
cause the magnetizations near the permanent magnetic
films rotate less owing to the stray field from the perma-
nent magnetic film. The difficulty in the magnetization ro-
tation near the permanent magnetic film explains the result
shown in Fig. 8: the asymmetric profile with the permanent
magnetic film is not as obvious as with the antiferromag-
netic fdm, and the AR with the permanent magnetic film is
nearly half that with the antiferromagnetic f&n.
IV. CONCLUSION
The magnetization distribution in magnetoresistive
(MR) fdm under a longitudinal bias field used for domain
control was analyzed by micromagnetic calculations. The
spatial sensitivity along the track width in the MR lllm was
also calculated. The magnetization distribution due to the
transverse bias field is asymmetric with respect to reflection
about the track width midplane. This asymmetry is caused
by an inhomogeneous demagnetization field in the MR
film. The left/right asymmetry in the track sensitivity pro-
..::i;ii $+ib
FIG. 11. Distribution of the magnetization angle change A0 caused by
the stray field from the medium. The MR film has an attached permanent
magnetic film under a 6 l&/m transverse bias field.
5670 J. Appl. Phys., Vol. 74, No. 9, 1 November 1993 lshikawa et a/. 5670
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
130.70.241.163 On: Tue, 23 Dec 2014 14:39:37file is caused by three factors: an asymmetric magnetiza-
tion distribution under the transverse bias field, a difference
in A6’s in the left and right regions facing the recording
medium, and anisotropic tlux propagation. The resistance
change in the region where the stray field from the medium
has a direct effect is larger than that in the region where the
flux propagates. Further, the resistance change AR of the
MR film with the antiferromagnetic film is roughly twice
as large as that with the permanent magnetic film.
ACKNOWLEDGMENTS
The authors are indebted to Dr. Takayoshi Nakata and
Dr. Kouji Fujiwara of Okayama University for using the
3D field calculation program, Dr. Yasutaro Uesaka of the
Central Research Laboratory for his support, and Dr. Ka-
zuo Shiiki of Data Storage and Retrieval Systems Division
of Hitachi, Ltd. for his helpful discussion. We also wish to
5671 J. Appl. Phys., Vol. 74, No. Q, 1 November IQ93 thank Dr. Ryou Suzuki for his valuable support through-
out this work.
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‘S. L. Tomlinson, E. W. Hill, and J. I. Miles, IEEE Trans. Magn. 27,
4698 (1991).
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(1991).
‘S. W. Yuan and H. N. Bertram, Abstract of 37th Conf. on Magn. &
Magn. Mater. EB-07, 1992.
5J.-G. Zhu and Y. Guo, Abstract of 37th Conf. on Magn. & Magn.
Mater. EB-08, 1992.
6C Ishikawa, K. Suzuki, N. Koyama, K. Yoshida, K. Shinagawa, Y.
Nakatani, and N. Hayashi, .I. Magn. Sot. Jpn. 16, Suppl. No. Sl, 85
(1992) (in Japanese).
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(1989).
*T. Nakata, N. Takahashi, K. Fujiwara, and Y. Okada, IEEE Trans.
Magn. 24, 94 (1987).
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Chue, IEEE Trans. Magn. 27,470l (1991).
lshikawa et a/. 5671
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
130.70.241.163 On: Tue, 23 Dec 2014 14:39:37 |
5.0052150.pdf | APL Materials ARTICLE scitation.org/journal/apm
Mode-sensitive magnetoelastic coupling
in phononic-crystal magnomechanics
Cite as: APL Mater. 9, 071110 (2021); doi: 10.1063/5.0052150
Submitted: 29 March 2021 •Accepted: 5 July 2021 •
Published Online: 21 July 2021
D. Hatanakaa)
and H. Yamaguchi
AFFILIATIONS
NTT Basic Research Laboratories, NTT Corporation, Atsugi-shi, Kanagawa 243-0198, Japan
Note: This paper is part of the Special Topic on Phononic Crystals at Various Frequencies.
a)Author to whom correspondence should be addressed: daiki.hatanaka.hz@hco.ntt.co.jp
ABSTRACT
The acoustically driven spin-wave resonance in a phononic-crystal cavity is numerically investigated. The designed cavity enables confine-
ment of gigahertz vibrations in a wavelength-scale point-defect structure and sustains a variety of resonance modes. Inhomogeneous strain
distributions in the modes modify the magnetostrictive coupling and the spin-wave excitation susceptible to an external-field orientation. In
particular, a monopole- likemode in the cavity having a near-symmetrical pattern shows a subwavelength-scale mode volume and can pro-
vide a versatile acoustic excitation scheme independent of the field-angle variation. Thus, the phononic-crystal platform offers an alternative
approach to acoustically control the spin-wave dynamics with ultrasmall and inhomogeneous mode structures, which will be a key technology
to integrate and operate large-scale magnomechanical circuits.
©2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license
(http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0052150
The mutual coupling between acoustic waves and magnetic
moments in a ferromagnet has been intensively investigated over
the last several years.1–21Developing the acoustic means to drive
ferromagnetic and spin-wave resonances is increasingly important
to explore the possibility of phononics technology in the field of
spintronics. Previous magnomechanical systems were mostly con-
structed on a surface acoustic-wave (SAW) device in which excited
acoustic plane waves permit the dynamics of magnon–phonon inter-
action to be studied with a simple analytical model.1–15However,
SAW-based driving via the magnetostriction1,2and Barnett effect4,8
only work in restricted external-field orientations. In addition, the
SAW energy loss to a bulky substrate results in poor spatial con-
finement,22,23and the subsequent large vibrational mode volume
reduces the spatial power density and weakens the effective interac-
tion. Thus, an alternative and versatile platform for magnomechan-
ics is highly desired.
In this article, we propose to use a phononic-crystal (PnC)
acoustic cavity for controlling the magnomechanical interaction.
This study is based on a numerical investigation of the res-
onant modal structures and the characteristics of the acousti-
cally driven spin-wave resonance at various field angles. The
phononic bandgap formed by the periodic structure fully confines
gigahertz vibrations in the wavelength-scale point-defect structure.This structure resonates at specific frequencies, revealing a variety
of modal shapes. The resultant strains are inhomogeneously dis-
tributed so that the field response of power absorption due to the
spin-wave excitation is variant with respect to the resonant mode
structures. Dipole- and quadrupole- like modes clearly show dif-
ferent field-angle dependencies, whereas the field-angle variation
mostly vanishes in a monopole- likemode. To the best of our knowl-
edge, this is a first proposal to exploit such a PnC structure as a
magnomechanical platform instead of using the SAW device. This
alternative system enables us to spatially control spin waves via
hypersonic waves and tailor the spin-wave excitation efficiency with
respect to the field direction, which are impossible with the previ-
ous SAW one. Therefore, our theoretical achievement will expand
the possibility of the rapidly growing field of magnomechanics and
is promising for a magnomechanical circuit in signal processing
application.
The simulated PnC is constructed by a triangular lattice of
snowflake-shaped air holes formed in a suspended GaAs slab,
as shown in Fig. 1(a), and the original design was proposed by
Safavi-Naeini and Painter.24By designing the periodic structure in
Fig. 1(b), a complete bandgap is formed in the frequency ranges of
0.97–1.30 GHz and 1.37–1.53 GHz, and the dispersion relation is
shown in Fig. 1(c). A point-defect structure is created by removing
APL Mater. 9, 071110 (2021); doi: 10.1063/5.0052150 9, 071110-1
© Author(s) 2021APL Materials ARTICLE scitation.org/journal/apm
FIG. 1. (a) Illustration of PnC-based magnomechanics in which a ferromagnet
thin film (Ni) with thickness dNi=50 nm is formed on the surface of a point-
defect cavity in a suspended membrane. This phononic cavity can be driven by an
electromechanical transducer, such as an IDT (inter-digit transducer). The crystal
orientations of GaAs are denoted as [110], [ ¯110], and [001]. GaAs is chosen as the
PnC medium because the microfabrication technology for it is well developed and
it hosts the piezoelectricity, enabling on-chip driving of coherent vibrations through
the IDT. The inset shows the relationship between the xyz- andζηξ-coordinate
systems. (b) Schematic of the PnC geometry consisting of a triangular lattice of
snowflake-shaped air holes. The inset shows the unit structure having dimensions
a=2.0μm,b=1.7μm,w=0.5μm, and t=0.5μm. (c) Dispersion relation of
acoustic waves in the PnC, showing full bandgaps in the frequency range as high-
lighted in gray. The bandgaps are divided by a phonon branch sustaining edge
vibrations of the unit structure. All the resonance modes studied in this work exist
in the lower bandgap, and an alternative mode is also found in the higher one.
Further discussions about the dispersion relation are given in the supplementary
material.
one snowflake hole from the lattice, and it resonates at frequen-
cies within the bandgap. This geometry is chosen as a PnC cavity
because we have confirmed the fabrication and experimental pos-
sibilities previously, and the details on the mechanical properties
are found elsewhere.25A circular polycrystalline nickel (Ni) film
is formed on the defect. The finite-element method (FEM) calcu-
lation using COMSOL Multiphysics reveals that multiple Lamb-
and Love-type resonance modes are formed in the cavity. Here, we
focus on three asymmetric Lamb modes, labeled monopole-, dipole-
, and quadrupole- like modes, and investigate the variation in the
magnetostriction with changing external-field orientation.
Acoustic spin-wave excitation by the PnC cavity is simulated
through the strain distribution of the resonant modal shapes. Fromvarious strain components ϵij(i,j=x,y,z), effective fields ( hζ,hη) via
magnetostriction are given by2
μ0hζ=2bs(ϵxzcosϕ+ϵyzsinϕ), (1)
μ0hη=2blsinϕcosϕ(ϵxx−ϵyy)−2bsϵxycos 2ϕ, (2)
with the shear and longitudinal magnetostrictive coupling constants
bsandbl, respectively, and vacuum permeability μ0. An alternative
ζηξ-coordinate system is defined in such a way that the ξ-axis is
aligned to the magnetization whose angle is set to ϕwith respect to
thexaxis, as shown in the inset of Fig. 1(a). Applying time-varying
strains to the cavity via an IDT generates magnetization precession
acoustically without rf electromagnetic waves, which is the usual
driving technique for ferromagnetic resonance experiments. This
also results in the acoustic power being absorbed, which suppresses
and modulates the resonance amplitude and phase. Using (1) and
(2), the change in the power via the magnetostrictive driving ( ΔP) is
expressed by
ΔP=−ωμ0
2∫V0⎡⎢⎢⎢⎢⎢⎣(h∗
ζ,h∗
η)¯χ⎛
⎜
⎝hζ
hη⎞
⎟
⎠⎤⎥⎥⎥⎥⎥⎦dV, (3)
whereωis the angular frequency of acoustic waves and V0is the vol-
ume of the Ni film. The magnetic susceptibility ¯χis obtained from
the magnetic free energy G=−μ0Hex⋅m+Bdm2
zwith the external
static field μ0Hex, out-of-plane shape anisotropy for the thin film
Bd=μ0Ms/2, unit magnetization m=M/Ms, and saturation mag-
netization Ms. Here, we assume that the static field is applied in the
plane of the Ni film and ignore the in-plane anisotropy for simplicity
due to the polycrystalline and circular structures. Then, the avail-
able components of ¯χare given by χ11=γμ0Ms(γμ0Hex−iωα)/D,
χ12=χ∗
21=−iγμ0Msω/D, andχ22=γμ0Ms[γ(2Bd+μ0Hex)−iωα],
with D=[γ(2Bd+μ0Hex)−iωα](γμ0Hex−iωα)−ω2, whereγis
the gyromagnetic ratio and αis the Gilbert damping ratio. The
detailed derivation and explanation of the above formula are shown
elsewhere.2Thus, taking the real and imaginary parts of ΔPgives
variation in the dispersion ( Pd) and attenuation ( Pa) by the spin-
wave excitation. In the simulations, the acoustic parameters for
GaAs were density ρ=5360 kg/m3, elastic constants C11=111.8 GPa,
C12=53.8 GPa, C44=59.4 GPa, and mechanical (acoustic) loss-
factor Q−1=5×10−4. These acoustic parameters are experimen-
tally valid from previous works so that defect and randomness
in the structure and material property during device fabrication
are considered in the calculation.25,26For Ni, they were density
ρ=8900 kg/m3, Young’s modulus E=219 GPa, and Poisson’s ratio
ν=0.31. The magnetic parameters from previous work were used:
bs=bl=23 T (polycrystalline), Bd=0.2 T, Ms=370 kA/m, and
γ=2.185μB/hwith Bohr’s magneton μBand the reduced Plank
constant handα=0.05.
First, a dipole- like cavity mode is considered. The vibration
amplitude and phase of the modal shape are displayed in the left
and right panels of Fig. 2(a), respectively, showing that two anti-
nodes oscillate out of phase at ω/(2π)=1.160 GHz. This enables
the effective mode volume to be calculated by integrating vibra-
tion energy over the slab as Veff=∫dr2(∣u(r)∣
∣u(r)∣max)2
t=0.852μm3
≈0.57λ2t, where u(r)(∣u(r)∣max) is vibration (maximum) displace-
ment at position randλis the acoustic wavelength.25The result
APL Mater. 9, 071110 (2021); doi: 10.1063/5.0052150 9, 071110-2
© Author(s) 2021APL Materials ARTICLE scitation.org/journal/apm
FIG. 2. (a) Vibration amplitude and phase of the dipole mode at ω/2π
=1.160 GHz. (b) From left to right, spatial distribution of the strains ϵxx,ϵyy,
andϵxy, which are normalized by the maximum strain of ϵmax
xx. (c) Field depen-
dence of the attenuation (left) and dispersion (right) of the acoustic resonant
vibrations at various ϕbetween −90○and 90○absorbed by spin-wave excita-
tion via the magnetostriction. The maximum attenuation is ∣Pa∣max=5.0 nW in
ϵmax
xx=4.3×10−5.
indicates the strong confinement of gigahertz vibrations in the tiny
sub-wavelength-scale space. The surface strain distributions were
also calculated, and only the strain components available for the
magnetostrictive driving, ϵxx,ϵyy, andϵxyare shown in Fig. 2(b),
whose magnitudes are normalized by the maximum ϵmax
xx. In con-
trast to usual SAW-based magnomechanical systems where only
longitudinal strain such as ϵxxis assumed to be nonzero, this PnC
cavity has a two-dimensionally confined structure so that additional
longitudinal and shear strains such as ϵyyandϵxyare involved.
Out-of-plane shear strains ϵxzandϵyzvanish due to the surface
boundary condition.27As a result, μ0hζ=0 and only μ0hηcon-
tribute to the spin-wave excitation. The calculated strains are substi-
tuted into (2), and then, the acoustic power absorption is obtained
through (3). The resultant dispersion and attenuation, normalized
by the maximum values Pnorm
d=∣Pd(Hex,ϕ)∣/∣Pd(Hex,ϕ)∣max and
Pnorm
a=Pa(Hex,ϕ)/∣Pa(Hex,ϕ)∣max, are plotted as a function of the
field atϕranging from −90○to 90○as shown in the right and left
panels of Fig. 2(c), respectively. As in conventional ferromagnet-
SAW systems,1,2,6,7the largest attenuation and dispersion shift occur
atϕ=±45○. This PnC still yields small but observable spin-wave
absorptions around ϕ=0○and±90○, at which no absorption occursin SAW-based systems. This behavior can be understood from (2),
where the shear strain ϵxyvaries with cos 2 ϕ, and thus, the magne-
tostriction is available even at these angles. This investigation reveals
that the PnC cavity with the inhomogeneous strains is capable of
driving spin precession acoustically.
To investigate the variation in the field-angle response when
changing the modal shape, the spin-wave-induced absorption in
the quadrupole- like mode with ω/2π=1.156 GHz was calculated
in attenuation and dispersion. This mode has four vibration anti-
nodes, each of which oscillates out of phase with respect to neigh-
boring ones, as shown in Fig. 3(a). The effective mode volume is
calculated as Veff=1.007μm3≈0.68λ2t. The resultant strains ϵxx,
ϵyy, andϵxyshown in Fig. 3(b) induce the normalized acoustic atten-
uation and dispersion via the magnetostriction. The field dependen-
cies are shown in the left and right panels of Fig. 3(c), respectively.
Remarkably, the susceptibility to the field angle changes from that
of the dipole mode, resulting in the maximum absorption at ϕ=0○
and±90○, whereas the absorption is small at ϕ=±45○. The result
can be interpreted to mean that the shear strain ϵxydominates the
magnetostrictive field from (2), whereas the difference in the lon-
gitudinal strains, ϵxx−ϵyyin (2), is reduced. Thus, different modal
FIG. 3. (a) Vibration amplitude and phase of the quadrupole mode structure at
ω/2π=1.156 GHz. (b) From left to right, spatial distribution of the strains ϵxx,ϵyy,
andϵxy, which are normalized by the maximum strain of ϵmax
xx. (c) Field depen-
dence of the attenuation (left) and dispersion (right) of the acoustic resonant
vibrations at various ϕbetween −90○and 90○absorbed by spin-wave excita-
tion via the magnetostriction. The maximum attenuation is ∣Pa∣max=7.7 nW in
ϵmax
xx=4.4×10−5.
APL Mater. 9, 071110 (2021); doi: 10.1063/5.0052150 9, 071110-3
© Author(s) 2021APL Materials ARTICLE scitation.org/journal/apm
structures enable us to modify the magnetostrictive characteristics of
the PnC.
The point-defect structure in the cavity sustains the monopole-
likemode having a hexagonal modal shape with ω/2π=1.131 GHz
and the smallest Veff=0.281μm3≈0.19λ2t, as shown in Fig. 4(a).
This hosts nearly radial symmetry on the Ni thin film and thus
the generated strains shown in Fig. 4(b), although sixfold rotational
symmetry slightly remains in the vibrational structure close to the
defect edges. Therefore, the simulation reveals that the acoustic
modulations, namely, the acoustic driving efficiency of the spin-
wave resonance, are mostly insensitive to the variation in the field
angle, as shown in Fig. 4(c). Indeed, there is a little fluctuation in the
maximum of Pnorm
a whileϕis swept, as shown in the inset of Fig. 4(c),
but it is negligibly small compared to that of other modes and SAW-
based magnomechanical systems. This monopole mode inherent
to the point-defect cavity is useful for developing and operating
PnC-based magnomechanical circuitry because it offers compact
and versatile control of the spin precession regardless of the
external-field angles.
FIG. 4. (a) Vibration amplitude and phase of the monopole mode structure at ω/2π
=1.131 GHz as shown in the left and right panels, respectively. (b) From left to
right, spatial distribution of the strains ϵxx,ϵyy, andϵxy, which are normalized by
the maximum strain of ϵmax
xx. (c) Field dependence of the attenuation (left) and
dispersion (right) of the acoustic resonant vibrations at various ϕbetween −90○
and 90○absorbed by spin-wave excitation via the magnetostriction. The maximum
attenuation is ∣Pa∣max=1.9 nW inϵmax
xx=3.3×10−5. The inset in the left panel
indicates the angle dependence of Pnorm
aatμ0Hex=3.4 mT.In conclusion, we numerically investigated the magnomechan-
ical characteristics of a point-defect cavity in a suspended PnC slab.
A variety of resonant modal shapes emerge in the wavelength-scale
cavity sustained by the bandgap, resulting in the formation of inho-
mogeneous strain distributions. The spatial variation in the longi-
tudinal and shear strains in the modes determines the field-angle
dependence of the magnetostriction, namely, the excitation effi-
ciency of the spin-wave resonance. The opposite dependency with
respect to the orientation is found between the dipole and quadru-
ple modes. It is remarkable that the monopole mode is mostly
insensitive to the field orientation for the spin-wave excitation.
Thus, the PnC architecture is a promising platform for enhanc-
ing the directionality and versatility of the magnetostrictive driving
scheme, which holds promise for developing hybrid magnomechan-
ical circuitry.
See the supplementary material for the dispersion relation of
the phononic branch between the bandgaps and the magnetostric-
tive effect on the edge resonant mode.
The authors thank H. Okamoto, Y. Kunihashi, and H. Sanada
for fruitful discussion.
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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© Author(s) 2021 |
1.3596805.pdf | Micromagnetic study on microwave-assisted magnetic recording in perpendicular
medium with intergrain exchange coupling
Yukio Nozaki, Ayumu Kato, Kenji Noda, Yasushi Kanai, Terumitsu Tanaka, and Kimihide Matsuyama
Citation: Journal of Applied Physics 109, 123912 (2011); doi: 10.1063/1.3596805
View online: http://dx.doi.org/10.1063/1.3596805
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/109/12?ver=pdfcov
Published by the AIP Publishing
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128.235.251.160 On: Thu, 18 Dec 2014 06:59:09Micromagnetic study on microwave-assisted magnetic recording in
perpendicular medium with intergrain exchange coupling
Yukio Nozaki,1,2,a)Ayumu Kato,3Kenji Noda,3Y asushi Kanai,4Terumitsu Tanaka,3
and Kimihide Matsuyama3
1Department of Physics, Keio University, Hiyoshi 3-14-1, Kohoku-ku, Yokohama 223-8522, Japan
2JST, CREST, Sanbancho 5, Chiyoda-ku, Tokyo 102-0075, Japan
3ISEE, Kyushu University, Motooka 744, Nishi-ku, Fukuoka 819-0395, Japan
4Department of Information and Electronics Engineering, Niigata Institute of Technology, Fujihashi 1719,
Kashiwazaki, Niigata 945-1195, Japan
(Received 14 February 2011; accepted 2 May 2011; published online 21 June 2011)
The influence of intergrain exchange coupling on the properties of microwave-assisted
magnetization reversal has been investigated. The results of micromagnetic simulation suggest that
the microwave frequency realizing the magnetization reversal is increased as the exchange stiffnessconstant among the magnetic grains becomes larger than 1 /C210
/C07erg/cm if an inhomogeneous
magnetization reversal occurred. The numerically expected increase in the microwave frequency
was observed experimentally in perpendicularly magnetized Co/Pd multilayer with an exchangestiffness constant of /C241/C210
/C06erg/cm. The simulations of signal recording and reproducing
processes in a granular medium with assistance of a microwave field also suggest that an adequate
intergrain exchange coupling is required to inhibit the nucleation of small island domains that willsuppress the amplitude of the signal-noise ratio of the readback signal.
VC2011 American Institute of
Physics . [doi: 10.1063/1.3596805 ]
I. INTRODUCTION
Microwave-assisted magnetization reversal, so-called
MAMR, is the method that uses microwave pumping of
magnetization to lower the potential barrier for switching ofmagnetization. Dynamics of local magnetization are gener-
ally governed by the following Landau-Lifshitz-Gilbert
(LLG) equation,
dM
dt¼/C0 cM/C2Heff ðÞ þa
MsM/C2dM
dt/C18/C19
that consists of terms representing the Lamor precession tor-
que and the Gilbert damping torque. M,Heff,c,a, and Ms
represent the magnetization vector, the effective field vector,
the gyromagnetic constant, the Gilbert damping constant,
and the saturation magnetization, respectively. When another
external magnetic field is applied parallel to the temporalvariation of M, the cone angle of magnetization precession
can be enlarged against the damping torque. To realize a suc-
cessive growth of magnetization precession, the vector of theexternal magnetic field must be rotated in the plane perpen-
dicular to the precession axis with a frequency consistent
with that of the Lamor precession of magnetization. The ex-perimental evidence of frequency-dependent switching of
magnetization by MAMR was first reported in Co nanopar-
ticles,
1and after that other experiments were demonstrated
for NiFe in magnetic tunnel junctions,2patterned NiFe ele-
ments (hexagons,3rectangle,4rings,5strips,6,7and ellip-
soids8–10), and CoFe films.11Recently, the MAMR in theperpendicularly magnetized film was demonstrated for Co/
Pd multilayers12and CoPt-based films.13
MAMR is considered to be one of the promising candi-
dates in solving the serious writability problem expected in
future hard disk drives (HDDs). The microwave field
assisted recording head proposed by Zhu et al. consists of a
single-pole type writing head with a trailing return yoke and
a spin-torque oscillator (STO) as the source of the ac mag-
netic field.14In this head geometry, the ac magnetic field is
intensely applied below the field generating layer (FGL) of
the STO embedded between the main pole and the trailing
return yoke. An inhomogeneous magnetization reversalowing to the distribution of both the head field and the ac
magnetic field, therefore, appears in the MAMR-HDDs. As a
consequence, it is expected that the distributions of not onlythe magnetostatic field but also the exchange coupling field
caused by the inhomogeneous magnetization reversal will
affect both the amplitude and the frequency of ac magneticfield required for realizing the MAMR. Okamoto et al.
numerically investigated the effect of dipolar interaction on
MAMR properties and concluded that the dipole-dipoleinteraction among the neighboring particles makes the phase
of magnetization precession coherent.
15Furthermore, they
found that in-plane dipole fields from the neighboring par-ticles assist the magnetization reversal resulting in significant
reduction of the switching field. However, they did not con-
sider the effect of exchange coupling among the particles.
In this article, for the quantitative understanding of the
influence of intergrain exchange coupling on the MAMR
properties, the micromagnetic simulations of MAMR record-ing have been performed on a granular perpendicular me-
dium. Furthermore, we found that a signal to noise ratio for
the playback signal of the recorded bit pattern can be
a)Author to whom correspondence should be addressed. Electronic mail:
nozaki@phys.keio.ac.jp.
0021-8979/2011/109(12)/123912/6/$30.00 VC2011 American Institute of Physics 109, 123912-1JOURNAL OF APPLIED PHYSICS 109, 123912 (2011)
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128.235.251.160 On: Thu, 18 Dec 2014 06:59:09improved by selecting an adequate strength of exchange cou-
pling among the magnetic grains.
II. NUMERICAL MODEL
In this paper, the signal recording and reproducing pro-
cess simulations of MAMR-HDDs are designed from the
viewpoint of finding the optimum condition of intergrain
exchange coupling. The hard magnetic recording layer wasmodeled by two dimensionally packed hexagonal columnar
grains of 4.6 nm diameter with uniaxial anisotropy with the
easy axes randomly distributed within a 2.5
/C14cone angle per-
pendicular to the medium plane. The center-to-center dis-
tance between grains was 5.5 nm with a grain boundarythickness of 0.9 nm. Hkwas assumed to be 27.5 kOe. Two
media with different Msof 410 and 820 emu/cm3were
assumed, which correspond to 30 and 60 in the thermal sta-bility index at the room temperature, respectively. awas
fixed at 0.02 which is a typical value of microwave-assisted
magnetic recording for both the media. Magnetization dy-namics were investigated by solving the LLG equation. The
effective field vector is defined as a vector summation of the
external applied field, the exchange interaction field, the ani-sotropy field, and the magnetostatic field. Full magnetostatic
interactions among the grains and intergrain exchange inter-
actions between neighboring grains were included in thecalculation.
Figure 1represents M-H hysteresis curves for the media
with different M
s. The value of exchange stiffness constant A
is fixed at 0. Gradual slope of M-H curve for the medium
with Ms¼820 emu/cm3translates to a large switching field
distribution. This means that effective field distribution ofthe grains is also large, resulting in large distribution of
ferromagnetic resonance frequency. Microwave-assisted
magnetic recording utilizes ferromagnetic resonance phe-nomenon, so discrepancy between microwave frequency and
ferromagnetic resonance frequency for magnetic grains is a
serious problem; this discrepancy may generate “islandreversals”
16in recorded bits for signal recording, resulting in
low signal-to-noise ratio, SNR. This is one of the issues to be
addressed in microwave-assisted magnetic recording.
As shown in Fig. 2(a), two types of trailing shield per-
pendicular recording heads without down track tapering
were employed in this study. Head I is a narrow main polewith narrow head gap, and Head II is a wide main pole with
wide head gap.
17,18The FGL is located between the main
pole and the trailing return yoke with 1 nm distance from the
FIG. 1. M-H hysteresis curves in perpendicular direction calculated for the
media with different Ms.
FIG. 2. (a) Two types of trailing shield
perpendicular recording heads without
down-track tapering employed in this
study. The perpendicular component of
the head field distributions along (b)
down track and (c) cross track directions.123912-2 Nozaki et al. J. Appl. Phys. 109, 123912 (2011)
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128.235.251.160 On: Thu, 18 Dec 2014 06:59:09trailing edge of the main pole. Head field distributions were
estimated using finite element method calculation on the
assumption that a thick soft magnetic backlayer exists underthe recording layer. In order to compare the recording char-
acteristics of these heads, head field strengths were normal-
ized to be 12.5 kOe, although the head field strengths areabout 12.5 and 17 kOe for Head I and Head II, respectively.
Figures 2(b) and2(c) show the perpendicular component of
the head field distributions along the down track and crosstrack directions, respectively. Asymmetric field profiles
shown in Fig. 2(b) are due to existence of the trailing return
yokes. The broken lines indicate the positions of FGL. Headfield gradient at the magnetic grains underneath the FGL
range from 390 to 480 Oe/nm and from 240 to 330 Oe/nm
for Head I and Head II, respectively. It is well known that asteep head field gradient is preferable for obtaining superior
bit transition in conventional perpendicular recording. Field
angles of the recording head are about 45
/C14and 40/C14with
respect to the magnetization easy axis of the magnetic grains
for Head I and Head II, respectively.
Reading process was simulated using the reciprocity
theorem.19Three-dimensional magneto-resistive (MR) head
sensitivity was obtained assuming an MR head with 20 nm
wide element and 20 nm long gap shields. Figures 3(a) and
3(b) show the z-component of the sensitivity distribution
along the down track direction at track center and along the
cross track direction at the center of MR element, respec-tively. The readback waveform is shown in Fig. 4, which is
for recorded signals of all “1” patterns in non-return to zero
inverse (NRZ-I) code. Nonuniform amplitude of the repro-duced pulses may come from island reversals generated in
recorded bits. SNR was estimated by the following equation
SNR¼10 log
SÐNðfÞdf;
where Srepresents power of the signal and N(f) is noise
power at frequency f. In this study, integration range was
fixed up to 2 GHz.20III. RESULTS AND DISCUSSION
A. Recorded bit patterns
The MAMR process was simulated considering two
types of recording media with 410 and 820 emu/cm3of satu-
ration magnetization. Head I was employed in this simula-
tion. Saturation magnetization of FGL is 1920 emu/cm3
(/C242.4 T), which gives 1.4 kOe of field amplitude in the
y direction. Microwave frequency was optimized by consid-
ering SNR. The chirality switching of the magnetization rota-
tion of FGL was assumed to be infinitely fast, following the
reversal of the head field. The distance between air bearing
surface (ABS) and medium surface was 5 nm. The pulsedhead field had 0.2 ns of field rise time. Thermal fluctuations
of the media were ignored.
Figure 5shows recorded bit patterns for the media with
different M
s. Signal with a linear density of 340 kbpi was
recorded on 650 kbpi of background signal. The recorded
track width /C2424 nm is slightly greater than the width of FGL
for both the media. The resultant SNR is 17 dB, and several
island reversals appear in recorded bits for medium with
Ms¼410 emu/cm3. On the other hand, many island reversals
are generated for the medium with Ms¼820 emu/cm3.T h e
reason is considered to come from mismatch between micro-
wave frequency and the FMR frequency of the magneticgrains that is governed by the effective internal field strength.
The mismatch increases with increasing M
sbecause the effec-
tive field distribution increases. This is also presumed by con-sidering the gradual slope of the M-H hysteresis curve.
The relationship between SNR and track pitch, TP,i s
estimated regarding the recording heads. The SNR shown in
Fig.6is for a 850 kbpi of recorded track after shingle over-
lap recording of an adjacent track with TP. Linear density of
FIG. 3. (a) z-component of the sensitivity distribution along the down track
direction at track center and (b) along the cross track direction at the center
of MR element.
FIG. 4. Readback waveform calculated for recorded signals of all “1” pat-terns in NRZ-I code.
FIG. 5. (Color online) Recorded bit pat-
terns for the media with saturation mag-netization of (a) 410 emu/cm
3and (b)
820 emu/cm3.123912-3 Nozaki et al. J. Appl. Phys. 109, 123912 (2011)
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128.235.251.160 On: Thu, 18 Dec 2014 06:59:09the adjacent track is 1016 kbpi. In the case that the recorded
track is partially erased by an adjacent track, SNR decreases.
As described in Ref. 21, recorded track width is mostly
defined by special configuration of FGL even though main
pole width is much greater than that of FGL, because the dis-tribution of the microwave field confines the switching
region. However, Fig. 6shows that there is extension writing
beyond FGL width for signal recording with Head II. Head Iis preferable for high track density, however, because high
SNR is obtained at small TP. This is because the narrow head
field distribution sharply confines the switching region. Thedifference between SNRs for the media is due to generation
of island reversals as mentioned before. The reason that the
profiles for signal recording with Head I have the maximumSNR around 20 nm of TPmay imply existence of erased
bands.
As discussed before, the intergrain exchange interaction
effectively inhibits the generation of island reversals in
recorded bits and improves SNR. Figures 7(a) and7(b) plot
theSNR as a function of exchange constant, A, for signals
recorded with Head I and II, respectively. The M
swas fixed
at 820 emu/cm3. Overlap recording degrades SNR as seen in
TP¼20 nm. On the other hand, SNRs increase with increase
ofAin the range below 1 /C210/C07erg/cm. The identical pro-
files for the single track and TP¼25 nm in this range trans-late that the SNR is improved by decrease of island reversals
without considerable extension in recorded track width. Plot
ofTP¼25 nm, however, deviates from that of the single
track above 1 /C210/C07erg/cm, which implies that recorded
track width increases because of excessive intergrain
exchange interaction. SNR of 20 dB can be acceptable for
practical recording, Head I (narrow main pole) achieves 1
Mtpi of track density. On the other hand, similar profiles are
obtained in the case of signal recording with Head II,although track density capability is poor, as seen in Fig. 7(b).
Achievable track density is estimated to be 630 ktpi.
B. Influence of intergrain exchange coupling on the
optimum frequency for MAMR
In order to evaluate the increase in the optimum fre-
quency for MAMR caused by an intergrain exchange cou-
pling, the following numerical simulation was demonstrated.
ac magnetic field was locally applied to the perpendicularmedium by using a field-generating strip line with the width
of 20 nm. The strip line is passing through the center of the
medium. H
k,Msand cof the medium were assumed to be
18 kOe, 330 emu/cm3, and 1.5 /C2107rad/(Oe /C1s), respectively,
the values of which are consistent with those of perpendicu-
larly magnetized Co/Pd multilayer used in our previousMAMR experiments.
12The corresponding FMR frequency
f0is 33 GHz. The exchange stiffness constant was varied
from 1 /C210/C010to 7/C210/C07erg/cm. Figure 8(a) shows the
typical example of calculated magnetization configuration
after applying ac magnetic field. It is found that the switched
grains indicated by gray contrast appear below the strip line.Figure 8(b) shows the number of switched grains as a func-
tion of frequency. In the calculation, the ac field with an am-
plitude of 0.2 H
kwas applied to the medium with Aof
1/C210/C07erg/cm. The number of switched grains shows rapid
increase at 27 GHz followed by the gradual decrease with
increasing the frequency. Figure 8(c) documents the result,
showing the frequency of appearing MAMR as a function of
exchange stiffness constant calculated for different ac field
amplitudes. The frequency of appearing MAMR begins toincrease as the Abecomes larger than 1 /C210
/C07erg/cm. It is
noted that the decrease in MAMR frequency with increasing
microwave power is attributed to a nonlinear FMR property.As shown in Fig. 3(a) of Ref. 22, an asymmetric frequency
dependence of the precessional angle of magnetization
FIG. 6. SNR calculated for 850 kbpi of recorded track after shingle overlap
recording of an adjacent track with TP.
FIG. 7. SNR as a function of exchange con-
stant, A, for signals recorded with (a) Head I
and (b) Head II.123912-4 Nozaki et al. J. Appl. Phys. 109, 123912 (2011)
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128.235.251.160 On: Thu, 18 Dec 2014 06:59:09appears when the amplitude of pumping field Hacis
increased. Furthermore, the frequency showing a maximum
precessional angle decreases with increasing Hac. This is
attributed to the reduction in an effective anisotropy fieldowing to large-angle precession of magnetization. The
decrease in FMR frequency with increasing H
acmust cause
the decrease in MAMR frequency. Indeed, Okamoto et al.
calculated that the minimum frequency realizing the MAMR
decreases with increasing Hacwhen the amplitude of nega-
tive dc magnetic field is set at a fixed value.23
It should be noted that the numerical result can explain
the frequency realizing the MAMR in our previous experi-ment on perpendicularly magnetized Co/Pd multilayers.
12
In our previous study, we have demonstrated the MAMR inthe Co/Pd multilayer by using an ac magnetic field with anamplitude of 480 Oe generated by a coplanar waveguide.
The amplitude of the coercive field (1.2 kOe) is much
smaller than that of the anisotropy field of 18 kOe, implyingthat the nucleation dominates the switching of magnetiza-
tion. As shown in Fig. 2(b) of Ref. 13, the probability of
magnetization switching after ac field application is clearlyincreased at frequencies in the range from 26 to 29 GHz.
Those values correspond to 0.79 and 0.88 of the FMR fre-
quency at zero field, respectively. The frequency at whichMAMR appears is, however, somewhat higher than that
numerically expected in a single grain with uniform mag-
netization. Zhu et al. reported that the maximum reduction
of switching field along the easy axis is realized at 0.45 of
zero field resonance frequency.
14It should be noted that our
sample consists of an ensemble of magnetic grains stronglyexchange-coupled with one another. The strength of inter-
grain exchange coupling of the Co/Pd multilayer is approxi-
mately in the range from 0.7 to 1.1 /C210
/C06erg/cm that was
evaluated from the width of stripe domain configuration
observed using a magnetic force microscope. Considering
the numerical result shown in Fig. 8(c), the increase of the
MAMR frequency due to the exchange field is, therefore,
expected in the demonstrated experiments for the Co/Pd
multilayers. Furthermore, it is found that the MAMR fre-quency observed in the Co/Pd multilayer seems to be con-
sistent with that calculated assuming the ac field with
amplitude of 0.4 H
k, although the ac field amplitude of our
experiment was 0.4 Hc. The inconsistency in the amplitude
ofHacremains an open question.C. Dynamics of magnetization reversal with an
assistance of microwave field
Finally, the dynamics of magnetization reversal were
analyzed. Figure 9shows the footprint image calculated for
the medium with A/C240. In the calculations, a dc reversal
field was uniformly applied to the whole numerical areawhile a linearly polarized in-plane ac field with optimized
frequency is locally applied to the grains surrounded by the
broken line. Figure 9also shows the temporal variations of
in-plane and perpendicular components of magnetizations of
grain 1 and grain 2 indexed in the footprint image. Magnet-
izations of the all grains are initially arranged in the /C0z
direction. The footprint images are for 3 ns after application
of the magnetic fields. The in-plane component of the
FIG. 8. (Color online) (a) Typical example of calculated magnetization configuration after applying ac magnetic field. (b) The number of switched grai ns as a
function of frequency. (c) The frequency of appearing MAMR as a function of exchange stiffness constant calculated for different ac field amplitude.
FIG. 9. (Color online) Footprint image calculated for the medium withA/C240. The temporal variations of in-plane and perpendicular components of
magnetizations are also shown. The solid and dashed curves are the data for
grain 1 and grain 2 indexed in the footprint image.123912-5 Nozaki et al. J. Appl. Phys. 109, 123912 (2011)
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128.235.251.160 On: Thu, 18 Dec 2014 06:59:09magnetization of grain 1 increases with time, which means
magnetization precession angle increases with time.23The
behavior is very similar to well-known magnetization behav-
ior of typical MAMR process for single magnetization.24
Meanwhile, magnetization of grain 2, the neighboring grain,
behaves in the same manner as that of grain 1 until 0.07 ns,
then the precession angle decreases with time and the mag-netization finally stabilizes in the initial direction. This is at-
tributable to the magnetostatic interaction fields of the
surrounding grains. On the contrary, in the case that an inter-grain exchange interaction exists, magnetization behaviors
become slightly disturbed as shown in Fig. 10. The preces-
sion angle of grain 1 increases with time and decreases againat 0.09 ns, and then the magnetization rapidly switches. This
comes from the intergrain exchange interactions among the
surrounding grains, which fluctuates the precession axis ofmagnetization. The precession axis of the magnetization for
grain 2 also fluctuates according to the magnetization direc-
tion of the surrounding grains through intergrain exchangeinteractions even after the magnetization of grain 1 switched.
The intergrain exchange interactions attempt to arrange mag-
netizations for neighboring cells in parallel. This causesmagnetization switching of grain 2.
IV. CONCLUSIONS
The signal recording and reproducing process simulations
of MAMR-HDDs were demonstrated from the viewpoint offinding the optimum condition of intergrain exchange cou-
pling. In the medium with large saturation magnetization,
small island reversals tend to be nucleated in the recorded bitpattern. The nucleation of small island reversals can be inhib-
ited by enhancing the intergrain exchange coupling. Our nu-
merical result also suggests that the frequency of appearingMAMR is increased as the value of Abecomes larger than
1/C210
/C07erg/cm. From the analysis of the temporal variation
of magnetization in each magnetic grain, it is found that thesuppression of small island reversals is attributable to the fluc-
tuation of the precession axis of magnetization caused by the
intergrain exchange coupling field. However, the excessiveexchange coupling leads to overwriting of the neighboring
track. From the micromagnetic simulations, it is found that the
SNR larger than 20 dB at the track pitch of 25 nm can be
achieved by optimizing the intergrain exchange coupling.
ACKNOWLEDGMENTS
This research was partially supported by the Storage
Research Consortium (SRC), JST-CREST, and Grant-in-Aid
for Young Scientists (A), Grant No. 20686025, 2008, fromthe Ministry of Education, Culture, Sports, Science and
Technology, Japan.
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FIG. 10. (Color online) Footprint image calculated for the medium with
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1.5022452.pdf | Spin-wave-induced lateral temperature gradient in a YIG thin film/GGG system excited
in an ESR cavity
Ei Shigematsu , Yuichiro Ando , Sergey Dushenko , Teruya Shinjo , and Masashi Shiraishi
Citation: Appl. Phys. Lett. 112, 212401 (2018); doi: 10.1063/1.5022452
View online: https://doi.org/10.1063/1.5022452
View Table of Contents: http://aip.scitation.org/toc/apl/112/21
Published by the American Institute of Physics
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Applied Physics Letters 112, 213101 (2018); 10.1063/1.5023896Spin-wave-induced lateral temperature gradient in a YIG thin film/GGG
system excited in an ESR cavity
EiShigematsu, Yuichiro Ando, Sergey Dushenko, Teruya Shinjo, and Masashi Shiraishia)
Department of Electronic Science and Engineering, Kyoto University, 615-8510 Kyoto, Japan
(Received 15 January 2018; accepted 29 April 2018; published online 21 May 2018)
The lateral thermal gradient of an yttrium iron garnet (YIG) film under microwave application in
the cavity of the electron spin resonance system (ESR) was measured at room temperature byfabricating a Cu/Sb thermocouple onto it. To date, thermal transport in YIG films caused by the
Damon-Eshbach mode (DEM)—the unidirectional spin-wave heat conveyer effect—was demon-
strated only by the excitation using coplanar waveguides. Here, we show that the effect exists evenunder YIG excitation using the ESR cavity—a tool often employed to realize spin pumping. The
temperature difference observed around the ferromagnetic resonance field under 4 mW microwave
power peaked at 13 mK. The observed thermoelectric signal indicates the imbalance of the popula-tion between the DEMs that propagate near the top and bottom surfaces of the YIG film. We attri-
bute the DEM population imbalance to different magnetic dampings near the top and bottom YIG
surfaces. Additionally, the spin wave dynamics of the system were investigated using the micro-magnetic simulations. The micromagnetic simulations confirmed the existence of the DEM imbal-
ance in the system with increased Gilbert damping at one of the YIG interfaces. The reported
results are indispensable to the quantitative estimation of the electromotive force in the spin-chargeconversion experiments using ESR cavities. Published by AIP Publishing.
https://doi.org/10.1063/1.5022452
Spin caloritronics
1—young, but quickly developing
spintronic field—is in pursuit of the comprehensive under-
standing of the connection between heat and spin currents.
Following the discovery of the spin Seebeck effect,2plenty
of experimental demonstrations of spin caloritronic phenom-ena have been reported, such as the spin-dependent Seebeck
effect
3and the spin Peltier effect.4Especially, the heat trans-
port via spin waves and spin-phonon interaction has attractedattention after the unidirectional spin-wave heat conveyereffect was reported.
5,6In contrast to the conventional case of
heat transport against the temperature gradient, the Damon-
Eshbach mode (DEM) spin wave7,8induces heat transport in
the direction of the thermal gradient. Apart from this surpris-ing achievement, the unidirectional spin-wave heat conveyereffect has important implications in the field of the dynami-cal spin injection, also known as spin pumping, since they
often occur simultaneously in a studied system. Spin pump-
ing
9,10is a method of generation of pure spin current in the
material due to the coupling of the interface spins to the pre-cession magnetization of the adjacent ferromagnet layer. Itquickly gained popularity as a spin injection method that can
easily be used in any bilayer system consisting of nonmag-
netic/ferromagnetic materials,
11,12whereas the electrical spin
injection needs more elaborate surface treatment, formationof tunnel barriers and nanofabrication.
13–15Using the spin
pumping technique, the spin-to-charge conversion-related
properties of various heavy metals,11,16semimetals,17–19
semiconductors,20–22and even two-dimensional materi-
als23–25were unveiled, along with the spin transport proper-
ties of the materials.26–28The DEM is the surface spin wave that is excited under
the conditions close to the ferromagnetic resonance (FMR) and
propagates in opposite directions on the top and bottom surfa-
ces.29,30When the DEM reaches the end of the sample, its
energy is damped as heat, increasing the temperature near thesample edge. In the case of uniform excitation across the ferro-
magnet, the population of the DEM on the top and bottom sur-
faces is the same and the net quantity of the transported heatcancels out. However, when the equivalence of the populationof the two DEMs propagating in the opposite directions is bro-ken, the unidirectional thermal transport takes place. In the pre-
vious studies of the unidirectional spin-wave heat conveyer
effect,
5,31such inequivalence was shown to be present in the
case of the DEM excitation using the microstrip line wave-guides. In that case, the bottom surface of the ferromagnet is
located closer to the microstrip line than the top surface, and
thus difference in the intensity of the microwave AC magneticfield causes a population difference in the two DEM spinwaves. Induced unidirectional heat transport happens in thedirection of propagation of the dominant DEM. Importantly,
the direction of propagation of the dominant DEM (wave vec-
tork) can be reversed by reversing the direction of the external
magnetic field. Thus, voltage generated due to thermal effects(for example, the Seebeck effect) also reversed with the direc-
tion of the magnetic field. Incidentally, the spin pumping and
spin-charge conversion experiments rely heavily on the rever-sal of the magnetic field to exclude non-magnetic spuriouseffects, including the thermal ones: sign reversal of the gener-ated electromotive force with the external magnetic field is
usually taken as a proof of its spin-charge conversion origin.
Thus, to confirm the origin of the electromotive force in thespin-charge conversion experiments, it is crucial to preciselyestimate the unidirectional heat transfer induced by the DEM.
a)Author to whom correspondence should be addressed: mshiraishi@kuee.
kyoto-u.ac.jp
0003-6951/2018/112(21)/212401/5/$30.00 Published by AIP Publishing. 112, 212401-1APPLIED PHYSICS LETTERS 112, 212401 (2018)
While there are a few experimental5,31and theoretical32
studies on the unidirectional heat transfer effect under micro-
wave excitation using wave guides, there are no such reports
on the microwave cavities. In contrast, a broad variety of spinpumping and spin-charge conversion experiments are carriedout using the TE
011cavity of the electron spin resonance
(ESR) systems,11,33,34and the excitation of the DEM using
the ESR cavities was also demonstrated by measuring themicrowave absorption spectra in the literature
35,36(see section
Fo ft h e supplementary material for a detailed discussion).
Our study fills the experimental gap and reports the observa-tion of heat transfer by the DEM in the TE
011ESR cavity. We
also performed the micromagnetic simulations and discussed
the origin of the DEM imbalance observed experimentally.
We now proceed to the experimental details and results.
The 1.2- lm-thick yttrium iron garnet (YIG) film was grown
by liquid phase epitaxy on top of the gadolinium gallium
garnet (GGG) substrate and is available commercially(Granopt, Japan). We fabricated a thermocouple on top of
the YIG surface to measure the temperature difference gen-
erated due to the heat transport by the DEM. While there aremany types of thermocouples available commercially, themost common ones (types: E, J, K, and T) use ferromagnetic
metals nickel (Ni) and iron (Fe), or their alloys, which may
exhibit ferromagnetism due to insufficient uniformity of thealloy. In the spin pumping experiments, the lateral static
magnetic field is applied in the plane of the samples under
ferromagnetic resonance conditions. In this geometry, theanomalous Hall effect in the thermocouple may be induced
by the heating of the YIG film, which would add up to the
electromotive force generated by the lateral thermal gradientof the YIG film and prevent its quantitative estimation. Torealize a thermocouple that is composed of nonmagnetic
metals, we focus on the combination of copper (Cu) and anti-
mony (Sb) and use Cu wiring to make an electrical contactwith the sample. First, we formed a 50-nm-thick SiO
2insu-
lating layer on top of YIG to exclude the influence of spin
pumping, which was shown to decrease exponentially withthe thickness of the tunnel barrier.
37On top of it, a 50-nm-
thick Sb layer was deposited by resistance heating deposi-
tion. Finally, the third layer consisting of two Cu pads sepa-rated by a 1 mm gap was deposited. The sample with theformed thermocouple was set in the Seebeck effect measure-
ment system [Fig. 1(a)]. Room temperature acted as a base-
line level, while the hot and cold heat sinks—thetemperature of which was controlled by the Peltier ele-
ments—were attached to the opposite sides of the sample.
The lateral temperature difference and the thermoelectric elec-
tromotive force were monitored simultaneously. For the ferro-
magnetic resonance measurements, the sample was mounted
onto the TE
011cavity of the ESR system (JEOL JES-FA200).
The DC and AC magnetic fields were applied in the plane ofthe sample in DEM geometry as shown in Fig. 1(b).T h ef r e -
quency of the AC magnetic field was set to 9.12 GHz and the
a p p l i e dm i c r o w a v ep o w e rw a ss e tt o4m W .A ne s t i m a t e d
value of AC magnetic field applied to the sample was 4.4 lT.
The DC magnetic field was swept through the FMR field of
the YIG film, while the microwave absorption spectrum and
the electromotive force between Cu electrodes were mea-sured simultaneously. Since a bipolar electromagnet was
used, measurements at 0
/C14and 180/C14DC magnetic fields were
carried out without rotating the sample or changing its posi-
tion. All measurements were carried out at room temperature.
Figures 1(a) and2show the schematic layout and the
detected thermoelectric electromotive force in the Seebeck
effect measurement of the Cu/Sb thermocouple fabricated on
top of the YIG/GGG sample. We follow the conventional
definition of the Seebeck coefficient S
DV¼/C0SDT; (1)
where DVandDTare the thermoelectric electromotive force
and the temperature difference, respectively. From the linear
fitting (black solid line in Fig. 2), the Seebeck coefficient of
the fabricated sample was determined to be þ15 nV/mK.
This result is comparable to the Seebeck coefficient of the
amorphous Sb film reported in the literature.38
Next, the sample was placed in the cavity of the ESR
system for the measurement of the magnetic-field-dependent
heat transport induced by the DEM. The ferromagnetic reso-
nance measurements with simultaneous detection of the elec-tromotive force and FMR spectra were carried out for the
opposite directions of the DC magnetic field 0
/C14and 180/C14.
The wave vector kof the DEM is parallel to the cross prod-
uct of the DC component of magnetization of the YIG film
Mand the normal vector to the surface n. The direction of k
determines the direction fDTof the generated temperature
difference DTon the propagation surface5,7
FIG. 1. (a) A schematic image of the Seebeck effect measurement. The fabri-
cated sample was attached to two Peltier elements that controlled the temperature
difference between the edges of the sample. (b) A schematic image of measure-
ment of DEM heat transfer under FMR excitation in the ESR TE 011cavity.
FIG. 2. The thermoelectric electromotive force dependence on the appliedtemperature gradient for the Sb/Cu thermocouple fabricated on top of the
YIG film. The black solid line is a linear fitting and purple bars are the mea-
surement error bars.212401-2 Shigematsu et al. Appl. Phys. Lett. 112, 212401 (2018)k==fDT==M/C2n: (2)
Therefore, we extracted the magnetization-dependent com-
ponent of the observed thermoelectric signal by subtractingthe signals measured at 0
/C14and 180/C14directions of the external
magnetic field ( V0/C14and V180/C14, correspondingly). Figure 3
shows the thermoelectric signal generated by the DEM,which is given by V
m¼(V0/C14/C0V180/C14)/2, and the FMR spectra
at the DC magnetic fields of 0/C14and 180/C14. The coincidence of
the two FMR spectra confirms the identical resonance condi-
tions for the opposite directions of the DC magnetic field(also, see Fig. S6 in section F of the supplementary material
for highlighted DEM resonances in the spectrum).
Interestingly, the DEM thermoelectric signal shows reversalof the polarity when approaching the FMR condition.
Following the results of the Seebeck effect measurement of
the sample, a positive V
msignal corresponds to the þydirec-
tion of the thermal gradient fDT, which is due to the DEM at
the GGG/YIG interface, and the negative Vmto the /C0ydirec-
tion, which is due to the DEM at the SiO 2/YIG interface,
respectively. The amplitude of the negative peak of the ther-moelectric signal was measured to be /C0190 nV. Using the
Seebeck coefficient of the sample, the estimated temperature
difference between the Cu pads separated by the 1 mm gap(/C0ydirection) is 13 mK. Figure 4shows the schematic lay-
out of the DEM excitation in our measuring geometry for 0
/C14
direction of the external magnetic field. The thermal gradient
direction fDTof/C0y(þy) suggests the contribution of
the DEM from the YIG interface with the SiO 2(the GGG)
film. Note that the uniformly excited DEMs in the thinferromagnetic film has the same population of the þkand
/C0kmodes, and thus they transfer equal amounts of heat in
the opposite directions and the induced temperature differ-ences by the two modes cancel each other out. Therefore, thenegative peak of the thermoelectric electromotive force inthe DC magnetic field close to the FMR condition signifiesthat the magnitude of the DEM at the SiO
2/YIG interface is
superior to that on the GGG/YIG interface.
The previous analytical magnetostatic studies of the
DEM assumed that the ferromagnetic film was placed in thevacuum and did not treat the symmetry breaking of the topand bottom sides of the film.
7Furthermore, the influence of
the Gilbert damping on the DEM propagation and dampingwas not considered. We carried out numerical micromag-netic simulations that evaluate the effect of symmetry break-ing in our SiO
2/YIG/GGG system on the DEM population
using the program MuMax3.39GGG is known as a paramag-
netic material with substantially large magnetization. Theinfluence of the GGG layer attached to the YIG interface onthe Gilbert damping of the surface YIG layer was alreadypointed out.
31Thus, in the micromagnetic simulations, we
set the Gilbert damping parameter aof one marginal layer
next to the YIG/GGG interface (we refer to it as the bottomlayer) larger than the other layers. We performed MuMax
3
simulations using parameters close to the experimental val-ues. The AC magnetic field excitation frequency was set tof
0¼9.12 GHz. The saturation magnetization of the YIG layer
was set to 1.275 /C2105A/m and the exchange stiffness to
3.7/C210/C012J/m.40The calculated geometry is illustrated in
Fig.5(a). The Gilbert damping was 0.01 and 0.001 in the bot-
tom layer and the bulk, respectively. At the beginning of thesimulation, the DC magnetic field was set and the magnetiza-
tion of the whole system was relaxed. Following that, the
AC excitation of the magnetic field was applied and—afterthe magnetization precession reached the steady state—weextracted the zcomponent of the magnetization of each spin
cell in the slice of x¼50 (where a coordinate represents the
layer number in that direction). The magnetization motion inthe slice consists of a non-time-dependent bias, standingwaves, and traveling waves along the ydirection. We can eval-
uate the DEM by extracting a portion of the traveling waves.For this purpose, the Fourier transform was implemented as
m
zf;ky
2p/C18/C19
¼ðymax
yminðtmax
tminWtðÞ/C1WyðÞ/C1mzt;yðÞ e/C02pjft/C0ky
2pyðÞdtdy;
(3)
FIG. 3. Red and green lines in the top box are the electromotive forces
observed under the microwave excitation of 4 mW under the DC magnetic
fields of 0/C14and 180/C14, respectively. Purple line in the middle box is the halved
subtraction of the electromotive force ( Vm) and represents the electromotive
force contribution that is reversed together with the direction of the DC mag-netic field. Two overlapped lines in the bottom box are FMR spectra mea-
sured in the 0
/C14(red) and 180/C14(green) directions of the DC magnetic field.
FIG. 4. A cross-sectional illustration of thermal gradient generation by theDEM under the application of the external magnetic field in the 0
/C14direction.
The direction of the kwave vector of the DEM is locked to the direction of the
cross product of the YIG magnetization Mand the surface normal vector n.212401-3 Shigematsu et al. Appl. Phys. Lett. 112, 212401 (2018)where mz,W,f,a n d kydenote the zcomponent of the normal-
ized magnetization, the hamming window function, the excita-tion frequency, and the wavenumber of the magnetization in
theydirection, respectively. As we extracted the data in the
finite range of [[ y
min,ymax],[tmin,tmax]], we applied the ham-
ming function to reduce the obstructive sublobes in the result-
ing Fourier spectra. We focus on the Fourier spectra in the
frequency of excitation f0, which was set to 9.12 GHz. When
mzf0;ky
2p/C16/C17
is deconvoluted into mzf0;ky
2p/C16/C17
¼AþBja n d
mz/C0f0;/C0ky
2p/C16/C17
¼CþDj, the absolute amplitude with wave-
number kyleads to mabs
zky
2p/C16/C17
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
AþC ðÞ2þ/C0 BþD ðÞ2q
.
Then, we obtain the wave distribution as a function of the
wavenumber ky/2p. A plot of the amplitude mzabsvs. wave-
number at two different DC magnetic fields is shown in Fig.
5(b). The amplitudes in the top and bottom layers showed a
clear difference at 251 mT. Figure 5(c) shows the absolute
amplitude mzabsin the central layer (top box), which is analo-
gous to the FMR spectrum detected experimentally; the sub-
traction of the maximum of mzabsbetween the top and bottom
layers (bottom box), which characterizes the imbalance of the
DEM between them. The difference in the amplitude between
the two DEMs propagating in the opposite directions peaked
around the FMR resonance field. Thus, numerical simulation
of YIG indicated that the top-layer DEM is dominant over the
bottom-layer DEM due to the break of the reflection symmetry
because of different magnetic dampings at the surfaces. This
result explains the dominance of the heat generated by the top
DEM observed in the experiment. We also carried out similarsimulations with the parameters of NiFe (permalloy), and the
same effect was reproduced (see section E of the supplemen-
tary material ). We note that micromagnetic calculations for the
full-size sample are necessary for the precise quantitative simu-
lation of the experimental results, which was limited by the
computational power in this study. Additionally, the quantita-
tive determination of the heat drift velocity—a key parameter
in the thermal distribution induced by the DEM imbalance—
needs a more elaborate analysis of the spin-phonon interaction,which is left for further study. However, our experimental and
numerical results clearly show that reflection symmetry break-ing between the two YIG surfaces by the magnetic damping at
the interfaces induces the imbalanced DEM population and the
unidirectional heat transfer.
In conclusion, we observed the unidirectional spin-
wave heat conveyer effect in a 1.2- lm-thick YIG film under
uniform microwave excitation in the ESR cavity. The origin
of the DEM imbalance that led to the heat transport isexplained by the increased Gilbert damping at one of the
YIG interfaces. The micromagnetic simulations confirmed
the existence of the DEM imbalance in such system. Ourstudy fills the experimental gap that existed in the literature
on the unidirectional spin-wave heat conveyer effect gener-
ated in the ESR cavity. The reported results are indispens-able to the quantitative estimation of the electromotiveforce in the spin-charge conversion experiments using ESR
cavities.
Seesupplementary material for the following informa-
tion: microwave power dependence of the detected electro-
motive force, dependence of the detected electromotive
force on the direction and speed of the DC magnetic fieldsweep, reproducibility of the results, details of the MuMax
3
calculation, micromagnetic simulations of permalloy, anddiscussion on the existence of the DEM in excitation usingthe ESR cavity.
E.S. acknowledges the financial support from the JSPS
Research Fellowship for Young Researchers and JSPS
KAKENHI Grant No. 17J09520. This work was supported inpart by MEXT (Innovative Area “Nano Spin Conversion
Science” KAKENHI No. 26103003), a Grant-in-Aid for
Scientific Research (S) No. 16H06330, and a Grant-in-Aidfor Young Scientists (A) No. 16H06089. S.D. acknowledges
support by a JSPS Postdoctoral Fellowship and a JSPS
KAKENHI Grant No. 16F16064. The authors thank T.Takenobu and K. Kanahashi for the informative adviceregarding the Seebeck coefficient measurement.
FIG. 5. (a) A schematic illustration of
the structure used in micromagnetic
simulation. The magnetic film had 21
layers in the zdirection, and each layer
consisted of 101 /C2301 unit cells.
Gilbert damping was set to 0.01 in thebottom layer and 0.001 in the others.
(b) The absolute amplitude of the m
z
component of magnetization in the
x¼50 slice in the top (red) and bottom
(green) layers. The DC magnetic field
was set to 251 (left box) and 253 mT
(right box). (c) The upper box: TheFMR intensity (the absolute amplitude
of the z¼15 layer). The lower box: the
DEM imbalance between bottom and
top layers calculated from the micro-
magnetic simulation. Experimental
measurements indicated similar domi-
nance of the top DEM from theobserved heat transport.212401-4 Shigematsu et al. Appl. Phys. Lett. 112, 212401 (2018)1G. E. W. Bauer, E. Saitoh, and B. J. van Wees, Nat. Mater. 11, 391 (2012).
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1.345868.pdf | The initial susceptibility of ferrites: A quantitative theory
J. P. Bouchaud and P. G. Zerah
Citation: J. Appl. Phys. 67, 5512 (1990); doi: 10.1063/1.345868
View online: http://dx.doi.org/10.1063/1.345868
View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v67/i9
Published by the American Institute of Physics.
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Downloaded 01 Mar 2013 to 128.197.27.9. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsThe initial susceptibility of ferrites: A quantitative theory
J. P. Bouchauda) and P. G. Zerah
Centre d'Etudes de Limeil-Valenton, B. P. 27, 94190 Villeneuve St. Georges, France
In this contribution a firm foundation to the observed complex permeability spectrum of
ferrites (such as NiZn or MnZn spinels) in the gyromagnetic regime (100 MHz, 10 GHz) is
provided. Many attributes of the initial permeability of uniaxial ferrites are not correctly
reproduced by a straightforward application of the usual Landau-Lifschitz-Gilbert equation.
Among those one can cite the observed, non-Lorentzian shape of J-l' and J-l", and the
permanence of losses (J-l" (w) > 0) up to the frequency Wm = y41TM" above which they
disappear ("low field loss"). The problem is recast here in the general framework of effective
medium theory which provides a firm interpretation of the observed complex permeability
spectrum offerrites (such as NiZn or MnZn spinels) in the gyromagnetic regime (100 MHz,
10 GHz).
The susceptibility of a demagnetized ferrimagnet is mar
kedly different from that of a polarized sample. The reasons
are twofold: First, domain walls, which have a dynamics of
their own, are absent in the latter case. Second, domains of
opposite dc moment orientation ( + and -domains) do
not react the same way to an external osciIlating field: There
appears a nonzero magnetization perpendicular to the excit
ing field which is ouf o/phase from one type of domain to the
other. Hence, those different domains dynamically interact,
yielding a total sus~eptibility which is not the linear sum of
the contribution of each individual one, and which can in
deed differ significantly from it. A theory for these mixture
effects (usually c.aBed Polder-Smit resonances in magne
tism) is thus needed to interpret quantitatively the initial
susceptibility of ferrites, and to extract microscopic param
eters from the experimental curves (see also Refs. 1 and 2).
Let us suppose that the local magnetization is along
± z. Then, in the plane (x,y), the susceptibility (Polder)
tensor (obtained from the Landau-Lifschitz equation)
reads
, [/1
J-l = ±iK
where ± refers to the local orientation of the domain,
and , Hu + (iw + r-')r-'IHur Ii -1= y-41TM, -,-,-------
y-H: + r 2 -w2 + 2icur '
W
K = y41TM, ~----------r H ~ + r 2 -w2 + 2iWT .. , ' (1)
(2a)
(2b)
with M, the saturation magnetization, Hu the anisotropy
field, r the spin relaxation time related to the usual a param
eter by r I = yHua, and w the frequency of the external
field. Some exact results may be obtained for the average
susceptibility for thrce particular arrangements of domain
walls, which we quote here in the completely demagnetized
case:
.•• Also at Laboratoire de Physique Statistique. associe au CNRS. 24 rue
Lhomond. 75231 Paris Cede x 05. France. (i) The domain walls are all parallel to, e.g., the (y,z)
plane (lamellar configuration). Then [see, e.g., Ref. 2(a)}
~ [J-l J-l= o (3)
(ii) The domains are translationally invariant along the
z axis. [The local normal to domain walls is always in the
(x,y) plane.} If the domain configuration is statistically iso
tropic, then one has the following exact resu\t2(a).3 :
, [il /1= 0 ~J, (4)
with
ill = /12 -K2. (5)
This result coincides with that obtained from an effec
tive-medium approximation.3 Figure 1 shows the resulting
frequency-dependent susceptibility as compared to the
"bare" one or to that obtained in case (i). The most impor
tant point is that mixture effects are crucial3.4 when intrinsic
dissipation is small (r large) and negligible in the opposite
case.
(iii) The sample is still translationally invariant and the
local magnetization lies in the (x,y) plane. In the case of an
isotropic configuration, one has4
p = [~/1 ~I;]. (6)
(Formulas for partially polarized samples may be found in
Refs. 3 and 4.)
It is reasonable to think that the domain configuration
within one metallurgical grain is well described by the as
sumptions of case (ii) since the anisotropy field is coherent
within one grain. [Even if locally the domain structure is
lamellar, the average over directions will yield back Eq.
(5) }. Then an isotropic distribution of anisotropy axis from
grain to grain can be taken into account through an effective
medium treatment4 [with Eq. (4) as the local tensor 1, lead
ing to an averaged susceptibility which can directly be com
pared to experiments. For il ~ 1, one has
5512 J. Appl. Phys. 67 (9). 1 May 1990 0021-8979/90/095512-03$03.00 @ 1990 American Institute of Physics 5512
Downloaded 01 Mar 2013 to 128.197.27.9. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsJL" '000
900
900
700
V\
l \ 'n I •
i
I
If ~ .00
300
/1 [7\\'\
/// \\\ I"
~, '" .......... ~ ~ _. ,~
w
'" f\
/ \
10" / \
II \
1/ \ • 00
--// "-~~ ., .. ,,,",
w
FIG. 1. Influence of the domain geometry on the imaginary part of the sus
ceptibility for (a) small intrinsic damping (T 1 = 5 MHz), and (b) large
intrinsic damping (T-1 = 50 MHz). The curve with the highestll" rna. cor
responds to the lamellar configuration, while that with the lowest Il" rna' is
the two··dimensional isotropic case [Eq. (5)]. The intermediate one is the
bare behavior [Eq. (2a)].
o
p
o
with ~ given by Eq. (5).
This last expression allows us to devise a fitting proce
dure: for large frequencies (w> -10 Ha), one has
p'w2_cte = r41TMsr-l~ and p"w-cte
= r41TMs~ 1 + a2 [a is the usually defined Landau-Lif
shitz parameter a = (rHa r) -I]. Indeed, many ferrite sam
ples behave precisely in that way. Then, from independent
5513 J. Appl. Phys., Vol. 67, No.9, 1 May 1990 10 100 w 1000
L II I 1IIIil TTTIIII
/J~2' I
I
II ... ,
1001~-------+~----
FIG. 2. Log-log plot of the susceptibility of NiZnFe20. as a function of
frequency (Ref. 5), as compared to the theoretical prediction. The differ
ence between the two curves gives the domain-wall contribution. Note the
excellent agreement in the region U! > 40 MHz, where the latter is expected
to be small .
Ms measurements, one determines a and r. The susceptibil
ity can then be fitted in the whole frequency range. As is
obvious from Fig. 2, both pi and p" are very well accounted
for, except in the low-frequency region. This is of course due
to the domain-wall contribution, which can in fact be ex
tracted quite neatly by subtracting from the experimental
curve the theoretical contribution from gyromagnetism.
From investigations on different ferrite samples (MnZn,
NiZn of different compositions5) we obtain three very im
portant pieces of information:
First, the parameter a is found to be nearly composition
independent, equal to 1.7 ± 0.1 for all the samples studied.
Hence the inverse spin-relaxation time is ofthe same order of
magnitude as the anisotropy field, or more precisely of its
spatial fluctuations. This is not unexpected in depolarized
samples where no dipolar narrowing (due to long-range de
magnetizing fields) should occur.6 A precise theory for the
value of a is however needed. Note this (low-frequency)
determination of a strongly differs from the result obtained
in ferrimagnetic resonance experiments, where a _ 10 -3_
10 -2 (but r has the same order of magni\uoe in both cases).
This may be important to resolve the domain-wall mobility
paradox found in Refs. 7 and 8.
Second, the domain-wall contribution is found to be
well described by the classic harmonic oscillator model, with
J. P. Bouchaud and P. G. Zarah 5513
Downloaded 01 Mar 2013 to 128.197.27.9. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsa resonance around 10 MHz, and parameters which have the
expected order of magnitude.
And third, the domain-walI contribution is found to be
negligible only above -50 MHz. In particular, the domain
walls and the uniform spin rotation are found to contribute
equally to the static susceptibility. This contradicts Snoeck's
classical interpretation,9 which only retains the latter to esti
mate the anisotropy field from f.i' (0) -I = 81T'M, 13H". On
the example of Nio 35 ZnOM Fe20-t, we estimate
rH" -13 MHz, while Snoeck's result9 is a factor of 2
smaller (5.5 MHz).
We thank D. Lepoutre for giving us fulI access to her
experimental results.
5514 J. Appl. Phys., Vol. 67, No.9, 1 May 1990 I D. Polder and J. Smit, Rev. Mod. Phys. 25, 89 (1953); G. Rado, Rev.
Mod. Phys. 25, 81 (1953); G. Rado, Phys. Rev. 89,529 (1953).
'(a) E. Schlomann, J. App!. Phys. 41, 204 (1970); (b) J. Phys. (Paris)
Colloq. 31, 443 (1971); (c) for experimental fits, see also S. C Zhu, L.
Zhao, E. P. Wen, and L. Z. Meng, J. App!. Phys. 57, 3806 (1985); 61, 4139
( 1987).
.1 J. P. Bouchaud and P. G. Zerah, Phys. Rev. Lett. 62, 1000 (1989).
4 J. P. Bouchaud and P. G. Zerah (unpublished).
, D. Lepoutre (private communication).
6 C W. Haas and H. B. Callen, in Magnetism, edited by G. T. Rado and H.
Suhl (Academic, New York, 1963), Vol. 1.
7 R. W. Teale, J. Phys. C 13, 2061 (1980).
• M. Guyot, T. Merceron, V. Cagan, and A. Messekher, Phys. Status Solidi
106,595 (1988).
"J. Smit and H. M. J. Wijn, Ferrites (Phillips Technical Library, Eindho
ven, The Netherlands, 1959).
J. P. Bouchaud and P. G. Zarah 5514
Downloaded 01 Mar 2013 to 128.197.27.9. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions |
1.5049832.pdf | Magnetoelectric coupling and the manipulation of magnetic Bloch skyrmions
Zidong Wang , Zhao Ma , and Malcolm J. Grimson
Citation: Appl. Phys. Lett. 113, 102403 (2018); doi: 10.1063/1.5049832
View online: https://doi.org/10.1063/1.5049832
View Table of Contents: http://aip.scitation.org/toc/apl/113/10
Published by the American Institute of PhysicsMagnetoelectric coupling and the manipulation of magnetic Bloch skyrmions
Zidong Wang,1,a)Zhao Ma,2and Malcolm J. Grimson3,a)
1State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics,
Tsinghua University, Beijing 10084, China
2Department of Mechanical Engineering, The University of Western Australia, Perth 6009, Australia
3Department of Physics, The University of Auckland, Auckland 1010, New Zealand
(Received 25 July 2018; accepted 20 August 2018; published online 5 September 2018)
Chiral-magnetic/ferroelectric composite systems offer the possibility of electrically inducing
magnetic Bloch skyrmions [Wang and Grimson, Phys. Rev. B 94, 014311 (2016)]. They are
appealing for potential applications in spintronics due to their self-protection behavior. To realize
skyrmion-based spintronic devices, it is essential to control the motions of the skyrmions. In this
work, we propose a mechanical technique to manipulate skyrmions collinearly with a mobile exter-nal electric field that is imposed on the chiral-magnetic/ferroelectric system. The role of propaga-
tion velocity strongly impacts on the quality of magnetic skyrmions. Published by AIP Publishing.
https://doi.org/10.1063/1.5049832
Magnetic Bloch skyrmions are topological vortex-like
spin structures with a range of sizes from 10 nm to approxi-mately 100 nm, which have been introduced theoretically
1,2
and observed experimentally3in many studies. The existence
of these skyrmions has been detected in chiral-magnetic(CM) crystals, such as the B20 metallic MnSi,
3FeGe,4
MnGe,5Mn-Fe-Ge alloys,5Fe-Co-Si alloys,6and the multi-
ferroic Cu 2OSeO 3,7,8which has no inversion symmetry. This
can allow the emergence of magnetic skyrmions due to theirnon-centrosymmetric lattice structure which gives rise to aDzyaloshinskii-Moriya (asymmetric exchange) interaction(DMI).
1In micromagnetics, this phenomenon is caused by
the interplay of a weak nearest-neighbor collinear exchangeinteraction and the inherent Dzyaloshinskii-Moriya coplanarinteraction. The competition between the two stabilizes thehelicity of magnetic skyrmions.
2
Lately, magnetic skyrmions attract much attention within
the spintronics community for a possible application in mem-ory devices and computing.
9The inherent stability of these
topologically protected structures combined with their lowpinning probability to defects are intriguing characteristics. Touse skyrmions as information holders, it is necessary to con-trol their motion. Previous studies revealed several non-mechanical methods, such as by using electric currents,
10
spin-polarized currents,11microwave fields,12temperature
gradients,13,14and magnons.15However, a mechanical method
has not been explored. In this work, we investigate a mechani-cal technique to move magnetic Bloch skyrmions collinearlywith a mobile electric field source.
Multiferroism allows us to create magnetic skyrmions
by the electric polarization.
16–18This is a result of the small
magnetoelectric (ME) coupling between the magnetizationand electric polarization fields. Unfortunately, the multifer-roic insulators, e.g., Cu
2OSeO 3, have a low transition tem-
perature and a limited magnetic response, which is adversefor applications.
19But in composite systems, which are syn-
thetic heterostructures of magnetic and ferroelectric (FE)orders can produce a remarkable ME coupling due to the
microscopic mechanism of the strain-stress effect.20
Specifically, a mechanical strain can be internally generatedby applying an electric field in the FE film due to the piezo-electric effect, then this strain acts on the coupled CM filmand results in a net magnetization due to the inverse magne-
tostrictive effect, as shown in Fig. 1. It is known as the con-
verse ME effect.
21A previous study has demonstrated that
creation of skyrmions can be induced by the converse ME
effect in a composite system that consists of a coupled CM
and FE bilayer film.18
In a composite CM/FE bilayer, the magnetic skyrmions
are moved by an external electric field source along the
bilayer film as shown in Fig. 2(a) (Multimedia view). The
CM film is on the top of the bilayer and it is glued to a FE
film by a strong ME coupling. The CM film can support
magnetic skyrmions. A localized electric field source islocated above and below the bilayer. It can travel in a planeparallel to the film. Then, we carry out a spin dynamics
method to determine the time dependent behaviors of the
magnetic and electrical responses.
The magnetic component of the system having a CM
structure is described by the classical Heisenberg model on a
FIG. 1. The physical mechanism of the converse ME effect that forms mag-
netic Bloch skyrmions in the composite system. The processes of the reverse
piezoelectric effect and inverse magnetostrictive effect are represented in
the green and blue, respectively.a)Authors to whom correspondence should be addressed: wangzidong@tsinghua.edu.cn and m.grimson@auckland.ac.nz
0003-6951/2018/113(10)/102403/4/$30.00 Published by AIP Publishing. 113, 102403-1APPLIED PHYSICS LETTERS 113, 102403 (2018)
two-dimensional rectangular lattice. The magnetic spin Si;j
¼Sx
i;j;Sy
i;j;Sz
i;j/C16/C17
characterizes the local magnetic moments,
and it has a normalized length with kSi;jk¼1, and i;jlocates
the position. The total Hamiltonian HCMis given by
HCM¼/C0 JCMX
i;j½Si;j/C1ðSiþ1;jþSi;jþ1Þ/C138
/C0DX
i;j½ðSi;j/C2Siþ1;jÞ^xþðSi;j/C2Si;jþ1Þ^y/C138
/C0KX
i;jSz
i;j/C0/C12: (1)
The first term contains the familiar collinear nearest-neighbor
exchange interaction, and J/C3
CM¼JCM=kBTrepresents the
dimensionless exchange interaction coupling coefficient. Thesecond term shows the coplanar DMI in the two-dimensional
lattice system, and D
/C3¼D=kBTrepresents the dimensionless
DMI coefficient, where ^xand^yare the unit vectors along the
x-a n d y-axes, respectively. This describes the Bloch-type
DMI in a non-centrosymmetric crystal.22The third term
represents the perpendicular magnetic anisotropy, and K/C3
¼K=kBTis the dimensionless uniaxial anisotropic coefficient.
The dynamics of magnetic spins have been studied by the
well-known Landau-Lifshitz-Gilbert equation, which numeri-
cally solves the rotation of a magnetic spin in response to itstorques
@Si;j
@t¼/C0 cSi;j/C2Heff
i;j/C16/C17
/C0kCMSi;j/C2Si;j/C2Heff
i;j/C16/C17 hi
;(2)where cis the gyromagnetic ratio which relates the spin to its
angular momentum, and kCMis the phenomenological Gilbert
damping term of CM materials.23,24Heff
i;j¼/C0dH=dSi;jis the
effective field acting on each magnetic spin, which is the func-tional derivative of the system Hamiltonian.
We employ a pseudospin model to solve the dynamical
behaviors of electric polarization.
25,26The local electric
moments are replaced by electric pseudospins, shown as
Pk;l¼Px
k;l;Py
k;l;Pz
k;l/C16/C17
that is regarded as a continuous vec-
tor, and k;lcharacterizes the location. The distinction
between pseudospins and classical spins is the variable size
and absence of precession. The electric polarization isdefined as the dipole moment density in dielectric materials.
The dipole moment density pis proportional to the external
electric field, , as p¼/C15
0veEext. In the pseudospin model, the
size of each electric pseudospin is proportional to the magni-
tude of its effective field as kEeff
k;lk¼k dH=dPk;lk. Hence
kPk;lk¼ /C150NekEeff
k;lk, where Neis the dimensionless pseudo-
scalar susceptibility. Consequently, the electric pseudospin
has a variable size as does the behavior of an electric dipole.
We use a transverse Ising model to describe the Hamiltonian
of pseudospins,27as
HFE¼/C0JFEX
i;jPz
k;lPz
kþ1;lþPz
k;lþ1/C0/C1/C2/C3
/C0XX
i;jPx
k;l/C0/C1/C0/C150veEextX
i;jPz
~k;~l/C16/C17
; (3)
where J/C3
FE¼JFE=kBTrepresents the dimensionless electric
exchange coupling along the Ising zdirection. X/C3¼X=kBT
represents the dimensionless transverse field along the x
axis, which is an in-plane field and perpendicular to the Ising
zdirection. E/C3
ext¼/C150veEext=kBTrepresents the dimensionless
applied electric field along the zdirection, where /C150is the
electric permittivity of free space, and veis the dielectric sus-
ceptibility. A site of location ~k;~lpresents the pseudospin
which in the presence of electric field. Note that the appliedelectric field is mobile, localized, and located to relative
bilayer film to reduce edge effects (i.e., edge-merons) in the
simulations.
28The time evolution of pseudospins is expected
to perform a precession free trajectory.26Because of the
electric dipole moment is a measure of the separation of pos-
itive and negative charges along the Ising zdirection. It is a
scalar. Therefore, only the zcomponent of pseudospin repre-
sents the real polarization. A modified Landau-Lifshitz-
Gilbert equation is used to describe the pseudospin dynam-
ics, as
@Pk;l
@t¼/C0 kFEPk;l/C2Pk;l/C2Eeff
k;l/C16/C17 hi
; (4)
where kFEis the phenomenological damping term in the FE
structure.
The interfacial effects between the CM and FE layers
are caused by a ME coupling. The analytic expression forME effect can be bilinear or nonlinear, particularly with
respect to the thermal effect.
29In this paper, we only account
for low-energy excitations between the CM and FE layersand so we restrict ourselves to the bilinear expression of MEinteraction, as
FIG. 2. (a) A schematic illustration of a composite bilayer consisting of a CM
layer and a FE layer stacked at the interface. A localized electric field source
can be moved longitudinally along the bilayer. (b) Sequential snapshots pre-
sent the generation of a skyrmion in the bulk of the CM layer, as the localized
electric field is statically applied at the initial position. (c) Propagating a sky-
rmion by moving the electric field source with a velocity of v/C3¼0:02 spin-
site/step. The color scale represents the magnitudes of the local zcomponent
magnetization and polarization. Multimedia views: https://doi.org/10.1063/
1.5049832.1 ;https://doi.org/10.1063/1.5049832.2102403-2 Wang, Ma, and Grimson Appl. Phys. Lett. 113, 102403 (2018)HME¼/C0gX
i;jðÞ k;lðÞSz
i;jPz
k;l/C0/C1; (5)
where g/C3¼g=kBTrepresents the dimensionless strength of
the ME coupling. Nonlinear forms have not been studiedhere for simplicity and due to their minor effects in themicromagnetic modelling.
To investigate the dynamical manipulation of magnetic
Bloch skyrmions, we implement dimensionless parametersJ
/C3
CM¼D/C3¼1,K/C3¼0:1,J/C3
FE¼0:8,X/C3¼0:1,g/C3¼0:4,
c/C3¼1, and k/C3
CM¼k/C3
FE¼0:1. Note that “ /C3” characterizes
dimensionless quantity. The number of magnetic spins andelectric pseudospins was set to N¼30/C290 in each layer.
Free boundary conditions and a random initial state wereapplied. Landau-Lifshitz-Gilbert equations are solved by afourth-order Range-Kutta method. A marginal electric fieldwas applied to order the FE and CM domain walls, and thenwe apply the localized electric field with a dimensionlessmagnitude of E
/C3
ext¼10 perpendicular to the bilayer surface.
The electric pseudospins quickly complete realignment, butthe responses of magnetic spins have a delay. The generationprocess of a magnetic skyrmion in the bulk of the CM layeris summarized in Fig. 2(b). Subsequently, this field source is
moved along the bilayer with a constant velocity. The velocity
is measured as v
/C3¼DN=Dt/C3,w h e r e DNcorresponds to spa-
tial movement to equivalent locations (i.e., spin-sites), and Dt/C3
is a dimensionless time step. Figure 2(c) shows a series of
diagrams that show the skyrmion transport in the CMlayer (Multimedia view). The skyrmions follow the polarizedpseudospins in the FE layer for a velocity of v
/C3¼0:02
spin-site/step.
In this propagation process, we can see the skyrmion
track deflecting to the bottom edge due to the skyrmion Halleffect.
30The behavior of a skyrmion is topologically like a
spinning disk and it generates a Magnus force when travelinglongitudinally. So, it induces a transverse force during thetranslational motion of the skyrmion. The figure furthershows the electric polarization reflecting the passage of field
source. But the magnetization has a component that is non-
collinear with the electric response and shows a spin spiralalignment due to the existence of a finite DMI. CM crystalshave a non-centrosymmetric structure that enables the mag-netic ordering to be broken.
The movement of the field source is externally controlla-
ble. We therefore explored two results of the effects ofhigher velocity on the skyrmion transport. Figure 3(a)shows
a trial with a propagation velocity of v
/C3¼0:05 spin-site/step
(Multimedia view). The skyrmion barely struggles to followthe motion of field source during the propagation process.Eventually, the system becomes more complicated, becauseanother two skyrmions are formed from edge-merons tocomplement the energy contribution. In Fig. 3(b), we set the
velocity to v
/C3¼0:1 spin-site/step and note the skyrmions are
lost immediately (Multimedia view). Furthermore, the sky-rmion Hall effect acts in the high-speed operation and thetransverse motion of skyrmion transport may result in itsannihilation at the boundaries.
The magnetization processes in the CM bilayer are con-
sistent with the magnetic phase diagrams calculated ear-lier.
31,32Furthermore the physics of isolated chiral skyrmionshave been investigated33and they show that the formation of
isolated skyrmions, their structure, and stability limits are con-sistent with the results shown here.
In summary, we have investigated a mechanical method
to control magnetic Bloch skyrmions by moving an electricfield source parallel to the composite CM/FE bilayer system.Skyrmions are supported by the electric polarization throughthe converse ME effect. The results demonstrate that the sky-rmion is moved collinearly with the field source at a slowspeed. But higher speeds may break the stability of skyrmiontransport and lead to annihilation of the skyrmions at theedges.
We would like to thank Xichao Zhang, Yan Zhou, and
Wanjun Jiang for the discussion. Z.M. gratefully acknowledgesBingjing Zhao and Yuhua Wang for support.
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1.4937749.pdf | Extraction and analysis of body-induced partials of guitar tones
Vincent Fréour, , François Gautier , Bertrand David , and Marthe Curtit TRM
Citation: J. Acoust. Soc. Am. 138, (2015); doi: 10.1121/1.4937749
View online: http://dx.doi.org/10.1121/1.4937749
View Table of Contents: http://asa.scitation.org/toc/jas/138/6
Published by the Acoustical Society of America
Extraction and analysis of body-induced partials of guitar tones
Vincent Fr/C19eour,1,a)Franc ¸oisGautier,1Bertrand David,2and Marthe Curtit3
1Universit /C19e du Maine, LAUM, UMR CNRS 6613, Avenue Olivier Messiaen F-72085, Le Mans Cedex 9,
France
2LTCI, CNRS, T /C19el/C19ecom ParisTech, Universit /C19e Paris-Saclay, 75013, Paris, France
3Institut Technologique Europ /C19een des M /C19etiers de la Musique, 71 Avenue Olivier Messiaen F-72085,
Le Mans Cedex 9, France
(Received 18 June 2015; revised 22 October 2015; accepted 23 November 2015; published online
24 December 2015)
Guitar plucked sounds arise from a rapid input of energy applied to the string coupled to the instru-
ment body at the bridge. For the radiated pressure, this results in quasi-harmonic contributions,
reflecting the string modes coupled to the body, as well as some transient and quickly decaying
components reflecting the excitation of the body modes of the instrument. In order to evaluate therelevance of this transient body sound, a high resolution analysis-synthesis method is described for
the extraction of the body-mode contribution from the radiated pressure measured in the near field
of the guitar top plate. This analysis scheme is first tested on synthesized signals. Some body-soundemergence indicators are then proposed and computed over a pool of instruments. The influence of
the conditions of excitation on the body-sound emergence is investigated, and the instruments cate-
gorized according to these objective descriptors. Results show a larger range of body-sound emer-gence with variations of the plucking position in hand-made guitars compared to industrial
instruments. This suggests that these particular hand-made instruments are more sensitive to varia-
tions in the control from the player and hence allow a wider range of timbres with respect to thetransient coloration of the body modes.
VC2015 Acoustical Society of America .
[http://dx.doi.org/10.1121/1.4937749 ]
[TRM] Pages: 3930–3940
I. INTRODUCTION
Significant research work has been devoted to the char-
acterization of music instruments thanks to the extraction of
objective indicators from acoustic or vibration measure-ments, and their correlations with subjective evaluations.
1–4
More recently, a number of studies have focused on the anal-
ysis of the vibroacoustic responses of string instruments and
their relation to manufacturing and design characteris-
tics,5–10as well as to the subjective evaluation of instrument
attributes or “quality.”11–13
In the case of the guitar, although the low-frequency
behaviour of the sound box is relatively well understood andrepresented through coupled oscillators models,
14–17the corre-
lation between a modal description of the body and a quantita-
tive or qualitative description of the produced sound over alarge frequency range is less clear. The analysis of the mechan-
ical admittance measured at the bridge, where energy is trans-
mitted from the string to the body of the instrument, has beenthe object of several studies aiming to identify objective fea-
tures possibly correlated to structural or perceptual attributes of
one or several instruments.
7,8,10,18–25The objective categoriza-
tion provided by these analyses show interesting potential,
while also revealing difficulties in identifying the proper indi-
cators from vibroacoustic measurements to be correlated tosubjective attributes of the instr ument. Alternatively, numerical
simulations can be used to investigate the response of aninstrument to a controlled excitation and for a set of design
parameters.
26,27
Plucked sounds are characterized by a sudden input of
energy, transmitted to the string after it has been movedfrom its equilibrium position, by either a finger or a plec-
trum, and released. According to the exact nature of the exci-
tation, the two polarizations of the string are excited to a
different extent. The out-of-plane polarization being strongly
coupled to the top plate, the energy transmitted to the bodywill result in a faster decay if this polarization is predomi-
nantly excited, and vice-versa for the in-plane polarization,
the latter of which will influence the overall energy decay ofthe vibrating string and ultimately of the radiated sound. In
order to control with a reasonable degree of confidence the
amount of plucking energy injected into the system and the
initial angle of the string excitation, an experimental method
using thin copper wires has been proposed.
28This technique
consists of applying an initial displacement at a given point
of the string using a calibrated copper wire until spontaneous
rupture. It thus allows both the plucking direction and theinitial displacement of the string to be adjusted in a control-
lable and repeatable manner and is therefore a suitable tech-
nique to compare the response of different instruments, as it
enables similar conditions of excitation to be reproduced.
Figure 1presents the spectrum of the first 200 ms of the
acoustic pressure of a E4 recorded 18 cm from the guitar
soundboard plane, in front of the tone hole. The harmonic
comb representing the contribution of the string mode series
can be clearly identified, with a fundamental frequency
around 330 Hz. On the top of this harmonic spectrum, some
a)Electronic mail: vincent.freour@mail.mcgill.ca
3930 J. Acoust. Soc. Am. 138 (6), December 2015 VC2015 Acoustical Society of America 0001-4966/2015/138(6)/3930/11/$30.00
weakly resonating components, interleaved with the har-
monic string components, are also clearly visible, as it hasalso been observed in piano.
29
From the observation of the input admittance measured
at the bridge, it is easy to conclude that these componentsoriginate from the excitation of the structural modes of theinstrument body. The higher damping of these modes rela-tive to the string modes explains the rapid energy decay ofthese body-induced partials. Nevertheless, we may reason-ably assume that, at tone onset, the contribution of thesemodes to the radiated sound may be strong enough to partici-pate to the overall tone color of the plucked sound.
In this paper, the acoustic response of classical guitars is
investigated with regards to the emergence of these bodymodes from which what we will further name a “bodysound” originates. The perceptual and categorizing relevanceof the body-sound emergence is evaluated over a pool ofclassical guitars of different manufactures, through the cal-culation of objective indicators describing the body-soundcontribution. The paper is organized as follows: the analysismethodology is described in Sec. IIand validated on syn-
thetic signals in Sec. III. Results obtained from measure-
ments over a pool of instruments are presented in Secs. IV
andVwhere body-mode emergence indicators are calcu-
lated. A discussion is provided in Sec. VIbefore concluding
on the main outcomes of the study.
II. ANALYSIS METHODOLOGY
A. High-resolution analysis of plucked sounds
Since our objective is to isolate the contribution of the
string from the contribution of the body modes in the
recorded sound, a robust and precise signal processing toolis needed to accurately estimate the frequency, amplitude,and damping of these body-induced partials. This will allowfor the reconstruction of the plucked sound without the effectof the body modes, as well as the synthesis and analysis ofthe residual sound arising from the excitation of the bodymodes only.
High resolution methods have been proposed in 1989.
30
More recently, they have been the object of further interest
for the modal analysis of piano and guitar mechanical admit-tances,
7,31–34as well as for the analysis of guitar sounds.35The method proposed in this study is adapted from pre-
vious analyses of guitar impulse responses.7In the present
case, we consider that the recorded signal spectrum can beexpanded on the basis of a series of eigenmodes character-ized by their frequency f
k, modal damping factor ak, modal
mass mk, and modal shape Uk. The modal parameter estima-
tion technique is based on the time-domain representation ofthe recorded signal s(t) as a sum of complex exponentials.
The discrete-time representation of a signal s[n] of size Nis
then given as follows:
s½n/C138¼X
K
k¼1ake/C0a0
knejð2pnf0
kþukÞ; (1)
where akejukis the complex amplitude of the kth component,
f0
kis the normalized frequency of the kth component
(f0
k2½ /C01
2;12/C138),zk¼e/C0a0
kþj2pf0
kis the corresponding pole with
normalized damping factor a0
k, and Kis the modeling order
(i.e., the number of complex modes considered). The estima-tion procedure consists in estimating the poles z
kfrom which
a collection of frequencies and damping factors can be
obtained:
fk¼argzkðÞ
2pFsand ak¼/C0Fslnjzkj; (2)
where Fsis the sampling frequency.
In order to reconstruct the original signal, the complex
amplitudes bk¼akejukof the Kpoles of the signal can then
be calculated through a least-square fitting in the time do-main as follows:
b¼V
†
es; (3)
where bis a vector formed by the bkvalues, sis the meas-
ured signal, and V†
eis the pseudo-inverse of the
Vandermonde matrix of dimension N/C2Kformed by the esti-
mated poles.31
The pole estimation is performed using the ESPRIT
algorithm.30This method is based on the decomposition of
the input vector space onto two orthogonal subspaces,
namely, the signal and noise subspaces. The signal subspace
Sis formed by the sinusoidal components corresponding to
the family of the Vandermonde vectors vk¼1:::K, where vk
¼eðj2pfk/C0akÞn=Fsandn¼0…(N/C01). The noise subspace N
is the orthogonal complement of S, such as S/H20003N¼/C15
where /C15is the vector space formed by the input vector. A ba-
sisW(K) of the signal subspace Sis then obtained by com-
puting the singular value decomposition of the Hankel datamatrix calculated from the input signal samples. Because thesignal subspace verifies the so-called rotational invarianceproperty, it remains invariant from one sample to the next,which leads to the following property:
W
"ðKÞ¼W#ðKÞRðKÞ; (4)
where R(K)i sa K/C2Kmatrix whose eigenvalues are the
poles zk,W#(K) is the matrix W(K) where the last row has
been deleted, and W"(K) is the matrix W(K) where the firstFIG. 1. Spectrum of the first 200 ms of a plucked guitar sound showing the
presence of weakly resonating components due to the body vibrations in the
harmonic comb of the string contributions.
J. Acoust. Soc. Am. 138 (6), December 2015 Fr/C19eour et al. 3931
row has been deleted. The estimation of the poles zkis finally
achieved through an eigenvalue decomposition of the matrix
R(K). For more details about the application of the ESPRIT
method to string instruments, the reader is invited to refer to
a previous study from Elie et al.7
B. ESTER criterion
To overcome the problem of estimating the correct
model order, several methods have been proposed. Amongothers, the ESTER (ESTimation of ERror) criterion designed
by Badeau et al.
31has been applied to the analysis of vibrat-
ing structures.7,34,35This criterion is based on the evaluation
of the rotational invariance property of the signal sub-space
through the error function
EðpÞ¼W"ðpÞ/C0W#ðpÞRðpÞ; (5)
where pis the modeling order to be determined. This allows
the function J(p) to be computed such as
JpðÞ¼1
kEpðÞk2
2: (6)
Denoting Kas the correct number of components in the orig-
inal signal, large values of J(p) should be obtained for
p<K, while smaller values will be obtained when p>Kas
the subspace spanned by W(p) will include noise compo-
nents. Note that for real signals, the number of components
Kis twice the number of physical eigenmodes.
One issue encountered using the ESTER criterion relies
on the proper adjustment of the threshold allowing to iden-tify the optimal model order. This threshold is, for instance,
set as a fraction of the global maximum of J(p).
C. Body-mode selection criteria
In the context of our study, the ESPRIT method and
ESTER criterion are used to estimate the modal parameters
of the identified components in the plucked response of theguitar. Because we are interested in quantifying the influence
of body modes on the sound at tone onset, the following con-
ditions are then applied to classify the identified modes intoeither the body or string categories.
1. C1: Quality factor
Body modes are clearly more damped than string
modes. A first criteria is hence based on the values of the
quality factors: Qk¼xk/2ak. To be considered as a body
mode, the value of the quality factor must satisfy the condi-
tion Qinf/C20Qk/C20Qsup, where Qsupis an arbitrarily set con-
stant value (e.g., Qsup¼500). Furthermore, a lower limit Qinf
is fixed to 2 in order to avoid selecting very damped modes
possibly arising from mechanical couplings with the guitar
stand and outside structures.
2. C2: Distance to the harmonic comb series
String modes are nearly harmonically distributed: fk
’kf0. Therefore, all modes that do not belong to thisharmonic series can be considered as body modes. The dis-
tance between one body mode eigenfrequency and the clos-
est harmonic partial is denoted dk[Eq.(7)],
dk¼minjfh/C0fkj: (7)
Ifdkis greater than a given threshold value Dlim, the
corresponding mode will be classified as a body mode as
opposed to a string mode.
3. C3: Frequency range of the body modes
In addition to criteria C1 and C2, as the body modes are
expected to contribute to the body sound in a relatively low
frequency range, these are only considered above 20 Hz and
below the frequency limit flim¼3 kHz.
III. TEST OF THE EXTRACTION PROCEDURE USING
SYNTHESIZED BRIDGE VELOCITIES
One of the challenges in this work is to achieve a proper
extraction of the modal parameters of the body modes
(within a given frequency range) among a collection ofweakly damped string modes. Therefore, in order to validate
the use of the ESPRIT–ESTER method applied to plucked
guitar sounds, we first propose to apply it to synthesized
sounds.
A. Hybrid frequency-domain synthesis scheme
The single-polarization synthesis scheme used in this
study was proposed by Woodhouse.27It is based on the
frequency-domain synthesis of the velocity at the point
where the string and the body of the instrument are assumedto be rigidly coupled. At that particular point, the coupled
system has an admittance Ysuch as
1
Y¼1
Ysþ1
Yb; (8)
where Ysis the admittance of the string alone and Ybis the
admittance of the body of the instrument alone at the cou-
pling point.
This section is organized into three parts: first the com-
putation of the string admittance is treated, second the com-
putation of the body admittance synthesized from
measurements is presented, and third we will concentrate on
the synthesis of the velocity at the bridge.
1. String admittance calculation
Let us consider a string of length Lfixed at x¼0 and
having an imposed harmonic displacement weixtimposed at
x¼L. It can be shown that, for small damping, the expres-
sion of the string impedance at the bridge can be written as
follows:27
1
Ys¼/C0iT
L1
xþX1
k¼11
x/C0xk1þigsk=2 ðÞ/C26"
þ1
xþxk1/C0igsk=2 ðÞ/C27/C21
; (9)
3932 J. Acoust. Soc. Am. 138 (6), December 2015 Fr/C19eour et al.
where Tis the string tension, gskis the loss factor of the kth
string mode, and xkis the angular frequency of the kth string
mode such as xkL/c¼kp(with cthe wave velocity in the
string). A model for the calculation of the loss factor of thekth mode g
skwas proposed by Woodhouse.27It involves
three loss factor terms gf,ga, and gbrelative to, respectively,
the friction between the string turns, the air viscosity, andthe visco and thermo-elastic effects
g
sk¼Tgfþga=xk/C0/C1þBgbkp=LðÞ2
TþBkp=LðÞ2; (10)
where B¼ðEpa4=4Þis the string bending stiffness, Eand
abeing the string Young modulus and string radius,
respectively.
Once Yscalculated, let us now concentrate on the calcu-
lation on the modeling of the body admittance Yb.
2. Body admittance modeling
In order to evaluate the performance of the ESPRIT–
ESTER method, sound synthesis must be performed usingknown body modal parameters. We hence chose to synthesizean admittance ^Y
bfrom a measurement perform on a real
instrument. The admittance of guitar A measured at the bridgeusing a miniature accelerometer (PCB 352C23) glued withwax on the bridge and excited using an impact hammer (PCB086E80) is presented in Fig. 2. The corresponding impulse
response his computed using the deconvolution technique
described by Ege et al.
34and Elie et al.7This method consists
in estimating the impulse response of a filter ^gin the mean
least square sense so that ^g/C3F¼d(Fbeing the force
injected into the system, and dbeing the Dirac function). The
estimated impulse response of the system ^his hence given by
^h¼^g/C3c(with cthe acceleration measured at the bridge).
Contrary to the procedure described in Sec. II A(which
will be further applied to synthesized velocities and measuredpressure signals), the calculation of the complex amplitudesb
kof^his here performed in the frequency domain under theassumption that the body mode shapes Ubkare real. This hy-
pothesis leads to the following expression for the admittance[Eq.(11)]:
^Y
bxðÞ¼jxXK
k¼1U2
bkAðÞ
mbkx2
bkþjgbkxbkx/C0x2/C0/C1 ; (11)
where Ais the point of observation, and xbk,mbk, and gbk
denote the angular frequency, modal mass, and loss factor of
thekth mode, respectively.
The modeling order Kis selected as the integer so that
the ESTER criteria J(p) remains below a given threshold
forp>K. This threshold can be automatically set as a frac-
tion of the global maximum of J(p) but is here adjusted
manually from the observed level of ESTER criterion asvarying with p.
Because an overestimation of the number of modes of-
ten leads to unrealistic modal parameters, the aberrant valuesare discarded by thresholding of the extracted modal parame-
ters. Furthermore, because of the low level of energy injected
by the hammer above 5 kHz, only modes below this fre-quency limit are considered for the modeling of Y
b. In the
present case, the following selection criteria are applied:
2<Q<500, mode frequencies fksuch as 20 Hz <fk<3 kHz.
The amplitude and phase of the synthesized admittance, ^Yb,
along with the original measured admittance, Yb, are pre-
sented in Fig. 2.
3. Synthesis of the velocity
No radiation model is here considered. The synthesized
signal is thus the velocity at the coupling point, which is at
the bridge. The transfer function allowing to determine the
velocity at the bridge ( x¼L), resulting from an impulse
force Fapplied at the string plucking point xcan be derived
such as27
H¼yxðÞ
w¼c
Lsinxx
c"
1
xþX1
k¼1/C01ðÞk 1
x/C0xk1þigsk=2 ðÞ/C26
þ1
xþxk1/C0igsk=2 ðÞ/C27#
; (12)
where cis the wave velocity in the string and w(x) is the
displacement at the plucking point. The frequency-domain
velocity Gis then computed from the expression of 1 =Y
¼1=Ysþ1=^YbandHsuch as
G¼Y/C1H¼_yxðÞ
FLðÞ; (13)
where _yðxÞis the string velocity at x. Making use of the reci-
procity property, we then obtain
G¼_yLðÞ
FxðÞ: (14)
The time-domain velocity gis obtained through the
inverse Fourier transform applied to G,FIG. 2. Measured admittance Yb(gray line) and resynthesized admittance
^Yb(black line) using ESPRIT–ESTER method for guitar A.
J. Acoust. Soc. Am. 138 (6), December 2015 Fr/C19eour et al. 3933
g¼realfFFT/C01ðGÞg; (15)
where Gð/C0xÞ¼GðxÞ.
The number of string components is set to the nearest
smallest integer of the ratio 5000/ f0. To make the synthe-
sized signal more realistic, measurement noise from themeasurement chain can be added to the gsignal. The spec-
trum G(x) of the synthesized velocity using the parameter
values summarized in Table I, including 15 string modes, 76
body modes and /C040 dB signal-to-noise ratio (SNR) added
measurement noise is presented in Fig. 3.
B. HR analysis and performance
The high-resolution and body-mode selection scheme is
applied to the synthesized velocity with and without the additionof measurement noise. ESPRIT analysis is conducted over a
200 ms segment of the signal with an initial order K
i¼300 and
ESTER threshold is set manually from the ESTER criterion.Selection criteria for the iden tification of body modes are para-
meterized as follows: f
lim¼3k H z , Qsup¼500,Dlim¼8H z .T h e
ESTER criterion calculated in the no-noise and /C040 dB SNR
conditions is presented in Fig. 4.
The identified body components in the /C040 dB SNR
condition are represented in the ( f,Q) plane in Fig. 5. This
representation highlights the efficiency of the method in
extracting body-mode parameters in the presence of overlay-
ing harmonic components with a good level of accuracy.
The number of successfully identified body modes Nid,
out of the total number of body modes identified NT, can be
assessed in the ( f,Q) space: a mode is considered as success-
fully identified if the smallest distance to a mode of ^Ybin the
(f,Q) plane is such that both frequency and Qfactor are esti-
mated with an error smaller than 10%. From the value ofN
id, and given the theoretical number of body modes Nth
located below 3 kHz, the recall (or sensitivity) r¼Nid/Nth
and the precision p¼Nid/NTcan be calculated. In the
/C040 dB SNR case, the recall and precision are found such as
r¼p¼92.5%.
IV. APPLICATION TO REAL INSTRUMENTS
Measurements are performed using the following proto-
col. The guitars are maintained in fixed positions using acommercial guitar stand. The instruments are hung by the
head and lie on two of the stand feet covered with foam sothat the contact between the stand and the body only occursat two points. Because the guitar is not totally vertical
but slightly tilted, this mounting can be considered as a
good compromise by allowing wire breaking maneuverswithout inducing parasitic solid-body motions. Althoughdifferent from “real” playing conditions, this setup enables
TABLE I. Synthesis parameters from Ref. 28.
Parameter Variable Value
fundamental frequency f0 330 Hz
wave velocity in the string c 107.3 m/s
string length L¼c
2f0
string tension T 71.6 N
string torsional stiffness B 57/C210/C06N/m
friction coefficient gf 2.10/C05
stiffness coefficient gb 2/C210/C02N/m
drag coefficient ga 1.2
number of string modes Nh¼5000
f0
plucking distance to the bridge d 10 cm
sampling frequency fs 22 050 HzFIG. 3. Amplitude spectrum of the synthesized velocity Gwith /C040 dB
SNR added measurement noise.
FIG. 4. ESTER criterion in the synthetic velocity with no-noise (top), syn-
thetic velocity with /C040 dB SNR added noise (middle), and measured
acoustic pressure in real conditions (bottom). The horizontal dashed line
indicate the ESTER threshold.
3934 J. Acoust. Soc. Am. 138 (6), December 2015 Fr/C19eour et al.
repeatable fixing conditions and guarantees reliable compari-
sons between the different instruments.
A PCB microphone is set 18 cm from the top plate
pointing toward the lower edge of the tone hole. Wire break-ing measurements are performed using a 56 lm copper wire.
Each excitation is repeated 3 times for three different pluck-ing positions (at 3 cm from the bridge for position P
1,9 c m
forP2, 15 cm for P3) as described in Fig. 6. The other unex-
cited strings are damped to avoid sympathetic vibrations.Signal conditioning and acquisition is performed using aPCB pre-amplifier (482C) and National Instrument I/O inter-face running at 44.1 kHz. The tuning of the guitars waschecked using a digital tuner before each measurement.
A. Identification results
Results from first, the estimation of poles (frequencies and
Qfactors), and then the classification (body vs string modes)
presented in Sec. II, applied to the radiated pressure for a
plucked E4 ( f0¼330 Hz) by exciting the E4 string in plucking
position P1in the out-of-plane direction, are presented in Fig.
7. The ESPRIT algorithm is applied using the following pa-
rameters: the maximum order used to compute ESTER criteriais set to K
i¼300, the duration of the analyzed signal is fixed to
200 ms and the body-mode selection criteria are set to the fol-lowing values: D
lim¼8H z , Qsup¼500. The ESTER criterion
calculated for guitar A (see Sec. Vfor the definition of guitar
A) is presented in Fig. 4. The amplitude spectra of the recon-
structed signals considering e ither body- or string-mode com-
ponents are presented in Fig. 7, along with the original sound
spectrum. The result of this reconstruction shows the capabilityof the algorithm to separate both contributions. Because of thehigh value of f
0relative to the location of the first body modes,
the first “signature” modes (between 80 and 300 Hz) are clearlyidentified and well defined in the pressure spectrum. To furtherillustrate this result, the sound files corresponding to the origi-nal recording (gray trace in Fig. 7), the additive synthesis of
the string sound where only the string-mode contributions areconsidered (black trace in the top plot of Fig. 7), and the syn-
thesis of the body sound where only body modes are consid-ered (black trace in the bottom plot of Fig. 7), are provided in
theMm. 1 ,Mm. 2 ,a n d Mm. 3 files, respectively. Thereconstruction of the original si gnal (additive synthesis of both
string and body modes) is provided in Mm. 4 .
Mm. 1. Original sound recording, wav file (130 Kb).
Mm. 2. Reconstructed string sound, wav file (130 Kb).
Mm. 3. Reconstructed body sound, wav file (130 Kb).
Mm. 4. Reconstructed original sound, wav file (130 Kb).FIG. 5. Body-modes in ^Yb(circle) and identified from the synthesized veloc-
ity with /C040 dB SNR measurement noise (cross). The vertical dotted line
indicates the frequency limit flim.
FIG. 6. (Color online) Plucking positions, directions, and initial angles dur-
ing experiments. E4 (330 Hz) string, normal to the top-plate plane at three
plucking positions, E4 string in P1tangent to the top-plate plane, E2 (82 Hz)
string in P1normal to the top-plate plane.
FIG. 7. Original (gray line) and reconstructed amplitude spectra (in dB) of
the string-sound component (top) and body-sound component (bottom).
J. Acoust. Soc. Am. 138 (6), December 2015 Fr/C19eour et al. 3935
B. Body-mode emergence
In order to evaluate the perceptual relevance of body
modes in the radiated sound, the masking effect of theharmonic string partials should be quantified. The compu-tation of the masking threshold of the first ten stringresonances is performed using a method described by
Painter and Spanias
36and detailed in the Appendix . This
masking threshold is calculated over the first 20 ms of thepressure signal as classically performed for audio-maskingestimations. Emerging bands are then defined as the fre-quency bands where the Power Spectral Density of thebody-sound signal is greater than the masking threshold in
magnitude. The perceived body sound s
p
bis hence derived
by combining the contribution of the modes locatedwithin the identified emerging bands through additivesynthesis.
The Power Spectral Density of the body sound and
string sound computed over the first 200 ms of a E4
(330 Hz) plucked in position P
1normal to the top plate
plane is represented in Fig. 8. In the same figure, the global
masking threshold of the harmonic string series is also rep-resented, allowing to indicate emerging bands using coloredareas.An index of emergence I
e, defined as the ratio of the
energy of the perceived body sound sp
bto the energy of the
original sound, can then be computed as follows:
Ie¼ðT20
0jsp
bj2dt
ðT20
0jsbþssj2dt; (16)
where sbis the sound arising from the body components
(body sound), ssis the sound formed by the string compo-
nents, and the integration time T20is the decay time of the
perceived body sound defined by Eq. (17), where teis the
total duration of the signal. In the example of Fig. 8, the
index of emergence over the decay time T20(here around
240 ms) is found to Ie¼42%, illustrating how significant is
the body sound in the early phase of the sound
/C020¼10/C1log10ðT20
0jsp
bj2dt
ðte
0jsp
bj2dt8
>>><
>>>:9
>>>=
>>>;: (17)
C. Factors influencing body-sound emergence
The emergence of the body components relative to the
string contribution may significantly vary according to the
conditions of excitation.
Considering the “nominal” conditions of excitation of
Fig.8(plucked E4 in position P1, in the direction normal to
the top-plate plane), the plucking position, plucking direction(or angle of release), and fundamental frequency f
0(plucked
string) are varied independently from each other in order toinvestigate their influence on the value of the index of emer-gence I
e. The variations of emergence in these three condi-
tions are illustrated in Fig. 9. The emerging character of
the body modes appears to be significantly reduced in thethree conditions compared to the “nominal” case. Particularly,the plucking position seems to significantly influence theemergence of body modes. These considerations are furtherdetailed in the following.FIG. 8. (Color online) Reconstructed power spectral density of the string-
sound component (black line), body-sound component (gray line), masking
threshold (thick black line), emergence area (filled area) for a plucked E4
(330 Hz) in plucking position P1and normal direction ( Ie¼42%).
FIG. 9. (Color online) Illustration of the body-sound emergence for a plucked E4 (330 Hz) in position P3(a) (Ie¼4.1%), plucked E4 (330 Hz) in position P1
and in the direction parallel to the top-plate plane (b) ( Ie¼3.7%), plucked E2 (82 Hz) in position P1(c) (Ie¼15%).
3936 J. Acoust. Soc. Am. 138 (6), December 2015 Fr/C19eour et al.
1. Dependence on the plucking distance to the bridge
A clear drop in the value of Ieis observed when pluck-
ing the string further away from the bridge: Ie(P1)¼42%
(see Fig. 8) and Ie(P3)¼4.1% [see Fig. 9(a)]. This observa-
tion may be explained qualitatively. The force applied at thebridge when plucking the string is the projection of the ten-sion in the normal direction. Assuming that the wire willbreak for an exact same effort applied to it, the anglebetween the string and top-plate plane at the instant whenthe wire breaks decreases with increase in the distance to thebridge. The magnitude of the projection of the tension in thenormal direction to the top plate is thus decreasing when
plucking further away from the bridge. This may hence
result in a smaller level of excitation of the body modes rela-tive to the string modes in position P
3.
2. Dependence on the angle of release
When the plucking excitation is normal to the top-plate
plane (Fig. 8), the initial out-of-plane polarization is predom-
inantly excited. The component of the force applied at thebridge and normal to the top plate is thus maximized and the
excitation of body modes is also maximum. In the case of a
plucking direction tangent to the top plate, the initial in-plane polarization is predominantly excited. The projectionof the force normal to the top plate is therefore significantlyreduced [see Fig. 9(b)]. This may contribute to explain a
much lower value of I
ein this condition ( Ie¼42% for a nor-
mal excitation and Ie¼3.7% for a tangent excitation). The
angle of release hence significantly influences the emergenceof the body modes in the radiated sound.
3. Dependence on the plucked string
Depending on the plucked string, or to some extent to
the fundamental frequency, the masking effect of the har-monic string contribution over the body-mode componentswill vary. A decrease of I
eis observed when changing the
plucked string to E2 (82 Hz): Ie(E4)¼42% (see Fig. 8) and
Ie(E2)¼15% [see Fig. 9(c)]. For this string, the lower fun-
damental frequency and the higher modal density empha-
sizes the masking effect from the string contribution: the
higher density of string modes and lower fundamental fre-quency close to the frequency of the first body modes con-siderably mask the body-induced partials. This results indecreasing the emergence of body modes in the radiatedpressure.
V. COMPUTATION OF INDICATORS FOR INSTRUMENT
CATEGORIZATION
In this section, we propose to explore the emergence of
body sounds over a range of instruments of different origins.Two series of measurements were conducted using the proto-
col described in Sec. IV: the first set of measurements con-
cerns height hand-made nylon-stringed classical guitars fromvarious guitar makers: Paulino Bernabe (es), Yulong Guo(cn), Jean-Marie Fouilleul (fr), Kazuo Sato (de), Hopf/Adalid (uk), Felipe Conde (es), Red Gate (au), StephanSchlemper (de), and labeled A to G in a different order thanthe above listed names to avoid associations. These instru-
ments are designed by instrument makers to answer profes-
sional players’ needs. The measurements were performed in
a relatively low-reverberant room with no particular acoustic
treatment. The second set of measurements was conducted
on six industrial low-cost nylon-stringed classical guitars of
the same brand and same model noted K1 to K6. These
measurements were performed using the same experimental
setup in an acoustically anechoic room.
A. Objective indicators of body-sound emergence
In light of previous experimental observations, three
main features related to body-sound emergence can be of in-
terest to allow objective comparisons between instruments:
1—the decay-time of the body-sound T20; 2—the index of
emergence Iefor a given string, at a given standard plucking
position, and with a given angle of release; 3—the degree of
variations of body-sound emergence (or range of body-
sound emergence) with variations of the plucking position
DIedefined by Eq. (18),
DIe¼IeðP1Þ/C0IeðP3Þ; (18)
where P1andP3are the plucking positions located 3 and
15 cm from the bridge, respectively (see Fig. 6), and where
Ie(Pj) is computed using Eq. (16).
B. Guitar categorization
The indices of emergence calculated at the three posi-
tions for all the instruments are summarized in Fig. 10. Each
instrument is represented by a rectangle: the left boundary of
the rectangle corresponds to Ie(P3), the right boundary to
Ie(P1), and the thick vertical dash to Ie(P2), considered here
as an arbitrary nominal plucking position. Each value of Ieis
obtained through averaging over three repetitions of a same
measurement. For each value of Ie, the measurement range
(indicating the minimum and maximum values) is calculated
over the three repetitions and represented by horizontal lines
FIG. 10. (Color online) Representation of the body-sound emergence of a
pool of guitars. Industrial guitars (K1 to K6) and hand-made guitars (A to
H). The thick dashed line indicates the value of Iein position P2. Horizontal
lines indicate Ierange of measurement (minimum and maximum values) at
the three plucking positions.
J. Acoust. Soc. Am. 138 (6), December 2015 Fr/C19eour et al. 3937
in Fig. 10. The vertical position of the rectangle is given by
the width of the corresponding rectangle, i.e., by the value of
DIecalculated from the averaged values of Ie(P1) and Ie(P3).
In this two-dimensional (2D) plane, two clustering
regions appear: a first region ( DIe<0.3) where all industrial
instruments and three hand-made guitars are found, and a
second region ( DIe>0.3) that clusters five of the hand-made
instruments. Within the pool of instruments made available
for this study, it then appears that hand-made instrumentsshow overall larger ranges of emergence than the industrial
guitars. The larger index of emergence is observed for guitar
K5 at 58% while the smallest emergence is observed at 5%
for guitar F. The largest range of emergence is found at 43%
in guitar E, while the smallest range is observed for guitarK2 at 12%. Looking at all instruments, the maximum level
of emergence in position P1 seems uncorrelated to DI
e,
while the smallest values of Iein position P3 are observed
for the four guitars presenting the largest ranges of
emergence.
The decay times T20ofsp
b, averaged over the three repe-
titions for the three plucking positions, are represented
against the associated ranges of emergence DIefor all instru-
ments in Fig. 11.T20measurement ranges, calculated over
the nine repetitions (three repetitions for each plucking posi-
tion) is represented by vertical lines. The smallest T20is
found for guitar D at 177 ms, while the largest value is
observed for guitar E at 396 ms. Overall, it is observed that
industrial instruments spread over a slightly smaller range of
T20values (205 to 323 ms) than hand-made instruments,
although no clear correlation between the two groups ofinstruments and the values of T
20can be derived.
Qualitatively, the range of emergence (horizontal
dimension in Fig. 11) may be associated to the sensitivity of
the instrument to the plucking position in terms of perceived
tone onset: a larger variety of attacks may be possible for a
given string according to the plucking position when DIeis
large, while the decay time T20(vertical dimension) indicates
how far the body sound is overlapping with the harmonic
string contribution at tone onset. According to thisrepresentation, an instrument allowing a large range of tim-
bre variations and strongly colored by the body modes attone onset will lie in the top right corner of the 2D space ofFig.11, while an instrument allowing less timbre variations
with respect to the transient emergence of body modes(more uniform along the string) and with a rather “damped”body-sound component will be located in the bottom leftcorner of this 2D space.
VI. DISCUSSION
The subjective evaluation of an instrument by a musi-
cian results from a very complex combination of acousticand tactile feedback. Among other features, the transientemergence of body-induced partials arising from the excita-tion of the body modes of the guitar may be of great impor-tance. At first, the importance of the body sound at the verybeginning of the tone is characterized by two descriptors: theindex of emergence I
eand the body-sound decay time T20.
Variations of these two descriptors are strongly linked totimbre modifications. Furthermore, the nature of the interac-tion with an instrument is crucial in the process of evalua-
tion. In particular, for a given instrument, its capability to
produce a wide variety of timbres when the characteristics ofthe control gesture are varied is an important attribute. Thisproperty can be further defined as the “sensitivity” or“transparency” of an instrument to the player’s control ges-ture. For a skilled player whose mastering of the playingtechnique enables her/him to control very finely the excita-tion conditions, a large sensitivity of the instrument might beof great importance. This would correspond to an instrumentlikely to reflect the subtleties of the player’s inputs thenoffering a large pallet of sound timbre, while being maybe
more demanding in terms of accuracy in the control. On the
contrary, a beginner player might be more comfortable withan instrument less sensitive (or more tolerant) to control ges-ture variations.
Among the tested instruments, industrial guitars show
smaller ranges of body-sound emergence while the greatestranges of emergence are obtained for the hand-made instru-ments. As discussed above, this result further suggests that the
industrial guitars are overall less sensitive to the plucking
position and offers a narrower range of timbres with respectto the influence of body modes at tone onset. On the contrary,the hand-made guitars may offer more versatility in terms oftimbre at tone onset, but possibly also require a higher degreeof control and precision from the player to reach a given targettimbre. During the measurement session (and before conduct-ing any analysis on the recorded signals), one guitar playerwas asked to evaluate each instrument and to report on a listof characteristics. These interviews were recorded but werenot analyzed systematically and are therefore not reported in
this paper. However, and very interestingly, it should be indi-
cated that three guitars (A, B, and F) were judged spontane-ously as “very sensitive to the gesture” and offering a largepallet of timbre compared to the others. Although this obser-vation should be taken with care, since we only consideredthe evaluation from one player, one should note that these gui-tars show large ranges of the emergence of DI
e¼35%,FIG. 11. (Color online) Guitar categorization according to the body-sound
decay time T20and the range of body-sound emergence DIe. Industrial gui-
tars (K1 to K6) and hand-made guitars (A to H). Vertical lines indicate the
T20range of measurement (minimum and maximum values).
3938 J. Acoust. Soc. Am. 138 (6), December 2015 Fr/C19eour et al.
DIe¼37%, and DIe¼38%, respectively, which possibly sup-
port the relevance of this descriptor for objective categoriza-
tion. Unfortunately, guitar E could not be evaluated by theplayer for time reasons.
The body sound depends mostly on the very first modes
of the guitar. Therefore, we may reasonably assume that thevalues of T
20are highly correlated to the quality factors of
these first modes. Despite the result of Fig. 11which suggest
that T20is not necessarily categorizing with respect to the
two class of instruments (both industrial and hand-madeinstruments show low and high T
20values), the higher T20
values observed for some hand-made instruments (B, E, and
G) suggest that these guitars show a greater tendency of the
body modes to color the early stage of the tone than the rest
of the instruments. This feature is possibly linked to lowerdamping factors for the first modes of these instruments.
VII. CONCLUSIONS
In this paper, an analysis-synthesis method for the extrac-
tion of body-induced transient components in guitar pluckedsounds is described, tested on synthetic signals, and applied toa pool of classical guitars. Some indicators of body soundemergence are proposed and the guitars classified according
to these descriptors. This analysis, while studying the masking
effect of string partials, reveals the perceptual significance ofthe body-sound components on the early phase of the tone, aswell as the strong dependence of the body-sound emergenceto the conditions of excitations. The main result of this study
shows that the studied hand-made instruments are overall
more sensitive to the plucking position than industrial instru-ments with respect to the transient emergence of the body-sound components. This outcome further suggests that thesehand-made instruments, that show a large range of emergence,
are more likely to provide a large pallet of attack sonorities
and are therefore more sensitive to the player’s input gesturewith regards to this attribute. We can hypothesize that this iscertainly an expected feature for high-end instruments; aninstrument capable of transcribing the small nuances in the
control of advanced player.
Beyond the potential interest of these descriptors for
instrument categorization, these results invite to further
investigate how the emergence of body sounds relate to the
perceptual evaluation of the instruments. In addition, furtherwork should be conducted in order to understand how thesecharacteristics of body-sound emergence may be inferredfrom the analysis of the dynamic properties of the body,
such as the mechanical admittance at the coupling point
between the string and body.
ACKNOWLEDGMENTS
The authors would like to acknowledge the Institut
Technologique Europ /C19een des M /C19etiers de la Musique
(ITEMM) and the guitar store La Guitarreria in Paris forsupporting this study by making instruments available forthe experiments. The authors would also like to thank theinstrument makers Jean-Marie Fouilleuil, Dominique
Chevalier, and Frederic Pons for their collaboration and for
fruitful discussions.APPENDIX
For the masking threshold estimation, a spread function
SF(z) is at first calculated using the following expression:36
SFðzÞ¼ð 15:81/C0iÞþ7:5ðzþ0:474Þ
/C0ð17:5/C0iÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þðsþ0:474Þ2q
; (A1)
where zis the frequency in barks, i¼minð5/C1j Fð fÞj
/C1BWðfÞ;2:0Þ,fis the frequency in Hertz, BW( f)i st h ec r i t i c a l
bandwidth at f,a n d jFðfÞjis the magnitude of the Fourier
transform of the input signal at f.
The model for the masking threshold M(z) induced by a
single masker is written as follows:
MðzÞ¼PðzmÞþSFðz/C0zmÞ/C0kðtyÞzm/C0lðtyÞ; (A2)
where zmis the frequency of the masker in barks, zis the
masked frequency in barks, and P(zm) is the power of the
masker in dB. The tonality tyof the input signal is taken con-
stant and set to 0.8, and the functions k(t) and l(t) are given
by the following expressions:
kðtÞ¼0:3tþ0:5ð1/C0tÞ (A3)
and
lðtÞ¼34tþ20ð1/C0tÞ: (A4)
The global masking threshold S(z) arising from Nhhar-
monics is then obtained by summing the amplitude of themasking thresholds of the maskers considered
SðzÞ¼20 logX
Nh
i¼110MiðzÞ=20 !
: (A5)
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|
1.3611424.pdf | Synthesis and processing of pseudo noise signals by spin precession in Y3 Fe5O12
films
Oleg V. Kolokoltsev, César L. Ordóñez-Romero, and Naser Qureshi
Citation: Journal of Applied Physics 110, 024504 (2011); doi: 10.1063/1.3611424
View online: http://dx.doi.org/10.1063/1.3611424
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/110/2?ver=pdfcov
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155.33.120.167 On: Fri, 12 Dec 2014 05:22:20Synthesis and processing of pseudo noise signals by spin precession
in Y 3Fe5O12films
Oleg V. Kolokoltsev,1,a)Ce´sar L. Ordo ´n˜ez-Romero,2and Naser Qureshi1
1Centro de Ciencias Aplicadas y Desarrollo Tecnolo ´gico, Universidad Nacional Auto ´noma de Me ´xico,
Ciudad Universitaria, 04510, Mexico
2IFUNAM, Universidad Nacional Auto ´noma de Me ´xico, Ciudad Universitaria, 04510, Mexico
(Received 30 March 2011; accepted 17 June 2011; published online 21 July 2011)
A simple method for synthesis of phase shift keying (PSK) signals in the microwave frequency
range is presented. It is shown that the signal coding and processing can be efficiently realized by
spin excitations in thin ferrite films. PSK signals are constructed through control of magnetization
precession in a magnetic material by a pulsed magnetic field, and their compression is performedby a spin-wave based correlator, eliminating the need for semiconductor circuitry.
VC2011
American Institute of Physics . [doi: 10.1063/1.3611424 ]
I. INTRODUCTION
Electron spin excitations in the form of vibrational
modes or spin waves in thin-film ferromagnetic materials
have been a subject of intense study for decades due to theirunique properties, which have found wide application in
microwave signal processing.
1,2There is currently a growing
interest in using spin waves in the design of a nano-scale busand in logic elements
3,4proposed for a new class of digital
processors.5In the classical approach, magnetic excitations
can be treated as a uniform or nonuniform magnetizationprecession. In all of the planar spin-wave elements realized
to date, the precession was excited by microwave currents
flowing through microstrip line electrodes. Significantly lessattention was paid to shock excitation of precession by fast
magnetic pulses. Most works on this topic report experi-
ments associated with magnetic recording in such ferrome-tals as permalloy and cobalt. In the earliest works,
6–10as
well as in the most recent studies on fast coherent spin rever-
sal in nanomagnet pixels,11,12the main interest was in resid-
ual precessional oscillations, which take place when a pulsed
control field switches a spin ensemble from one equilibrium
direction to a new energy minimum direction. As a result,magnetization undergoes free damped precessional oscilla-
tions about a new equilibrium position, an effect known as
spin ringing. A number of schemes were proposed to sup-press precessional ringing, since it limits the recording speed
in magnetic storage systems. The first successful experimen-
tal result was reported by Wite et al.,
7where spin ringing in
a yttrium iron garnet (YIG) film was suppressed by using a
two-step control field. Crawford et al. then applied a similar
technique to a long permalloy stripe.13At the same time,
Bauer et al. showed theoretically14and experimentally15,16
that ringing can be suppressed by proper adjustment of the
control pulse duration or field magnitude. This technique hasallowed the authors of Ref. 17to reach a temporal limit inmagnetization reversal. More detailed pulsed-field experi-
ments have revealed that, in micro-scale and sub-micron
sized samples, the coherence of spin precession somewhatdeteriorates because of shape anisotropy effects, leading to
the appearance of multi-mode spin-wave excitations.
18–24
Nevertheless, even in this case, the coherent suppression of
ringing has been demonstrated.25
From a different point of view, spin ringing with a large
lifetime in a material with high magnetic Q-factor likeyttrium iron garnet (YIG) can be more of a benefit than a
problem. In this work, we report results on the possibility of
using this to synthesize a microwave pseudo noise (PN)sequence by taking advantage of shock excitation of spin
precession in thin-film YIG samples. The results are pre-
sented for traveling wave and quasi-uniform precession con-figurations. The traveling wave regime, to the best of our
knowledge, has been studied only in Ref. 26(YIG) and in
Refs. 19,23, and 30(permalloy). Here, we show that mag-
netization precession, driven by a train of fast magnetic
pulses, can effectively generate an arbitrary phase shift key-
ing (PSK) microwave signal, for example, in the form of aBarker code. We also show in this work that a spin-wave
generated PN signal can be compressed by a simple spin-
wave-based correlator. From a practical point of view, suchall-spin devices can be useful for protected wideband com-
munication channels, which require simple and practical sol-
utions for the synthesis and compression of PSK signals. Theexperimental data presented here are also complemented by
simulations.
II. THEORY
Dynamics of nonuniform magnetization ~M~r;tðÞ in fer-
rites can be modeled by using the Gilbert form of the Lan-
dau-Lifshitz equation (LLG).27To integrate LLG, we used a
time-domain micromagnetic approach based on an explicitnumerical scheme, the 4th order Runge-Kutta method.
28In
r-space, the discretization of LLG was done under the point-
dipole approximation.5,29This means that, at the center ofa)Author to whom correspondence should be addressed. Electronic mail:
oleg.kolokoltsev@ccadet.unam.mx.
0021-8979/2011/110(2)/024504/6/$30.00 VC2011 American Institute of Physics 110, 024504-1JOURNAL OF APPLIED PHYSICS 110, 024504 (2011)
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155.33.120.167 On: Fri, 12 Dec 2014 05:22:20each elementary volume DVi, the behavior of the magnetic
moment ~Miis described by LLG:
d~Mi~r;tðÞ
dt¼/C0c~Mið~r;tÞ/C2 ~Hi
Effð~r;tÞhi
þa
MS~Mið~r;tÞ/C2d~Mið~r;tÞ
dt"#
; (1)
with the effective magnetic field ~Hi
Eff¼~Hi
Extþ~Hi
SAþ~HCAþ
~HUAþ~Hi
Dip, where ~HDip¼/C0 r Udetermines the dipole cou-
pling between spins and is responsible for spin-wave forma-
tion. Here, the scalar potential Uis the solution to the
Poisson equation r2U¼/C04pr~M. The explicit expression
for the dipole field that interacts with local magnetization ~Mi
is:
Hi
Dip¼X
Allcells ði6¼jÞDijMj;
^D¼DVj
4pr3
ij3e1e1/C013 e1e2 3e1e3
3e2e1 3e2e2/C013 e2e3
3e3e1 3e3e2 3e3e3/C012
643
75
where r3
ijis the distance between centers of DViandDVjand
emare the components of unit vector directed from DVito-
wardDVj. In Eq. (1),c/2p¼2.8 MHz/Oe is the gyromag-
netic constant, ais the damping coefficient, and MSis the
saturation magnetization ( ~Mi/C12/C12/C12/C12/C17MS);~Hi
Effincludes the
external field ~HExt¼~H0þ~hið~r;tÞwith the uniform bias ~H0
and the nonuniform control pulsed-field ~hið~r;tÞ. For the
shape anisotropy ~Hi
SA¼/C0 ^N~Mi(^Nis the shape demagnetiza-
tion tensor), the magnetocrystalline anisotropy ~HCA, and the
uniaxial anisotropy ~HUAanalytical expressions are well
known and can be found, for example, in Ref. 27.
III. EXPERIMENTAL GEOMETRY
Figure 1shows two configurations used both in our
experiments and simulations of PSK signal generation. All
the results were obtained for an yttrium iron garnet (YIG)film with thickness d¼7.3lm grown on a gallium gadolin-
ium garnet (GGG) substrate by the standard liquid phase epi-
taxy technique (from one side of the sample, YIG film wasremoved by polishing). The ferromagnetic resonance half-
power linewidth ( DH) of the film was estimated to be about
0.3 Oe at 5 GHz, and M
s¼140 G.
Figure 1(a)presents a quasi-uniform precession configu-
ration. In this case, a narrow strip of YIG/GGG sample, 10
mm-long and 1 mm-wide, was placed on a 20 mm long, 0.5mm wide microstrip line excitation structure, terminated
with a 50 Xload. The film magnetization was saturated by a
static magnetic field H
0/C2150 Oe applied along the micro-
strip line. The shock excitation of YIG magnetization preces-
sion was realized by the control magnetic field pulses h(r,t)
applied perpendicularly to H0. This field, h(r,t), was induced
by DC current pulses i(t) injected in the microstrip line by a
HP8161A square waveform pulse generator with a peak volt-
age of 5 V, a 1 ns rise/fall time, and variable pulse-width ( s).
For convenience of spin signal detection, the sample was ori-
ented so that the polished GGG surface was in contact withthe microstrip-line electrode, as shown in Fig. 1. The output
microwave signals were detected by a 0.3 mm-diameter in-
ductive loop probe, shown in Fig. 1, and the signal was
amplified by a 10 dBm-gain amplifier and recorded by an os-
cilloscope. The inductive loop probe was attached to 3-axis
translation stage to displace the probe over the YIG film.This allowed us to measure the attenuation of spin-wave
excitations in the traveling wave configuration shown in Fig.
1(b). In the last case, we used a 1 mm-wide YIG sample of
length 3 cm.
It is important to note that shock excitation of the spin
precession takes place only when the rise time of h(t) is less
than half the period of the Larmor precession. This means
the spectral components of h(t) have to overlap with the Lar-
mor frequency f
0/C25(c/2p)Heff. In an ideal case of so-called
uniform precession, i.e., when in-plane sample dimensions
are infinite and h(t) is a spatially uniform Dirac- dor Heavi-
side-like function, the lifetime of the spin ringing is deter-mined only by the intrinsic loss mechanisms of the YIG/
GGG structure, usually expressed through the DH. Neverthe-
less, in practical samples, the ringing waveform can be con-trolled strongly by the sample dimensions as well as by the
spatial nonuniformity and waveform of h(t). In our
FIG. 1. (Color online) Experimental configurations for the proposed PSK
signal generators: a) uniform precession geometry and b) traveling wavegeometry.024504-2 Kolokoltsev, Ordo ´n˜ez-Romero, and Qureshi J. Appl. Phys. 110, 024504 (2011)
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155.33.120.167 On: Fri, 12 Dec 2014 05:22:20experiments on quasi-uniform precession geometry, the
ringing duration was varied from 100 ns to /C25200 ns at f0of
0.6–3 GHz (this frequency range was limited only by theavailable equipment).
IV. RESULTS AND DISCUSSIONS
A. Uniform precession
Figure 2and Fig. 3show the results of the simulations
based on Eq. (1)obtained for the uniform precession geome-
try (Fig. 1(a)). Figure 2shows a typical reaction of the spin
system to the pulsed control field with amplitude h¼30 Oe
and pulse width of s¼400 ns. The oscillations of magnetiza-
tion with a lifetime of approximately 200 ns were obtained
atH0¼50 Oe, with a damping coefficient a¼2:6/C210/C05
that corresponds to aof the experimental YIG sample. The
experimental response of the spin system in these conditions
is shown in Fig. 4(a). Note that, although the YIG magnet-
ization oscillates about different vectors ~H1¼~H0þ~hand
~H0after the leading and trailing edges of the control pulse
respectively (Fig. 2), the voltage induced in the loop probe in
both cases is identical (Fig. 4(a)). This was determined by
angular adjustment of the probe. The calculated set of curves
“A” in Fig. 3illustrate what happens when the signal pulse
width becomes significantly shorter than the ringing lifetime(s¼5–10 ns /C28200 ns) and satisfies s¼NT, where T ¼1/f
0
and N is the number of the precession cycles. As seen in the
figure, the above conditions, used in Ref. 16for NiFe, also
lead to effective suppression of ringing in YIG. The suppres-
sion takes place because of destructive interference between
the oscillations induced by the leading and trailing signalpulse edges when the oscillations are temporally overlapped.
The experimental confirmation of this is presented in Figs.
4(b)and4(c). The benefit in this for signal processing is evi-
dent. Microwave oscillations with a desired number of peri-
ods shown in Figs. 4(b) and4(c) can be considered as a
building block (segment) for the synthesis of encoded PSKwaveforms. Such segments can be attached with a required
phase shift between them. For example, to obtain the PSK
sequence 0, p,0 ,p,…, one needs to apply h(t) with an AC
square meander waveform.Here, we propose a somewhat modified solution for
ringing suppression. It is based on short control pulses with
s/C20T/4, as shown in the curves “B” in Fig. 3. The first pulse
excites and the second pulse stops the precession under the
condition that both pulses are equivalent with s
1:s2.I n
contrast to the solution shown in Fig. 3“A”, this case is char-
acterized by significantly less power consumption for h(t).
The case “C” in Fig. 3shows a waveform of h(t) that gener-
ates adjacent microwave segments with phase shift ( D/)o f
prad. This solution can be used to obtain ultra-wideband
encoded PSK waveforms with a small N, the so-called bi-
phase PSK signals (BPSK). Any other value of D/is also
possible.
FIG. 2. Simulation of magnetization dynamics excited by uniform pulsed
field in YIG/GGG.
FIG. 3. The quasi-uniform precession geometry. Possible forms of the con-
trol pulses for 1) suppression of spin ringing (the curves A and B) and 2)generation of PSK signal (curves C).
FIG. 4. Experimental signals obtained in the quasi-uniform precession ge-ometry. a) Magnetic oscillations with /C25200 ns lifetime formed by the lead-
ing and trailing edges of a control pulse and, b) and c) with overlapping
control signals, one can control the number of oscillations to form a coded
pulse.024504-3 Kolokoltsev, Ordo ´n˜ez-Romero, and Qureshi J. Appl. Phys. 110, 024504 (2011)
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155.33.120.167 On: Fri, 12 Dec 2014 05:22:20B. Traveling wave configuration
Simulations and experimental results for the traveling
wave configuration (Fig. 1(b)) are presented in Figs. 5–11.I n
this regime, the locally induced magnetization precession
propagates in the form of spin-wave harmonics, the superpo-sition of which form a short spin wave envelope. The propa-
gation distance of this envelope is mainly determined by DH
and by the spin-wave group velocity ( v
g). The configuration
shown in Fig. 1(b) corresponds to the excitation of magneto-
static surface waves (MSSW).27First, let us consider propa-
gation characteristics of the MSSW packet important both forgeneration and compression of PSK spin-wave signals. Fig-
ure5shows a calculated MSSW envelope at H
0¼50 Oe and
with the sample at a distance of y ¼3 mm from the exciting
antenna that induces the nonuniform field hðr;tÞ/FðtÞFðyÞ,
where F(t) is the step-like function and F(y) determines spa-
tial profile of hnear the antenna. The experimental wave-
packet obtained under the same conditions is presented in the
curve a) in Fig. 6, and its experimental spectrum is shown in
Fig. 7. The experimental data and calculations were com-
pared in the frequency domain. As seen in Fig. 7, the Fourier
transform of the calculated signal agrees quite well with the
experimental spectrum. It should be noted that our data arealso in a good agreement with a spectral profile obtained in
Ref. 26. In addition, we measured the spatial attenuation of
the wave-packet. For this, the signals were recorded at differ-ent distances from the excitation antenna by displacing the
probe along the sample with the help of a micromechanical
system. These results are plotted in Fig. 6and Fig. 8. Their
analysis provides us such important parameters as the tempo-
ral width, delay, and attenuation of the wave packets versus
the propagation distance needed for proper design of a spin-wave-based correlator described below.
C. Synthesis and compression of PSK spin-wave
signals
Although a PSK encoder based on uniform precession
configuration is capable of providing almost perfect phasecoded signals, its realization requires AC control pulses (Fig.
3“C”). With the pulse source available to us, however, we
could only use two pulses with fixed polarity and equal dura-
tion. For this reason, PSK signals were constructed on the
basis of spin-wave packets. The 20 ns microwave envelopesshown in Fig. 9(a)represent a 4-bit BPSK signal, induced in
the probe by spin wave packets formed by two electrical con-
trol pulses. The phase shift of pradians between the adjacent
oscillations was realized by simple adjustment of the pulse
width sand delay between the two control pulses. Figure
9(b)demonstrates the possibility of setting up any phase shift
between adjacent oscillations by varying of s.
A spread over time of the energy of any phase-coded
sequence can be compressed by using a correlator. Thiswidely used technology provides security protection and
enhancement of the signal-to-noise ratio in communication
channels. To compress the 4-bit signal generated at H
0¼50
Oe by the traveling wave encoder (Fig. 1(b)), we have used a
simple spin-wave based correlator, shown in Fig. 10.I ti s
based on a 4.2 cm-long, 1 mm-wide YIG sample that oper-ates as a multi-port delay-line, at H
0/C2550 Oe, with the me-
ander shape input and straight-line output shorted electrodes.
The 500 lm-wide electrodes of 50-Ohm impedance were
formed on a duroid substrate.
FIG. 5. The traveling wave configuration. The result of an LLG simulation
on a spin-wave packet formed by the superposition of MSSWs excited by a
0.5 mm-wide microstrip line.
FIG. 6. Experimental spin-wave packets recorded at different distances fromthe excitation electrode a) y ¼3.5 mm, b) 11.5 mm, c) 19 mm, and d) 25 mm.
FIG. 7. Experimental ( -^-) and calculated ( -O-) spectra of the signals
shown in Fig. 5. and Fig. 6(a).024504-4 Kolokoltsev, Ordo ´n˜ez-Romero, and Qureshi J. Appl. Phys. 110, 024504 (2011)
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155.33.120.167 On: Fri, 12 Dec 2014 05:22:20The operation principle of the device is the following:
the signal bits of an input microwave sequence (Fig. 11(a) )
appear simultaneously in all four active sections of the mean-
der (the phase delay along the electrode is negligibly small
at the frequency used). Each of the input bits excites fourMSSW packets in the YIG sample (Fig. 11(b) ). We will call
these packets a “set”. Four sets, therefore, propagate in the
YIG film simultaneously. The set induced by the ( jþ1)-th
input bit is delayed with respect to a set induced by jth input
bit. Finally, the output electrode detects a microwave signal
that is the result of superposition of all sets (Fig. 11(c) ). The
signal to be compressed in this case, shown in Fig. 11(a) ,
was a simple {0, p,0 ,p} sequence with a delay between sig-
nal bits of s/C2523 ns. Note that, in the active electrode sec-
tions of the meander, a phase of microwave current also
changes as {0, p,0 ,p}. It is easy to show that the output sig-
nal with optimal waveform (i.e., that with highest amplitude)can be obtained if the meander is a strictly periodical struc-
ture. The period Kof the meander was estimated with the
help of the data in Fig. 8,K¼s
distance
delay¼svg¼10.1 mm. In
FIG. 8. Experimental (- h-) spin-wave packet delay vs the distance (y) from
the input electrode, and experimental (- D-) and theoretical (LLG: - O-, Eq.
(2):-^-) delay vs H0at y¼10 mm.
FIG. 9. a) Phase modulation of the precession achieved by two control
pulses, resulting in a 4-bit BPSK sequence. The figure shows both unshifted
andp-shifted signals in the 2nd and 4th bits; b) The 2nd bit was shifted by
p/4,p/2, 3p/4, and prad., with respect to the 1st bit, by varying s.
FIG. 10. (Color online) A schematic view of the correlator a) and its cross-
ection b): angle b/C253 deg.
FIG. 11. a) Experimental input PSK signal formed by the spin-wave coder.
b) Schematic representation of spin wave packets superposition (the experi-
mental sets recorded by an oscilloscope). c) Output signal of the correlator.024504-5 Kolokoltsev, Ordo ´n˜ez-Romero, and Qureshi J. Appl. Phys. 110, 024504 (2011)
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155.33.120.167 On: Fri, 12 Dec 2014 05:22:20this case, the convolution of the delay line transfer function
and the input signal gives the output signal waveform: { A,–
2A,3A,– 4A,3A,– 2A,A}, shown in Fig. 11(c) , where Ais
the amplitude of the signal bit. The ideal result, { A,0 ,– A,
4A,–A,0 ,A}, can be obtained when using the input signal in
the form of Barker code {0, 0, p, 0} and the meander electro-
des code {0, p, 0, 0}.
It should be noted that, to account for attenuation and to
equalize the amplitudes of MSSWs propagating from differ-ent sections of the meander electrode, we have assumed a
linear increase of the distance between the sections and the
film (Fig. 10(b) ). Also, the important point is the nonrecipro-
cal character of MSSW.
30The amplitude difference between
the waves propagating in the –Y and þY direction was
measured to be /C2521 dB. This provides a minor interference
between the counter-propagating waves. In principle, the de-
vice can operate in a wide range of microwave frequencies.
To evaluate the function K(H0), we measured the delay of a
MSSW packet versus H0in the configuration of Fig. 1(b)
using a “Picosecond Lab” monopulse generator with rise
time of 250 ps and falltime of 3 ns. The experimental andcalculated Delay (H
0), shown in Fig. 8, were obtained at the
distance y¼1 cm from the input electrode. The theoretical
curves in the figure were calculated in two ways: a) in thetime domain from Eq. (1)and b) in the frequency domain by
using the experimental spectrum S(f) shown in Fig. 7. In the
latter case, the wave-packet is formed by the superpositionof MSSW
S:
Aðt;H0Þ¼ðkmax
kminSð~fÞexp 2 pfðk;H0Þt/C0ky ½/C138 dk; (2)
with wavenumbers from kmintokmax, which can be
excited by the electrode.26In Eq. (2),fðk;H0Þ¼cjj
2p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
H0ðH0þ4pMSÞþð 4pMSÞ21/C0e/C02kd
4/C0/C1q
is the well-known
dispersion relation for MSSW S. As seen in Fig. 8, the delay
of the MSSW packet at H0¼500 Oe ( f0/C253 GHz) is of 65
ns per centimeter of propagation length, i.e., the packet prop-agates /C253 times slower compared to the case where H
0¼50
Oe. This means that Kdecreases to /C253.3 mm as the fre-
quency increases to 3 GHz, and the longest 13-bit Barkercode can be compressed by using the same 4.2 cm-long YIG
sample.
V. CONCLUSIONS
In conclusion, a simple solution for PSK encoding and
decoding in the microwave frequency range is presented.
The all spin-wave technology described does not require
semiconductor switching circuits to modulate the carrier fre-quency. The necessary phase modulation is effectively per-
formed through control of magnetization precession in amagnetic material by a pulsed magnetic field. To form a
PSK output, the only electronic modulation necessary is an
input data bit stream.
ACKNOWLEDGMENTS
This work was supported by grants from ICyTDF
PIUTE10-71, PAPIIT-UNAM IN114909, and CONACyT
81532, Mexico.
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1.373695.pdf | Nanocrystalline Si thin films with arrayed void-column network deposited by high
density plasma
A. Kaan Kalkan, Sanghoon Bae, Handong Li, Daniel J. Hayes, and Stephen J. Fonash
Citation: Journal of Applied Physics 88, 555 (2000); doi: 10.1063/1.373695
View online: http://dx.doi.org/10.1063/1.373695
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134.71.135.191 On: Tue, 25 Nov 2014 10:36:21Nanocrystalline Si thin films with arrayed void-column network deposited
by high density plasma
A. Kaan Kalkan,a)Sanghoon Bae, Handong Li, Daniel J. Hayes, and Stephen J. Fonash
Electronic Materials and Processing Research Laboratory, The Pennsylvania State University,
University Park, Pennsylvania 16802
~Received 4 January 2000; accepted for publication 3 April 2000 !
High porosity nanocrystalline Si thin films have been deposited using a high density plasma
approach at temperatures as low as 100°C. These films exhibit the same unique properties, such asvisible luminescence and gas sensitivity, that are seen in electrochemically etched Si ~i.e., porous
Si!. The nanostructure consists of an array of rodlike columns normal to the substrate surface
situated in a void matrix. We have demonstrated that this structure is fully controllable and havevaried the porosity up to ;90% ~as derived from optical reflectance !by varying the deposition
conditions. In particular, the impact of plasma power has been found to reduce porosity byincreasingthenucleidensityandthereforethearealdensityofcolumns.Humiditysensorshavebeendemonstrated based on the enhanced conductivity of our films ~up to 6 orders of magnitude !in
response to increase in relative humidity. Depending on the porosity, the conductivity-relativehumidity behavior of our films shows variations which can be correlated with the nanostructure.Also, these variations indicate that the dominant charge transport is limited by the dissociation ofwater into its ions at the column surfaces. © 2000 American Institute of Physics.
@S0021-8979 ~00!07813-0 #
INTRODUCTION
Intense research activity in porous semiconductors has
been stimulated over the last decade by the discovery ofroom temperature visible light emission from electrochemi-cally prepared porous Si in 1990 by Canham.
1Soon after
Canham’s discovery, further intriguing properties and appli-cations of porous semiconductor materials were also real-ized. These include gas and vapor sensitivity,
2biocompati-
bility,3desorption mass spectroscopy,4and ease of micro-
machining.5
In this article, we describe a new type of high void den-
sity crystalline Si film produced using a unique low tempera-ture~,200°C !, high density plasma ~HDP!approach. Rather
than using wet-chemistry, electrochemical-etching process-ing, this high void density material is produced by plasmadeposition. Unlike porous materials produced by electro-
chemical etching with voids defined in the bulk, this thin filmmaterial is composed of nanometer-sized oriented columnsdefined normal to the substrate in a void matrix resulting inan interconnected, continuous void volume. The films can beobtained on a variety of substrates such as glass, metal foils,and plastics as well as on the more conventional substrates.Here, we report on tailoring the ~areal and volumetric !void
density and nanostructure of these void-column Si films us-ing the variation of the plasma deposition parameters. Wealso demonstrate that these arrayed void-column crystallineSi thin films exhibit potentially very useful properties similarto those seen for the electrochemically etched porous Si,such as visible luminescence and sensitivity to water vapor.EXPERIMENTAL PROCEDURE
In our HDP approach for producing high porosity crys-
talline films, we employed an electron cyclotron resonanceplasma-enhanced chemical vapor deposition ~ECR-PECVD !
tool~PlasmaTherm SLR-770 !with hydrogen diluted silane
(H
2:SiH4) as the precursor gas at substrate deposition tem-
peratures less than 200°C. Our deposition apparatus consistsof two coaxialy connected plasma chambers; an ECR cham-ber and a deposition chamber which are 14.9 and 35.6 cm indiameter and 13.3 and 29.2 cm in height, respectively. Themicrowave power, with an excitation frequency of 2.45 GHz,is introduced into the ECR plasma chamber through a rect-angular waveguide and a fused quartz window. The reflectedpower is minimized by using a three stub tuner. A dc elec-tromagnet is located coaxialy around the ECR chamber to setup an axial magnetic field of 875 G ~in the direction towards
the substrate !within the ECR chamber to achieve the ECR
condition. H
2is introduced into the ECR chamber through a
gas distribution ring close to the quartz window. SiH 4is
injected into the deposition chamber through a gas inlet ringabout 1.3 cm above the location of the substrate ~27.9 cm
below the ECR chamber !. A secondary dc electromagnet is
placed just below the substrate level to establish an axialmagnetic field of 100 G ~in the direction towards the ECR
chamber !. A high throughput turbomolecular pump has been
used to achieve a base vacuum pressure of 2 310
27Torr.
Table I lists the range of the plasma deposition parametersexplored. The films in this article were deposited on Corning1737 glass substrates coated with a 800 Å thick silicon ni-tride barrier layer ~unless otherwise stated !. The film thick-
ness was measured with a Tencor-500 profilometer. The
nanostructure was studied with cross sectional transmission
a!Author to whom correspondence should be addressed; electronic mail:
akk105@psu.eduJOURNAL OF APPLIED PHYSICS VOLUME 88, NUMBER 1 1 JULY 2000
555 0021-8979/2000/88(1)/555/7/$17.00 © 2000 American Institute of Physics
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134.71.135.191 On: Tue, 25 Nov 2014 10:36:21electron microscopy ~TEM !~Hitachi HF-2000 cold field
emission !and atomic force microscopy ~AFM !~Digital In-
struments Multimode AFM system with NanoScope IIIaController !. The AFM was used in the TappingMode™ with
an Olympus cantilever and etched silicon tetrahedral probewith a tip angle of ;35° and radius of 5–10 nm. An x-ray
diffractometer ~Phillips X’pert !with glancing incidence op-
tics was employed to investigate the crystallinity. Opticalreflectance of the films was obtained with a Perkin–ElmerLambda 9 spectrometer to assess the level of porosity. Pho-toluminescence was measured with an ISA U-1000 Ramanspectrometer using a 488 nm Ar laser excitation of 5 mW.
To monitor the conductivity variation in response to
changing relative humidity ~RH!, Al or Au contacts were
defined on the film surface in a parallel stripe ~19 mm in
length !configuration ~with 0.4 mm spacing !. During this
measurement the samples were situated in a glass tube on aTeflon stage with two Au probes in contact with the stripeelectrodes. The RH inside the glass tube was graduallyramped from ;2% to ;97% at 20°C by flowing N
2as the
humidity carrier through a heated bubbler and then into thetube. A capacitive type sensor was used as the reference tomeasure RH ~TI-A from TOPLAS CO., Japan !with an HP-
4284ALCRmeter, while the enhancement in current for a
fixed voltage was traced for our sensor structure with anHP-4140B pA meter/dc voltage source.
RESULTS AND DISCUSSION
The nanostructure of our high porosity Si films, depos-
ited by HDP, consists of an array of rodlike columns normalto the substrate surface and situated in a continuous voidmatrix. Figure 1 shows the TEM view of the nanostructure ofa film deposited on glass at a temperature of 120°C, processpressure of 7.8 mTorr, and microwave power of 500 W.Here, a typical column diameter is 100 Å and a typical col-umn separation is 30 Å. The undulations seen in the clear-ance between the columns ~i.e., changes in the intercolumn
void size !are considered to be due to statistical variations inthe size of the crystallites building up the columns. Before
the columnar growth initiates, a void free amorphous-likelayer is seen to form at the substrate interface. The thicknessof this transition region is found to be up to ;20 nm. The
x-ray diffraction ~XRD !patterns in Fig. 2 identify films, de-
posited in this case for 35 min at 100°C and various powersof 400, 490, and 600 W, to be crystalline. Taking into ac-count the film thickness as given in the inset of Fig. 2, areduction in XRD peak intensity per unit thickness is ob-served with decreasing power, which becomes more pro-nounced toward 400 W. On the other hand, the peak inten-sity drop is accompanied by only a slight increase of the fullwidth at half maximum of the ~111!peak from 0.76° to 0.78°
as power decreases from 600 to 400 W. Using the Scherrerformula,
6an average crystallite size of 110 Å is calculated
for these peak widths after Warren’s correction6for instru-
mental broadening ~i.e., 0.27° !. Since the XRD peak inten-
sity decrease is not due to any peak broadening, it mustresult from a reduction in crystalline volume fraction. Morespecifically, if negligible variations in crystallite size are as-sumed, as implied by the Scherrer formula, we infer a reduc-tion in the number of crystallites per unit volume with de-creasing power. Equivalently, these results mean an increasein the fraction of void or amorphous volume with decreasingpower. Using optical reflectance measurements we will es-tablish that the increase is in void fraction.
When varying power, one also has to consider the pres-
sure in our process. We have found that, for a given processpressure, the H
2:SiH4ECR plasma is only stable if the inci-
dent microwave power is above a certain threshold level. Atthis threshold level or just above all the incident power isabsorbed by the plasma. If the power level is further in-creased above this threshold level, the absorbed powerchanges only a little and the excess power is reflected. There-fore, the net absorbed microwave power and process pres-sure are strongly coupled as shown in Fig. 3. Here, the flowrates are as given in Table I. Accordingly, all referencesrelated to power in this article are the net absorbed micro-wave power.
FIG. 1. Cross sectional TEM micrograph of a void-columnar network Si
thin film on glass.
FIG. 2. XRD spectra of the void-columnar films deposited at 100°C andvarious powers of 600, 490, and 400 W which correspond to process pres-sures of 6, 8, and 10 mTorr, respectively. As will be shown in Fig. 3, thepower and process pressure cannot be varied independently. The inset showsthe corresponding film thickness variation for a deposition time of 35 min.TABLE I. Growth conditions of the void-columnar Si films explored.
Substrate temperature 100–200°C
Microwave power 340–640 WProcess pressure 5–12 mTorrSiH
4flow rate 2 sccm
H2flow rate 40 sccm556 J. Appl. Phys., Vol. 88, No. 1, 1 July 2000 Kalkanet al.
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134.71.135.191 On: Tue, 25 Nov 2014 10:36:21Figure 4 shows the reflectance spectra of our films de-
posited at 100°C. As just noted, various process pressureshere correspond to certain power levels as given in Fig. 3.The peak in these spectra at ;275 nm characterizes our films
as crystalline.
7The spectra show interference above ;400
nm due to the reflected light from the substrate which isdamped out toward lower wavelengths with increasing opti-cal absorption. Based upon dielectric mixing theory one mayinfer, using the interference-free reflectance regime for ourhigh porosity films seen in Fig. 4, that porosity increasesmonotonically with the process pressure ~hence with de-
creasing power !when other deposition parameters remain
constant. The interference free reflectance Rat normal inci-
dence should equal
R5
~n21!21k2
~n11!21k2,
wherenandkare the refractive index and the extinction
coefficient, respectively. If kis sufficiently low compared to
n, then one may relate the reflectance to nonly. We use these
conditions, and assume the effective permittivity of the void~space !column ~Si!mixture phase can be approximated by
linear dielectric mixing. Consequently, for normal incidence,the porosity Pmay be evaluated as
P’eSi2~~11AR!/~12AR!!2
eSi21,
where eSiis the permittivity of the crystalline Si. The inset
shows the calculated porosities for the films of Fig. 4 basedon the reflectance at 380 nm. At 380 nm k
2/(n21)2for
crystalline Si is found to be ;0.025,8and hence the approxi-
mation above is considered to be valid. To make our estimate
forPmore accurate we have adopted AeSi’nSi54.44 at 380
nm in our calculations, which corresponds to a void freehydrogenated nanocrystalline film deposited by ECR-PECVD,
9instead of single crystalline Si.
According to the structural zone model ~SZM!, our films
would be expected to display what is termed zone 1 mor-phology consisting of tapered columns ~typically tens of nm
in diameter !separated by voids ~typically a few nm
across !.
10–12Zone 1 structure is a consequence of the low
mobility and therefore low diffusion length of the depositionspecies compared to the average distance between the phys-isorption sites. This zone 1 situation is expected due to ourlow substrate temperature in addition to the low kinetic en-ergy of the impinging ions on the surface ~less than 50 eV
for an ECR plasma !.
13,14At these conditions two factors are
responsible for the generation of voids in the SZM model:statistical roughening and self-shadowing. However, thesemechanisms lead to randomly shaped voids and tapered col-umns, resulting in a column width which increases with filmthickness.
11,12Within the range of film thicknesses we ex-
plored in this work which is from 0.2 to 1 mm, we observed
only a slight increase in the interference free reflectance ofour films with increasing thickness. Therefore, in our casethe void density and column width persist to be uniform ~i.e.,
untapered columns !.
Void networks and their evolution with film thickness in
amorphous-Si films deposited by rf sputtering have been ex-tensively studied by Messier and co-workers.
15The Messier
et al.work dealt with amorphous silicon and did not show
how to achieve the degree of porosity our technique canyield unless postdeposition etching is carried out.
16In HDP,
with the very low processing temperature approach we use,we believe H radical etching does not only account for thecrystallinity of our films but also plays an important role inthe formation of our unique nanostructure. Elimination of theweak or strained bonds by H radical etching to promote theatomic ordering to an equilibrium or crystalline structure isthe dominant model for explaining microcrystalline Si(
mc-Si:H) film growth at low temperatures.17,18With a con-
ventional rf PECVD system ~13.56 MHz !mc-Si:H films can
be deposited from H diluted SiH 4at dilution ratios equal to
or in excess of 20 at substrate temperatures higher than250°C. On the other hand, here we demonstrate crystallinefilms grown at temperatures as low as 100°C. Furthermore,even thoug h a H dilution ratio of 20 is employed in this
study, our previous work with ECR-PECVD films deposited
FIG. 3. The relationship between the absorbed microwave power and the
process pressure.
FIG. 4. Control of the porosity of our void-columnar Si films deposited at100°C as monitored from their reflectance spectra. Here the reflectance iswith respect to a barium sulphate reflector. For the calculation of porosity,which is shown in the inset, the reflectance is corrected for the absolutevalues.557 J. Appl. Phys., Vol. 88, No. 1, 1 July 2000 Kalkan
et al.
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134.71.135.191 On: Tue, 25 Nov 2014 10:36:21at 120°C has shown no significant decrease in crystallinity
~as measured by XRD !when dilution ratio was reduced to as
low as 3.19Hence, with an ECR plasma one does not need a
high hydrogen dilution to achieve effective H radical etchingand crystallinity. Kitagawa et al.have shown with optical
emission spectroscopy that H radical concentration in anECR plasma is remarkably higher compared to that in a con-ventional rf plasma.
20This is because in an ECR plasma
dissociation of gas molecules into their fragments can yieldconcentrations typically two orders of magnitude higher thanfound in a conventional rf plasma.
21Therefore, our deposi-
tion process involves a more severe H radical etching com-pared to
mc-Si:H film growth by conventional PECVD. As a
result, crystallinity can be achieved at lower temperatures. Inaddition, the strained regions forming inbetween the crystal-lites cannot withstand this severe H radical etching, leadingto formation of a void volume. However, this etching of thestrained matrix is not uniform as crystallites shelter the re-gions below themselves from being etched, whereas thestrained regions which form laterally inbetween crystallitesare subject to being etched off. Consequently, lateral coales-cence of the crystallites does not occur, resulting in forma-tion of vertically protruding columns in a void matrix. An-other distinct feature is the grain upon grain composition ofthe columns with an average grain size of ;100 Å. This may
be related to a very high nucleation rate in our process. Thenanostructure shown in Fig. 1 approximately corresponds toa growth nuclei density of ;6310
11cm22, which is much
higher than encountered in mc-Si:H deposited by conven-
tional rf PECVD; i.e., ;331010cm22.22In summary, we
attribute our unique nanostructure to the following factors:~1!low surface mobility caused by low temperature and re-
duced ion bombardment, ~2!severe H radical etching, and
~3!high nucleation rate.
We also studied the impact of the substrate temperature
on porosity and observed a systematic drop in porosity withincreasing temperature as expected from SZM. On the otherhand, the ability of increased power to decrease porosity,which we observe, is not obvious from any plasma deposi-tion theory. From the nanostructure of our films it followsthat nuclei density determines the spacing of the columns,
whereas the column width is considered to be determined bythe grain size due to grain-upon-grain composition of thecolumns. Therefore, in our material porosity is determinedby the nuclei density and the grain size, which can be con-trolled by the deposition conditions.
In Fig. 5, AFM surface images of the three films of Fig.
2, deposited at 100 °C and various pressures ~6, 8, and 10
mTorr !, are shown. At this resolution the nanometer-sized
columns observed in the TEM micrograph of Fig. 1 are seento be arranged in clusters appearing as thicker columns. It isalso seen that the separation between these clusters of col-umns is increasing with decreasing power ~i.e., increasing
pressure !. As we have deduced from the XRD data, at a
given temperature, power has no impact on crystallite sizebut controls the density of crystallites ~i.e., nuclei density !.
Thus, power determines the average spacing between thecrystallites or nanocolumns and hence the porosity. The in-creasing porosity of our films with decreasing power is,therefore, caused by increasing average spacing between the
crystallites which is equivalent to increasing intercolumnspacing, increasing intercluster spacing, or both.
As is the case for conventional electrochemically etched
porous Si films, our arrayed void-column films exhibit pho-toluminescence. The photoluminescence spectra of the threefilms deposited at 100°C and 6, 8, and 10 mTorr are given inFig. 6. Here, the photoluminescence band peaks at ;1.8 eV
with a full width at half maximum of ;0.3 eV, similar to
that observed in porous Si.
23
Our films also display the gas and vapor sensitivity re-
FIG. 5. AFM surface images of our void-columnar films deposited at
100°C and at: ~a!6,~b!8, and ~c!10 mTorr.558 J. Appl. Phys., Vol. 88, No. 1, 1 July 2000 Kalkanet al.
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134.71.135.191 On: Tue, 25 Nov 2014 10:36:21ported for electrochemically etched porous Si.2As a prelimi-
nary investigation, we have studied the sensitivity of ourfilms to water vapor. Figure 7 illustrates the typical behaviorof our films in response to variations in RH. A weak re-sponse is found up to a threshold humidity level above whicha steep ~generally exponential !increase is observed. We at-
tribute this very strong variation in conductivity to the cap-illary condensation of water inside the voids. Similar behav-ior is observed for ceramics and the charge transport hasbeen considered to be via protons H
1, hydronium ions
H3O1, or hydroxyl ions OH2.24,25Once the Si surface inter-
acts with humidity the formation of chemically bondedhydroxyl ions induces a high charge density and a strongelectrostatic field. Hence, additional water molecules phys-isorbed on this chemisorbed layer can easily dissociate intoions (2H
2O$H3O11OH2) because of this high electro-
static field.24,25The proton transport occurs when a H 3O1
releases a proton to a neighboring H 2O transforming it into
H3O1, and so forth ~Grotthuss chain reaction !.24,25The steep
rise in conductivity is possibly associated with growth ofmolecular layers of water with increasing RH.
26Finally, a
saturation follows the steep increase in conductivity with in-creasing RH. At this point the complete filling of the voidvolume with condensed water is considered to occur. There-fore, for a given surface tension or adsorption coefficient, thesaturation is expected to occur at higher RH for larger void
or pore size, as also addressed by Kelvin’s relation for cap-illary condensation.
24,26
We have successfully confronted a very serious problem
in the lack of repeatability of the behavior of our films intheir response to humidity. This lack of repeatability canmanifest itself in changes as large as 2 orders of magnitudein sequential measurements. At first we attributed this prob-lem to changes in surface chemistry similar to that occurringin conventional porous Si. This problem with etch-producedporous Si has been the subject of a serious challenge over thelast decade since it accounts for degradation of the visibleluminescence.
27Accordingly, we followed the porous Si sur-
face passivation procedure reported by Buriak and Allen.28
Their passivation involves the Lewis acid ~1 M solution of
EtAlCl 2in hexane !mediated functionalization of porous Si
surface with hydrosilylation of 3-butyn-l-ol. As a result, wehave achieved a remarkable gain in repeatability. We notethat, even though the Lewis acid is considered only to be amediator in this reaction,
28our control samples treated with
EtAlCl 2solution only ~the same concentration as above !
yielded the same success.
Figure 8 shows the stabilized humidity response of the
three films deposited at 100°C and various pressures ~6, 8,
and 10 mTorr !after EtAlCl 2treatment of 30 min. Among
these films, the one deposited at 10 mTorr ~the highest po-
rosity film !shows the most regular behavior and is the most
promising for humidity sensor applications. Figure 9 showsseveral measurements taken from this high porosity film insequence. Obviously, the repeatability is significant.
Despite extensive surface analysis utilizing infrared ab-
sorption spectroscopy ~on films deposited on Si wafers !no
passivation species attached to the Si surface were detectedafter straight EtAlCl
2treatments. An interesting observation
about the samples untreated with EtAlCl 2, and thus showing
a lack of repeatability, was the accumulation of condensedwater under the metal contacts which was visible from theback of the sample through the glass substrate. We deter-mined that this water, which accumulates under the contactsduring a measurement as RH is ramped up to 97%, could
FIG. 6. Photoluminescence spectra of the void-columnar Si films deposited
at 100°C and various pressures.
FIG. 7. Conductivity enhancement of a void-columnar Si film in response toincreasing RH. The film was deposited at 100°C and a pressure of 8 mTorr.The measurement was carried after a deposition of 100 Å thick Pd layer anda subsequent annealing at 600°C fo r 1 h for improved long term stability.
The applied voltage is 10 V.
FIG. 8. Conductivity behavior of void-columnar Si films of different porosi-ties in response to varying RH. The films were deposited at 100°C and threedifferent pressures of: 6, 8, and 10 mTorr. The applied voltage is 50 V. The
measurements were taken after EtAlCl
2treatment. The behavior is repeat-
able.559 J. Appl. Phys., Vol. 88, No. 1, 1 July 2000 Kalkanet al.
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134.71.135.191 On: Tue, 25 Nov 2014 10:36:21remain trapped under the contacts even after N 2purging of
hours. However, no water was visible under the contacts ofEtAlCl
2treated samples. The effect of EtAlCl 2on Al con-
tacts was found to be inhomogeneous etching, leading toformation of micropores and microcracks which render thecontacts permeable to water vapor so that the condensed wa-ter under the contacts could evolve out. With untreatedsamples the porous volume under the contacts is not exposedto the ambient directly and communicates with the ambientvery slowly. So when RH is varied during a measurement,these regions are never in thermodynamic equilibrium withthe ambient nor with the channel region between the con-tacts. Consequently, these regions deplete or feed the chan-nel region with water rendering variations in the measuredcurrent through the channel.
To test the above conclusion, we prepared high porosity
samples with ultrathin Au contacts ~100 Å thick !which
cover the Si surface only partially so as to obtain permeablecontacts to water. The result was repeatable sample responseto RH.
Maresˇet al.
26have reported a conductivity enhancement
~of 3 orders of magnitude !for electrochemically prepared
porous Si upon variation of RH from 0% to 100% yieldingthe similar S-shaped curve as in Fig. 7. However, their mea-surements involved a high level of instability and requireddays for stable readings. This is because of their use of elec-trochemically etched porous Si with its need for a conductiveSi wafer underlying the electrochemically prepared porous Silayer. In this case the electric current can only be transportedacross the thickness of the porous Si layer. Hence, a sand-wich monitoring structure with the conventional porous sili-con located between two electrodes is required for ~conven-
tional !electrochemically etched, porous Si. This is a major
disadvantage as the water vapor has to pass through the topelectrode ~contact !before reaching into or evolving out from
the porous Si layer. As a result, the sensitivity as well as theresponse time is strongly degraded.
On the other hand, since our porous films are on an
insulator, the electrical properties can easily be monitoredlaterally with a lateral contact configuration. This structureallows direct interaction between the ambient and the moni-tored film area inbetween the contacts. As a result, our filmsexhibit response times within seconds when RH is changedsuddenly from 0% to 100% or vice versa. Furthermore, in
our morphology the voids are interconnected and thereforethe columns are isolated even for lower porosities, resultingin a continuous void network aligned normal to the film/ambient interface throughout the film. This facilitates a uni-form and rapid molecular diffusion in and out of our films.Electrochemically prepared porous Si films lack this featureas the voids are not necessarily interconnected.
In Fig. 8 the shift of the RH value producing a saturated
response is indicative of different void sizes for the threefilms. According to the discussion of capillary condensationabove, this shift in the saturation response can be attributedto an increasing void size ~increasing column separation !
with increasing process pressure ~decreasing power !. This is
in full agreement with our conclusions from the AFM andXRD data. We note that, because of its highest porosity andthickness, the 10 mTorr sample might be expected to yieldthe highest current due to the void volume that can be filledwith condensed water. However, what actually occurs is justthe opposite in that the saturation current decreases with thevoid volume ~which is proportional to porosity times film
thickness !as seen from Fig. 8 ~from 6 to 10 mTorr !. This is
probably because, for our films, the dominant charge trans-port does not take place in the bulk liquid, but at the Sisurface where water most effectively dissociates due to thesurface charge. The ionization fraction at the surface is esti-mated to be only 1%, but this is 6 orders of magnitude largerthan that in liquid water.
25Accordingly, the 6 mTorr sample,
which accommodates the highest density of columns, showsthe highest saturation current due to its highest inner surfacearea. On the other hand, the 10 mTorr film possesses thelowest inner surface area, and consequently the lowest valueof the saturation.
CONCLUSIONS
We have developed a unique arrayed void-column thin
film crystalline Si by using HDP-based deposition fromH
2:SiH4at temperatures as low as 100°C. Morphology of
our films consists of an array of nanometer-sized rodlike col-umns normal to the substrate surface and situated in a con-tinuous void matrix. This nanostructure is considered to be aconsequence of low surface mobility, severe hydrogen radi-cal etching, and high nucleation rate. The porosity is control-lable with substrate temperature and microwave power. Theimpact of increasing power is increasing growth–nuclei den-sity leading to decreasing porosity. The power and processpressure are coupled and cannot be varied independently.The photoluminescence of these arrayed void-column filmsis similar to that of electrochemically etched porous Si. Inaddition, up to 6 orders of magnitude enhancement in con-ductivity of our films was found in response to increasingRH, which induces capillary condensation. Within this rangethe response time is observed to be in seconds due to ourlateral contact configuration, which is a major advantage forsensor structures offered by our films compared to electro-chemically prepared materials. This advantage arises out ofthe fact that electrochemically etched materials must sit on aconducting base needed for the electrochemical processing.
FIG. 9. Several current vs RH measurements indicative of repeatability
taken from the void-columnar film deposited at 100°C and 10 mTorr after
EtAlCl2treatment. The applied voltage is 50 V.560 J. Appl. Phys., Vol. 88, No. 1, 1 July 2000 Kalkanet al.
[This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:
134.71.135.191 On: Tue, 25 Nov 2014 10:36:21On the other hand, our arrayed void-column network films
can be easily deposited on insulators such as plastics orglass. The conductivity behavior of our porous films in re-sponse to varying RH also yields valuable structural infor-mation, which is in agreement with our AFM, XRD, andoptical reflectance results.
ACKNOWLEDGMENTS
The support of this work through Defense Advanced Re-
search Projects Agency under Contract No. F33615-98-1-5164 is gratefully acknowledged. The authors would like tothank Juergen Sindel for his valuable support in AFM. TheyalsoappreciatethetechnicalassistanceofWilliamDrawlandShang-Cong Cheng in Raman spectroscopy and TEM, re-spectively.
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1.5030342.pdf | 180°-phase shift of magnetoelastic waves observed by phase-resolved spin-wave
tomography
Yusuke Hashimoto , Tom H. Johansen , and Eiji Saitoh
Citation: Appl. Phys. Lett. 112, 232403 (2018); doi: 10.1063/1.5030342
View online: https://doi.org/10.1063/1.5030342
View Table of Contents: http://aip.scitation.org/toc/apl/112/23
Published by the American Institute of Physics180/C14-phase shift of magnetoelastic waves observed by phase-resolved
spin-wave tomography
Yusuke Hashimoto,1Tom H. Johansen,2,3and Eiji Saitoh1,4,5
1WPI-Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
2Department of Physics, University of Oslo, 0316 Oslo, Norway
3Institute for Superconducting and Electronic Materials, University of Wollongong, Northfields Avenue,
Wollongong, NSW 2522, Australia
4Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan
(Received 20 March 2018; accepted 18 May 2018; published online 5 June 2018)
We have investigated optically excited magnetoelastic waves by phase-resolved spin-wave tomog-
raphy (PSWaT). PSWaT reconstructs the dispersion relation of spin waves together with theirphase information by using time-resolved magneto-optical imaging for spin-wave propagation fol-
lowed by an analysis based on the convolution theorem and a complex Fourier transform. In
PSWaT spectra for a Bi-doped garnet film, we found a 180
/C14-phase shift of magnetoelastic waves at
around the crossing of the dispersion relations of spin and elastic waves. The result is explained by
a coupling between spin waves and elastic waves through the magnetoelastic interaction. We also
propose an efficient way for the phase manipulation of magnetoelastic waves by rotating the orien-tation of magnetization less than 10
/C14.Published by AIP Publishing.
https://doi.org/10.1063/1.5030342
Spintronics is the research field aiming to develop novel
devices based on spin degrees of freedom,1–5which attracts
great attention due to the potential for an invention of new
solid-state devices with low energy consumption4and a
THz-working frequency.6,7In these devices, data may be
transferred by spin waves, which are the collective excitation
of magnetization, M. So far, logic devices for NOT, XNOR,
and NAND gates using the phase of spin waves have been
demonstrated.8,9
Recently, we developed spin-wave tomography
(SWaT): reconstruction of the dispersion relation of spin
waves.10SWaT is based on the convolution theorem, a com-
plex Fourier transform (FT),11and a time-resolved magneto-
optical imaging method for spin-wave propagation.12SWaT
realizes the direct observation of the dispersion relation of
spin waves in a small- kregime, so-called magnetostatic
waves.10We then developed an advanced application of
SWaT, named phase-resolved SWaT (PSWaT).13PSWaT
obtains the phase information of spin waves by separating
the real and the imaginary components of the complex FT in
the SWaT analysis.
Spin waves hybridized with elastic waves through mag-
netoelastic coupling are called magnetoelastic waves.14–22
The amplitude of magnetoelastic waves is resonantly
enhanced at the crossing of the dispersion relations of spin
waves and elastic waves, where the torque caused by elastic
waves via magnetoelastic coupling is synchronized with the
precessional motion of Min spin waves.14The excitation and
the propagation dynamics of magnetoelastic waves have beeninvestigated by SWaT,
10,21while then their phase information
was disregarded. In the phase of magnetoelastic waves, one
can expect contributions by the phase of elastic waves, which
are the source of magnetoelastic waves.
In this study, we investigated the phase of magnetoelastic
waves by PSWaT. In the PSWaT spectra, we found180/C14-phase shift of magnetoelastic waves at around the cross-
ings of dispersion relations of spin waves and elastic waves.
This feature is explained by a model based on the convolution
theorem for optically generated elastic waves and magnetoe-
lastic coupling. Finally, we propose an efficient way for the
phase manipulation of magnetoelastic waves.
We used a 4- lm thick Lu 2.3Bi0.7Fe4.2Ga0.8O12film
grown on a Gd 3Ga5O12(001) substrate. The film has a satura-
tion magnetization of 780 Oe, which was aligned along the
[100] axis by applying an external magnetic field of 240 Oe.
The propagation dynamics of optically excited spin
waves was observed with a time-resolved magneto-optical
imaging system based on a pump-and-probe technique and a
rotating analyzer method using a CCD camera.12A pulsed
laser with the 800-nm center wavelength, 100-fs time dura-
tion, and 1-kHz repetition frequency was used as a light
source. This beam was divided into pump and probe beams.
The center wavelength of the probe beam was changed to
630 nm, where the sample shows a large Faraday rotation
angle ð5:2/C14Þand a high transmissivity (41%),23,24by using
an optical parametric amplifier. The pump beam was circu-
larly polarized and then tightly focused on the sample sur-
face with a radius, r0,o f1 lm. The linearly polarized probe
beam was weakly focused on the sample surface with a
radius of roughly 1 mm. Optically excited spin waves were
observed with a time-resolved magneto-optical imaging sys-
tem based on a rotating analyzer method using a CCD cam-
era.12The time delay between the pump and the probe pulses
was scanned from /C01 ns to 13 ns. All the experiments were
performed at room temperature. In Fig. 1, we define the
coordinates ( x,y,z) and /, i.e., the angle between Mand the
wavevector, k. The details of the experimental setup were
reported in Ref. 12.
We investigated the excitation and the propagation
dynamics of spin waves by SWaT10,21and PSWaT.13Since
0003-6951/2018/112(23)/232403/5/$30.00 Published by AIP Publishing. 112, 232403-1APPLIED PHYSICS LETTERS 112, 232403 (2018)
spin waves were observed through the Faraday effect repre-
senting the magnetization along the sample depth direction[m
z(r,t)], the SWaT spectrum is denoted as jmzðk;xÞj,
where xis an angular frequency. The PSWaT spectrum is
denoted as mpqs(k,x), where p,q, and slabel the real ( r) and
imaginary ( i) components of FT along the x,y, and time
axes, respectively.13The phase of spin waves determined by
the temporal FT is defined by /pqðk;xÞ¼tan/C01½mpqiðk;xÞ=
mpqrðk;xÞ/C138.
Let us first demonstrate the angular dependence of the
SWaT spectra in Fig. 2(a). The lines in Fig. 2(a) show the
dispersion relations of spin waves and elastic waves calcu-lated with the Damon-Eshbach theory
25and the parameters
obtained in Ref. 10. At around the crossing of the dispersion
relations of spin waves and elastic waves, we see signalsascribed to the spin-wave components of magnetoelasticwaves. The excitation of the magnetoelastic waves has been
attributed to the coherent-energy transfer from the optically
excited pure-elastic waves to spin waves through magnetoe-lastic coupling. This is supported by the systematic SWaT
experiments in terms of the external magnetic fields as dem-
onstrated in the previous studies of Refs. 10and 21.B yanalyzing the obtained SWaT spectra, the dispersion rela-
tions of the volume and the surface modes of magnetostatic
waves were reconstructed as shown in Fig. 2(b).
Next, we demonstrate the observation of the phase of
magnetoelastic waves by PSWaT. In this study, we discuss
the magnetoelastic waves generated by the optically excited
longitudinal mode of elastic waves.
10,21This mode of mag-
netoelastic waves is observed in the miirand miiicompo-
nents of the PSWaT spectra representing signals having
odd symmetries for both xandyaxes [Figs. 3(a)–3(d) ].13
Interestingly, we found that magnetoelastic waves show
sign reversal at around the crossing of dispersion relationsof spin waves and elastic waves. At this point, the phase of
magnetoelastic waves changes by 180
/C14as shown in the
plots of /iiin Figs. 3(e) and3(f). The explanation of the
180/C14shift of the phase of the magnetoelastic wave observed
in the PSWaT spectra is the main topic of the following
discussion.
Since the precession angle of Min the observed magnetoe-
lastic waves is small (several degrees), the magnetic compo-
nents of magnetoelastic waves may be written as m¼vh,
where vis a dynamical susceptibility26andh(¼hxþihy)r e p r e -
sents an internal field applied to Mfor the spin-wave excitation.
By solving the Landau-Lifshitz-Gilbert equation,26we write
vðk;xÞ¼vðk;xÞþijðk;xÞwith vðk;xÞ¼xsðkÞ6x
ðxsðkÞ6xÞ2þa2sx2
andjðk;xÞ¼asx
ðxsðkÞ6xÞ2þa2sx2,w h e r e xsðkÞrepresents the dis-
p e r s i o nr e l a t i o no fs p i nw a v e sa n d asis the Gilbert damping
constant. The torque caused by the longitudinal mode of elasticwaves through magnetoelastic coupling can be treated as inter-
nal fields induced by elastic waves given by
l
0hLM
yðk;xÞ¼b1ksin 2//C15LM
kðk;xÞ,17where /C15LM
kðk;xÞis the
strain accompanied by the longitudinal mode of elastic waves
along its kdirection, and b1is the magnetoelastic coupling con-
stant. Because of the spatial symmetry of hLM
y, which has mir-
ror symmetries for both xandyaxes, the longitudinal mode of
magnetoelastic waves appears in the miirandmiiicomponents
of the PSWaT spectra. This is consistent with our observations.
We have ascribed the optical excitation of such elastic
waves to photo-induced charge transfer transition by two-
photon absorption of the pump beam with the resonance at
around 400 nm.21All the waveforms observed in our
FIG. 1. Schematic illustration of the experimental configuration and the rect-
angular coordinates ( x,y,z). The sample surface is in the x-yplane with the
x-axis along the orientation of M, which is parallel to the [100] axis. The
orientation of Mwas controlled by an external magnetic field ( H). The sam-
ple normal is along the z-axis. The pump pulse is focused on the sample sur-
face at the origin of the coordinate to excite elastic waves, which is the
source of the magnetoelastic waves investigated in this study. The wavevec-
tor of elastic and magnetoelastic waves is denoted as k. The angle between
kandMis defined as /.
FIG. 2. (a) The angular dependence of the SWaT spectra for various kdirections with an angle ( /) between Mandk. The color indicates the SWaT amplitude
as shown in the color code. The red dashed and the blue dotted lines are the dispersion relations of the surface and the volume modes of magnetostatic wav es,
respectively, calculated with the Damon-Eshbach theory25and the parameters obtained in Ref. 10. The white dashed and dotted lines are the dispersion rela-
tions of the longitudinal and the transversal modes of elastic waves, respectively. (b) The dispersion relations of the surface (red plane) and the vo lume (blue
plane) modes of the magnetostatic waves shown in (a). The direction of Mis indicated by the green arrow.232403-2 Hashimoto, Johansen, and Saitoh Appl. Phys. Lett. 112, 232403 (2018)experiments show strong magnetic field dependences and
thus are attributed to spin waves. The direct observation of
elastic waves through photoelasticity has not been obtained
in our experiments due to the limited sensitivity of oursystem. In this study, we assume a response function of the
optically excited elastic waves to be
gðr;tÞ¼2g0HðtÞsinfxpðkÞt/C0k/C1rgexpð/C0aptÞ; (1)
where g0represents the strain accompanied by the longitudi-
nal mode of elastic waves, HðtÞis a Heaviside step function,
xp(k) is the dispersion relation of elastic waves, and aprep-
resents the damping of elastic waves. This assumption was
employed to explain the phase of the magnetoelastic waves
observed in the PSWaT spectra, although the excitation
mechanism of the elastic wave is out of the scope of this
study. With a model for the displacive excitation of coherent
phonons,27this assumption can be interpreted as the genera-
tion of elastic waves by a photoinduced change in the equi-
librium position of the lattice by photoexcited electrons.28
With this assumption, the elastic waves excited by the illu-mination of the focused pump pulse via two-photon absorp-
tion may be given by
/C15ðr;tÞ/ððð
gðr/C0a;t/C0sÞi
pða;sÞ2dads; (2)
where ipða;sÞrepresents the fluence of the pump pulse at the
sample surface, assumed to be a Gaussian function with
ipðr;tÞ¼exp½/C0jrj2=ð2r2
0Þ/C138dðtÞ. A Dirac delta function in
time [ dðtÞ] was used since the duration of the pump pulse
(sub-ps) is much shorter than the precession period of M
(/C24ns). Thus, the time-space FT of gðr;tÞgives gðk;xÞ¼gr
ðk;xÞþigiðk;xÞwith grðk;xÞ¼g0x6xpðkÞ
½x6xpðkÞ/C1382þa2pand gi
ðk;xÞ¼g0ap
½x6xpðkÞ/C1382þa2p.29By using the convolution theo-
rem,29the time-space FT of /C15ðr;tÞis/C15ðk;xÞ/ipðk;xÞ2
gðk;xÞ, where ipðk;xÞ¼expð/C0jkj2r2
0=2Þis the time-space
FT of ipðr;tÞ.
We then calculate the PSWaT spectra of the longitudinal
mode of magnetoelastic waves. By using the equations
shown above, we can write
mLMðk;xÞ/ib1ksin 2/ipðk;xÞ2gLMðk;xÞvðk;xÞ:(3)
Since kandipare real numbers while gandvare complex
numbers, we write
mLM/ki2
pgv¼ki2
pðgrþigiÞðvþijÞ
¼ki2
pðvgr/C0jgiÞþiðvgiþjgrÞ ½/C138 :(4)
The notation of ðk;xÞis omitted for clarity. The real ( R)
and the imaginary ( I) components of mcalculated with Eq.
(4)are shown in Figs. 4(a) and4(b), respectively. Since the
maximum time delay between the pump and probe pulses
(Tmax¼13 ns) is much shorter than the relaxation time of
spin waves and elastic waves in garnet films, we used alim-
ited by Tmaxgiving a/C241/(xTmax)¼0.03 for both apandas.
In both RfmgandIfmg, we can see the sign reversal of
magnetoelastic waves at around dispersion relations of spin
waves and elastic waves. These trends are in good agreement
with the experimental results shown in Figs. 3(a) and3(b).
The sign reversal of the magnetoelastic waves is attributed to
FIG. 3. (a) and (b) The cross-sectional views of the miir(a) and the miii(b)
components of the PSWaT spectra along the /¼15/C14direction. (c) and (d)
The cross-sectional views of the miir(c) and the miii(d) components of the
PSWaT spectra at the frequency of 1.57 GHz. In (a)–(d), the color indicates
the PSWaT intensity as shown in the color code. The dispersion relation of
the volume mode of magnetostatic waves calculated with the Damon-Eshbach
model10,25is shown by the green dotted lines. The dispersion relation of the
longitudinal mode of elastic waves is shown by the black dashed lines. (e)The frequency dependence of /
iialong the dispersion relation of elastic waves
shown by the black dashed lines in (a) and (b). (f) The angular dependence of
/iialong the dispersion relation of elastic waves shown by the black dashed
lines in (c) and (d). 180/C14-phase shift of magnetoelastic waves is obtained by
rotating the orientation of the magnetization, which changes the phase of spin
waves from the phase of point A to that of point B.232403-3 Hashimoto, Johansen, and Saitoh Appl. Phys. Lett. 112, 232403 (2018)the sign reversal of vandgrat the frequencies of the disper-
sion relations of spin waves ( fs) and elastic waves ( fp),
respectively. This is confirmed in the cross sections of
RfmgandIfmgalong at k¼1.0/C2104rad cm/C01[Fig.
4(c)], calculated by Eq. (4)andv,j,gr, and gi[Fig. 4(d)].
By comparing Figs. 4(c)and4(d), we find that the sign rever-
sals in the data of Rfmgatfsandfpare caused by the sign
reversal of vandgratfsandfp, respectively.
Finally, we propose an efficient way for 180/C14-phase
manipulation of the spin-wave components of magnetoelas-
tic waves. Let us consider the case where magnetoelasticwaves with kandxat the point A in Fig. 3(f)are selectively
excited by using, for instance, an interdigital transducer.17
Then, by rotating Mless than 10/C14, the phase of magnetoelas-
tic waves is shifted from the phase of point A to that of point
Bi nF i g . 3(f). This realizes 180/C14-phase manipulation of mag-
netoelastic waves. Since the garnet film used in our experi-
ments is magnetically soft to the magnetic field along the in-
plane direction, we can easily rotate the orientation of the
magnetization by rotating the direction of the external mag-
netic field. Therefore, 180/C14-phase manipulation of magnetoe-
lastic waves by slightly rotating Mmay give us great
potential for the development of future spin-wave deviceswith low-energy consumption.
In summary, we investigated phase of magnetoelastic
waves in a Bi-doped garnet film by phase-resolved spin-
wave tomography (PSWaT). In the PSWaT spectra, we
observed 180
/C14-phase shift of magnetoelastic waves at around
the crossing of the dispersion curves of spin waves and elas-
tic waves. This feature is consistent with a model based on
the convolution theorem for spin waves excited by elastic
waves through magnetoelastic coupling.
We thank Mr. T. Hioki, Dr. K. Sato, and Dr. R. Ramos
for fruitful discussions. This work was financially supported
by ERATO Spin Quantum Rectification Project (Grant No.
JPMJER1402) from JST, Grant-in-Aid for Scientific
Research on Innovative Area Nano Spin Conversion Science
(Grant No. JP26103005) from JSPS KAKENHI, JSPS Core-
to-Core program International research center for new-
concept spintronics devices, and World PremierInternational Research Center Initiative (WPI) from MEXT,
Japan.
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FIG. 4. (a) and (b) show the real ( R) and the imaginary ( I) components of
m, respectively, calculated with Eq. (4). We used xsðkÞandxpðkÞobtained
by the SWaT spectra shown in Fig. 2(a) anda¼0.03, determined by the
maximum time delay between pump and probe beams in our experimental
setup ( Tmax¼13 ns), for both asandap. (c) The real and the imaginary com-
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blue dashed lines, respectively. (d) The spectra of v,j,gr, and giused to cal-
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1.4709188.pdf | Magnetization dynamics, gyromagnetic relation, and inertial effects
J.-E. Wegrowe and M.-C. Ciornei
Citation: Am. J. Phys. 80, 607 (2012); doi: 10.1119/1.4709188
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Downloaded 16 Mar 2013 to 136.159.235.223. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permissionMagnetization dynamics, gyromagnetic relation, and inertial effects
J.-E. Wegrowe and M.-C. Ciornei
Ecole Polytechnique, LSI, CNRS and CEA/DSM/IRAMIS, Palaiseau F-91128, France
(Received 21 September 2011; accepted 14 April 2012)
The gyromagnetic relation—that is, the proportionality between the angular momentum ~Land the
magnetization ~M—is evidence of the intimate connections between the magnetic properties and the
inertial properties of ferromagnetic bodies. However, inertia is absent from the dynamics of a
magnetic dipole: The Landau–Lifshitz equation, the Gilbert equation, and the Bloch equationcontain only the first derivative of the magnetization with respect to time. In order to investigate
this paradoxical situation, the Lagrangian approach, proposed originally by Gilbert, is revisited
keeping an arbitrary nonzero inertia tensor. The corresponding physical picture is a generalizationto three dimensions of Ampe `re’s hypothesis of molecular currents. A dynamic equation
generalized to the inertial regime is obtained. It is shown how both the usual gyromagnetic relation
and the well-known Landau–Lifshitz–Gilbert equation are recovered in the kinetic limit, that is, fortime scales longer than the relaxation time of the angular momentum.
VC2012 American Association of
Physics Teachers .
[http://dx.doi.org/10.1119/1.4709188]
I. INTRODUCTION
The precession of a magnetic vector is the basis of nuclear
magnetic resonance, electron paramagnetic resonance, andferromagnetic resonance. It is also present in many othercontexts, like that of Rabi experiments or neutron interfer-ometry. On the other hand, precession is also present in themovement of a spinning top or a gyroscope.
1
The analogy between the dynamics of the magnetization in
a magnetic field on one hand and the dynamics of a symmetri-cal spinning top in a gravitational field on the other hand is of-ten exploited in introductory courses on magnetism. Theprecession effect (i.e., the rotation of the direction of a vectorof constant modulus) is indeed easy to observe in a spinningtop, while it is difficult to see with a ferromagnet because itwould require observations at sub-nanosecond time scales.
2
However, the analogy seems to be incomplete because thedynamics of the spinning top implies inertial effects (e.g.,nutation) while for a uniformly magnetized body, the dynam-ics of the magnetization is described by the time variation ofthe magnetization d~M=dt(i.e., the velocity) and does not
include the second derivative d
2~M=dt2(i.e., the acceleration).3
In other words, there is no inertia in the dynamic equation.
The aim of this paper is to push the analogy to its logical endwith the introduction of inertia4,5in the dynamics of uniform
magnetization within the Lagrangian formalism.
The precession of a uniform magnetic moment ~M¼Ms~e3
(Msis the magnetization at saturation and ~e3is the radial unit
vector) under an effective magnetic field ~His often presented
as a consequence of the gyromagnetic relation ~M¼c~Lthat
links the magnetization to the angular momentum ~L. The
constant cis the gyromagnetic ratio. The gyromagnetic rela-
tion and the value of the constant c¼q=ð2mÞcan be justified
in a basic atomic model of an electron of charge qand mass
morbiting around a nucleus. This well-known model (see
Sec. V) constitutes the hypothesis of the Ampe `re molecular
currents,6validated by Einstein and de Haas in their famous
experiments of 1915–1916.7–9In the general case, with both
spin and orbital contributions in condensed material, thegyromagnetic ratio is written as c¼gq=ð2mÞ, where the g
factor accounts for the fact that the electron in a ferromagnetis a complex quasiparticle.
2,10Using the gyromagnetic relation, the application of
Newton’s second law, d~L=dt¼~M/C2~H, leads directly to the
precession equation: d~M=dt¼c~M/C2~H. However, the appli-
cation of Newton’s law—or equivalently the Lagrange equa-tion—to a rigid rotating body (typically the spinning top in a
gravitational field) leads to a more complex gyroscopic equa-
tion that contains inertial terms. As will be shown, a deriva-tion based on a modified gyromagnetic relation [Eq. (11)]
leads us to consider inertial terms for the dynamics of themagnetization. The well-known equation of the dynamics isrecovered by considering the equation with inertia and ex-plicitly going to the kinetic limit.
It is first useful to come back to the short history of the
dynamic equation of the magnetization, especially with theintroduction of the dissipation, since the precession equation
d~M=dt¼c~M/C2~Hcannot account for the rapid relaxation
toward the equilibrium state of the magnetization (typicallyafter a couple of precession cycles, i.e., after some nanosec-onds, in usual ferromagnets). In 1935, Landau and Lifshitzproposed an equation for the dynamics of the magnetizationthat takes into account both the precession and the relaxa-
tion along the magnetic field: d~M=dt¼~c~M/C2~Hþh
0~M
/C2ð~M/C2~HÞ,w h e r e h0is a damping term (defined below) and
~c¼c.11The basic argument used to derive the equation was
to keep the modulus of the magnetization constant. The de-
rivative d~M=dtis hence perpendicular to the vector ~M.
Two decades later, after the development of ferromagnetic
resonance (FMR) experiments12and motivated by the obser-
vation of systematic deviations from the above equation for
high damping, Gilbert derived the equation that bears his
name using a Lagrangian formalism.13,14The dynamics of
the magnetization is then described by the equation
d~M
dt¼c~M/C2 ~H/C0gd~M
dt !
; (1)
with the introduction of the damping coefficient g.
The Landau–Lifshitz equation and the Gilbert equation
are equivalent15provided that h0¼ac=ð1þa2Þand
~c¼c=ð1þa2Þ, where a¼cgMsis the dimensionless Gilbert
607 Am. J. Phys. 80(7), July 2012 http://aapt.org/ajp VC2012 American Association of Physics Teachers 607
Downloaded 16 Mar 2013 to 136.159.235.223. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permissiondamping. Note that ~c6¼c: this was the decisive improvement
brought by Gilbert to the Landau–Lifshitz proposition.
In line with previous works performed by Do ¨ring,3Gilbert
introduced the Lagrangian of a uniform ferromagnet with akinetic energy T¼ L
2
1=2I1þL2
2=2I2þL2
3=2I3, where
fI1;I2;I3gare the principal moments of inertia and
fL1;L2;L3gare the components of the angular momentum
defined with respect to the principal axis. Gilbert chose anad hoc inertia tensor in such a way that the inertial terms dis-
appear from the dynamic equation (i.e., such that theLandau–Lifshitz equation is recovered in the low dampinglimit). To do that, a sufficient condition is to set to zero thetwo first principal moments of inertia: I
1¼I2¼0 (but keep-
ing a non-zero kinetic energy: I36¼0). As pointed out by
Gilbert himself,14this puzzling condition does not seem to
correspond to any realistic mechanical system (see Gilbert’s
footnote 7: “I was unable to conceive of a physical object
with an inertial tensor of this kind”). In the subsequent reportabout Gilbert’s derivation, the ad hoc and puzzling condition
I
1¼I2¼0 is explicitly stated despite its problematic char-
acter. In his presentation of the Gilbert equation published in1960 in the American Journal of Physics ,16Brown wrote,
“We treat the rotating moment system as a symmetric top,with principal moments of inertia A¼B¼0;C>0. For a
top made of classical mass particles, A¼B¼0 implies
C¼0; but this top is not made of classical mass particles.”
(In our notation A/C17I
1;B/C17I2;C/C17I3.) In the reference
textbook of Morrish17(edited from 1965 to 2002), we can
read: “A Lagrangian function, L, consistent with the
accepted equation of motion [equation (10–3.2)] can beobtained by considering the magnetic system as a classicaltop with principal moments of inertia ð0;0;CÞ….” In
our notation, Morrish’s equation (10)–(3.2) is d~M=dt
¼c~M/C2~Hþðdamping Þ. Accordingly, the mechanical
approach is not presented as a realistic physical model (as itshould be, according to the gyromagnetic relation) but seemsto be introduced as a pedagogical analogy to an unspecifiednon-classical theory, which would give a physical interpreta-tion to Gilbert’s puzzling condition. Indeed, this strange con-dition is presented as a specific property of the magneticmoments that would be due to the fact that “this top is notmade of classical mass particles.”
This is probably the reason why, after more than half a
century of intensive use of Gilbert’s equation, no full deriva-tion following Gilbert’s approach—with the complete set ofprincipal moments of inertia (i.e., without the ad hoc
assumption)—has been proposed (see, for example, Refs. 18
and15for recent presentations of the Gilbert’s derivation).
However, as will be shown below, such a derivation can be
performed at an elementary level, as a direct application of
the Lagrangian formalism. Although straightforward, thisderivation is instructive because it shows that the puzzlingcondition I
1¼I2¼0 is not necessary to obtain the Gilbert
equation. Instead, Gilbert’s condition is replaced by the nec-essary physical condition under which a diffusion processcan be described by a non-inertial diffusion equation. Thiscondition is the usual kinetic limit expressing the require-ment that the typical measurement times should be longerthan the relaxation time sof the momentum [here for the
angular momentum s¼I
1=ðgM2
sÞ].19In this picture, the pre-
cession with damping is simply a diffusion process in a fieldof force, for which the angular momentum has reached itsequilibrium. This change of paradigm has two consequences.A first important consequence is that an inertial regime ofuniform magnetic dipoles is expected and should be
observed at short enough time scales. Second, the classicalmechanical approach is much more than a pedagogical anal-ogy, and it could be used (beyond the gyromagnetic relation)for a deeper understanding of non-equilibrium magnetome-chanics and related processes.
II. THE MECHANICAL ANALOGY
The mechanical model is sketched in Fig. 1. A rigid cylin-
drical stick of length M
s, with one end fixed at the origin, is
pointing in a direction described by the angles handu. The
stick is precessing around the vertical axis at angular velocity
_uand is spinning around its own symmetry axis at angular
velocity _w. The phase space of this rigid rotator is defined by
the angles fh;u;wgand the three components of the associ-
ated angular momentum ~L. The relation between the angular
momentum and the angular velocity ~Xis~L¼/C22/C22I~X, where/C22/C22Iis
the inertia tensor.
A. The rotating frame
In the rotating frame, or body-fixed frame f~e1;~e2;~e3g, the
inertia tensor is reduced to the principal moments of inertiafI
1;I2;I3g. The symmetry of revolution of the spinning stick
imposes furthermore that I1¼I2
/C22/C22I¼I100
0I10
00 I30
@1
A: (2)
In the fixed body frame, the angular velocity reads (see
Fig.1):
X1¼_usinhsinwþ_hcosw;
X2¼_usinhcosw/C0_hsinw;
X3¼_ucoshþ_w:(3)
Fig. 1. Illustration of the magnetomechanical analogy of a spinning stick
that precesses around the zaxis. The coordinates of the stick in the space-
fixed frame are parameterized by the angles ðh;u;wÞand the radius of the
sphere is Ms. The body-fixed frame, denoted f~e1;~e2;~e3g, is spinning with
angular velocity _wand is precessing around ~ezwith angular velocity _u.
608 Am. J. Phys., Vol. 80, No. 7, July 2012 J.-E. Wegrowe and M.-C. Ciornei 608
Downloaded 16 Mar 2013 to 136.159.235.223. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permissionThe kinetic equation is obtained from the angular velocity:
For any vector ~Mof constant magnitude carried with the
rotating body, we have
d~M
dt¼~X/C2~M: (4)
This equation can be inverted by cross multiplication by ~M
and developing the double cross product. Since ~M¼Ms~e3,
we have
~X¼~M
M2
s/C2d~M
dtþX3~e3: (5)
B. The Lagrange equation
Following Gilbert and Do ¨ring, we introduce the Lagran-
gian of the system:
L¼1
2I1X2
1þX2
2/C0/C1
þI3X2
3/C0/C1
/C0Vðh;uÞ; (6)
where Vðh;uÞis the ferromagnetic potential energy that
defines the effective magnetic field ~H¼/C0 ~rV. The effective
field ~Hcomprises the applied field, the anisotropy field, the
dipolar field (or the demagnetizing field), the magneto-elastic contributions, etc.
Inserting Eq. (4)in Eq. (6), we see that the Lagrangian of
the system is independent of the variable w:L
¼
1
2I1_u2sin2hþ_h2/C16/C17
þI3ð_ucoshþ_wÞ2no
/C0Vðh;uÞ. The
Lagrange equations are20
d
dt@L
@_qi/C0@L
@qiþ@F
@_qi¼0; (7)
where qirefers to the three coordinates fh;u;wg, and the
components of the kinetic momentum are defined bythe three derivatives @L=@_q
i¼Li. The function Fis the
Rayleigh dissipative function. In a viscous environment, theRayleigh function is defined by the damping coefficent g
such that F¼
1
2gðdM=dtÞ2¼1
2gM2
sX2
1þX2
2/C0/C1
.
For the magnetomechanical model, the Lagrange equa-
tions read
d
dt½I1_h/C138/C0I1_u2sinhcoshþI3_usinh_ucoshþ_w/C16/C17
¼/C0@F
@_h/C0@V
@h;(8)
d
dtI1_usin2hþI3_ucoshþ_w/C16/C17
coshhi
¼/C0@F
@_u/C0@V
@u;
(9)
d
dtI3_ucoshþ_w/C16/C17hi
¼/C0@F
@_w¼0: (10)
Equation (10) is equal to zero because there is no damping
for spinning in the case of the usual viscous environment.21
This property can be verified in the case of a very thin wiremoving in a viscous fluid: the rotation of the stick around theaxes Oxor~e
z(see Fig. 1) is damped, but not the spinning of
the stick around its symmetry axis. The quantity L3¼I3X3is then a constant of motion, and it can be written L3¼Ms=c
without loss of generality.
III. KINETIC EQUATION AND GILBERT’S
ASSUMPTIONS
It is not trivial to see how to recover the Landau–Lifshitz
equation from Eqs. (8)to(10), even in the low damping limit
g!0. But it is clear that the inertial terms in the left-hand
sides of Eqs. (8)–(10) are not welcome from that point of
view and should be removed. The best way to consider the
inertial terms is to substitute Eq. (5)into ~L¼/C22/C22I~X(with the
relation L3¼Ms=c):
~L¼I1
M2
s~M/C2d~M
dt !
þMs
c~e3: (11)
It is then rather immediate to see that the gyromagnetic
relation ~L¼~M=ccannot be recovered without removing the
first term on the right-hand side. Gilbert’s idea was toassume that I
1¼0. This is indeed a sufficient condition to
kill the inertial terms, and the gyromagnetic relation is neces-sarily recovered with the definition c¼M
s=L3of the gyro-
magnetic ratio.
With both assumptions, the Lagrange equations become
0¼/C0Ms
c_usinh/C0@F
@_h/C0@V
@h; (12)
d
dtMs
ccosh/C20/C21
¼/C0@F
@_u/C0@V
@u; (13)
d
dtMs
c/C20/C21
¼0: (14)
Since in the rotating frame the effective field ~H¼/C0 ~rVis
H1¼/C01
Mssinh@V
@u; H2¼1
Ms@V
@h; (15)
inserting Eq. (4)into Eqs. (12)–(14) leads to
d~M
dt¼c~M/C2 ~H/C0gd~M
dt !
: (16)
This is the well-known Gilbert’s equation (1)obtained fol-
lowing the standard Lagrangian approach.13–18
However, the absence of inertia shows that the equation
should be derived in the configuration space instead of thephase space (this is performed, e.g., in Ref. 22). Indeed, the
dynamics is described by the two variables handuand not
in the phase space defined by the five variables h,u, and the
components of ~L. Accordingly, the gyromagnetic relation
is—in this approach—not necessary (nor sufficient) for thederivation of the Gilbert equation.
IV. BEYOND GILBERT’S ASSUMPTION
In order to take into account the gyromagnetic relation, it
is necessary to go beyond Gilbert’s ad hoc assumption and
to make I
1andI2nonzero. The generalization of Ampe `re’s
model from a quasi-one-dimensional atomic model (the elec-tric charge qof mass mdistributed along the circular orbit)
609 Am. J. Phys., Vol. 80, No. 7, July 2012 J.-E. Wegrowe and M.-C. Ciornei 609
Downloaded 16 Mar 2013 to 136.159.235.223. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permissionto a more realistic three-dimensional atomic model—for
which the electric charge is now distributed in three dimen-sions—imposes non-vanishing principal moments of inertiaI
1¼I2. Figure 2shows a simple generalization of Ampe `re’s
model with an ellipsoid of revolution of equatorial radius r
and polar radius c(along the zaxis), for which
I1¼2mr2=5¼I2and I3¼mðc2þr2Þ=5. The model can
straightforwardly be generalized to more realistic electronicorbitals, typically defined by combinations of sphericalharmonics.
Two parameters are now necessary to take into account
the amplitude of the magnetic moment: the usual magnetiza-tion at saturation M
s¼c=ðX3I3Þ, and the anisotropy of the
ferromagnetic atom (with the dimensionless parameter1/C0I
1=I3), which plays a role only for the inertial regime.
Let us take for the sake of simplicity _w¼0.23With the
relation L3¼Ms=c, Eqs. (8)–(10) give
_X1¼/C0X1
sþX31/C0I3
I1/C18/C19
X2/C0Ms
I1H2; (17)
_X2¼/C0X2
s/C0X31/C0I3
I1/C18/C19
X1þMs
I1H1; (18)
X3¼Ms
cI3; (19)
where the typical relaxation time s/C17I1=ðgM2
sÞhas been
introduced. The relaxation time sis the typical time scale for
which the diffusion approximation is valid, i.e., above whichthe angular momentum has relaxed toward the equilibriumstate.
4
Using_~X/C1~e3¼0, Eqs. (17)–(19) can be rewritten in vecto-
rial form
_~X¼/C01
s~X/C0X3~e3/C16/C17
þMs
c1
I3/C01
I1/C18/C19
~X/C2~e3/C16/C17
þMs
I1~e3/C2~H/C0/C1
:(20)Inserting Eqs. (4),(5), and the time derivative of Eq. (5),w e
have
~M
M2
s/C2d2~M
dt2þ1
cI3d~M
dt¼/C01
sM2
s~M/C2d~M
dt !
þ1
c1
I3/C01
I1/C18/C19d~M
dt
þ1
I1~M/C2~H; (21)
so that Eq. (21) can be reduced to the compact form
d~M
dt¼c~M/C2~H/C0gd~M
dtþsd2~M
dt2 !"#
: (22)
This is Gilbert’s equation for the dynamics of the magnetiza-
tion that includes the new inertial term /C0cgs~M/C2ðd2~M=dt2Þ.
V. TYPICAL TIME SCALES
The limit of Eq. (22) fort/C29s, where s¼I1=ðgM2
sÞ, leads
to the kinetic limit. Since the damping gcan be replaced by
the usual dimensionless Gilbert coefficient a¼cgMs,w e
have s¼ðI1=I3Þð1=aX3Þ. A rigorous study of the asymptotic
behavior (as a function of the parameters c~H;s/C01, and g)i s
beyond the scope of this work. However, it is sufficient toobserve that the limit s!0 leads straightforwardly to the
Landau–Lifshitz–Gilbert equation
d~M
dt!c~M/C2 ~H/C0gd~M
dt !
: (23)
In the same manner, the vectorial gyromagnetic relation is
recovered in the limit s!0. Equation (11) gives
d~L
dt¼gs ~M/C2d2~M
dt2 !
þ1
cd~M
dt!1
cd~M
dt: (24)
The sufficient condition of validity of the Gilbert equation
I1!0 is hence replaced by the condition s!0 (i.e., s/C28t
for the relevant range of the parameters).
An estimation of the value of sgives the typical time scale
for which inertial effects can be observed. Here, we comeback to the simplest argument for the justification of thevalue of cwith the Ampe `re molecular currents [Fig. 2(a)].
This is a quasi-one-dimensional atomic model, for which the
atomic orbital moment is defined by the electronic charge q
orbiting around a nucleus at the distance rwith a velocity v.
This system defines an electric loop that generates a mag-netic moment ~M¼IS~e
z, where I¼ qv=ð2prÞis the electric
current, S¼pr2is the surface enclosed by the loop, and ~ezis
the vector normal to the loop. This leads to the microscopicmagnetic moment M
s¼qvr=2. If we take the Bohr radius a0
and the electron velocity vwith the Heisenberg relation
mva0/C21/C22h=2, the Bohr magneton is obtained for the mini-
mum value of the atomic magnetic moment lB¼c/C22h=2. On
the other hand, the angular momentum of this system isL
3¼rmvand the ratio is M3=L3/C17c¼q=ð2mÞ. The angular
frequency X3is given by L3¼I3X3¼lB=c, that is,
X3¼lB=ðcI3Þwhere I3¼ma2
0.W eh a v e X3/C253/C21016rad/s
and an order of magnitude of the typical timess¼I
1=aI3X3ðÞ at which inertial effects should be observed
Fig. 2. (a) Illustration of Ampe `re’s model of molecular current. The electron
of mass m, charge q, and angular velocity X3is confined in a circular loop
of radius r. The angular momentum due to the rotating mass is
L3¼I3X3¼mvr and the magnetic moment due to the rotating charge is
Ms¼qvr=2. (b) Generalization of Ampe `re’s molecular current that corre-
sponds to the mechanical analogy shown in Fig. 1: The components of the
angular momentum are defined by the three principal moments of inertia
I1¼I2andI3.
610 Am. J. Phys., Vol. 80, No. 7, July 2012 J.-E. Wegrowe and M.-C. Ciornei 610
Downloaded 16 Mar 2013 to 136.159.235.223. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permissionis around a femtosecond (for a damping coefficient asuch
thatI1=ðI3aÞ/C251).
VI. CONCLUSION
The paradoxical role played by the angular momentum for
the dynamics of the magnetization has been studied in thelight of the model introduced by Gilbert for the derivation ofthe equation that bears his name. The demonstration hasbeen reconsidered without Gilbert’s puzzling assumption ofvanishing first moments of inertia I
1¼I2¼0 and I36¼0.
Instead, a general inertial tensor with non-vanishing princi-pal moments of inertia fI
1;I1;I3ghas been used. The corre-
sponding physical picture is a more realistic generalizationto three dimensions of Ampe `re’s model of molecular cur-
rents. A generalized expression of the equation of the dy-namics of the magnetization is obtained, which includes aninertial term. The mechanical analogy of the magneticmoment with the rigid rotator is complete. Both the usualexpression of the Landau–Lifshitz–Gilbert equation and theusual gyromagnetic relation are recovered provided that a ki-netic limit is performed for time scales much larger than therelaxation time of the angular momentum, s¼I
1=ðgM2
sÞ.
The typical time scale is found to be of the order of onefemtosecond.
1R. Feynman, R. Leighton, and M. Sands, The Feynman Lectures on
Physics (CIT, 1963), Vol. I, 20.6–7 for the description of the gyroscope
and Vol. II 34–2 to 34–5 for the magnetic precession.
2J. Sto ¨hr and H. C. Siegmann, Magnetism: From Fundamental to Nano-
scale Dynamics (Springer, Berlin, 2006).
3This is not the case for non-uniform magnetization (domain walls or anti-
ferromagnets), for which a magnetic mass is defined. See the pioneering
work of W. Do ¨ring: “Uber die tra ¨gheit der Wa ¨nde zwischen Weisschen
Bezirken” (On the inertia of walls between Weiss domains), Z. Natur-
forsch. 3a, 373–379 (1948). Do ¨ring introduced the magnetic Lagrangian in
this paper.
4M.-C. Ciornei, J. M. Rubı ´, and J.-E. Wegrowe, “Magnetization dynamics
in the inertial regime: Nutation predicted at short time scales,” Phys. Rev.
B83, 020410(R) 1–4 (2011).
5Fa¨hnle, D. Steiauf, and Ch. Illg, “Generalized Gilbert equation including
inertial damping: Derivation from an extended breathing Fermi surface
model,” Phys. Rev. B 84, 172403 (2011).
6L. P. Williams, “Why Ampe `re did not discover electromagnetic
induction,” Am. J. Phys. 54, 306–311 (1986).
7The gyromagnetic relation ~M¼c~Lhas been established through static
magnetomechanical measurements. See S. J. Barnett, “Gyromagnetic andelectron-inertia effects,” Rev. Mod. Phys. 7, 129–167 (1935); and A. Ein-
stein and W. J. de Haas, “Experimenteller Nachweis der Ampereschen
Molekularstro ¨me,” Verhandl. Dtsch. Phys. Ges. 17, 152–170 (1915).
8V. Ya. Frenkel’ “On the history of the Einstein-de Haas effect,” Sov. Phys.
Usp.22, 580–587 (1979).
9P. Galison, How Experiments End (The University of Chicago Press, Chi-
cago, 1987), Chapter 2.
10H. C. Ohanian, “What is spin?,” Am. J. Phys. 54, 500–505 (1986).
11L. Landau and E. Lifshitz, “On the theory of dispersion of magnetic per-
meability in ferromagnetic bodies,” Phys. Z. Sowjetunion 8, 153–169
(1935).
12N. Bloembergen, “On the ferromagnetic resonance in Nickel and super-
malloy,” Phys. Rev. 78, 572–580 (1950).
13T. L. Gilbert, “Formulation, foundations and applications of the phenome-
nological theory of ferromagnetism,” Ph.D. dissertation, Illinois Institute
of Technology, 1956, Appendix B.
14T. L. Gilbert, “A phenomenological theory of damping in ferromagneticmaterials,” IEEE Trans. Mag. 40, 3443–3449 (2004). The discussion
related to the assumption I
1¼I2¼0 is confined in footnotes 7 and 8.
Note that the original reference in Physical Review is only an abstract:T.
L. Gilbert, “A Lagrangian formulation of the gyromagnetic equation of the
magnetization fields,” Phys. Rev. 100, 1243 (1955).
15J. Miltat, G. Alburquerque, and A. Thiaville, “An Introduction to Micro-
magnetics in the Dynamics Regime,” in Spin dynamics in confined mag-
netic structures I , edited by B. Hillebrands and K. Ounadjela (Springer,
Berlin, 2002). The kinetic energy is introduced through the Lagrangian on
page 19, Eq. (37). The Lagrangian is such that (with our notation)
I1¼I2¼0 andX3¼_ucosh.
16W. F. Brown Jr., “Single-domain particles: New uses of old theorems,”
Am. J. Phys. 28, 542–551 (1960). See page 549.
17A. H. Morrish, The Physical Principles of Magnetism (John Wiley &
Sons, New York, 1965; reprinted by IEEE Press, New York, 2001). Seeend of page 551.
18T. F. Ricci and C. Scherer, “A stochastic model for the dynamics of classi-cal spin,” J. Stat. Phys. 67, 1201–1208 (1992), page 1204: “In order to
simulate the behavior of a classical spin, we take, for these equations, the
limit I
1!0,I3!0, and _w!1 , but maintaining I3_w¼SðtÞ¼finite.”
In our notation S/C17Ms.
19J. M. Rubı ´ and A. Pe ´rez-Madrid, “Inertial effects in non-equilibrium
thermodynamics,” Physica A 264, 492–502 (1999).
20H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA,
1980).
21D. W. Condiff and J. S. Dahler, “Brownian motion of polyatomic mole-cules: The coupling of rotational and translational motions,” J. Chem.
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22J.-E. Wegrowe, “Spin transfer from the point of view of the ferromagnetic
degrees of freedom,” Solid State Commun. 150, 519–523 (2010).
23The full calculation with _w6¼0 gives the same result. It is consistent with
that found with a different approach in M.-C. Ciornei et al. ,
“Magnetization dynamics in the inertial regime: Nutation predicted at
short time scales,” Phys. Rev. B 83, 020410(R) (2011).
611 Am. J. Phys., Vol. 80, No. 7, July 2012 J.-E. Wegrowe and M.-C. Ciornei 611
Downloaded 16 Mar 2013 to 136.159.235.223. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission |
1.384925.pdf | Asymmetric Doppler amplitudes in the surface scatter
channel for crosswind transmitter-receiver geometry
F. B. Tuteur, H. Tung, and J. G. Zornig
Department of Engineering and Applied Science, Yale University, New Haven, Connecticut 06520
(Received 18 June 1979; accepted for publication 10 July 1980)
Doppler asymmetries have been observed in model tank experiments in the signal received after being
scattered from a wind-driven surface. A distinct spectral shift was observed when the plane containing the
transmitter and receiver was perpendicular to the wind direction. This paper examines the conjecture that this
shift arises from the fundamental asymmetry observed in a wind-driven surface, i.e., from the fact that the
upwind slopes tend to be shallower than the downwind slopes. Three different situations are considered: (1) a
deterministic surface, (2) a random surface with large Rayleigh parameter, (3) a random surface with small
Rayleigh parameter. All three models show that the slope asymmetry results in Doppler asymmetry; however,
the amount of this asymmetry corresponding to realistic values of surface parameters is very small. Hence the
slope asymmetry does not completely explain the occasionally observed larger spectral shifts, and other
mechanisms must be considered.
PACS numbers: 43.30.Gv, 43.30.Dr, 43.20.Bi
INTRODUCTION
When a sinusoidal acoustic signal is scattered from
a moving rough surface the received signal is generally
spread in frequency. If the surface roughness is small
and if the surface deformation is roughly periodic,
the spreading results in a fairly distinct sequence
of sideband frequencies. This is the result of phase
modulation of the transmitted sinusoid by the moving
surface, and the separation between the spectral lines
at the receiver is equal to the surface frequency. If
the surface is rough and confused, the sidebands merge
together into a more or less continuous spreading of
frequencies around the transmitted frequency.
In several recent papers 1'2 E.Y. Harper and F. M.
Labianca showed that if the transmitter and receiver
are at different depths the upper and lower side-bands
may have different amplitudes. A similar result was
obtained more recently by Kuperman. 3 The amplitude
ratio depends on the direction of the surface motion
with respect to the transmitter-receiver geometry.
Specifically, if the transmitter is at a greater depth
than the receiver, and if the wind causes surface waves
to travel away from the transmitter and toward the
receiver, then the upper sideband amplitudes are
smaller.
Harper, Labianca, and Kuperman obtain their result
by use of a perturbation procedure for introducing the
surface boundary condition into the solution of the wave
equation. These solutions are therefore strictly applic-
able only in the limit of very small surface deforma-
tions. However, similar results have been obtained by
F. B. Tuteur and H. Tung 4'5 using a modified Fresnel-
corrected Kirchhoff integral method, which, in principle
applied also to relatively rough surfaces.
In order to confirm the existence of unequal sidebands,
we have mad e spectral measurements in a model tank. 6
[hese measurements showed the predicted asymme-
tries, butthey also produced a number of surprises. In
particular, it was found that there was some asymmetry
in the spectral spreading function when the transmitter
and receiver were at the same depth, and a line drawn
between transmitter and receiver was at right angles to the direction of the surface waves. All the current
theories would have predicted symmetrical frequency
spreading for this arrangement. It seems, therefore,
that there may be features of the scattering mechanism
that have not been properly included in any of the cur-
rently available theories.
In order to motivate our analysis of this problem con-
sider first the following heuristic explanation. In a
wind-driven water surface, the slopes on the windward
side have a smaller magnitude than those on the down-
wind side (Fig. 1). In a cross-wind transmitter-re-
ceiver configuration, rays reflected from up-wind
facets have an upward Doppler shift, whereas those
reflected from down-wind facets have a downward
Doppler shift. Because of the asymmetry in the surface
slopes the area of up-wind slopes capable of reflecting
rays from the transmitter to the receiver is larger than
the area of the down-wind slopes. Hence we would ex-
pect upper Doppler sidebands to be larger than lower
Doppler sidebands.
We see that in order to investigate this phenomenon
analytically it is necessary that' the analysis take into
account details of the surface slopes. This is usually
not done. In particular, the Fresnel-corrected Kirch-
hoff integral method as used, 7'9 eliminates all slope in-
formation and implies that scattering effects can be
related solely to the up and down motion of the surface.
WIND DIRECTION
FIG. 1. Heuristic explanation for the observed Doppler asym-
merry.
1184 J. Acoust. Soc. Am. 68(4), Oct. 1980 0001-4966/80/101184-09500.80 (D 1980 Acoustical Society of America 1184
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 155.33.16.124 On: Thu, 27 Nov 2014 05:09:15I. THE CHANNEL TRANSFER FUNCTION
We consider only the cross-wind case in this paper.
The scattering geometry is then as illustrated in Fig. 2.
The x-y plane is the plane of the smooth surface. The
positive direction for the z axis is into the water, and
the origin of coordinates is taken to be at the smooth-
surface specular point. The source and receiver lie
in the x-z plane, as does the specular ray, which
makes an angie with the surface. For cross-wind
transmission the wind direction is along the y axis.
The channel transfer function can be derived from
the Helmholtz integral, which is an alternate form of
the wave equation. If the surface is regarded as a pres-
sure-release boundary, and if the water is assumed to
be a homogeneous transmission medium, then it is
shown in Appendix A that the channel transfer function
can be written in the form
x Be(x,y)expjk(ro +rt) dxdy , (1) T'0T' 1
where k = oo/c is the acoustic wavenumber, r 0 and r
are, respectively, the distances from the source and
the receiver to an arbitrary point (x,y ,z) on the sur-
face, [ = (x,y) is the displacement of the surface at
point {x,y) away from the smooth-surface reference
level (positive is upwards), and Be(x,y) is the pro-
jection of the source and receiver beam patterns on the
surface. Observe that Eq. {1) contains the surface
slope components a[/ax and a/y explicitly.
The derivation of Eq. (1) is based on a number of
simplifying assumptions referred to as the Kirchhoff
approximations. 9 Basically these are that the radius of
curvature of the boundary is large compared to the acous-
ticwavelength and that there is no shadowing or multiple
reflections. Note that the Kirehhoff approximations do not
necessarily require small surface heights. Also,
since the slope components O/x and a/y appear
explicitly, it is not necessary to assume that the slopes
are small.
The distances r 0 and r depend on the instantaneous
surface point (x,y,z) as shown in Fig. 3. Specifically,
we have
Y'o =[(xs + x) 2 +(Zo + [)2 +y211/2 ,
r 1 =[(x s -x) 2 +(z! + [)2 +y211/2 . (2)
In accordance with the usual procedure the expressions
for r 0 and ri in Eq. (2) are expanded in.a Taylor series
WIND DIRECTION
///
source Z receiver FIG. 2. Scatter geometry. /
f , / .x
,o -/ ' I ' $P.½u,^
'OU.CE y' ' SURAC REFLECTED
RAY
z
FIG. 3. Expansion of r0 and rl in terms of displacements along
the x, y, and z axes.
about a nominal value r00 and rio. The nominal value
chosen for this purpose is that given by reflection from
the specular point, i.e., r 0 and r are evaluated around
x =y =z= 0. This results in
ro+ri=roo+ro+R'(x2sinZg,+yZ)+2sin½+' ß ß , (3)
where ½ is the grazing angie and R = 2roroo/(ro + too ).
The beam pattern function is assumed to have the form
Be(x,y ) = exp(- A'x 2 - B ,y2) .. (4)
Following the usual procedure we argue that significant
scattering is confined to the first few Fresnel zones
and that therefore x and y are small. For this reason,
the variation of r 0 and ri affects mainly the exponent
in Eq. (1) and the denominator for can be replaced by
rooro and taken outside the integral. Also the expan-
sion given in Eq. (3) needs to be taken only to second
order in x and y and first order in . By putting all of
these arguments together we finally obtain H(co,t) in
the form:
H(oo, t) = exp[jk(røø + 4rr00r0
ax ax ay ay +
x exp(-Ax 2 -By 2 +jkF[) dxdy ,
where
A =-j(ksin2½/R)+A ' ,
B=-j(k/R)+B' , (5)
F = 2 sing, . (6)
II. A DETERMINISTIC SURFACE MODEL
As we observed earlier, we conjecture that the fea-
ture of the windblown surface responsible for sideband
asymmetry in an otherwise symmetric cross wind con-
figuration is that the magnitude of the surface slope on
the downwind side is greater than it is on the upwind
side (Fig. 1). In order to test this conjecture we con-
sider in this section a deterministic surface corrugation
which is a function of y and t only, moving in the y
direction. A wave having a steeper slope on one side
than on the other can be obtained from the proper com-
bination of a fundamental and a second harmonic of a
Fourier expansion. Specifically one can show that the
1185 J. Acoust. Soc. Am., Vol. 68, No. 4, October 1980 Tuteur eta/.' Asymmetric Doppler amplitudes 1185
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 155.33.16.124 On: Thu, 27 Nov 2014 05:09:15expression
z(t) =h 1 cos(cot + (p) + h 2 cos2cot , (7)
has the desired form if h2 =0.25 h i and (p=45 ø (Fig.4).
We accordingly evaluate Eq. (5)with ! given by
(x,y,t)=hlcos(py -sqt + qb) +h2 cos(2py - 2sqt) . (8)
This is a nonsinusoiclalwave of the form of Eq. (7) moving
in the y direction with a phase velocity of 2/p. The
fundamental period in the y direction is 2r/p and the
temporal surface frequency is Sq. In order to perform
all the operations indicated in Eq. (5) it is necessary
to assume small Rayleigh parameter so that the factor
exp(-jkF:) can be approximated by 1 -jkF[. Other
details of the computation are given in Appendix B.
After somewhat lengthy but straightforward manipula-
tion the result appears in the form
/(co0 t)= , 'AB/ 4rooro(
x Do + [D.exp(-jmt)+D.exmt) . (9)
This shows that the interaction of an acoustic sinu-
soidal wave of frequency w 0 with a moving surface
corrugation given by an expression such as Eq. (8)
results in the generation of frequencies w 0 m with
m ranging from 0 to at least 6. [The number 6 rises
from the first-order approximation of the term exp
(-jkF). Higher-order approximation of this term re-
suit in more frequencies.]
The coefficients of the first sideband frequencies
above and below w 0 are, respectively, D. andD1..
Examination of the formulas for ese coefficients
(as well as those for D . and D . for m up to 4) reveals
that these coefficients are in general t complex con-
jugates of each other. Their magnitudes are therefore
different, and the corresponding sideband amplitudes
e therefore equal. We find specificallyat tD.(Wo,t)I
and [Dx.(Wo, t)I can be eressed in the form:
iD,.(wo,t) 12 =(x + y2 + y sin2) exp(- pX/2B) ,
iD,.(,t)i=(X +y2 _y si½) ex-f/2) , (0)
where
X =hlk(sin½l - (px/41h + 8h -(1/k sin½ )]},
y =-pXhlhx . (11)
FIG. 4. z(t)=eos(O+ d))+ keos20 for various k and q. (1) k=0.2,
q=45ø; (2) k=0.2, qb=135ø; (3)
k=0.3, q=Oø; (4) k=0.3, q=90 ø. We observe that the asymmetry is most pronounced
when qb = 45ø which also represents the most asymme-
tric saw tooth wave (Fig. 4). There is no asymmetry
when qb =0 ø or (P =90ø; also if the amplitude h of the
second harmonic of the surface wavefunction is zero.
The upper sideband amplitudes are larger than the
lower sideband amplitudes for all 0<(p <90ø; i.e., for
wind driven waves moving in the direction of the steep-
er wave front. Values of (P outside the range (0,90*)
make little sense for hydrodynamic reasons. Hence the
upper sideband amplitudes are generally larger in the
cross-wind geometry considered here.
Although the surface generated in the model tank is
not deterministic, we use some of the random-surface
parameters to get an idea of the magnitude of the asym-
metry. Typical values for these parameters are given
in Appendix F. Here we use hi=0.1 cm, h2=hl/4 ,
(P=45ø,k=21.25 cm 'l, p=0.63 cm 'l o , g=17. With
these values kFh 1 = 0.62 < 1, and .
l D 1. [ -- 0.6223 exp(- p2 lB) ,
0.6207 exp(-p/B) . (12)
Thus the upper and lower sideband amplitades differ by
about 0.4ø under these conditions. A larger value for
h i and a correspondingly lower value for k (to keep
kFhi < 1) could result in a somewhat larger percentage
difference between and [Dl.[; for instance with
h i =0.2 cm, k = 10.63 the percentage difference would
be 0.8%.
III. RANDOM SURFACES-HIGH RAYLEIGH
PARAMETER
If the surface is random, the scatter channel can be
regarded as a randomly time-varying linear system
having a transfer function H(coo,t). This represents the
instantaneous amplitude and phase at the receiver due
to a sinusoidal signal with frequency co o at the transmit-
ter. The received signal is therefore
r(t) =H(co0, t) exp(-jcoo t) (13)
and its autocorrelation function is
r(t)r*(t + r)= H(coo,t)H*(coo,t + r) exp0%0 r) . (14)
We use the notation
oh(coo, r)=tt(coo,t)H*(coo,t + r) . (15)
Then the output power spectrum is obtained by Fourier
transforming Eq. (14)-
r(co, co0) = f. (co0,*)exP[j(co0-co)r] dr. (16)
This can be regarded as. the received signal power at
frequency co when the input frequency is coo.
The channel correlation function (co0, ) is obtained
by substituting Eq. (5) into Eq. (15). The result is
considerably simplified by neglecting the cross wind
1186 J. Acoust. Soc. Am., Vol. 68, No. 4, October 1980 Toteur eta/.' Asymmetric Doppler amplitudes 1186
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 155.33.16.124 On: Thu, 27 Nov 2014 05:09:15slope terms, i.e., by setting a/ax-O. This is area- on being studied here.
sonable simplification since the transmitter geometry
is assumed to be symmetric in the x direction and
therefore x-axis slopes should not affect the phenomen- by
I
= 2 X[A dy ayxexp(-By-B*y)[jkF(1-x)+2Bxyx][-jkF(1-g.)+2B*.y]exp[jkF(; 'k(wø'r) 16r00r0 I I
where A, B, and F are defined in Eq. (6), the asterisk
refers to complex conjugation l -= (Yi),[2 = (Y2), and
, and 2 are the surface slopes evaluated at y and Y2, respectively.
The averaging operation indicated by the overbar
in Eq. (17) is easily performed if the surface heights
are assumed to be Gaussian and uncorrelated from
the slopes. A possible basis for the second part of this
assumption is that the slopes depend largely on small
ripples of the surface, and that ripples of any slope can
occur at any surface level. The assumption of Gaussian
surface heights is often used and is based on measure-
ments of the marginal distribution of ocean surface
heights given in Kinsman ø as well as on experimental
evidence obtained in model-tank measurements by Zor- 6
nig.
If the slopes and heights are uncorrelated, the aver-
age shownin Eq. (17) breaks up into the product of the aver-
ages of the bracketed terms involving the surface slopes
and the exponential term involving surface displace-
ment. The latter is simply the two-dimensional char-
acteristic function of the surface. We use and r/ to
denote displacements along the x and y axes, respec-
tively. Then for a statistically stationary Gaussian
surface the characteristic function is given by
exp[jkF([,-[2)]' = exp{-k2F2q2[ 1 -I,(,/,r)]}, (18)
where rr 2 = - is the mean-square height and ,r(, 7, r)
= (1/( 2) (x,y ,t)[(x + , y + rl, t + r) is the normalized
surface autocorrelation function. In accordance with
a frequently used model, we let
- 2 r x ) ,r(,/,r)=exp -2-X- - cos9( -c,). (19)
This represents a surface wave motion with velocity c
in the y direction.
In this section, we consider the large Rayleigh param-
eter case. Hence we assume that
k2F2o'2 >> 1, (20)
and therefore the integrand in Eq. (17)will have signifi-
cant values only if (,r/, -) is close to unity. Thus if
we expand (,r/,r) in a Taylor series about
-0 only the first few terms of the series are signifi-
cant. The displacement does not need to be consid-
ered since the integration in Eq. (17) is only over the
y's. Thus an expansion of (.) correct to second order
is With this simplification we find that 4(cv0, r) is given
(17)
I
r] 2 -2 p2
ß (o,v, - (v ,
and then the characteristic function to be used in Eq.
(7) is
_
[_k2F2G2 2 T2 2)] = exp 2 [ + + Px(V -cr) .
The average of the bracketed terms in Eq. (19) in-
volyes moments of the surface slope. The first-order
moment is zero since the surface has zero average
slope. The second moment appears in the form [x.
For stationary statistical surface we assume that this
has the form (21)
(22)
',2 = 2 exp[ -(7 2/2A)] ß
The third-order moment appears in the form 2
+ t} 2 ' and we assume that this has the form (23)
(24)
The parameters e 2 and e,3 are generally positive, and
the appearance of the minus sign in front of is ex-
plained in Appendix C. Both of these slope moments
are, strictly speaking, also a function of ; however,
inclusion of the r dependence only complicates the
form of the final result without adding anything essen-
tial. In fact, even the V dependence turns out to be un-
important in the final result. We also neglect the fourth-
order moment l[z This moment is small because Y2'
l and x are normally much smaller than unity as a
result of hydrodynamic constraints. In any case, this
moment would not contribqte any asymmetry to the final
result.
It is now possible to evaluate Eq. (17) and then Eq.
(16). The computation is lengthy, but basically straight-
forward. Some of the details are contained in Appendix
D. An approximate expression for F(v,v0) obtained by
assuming a wide beam pattern [i.e., in Eq. (6) A' <<k
sin2/R ,B' << k/R] is
(27r) l/2exp{-I(co ' = +r00)
A + z zz' + , (25) kf ,; koFc
where a is a function of frequency, beam pattern, and
transmitter receiver geometry, whose precise form is
given in Eq. (DS).
1187 J. Acoust. Soc. Am., Vol 68, No. 4, October 1980 Tuteur et aL' Asymmetric Doppler amplitudes 1187
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 155.33.16.124 On: Thu, 27 Nov 2014 05:09:15Observe that if the second- and third-order slope
moments in Eq. (25) are small, r(co,co 0) is essentially
a Gaussian function that peaks at co = coo and that has
a width determined by a. The effect of the third-order
term is a skew and a shift of the maximum point away
from co o . The amount of the shift is proportional to ½3,
the third-order moment of slopes. As noted above,
e 3 >--0, and therefore the peak in r(,o,co 0) occurs for
co'co o.
To get an estimate of he magnitude of the asymmetry
we calculate r(co, coo) for co- coo- :gs2, where g-kFer
is the Rayleigh parameter, and S2-pc is the surface
frequency. By inspection of Eq. (DS) one can see that
a =gS2, especially for wide beam pattern (small B ');
hence lco- co01 -gz defines the approximate spreading
width. Values of the parameters are again taken from
the model-tank measurements; in particular values for
½. and ½3 can be obtained from slope measurements
presented in 12. Typical numerical values used in this
calculation are summarized in Appendix F. When these
are substituted in Eq. (25) we find that -
r(ag) - r(- g) _e( m
r0(g) = 1 - 2e z + (y2e2/A2 + e(y 2p2 =' 0.0011 ,
where F0(.) is the expression in Eq. (25) with ½3 = 0.
We see that the asymmetry for the random surface is
substantially smaller than for the deterministic surface,
a not unexpected result. However, more to the point,
whether random or deterministic the calculated asym-
metry is very small; in fact so small as to be neglig-
ible in practice. The experimentally observed asym-
metry is therefore apparently caused mostly by factors
other than the slope asymmetry analyzed here.
An example of measured spectra is shown in Fig. 5.
For details of the experimental procedure see Zorn-
ig. 13 Spectra computed from Eq. (25) with parameter
values matching those for the experimental results as
closely as possible (Appendix F) are also shown in this
figure. The theoretical results are seen to match the
experimental ones quite closely, and this agreement
can be regarded as good confirmation for the theoretical
model analyzed here. Although all of the spectra ap-
pear to peak for values of co above the transmitter fre-
quency, the spectral asymmetry is fairly small.
Expressions for ½. and ½3 corresponding to the deter-
ministic surface considered in Sec. II are derived in
Appendix C. The results are
- _2,2
e. =p n I + 2p2h , (26)
e = 2p3hh2 sin2(p . (27)
Of particular interest is the variation e with the phase
angle. Observe that (3 reaches a maximum for b = 45 ø
and goes to zero for b = 0 o or b = 9 0 o. Thus the re suit
obtained here for the random surface corresponds to
that obtained earlier for the deterministic surface.
IV. RANDOM SURFACE-SMALL RAYLEIGH
PARAMETER
If the Rayleigh parameter is small, then in Eq. (18)
k"F"cf' c 1. Under these conditions, one can approximate (HZ) !
FIG. 5. Measured and calculated spectra for two different
center frequencies. (a) f0 = 0.256 MHz (b) f0 = 0.512 MHz. The
dashed line is the result calculated by use of Eq. (25) with the
parameter values of Appendix F. The solid line is the experi- mental result.
the exponential function by the first few terms of the
Taylor series, so that Eq. (18) becomes
exp[jkF([ - 2] -- exp(- k2F2o-2) [ 1 + k2F2cr2(,tl, ')]
= exp(- k2F2(y 2) [1 + k2F2G 2
exp - cosp(-c .
We again use = 0 since the integration in Eq. (17) is
only over y.
For the second- and third-order slope moments we
assume that the space and time dependence is similar
to that of the surface height, i.e. oscillatory with the
same wavenumber and velocity, but with possibly a dif-
ferent damping constant. Thus
['12 =e2 exp[ -()2/2A)] cosp(tZ -c) (29)
2-;, '4- ;' ;:-= exp[-(/2/2A)] cosp(/ cf) (30) 2 - - '
In spite of the similarity in the form of the height and
slope moments, we regard the heights and slopes to be
statistically independent so that the averaging operation
in Eq. (17) breaks up into an average of slope terms
multiplied by the surface characteristic function. Ex-
amination of possible cross product terms indicates
that these terms should in any case be small so that
they would not cause any major change in the final
result.
1188 J. Acoust. Soc. Am., Vol. 68, No. 4, October 1980 Tuteur et aL' Asymmetric Doppler amplitudes 1188
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 155.33.16.124 On: Thu, 27 Nov 2014 05:09:15Substitution of Eqs. (28)-(30) in Eq. (17) is now again
possible. Details of this calculation are contained in
Appendix E. The interaction of the various cosp(-½-)
terms results.in the generation of terms of the form
coskp(r-½)withk-0, 1, 2. The terms withk-0 re-
present coherent or specular reflection, while those
with k---2 correspond to second-order Doppler shift.
We are primarily interested in the first-order Doppler
term. The result is again considerably simplified if
one assumes a large beam pattern. Under these condi-
tions we find that
exp[j(w o + pc ) - k272F 2]
2(too + ro) 2
X exp[_ ('p2/2 I ]2)] [k2F2c2(1 _ 2e2)
+e 2 2 2 pe/kF] 2/(k F A2) + . (31)
Since, by Eq. (16) the output spectrum is obtained by
multiplying (0, r) by exp(-j') and integrating over
we see that (i, l + represents the first upper sideband and
(l,l. the first lower sideband. Since e 3 > 0 we see, as
before, that the upper sideband has a larger amplitude
than the lower one. Also, since I 3 is numerically very
small the difference in sideband amplitudes is probably
even smaller than in the large-Rayleigh-parameter
case.
V. SUMMARY AND CONCLUSION
We have investigated three surface scattering models
to explain the Doppler-shift asymmetry observed in
what appears to be a symmetric cross wind configura-
tion; i.e., transmitter and receiver at the same depth.
The three models were (a) deterministic surface having
a second-harmonic component to simulate the differ-
ences between upwind and downwind slopes, (b) a ran-
dom surface with large Rayleigh parameter, and (c)
a random surface with small Rayleigh parameter.
Since the observed effect appears to be related to the
scattering slope, all three models used explicit slope
information in their formulation. In the deterministic
model, the calculated asymmetry was shown to be di-
rectly related to the phase between the second-harmonic
and fundamental of the assumed surface wavefunction.
Maximum asymmetry corresponds to a maximally
asymmetric saw tooth surface, no asymmetry is obser-
ved when the surface consists of symmetric pulses.
In the random surface model, asymmetry of the Dop-
pler sidebands was shown to be related to the magnitude
of the third-order moment of surface slopes. This mo-
ment has also been calculated for the deterministic
slope, and it was shown to have its maximum value
when the deterministic surface is maximally asymme-
tric and to vanish if the deterministic surface is sym-
metric. Thus the third-order moment appears to con-
rain the same kind of information as the phase shift
between the second harmonic and fundamental.
Although the theory shows a connection between the
asymmetry of surface slopes and the spectral asymme-
try in the frequency-spreading function, the magnitude of spectral asymmetry contributed by reasonable
amounts of surface asymmetry is too small to explain
the fairly large spectral asymmetries that have some-
times been observed in experiments. There is some
reason to believe that very small misalignments in the
measuring apparatus may result in fairly large spectral
asymmetries, and such misalignments may therefore
offer a better explanation of the observed effects. A
further analysis of this effect will be presented in a
forthcoming paper.
ACKNOWLEDGMENT
This work was supported by the Office of Naval Re-
search, Code 480, and by the National Science Founda-
tion, Engineering Division.
APPENDIX A. DERIVATION OF EQ. (1)
The Helmholtz integral describing the pressure field
in one of the half-spaces defined by the surface S is
given by
r /j
-p dS . (AX)
In this expression p(A) refers to the pressure at the
arbitrary point A below the surface, S is the surface,
n is the outward normal to the surface, and rx is the
distance from A to the point z =(x,y,t) on the surface.
Also k = w/c is the acoustic wavenumber.
The incident sound pressure at the surface point z
is given by
(A2) p o(X, y, z) = PoBe(x ,y )[ exp(jkro)/ro] ,
where Be(x,y) is the projection of the beam pattern on
the x,y plane, r 0 is the distance from the center of the
acoustic source to the point z, and where P0 is the
amplitude of the acoustic source.
One boundary condition at the surface is obtained by
assuming the surface to be a pressure-release boundary
so that
Po +P =0 on S . (A3)
Also if the surface is regarded as "locally flat"; i.e.,
if it can be approximated by a plane tangent to the act-
ual rough surface, then we have the
Kirchhoff app ro xi mation
apo
an= an onS. (A4)
Substitution of Eqs. (A2) through (A4) into Eq. (A1) then
results in
p,(A, Pø f a ½ exdjk +r,,]) =4rr e(x y ) (tO dS . ' r0rl (A5)
We convert from an integration over S to one over x,y
by using the transformations:
dS-- + + dx dy , (A6)
1189 J. Acoust. Soc. Am., Vol. 68, No. 4, October 1980 Tuteur etal.: Asymmetric Doppler amplitudes 1189
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a aL -/ aL aL a . (A7) an - + + - a- - y ay .
Furthermore we define the channel transfer function
H(w,t) by
H(o , t) =p(A )/P o . (AS)
Combining all of these equations then results in Eq. (1).
APPENDIX B. DERIVATION OF EQ. (9)
The surface model is that of Eq. (8), repeated here
for convenience:
L(x,y ,t)=h cos(py -t + (p) + h 2 cos(2py -2t) . (B1)
Thus
aL
ax -0 ,
a__L_
ay -- ph sin(py - t +d>) -2ph2sin(2py-2t) ß (B2)
Since aL lax =0 the differentiation with respect to in
Eq. (5) gives
H(o,t)=exp[jk(roo+r,o)]{. 4rr r00rl 0 f:f kF [1 - (,)
+ 2Byyy exp(-Ax 2 -By 2 +jkF) dxdy . (B3)
We now assume that kFL << 1 so that
exp(jkF) = 1 + jkF:. (B4)
We finally substitute Eqs. (B1) and (B2) into Eq. (B3),
using Eq. (B4) and perform the indicated integration.
The result is Eq. (9) withD 0 andDl as in Eqs. (10) and
(11). Higher-order D,s can formally be calculated, but
because Eq. (B4) is only a first-order approximation,
their use in predicting the amplitudes of the higher-or-
der sidebands cannot be justified.
APPENDIX C. SIGN OF THE THIRD-ORDER MOMENT
OF SLOPES
The wind is assumed to be blowing along the y axis.
We approximate the windblown surface wave by the sim-
ple sawtooth wave shown in Fig. 6. The peak-to-peak
wave height is h, the period is A, and A/2 + zx and A/2
-zx are the proje9tions of the two slope segments on the
horizontal axis. We take zx > 0 for wind blowing in the
positive direction as shown. For this wave we have
: + . h a
=(A/2+A)Z(A/2-A)z (-2AA)<0. (C1)
wind
FIG. 6. Idealized surface profile for estimating ½3- 3> 0 E3= -2L .
For a deterministic wave of the form
z(y ,t)=h cos(py -ot + (p) + h 2 cos(2py - 2ot) , (C2)
the slope in the y direction is
z(y ,t)= -ph sin(py - o.,t + (p) - 2ph 2 sin(2py - 2cot) . (C3)
Then
z2(y ,t)=p2h sin2(py -vt
+ 492hh2 sin(py -ot + (p)sin(2py -2ot)
+ 492h sinZ(2py - 2c)/) . (C4)
The average is most conveniently computed as a time
average over one period. This gives
e 2 z2(y t)dt 1-2'-2 292h (C5) , --P r I + ß
Similarly
ea= - 2z2(y ,t)
= 2pah h sin2(p , (CO)
for0 <qb<rr/2 ca >0.
APPENDIX D. THE OUTPUT POWER SPECTRUM
FOR LARGE RAYLEIGH PARAMETER
If surface heights and slopes are uncorretated, the
average indicated in Eq. (17) breaks up into the product
of two averages'
[jkF( 1 - 2 h) + 2BLhy ,][-jkF( 1 - 22) + 2B* 2y2]
x exp[jkF ( - .)].
The average of the exponential term is the two-dimen-
sional characteristic function of the surface, and for
large Rayleigh parameter we assume this to have [he
form of Eq. (22). Expansion of the term involving the
slopes results in
+j2kF[B2([2 - 21i2 ) - By i(, - l 2 )] Y2
+ 4 lB 12 L,i2y,y 2 . (D1)
As noted in the body of the paper [l = [2 = 0, and we
also assume that [-2 is negligible. Expressions for the remaining slope correlations are given in Eq. (23)
and (24).
The integration of Eq. (17) with Eqs. (22) to (24) and
(D1) substituted is facilitated by making the change of
variable r/=Y2 -Yi. The expression resulting from the
integration is very lengthy and no purpose is served in
reproducing it here. The expression can be consider-
ably simplified by assuming a large beam pattern; i.e.,
1190 J. Acoust. Soc. Am., Vol. 68, No. 4, October 1980 Tuteur eta/.' Asymmetric Doppler amplitudes 1190
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 155.33.16.124 On: Thu, 27 Nov 2014 05:09:15A' << (k/R) sin'p = IA I '
Then (I,(o0,,) can be put into the approximate form'
(Wo, 7) = (too + r10)2[(1 - 22) + (! --T2)E 2 (D2)
(D3) F(w,w 0) = f: $(r 0, *) exp[J(w0 - w),] d, , (D8)
and after a straightforward integration we obtain Eq. (25).
APPENDIX E. OUTPUT SPECTRUM FOR SMALL
RAYLEIGH PARAMETER
+J2 ] ex- ('x/2) ,
where
2 2 B,kF22[ 1 + (A/A)]}
2 2 2
-- + , (D4)
(m)
t . k oF r 'p'c . ( D7 )
To get Eq. (D5), it is assumed that p2A2,>> 1. The power
spectrum is obtained from As in Appendix D the assumption that surface heights
and slopes are uncorrelated permits us to separate the
average indicated in Eq. (17) into the product of the
average of the bracketed terms and of the exponential
term. The average of the exponential term yields the
characteristic function of the surface heights, which
for small Rayleigh parameter is given by Eq. (28). The
average of the bracketed terms involving the slopes has
the same form as in Appendix D; however we use Eqs.
(29) and (30) instead of Eqs. (23) and (24) for the second
and third moment of the slopes. As before, we make
the change of variable Yl =Y ,Y. =Y + r/, and the integra-
tion over y results in
= 2B' r/ "F'(1-2½0+ B; .' rl2) ½2 exp(- 2--) cosp(rl - c,r)
+jkFIBI B (El)
Upon multiplying out the integrand we obtain terms
containing zeroth, first, and second powers of cosp(g
-c,). The only term of interest is the one containing
the first power of cosp(r -,c) which represents the
first-order Doppler shift. This term is designated by
l(w0,7). The integration over r is simplified by neg-
lecting terms such as exp(- r/2/2A2), exp(-r2/2A),
exp(-r2/2A2). This is justified since we found in the
large Rayleigh-parameter case that for reasonably large
beam patterns, these terms are negligible compared with
with exp(- 2 lB {2/2B')o The final integration is facilit-
ated further by expanding the cosine as the sum of two
exponential terms. This results in two terms designa-
ted by l,(co0,, ) and l.(co0,, ) for upper and lower Dop-
pler-shift components, respectively. After using the
approximate definitions for [A [ and lB [ from (D2) and
(D3) we finally obtain Eq. (31).
APPENDIX F. PARAMETER VALUES USED IN
CALCULATIONS
The following set of parameters are typical values
measured in a model tank with a wind-driven surface.
The measurement technique used to obtain these num-
erical values is that described in Refs. 6, 12, and 13.
w = acoustic frequency = r x 106 rad/s = 0.5 MHz
c = velocity of sound = 1478 m/s
k = w/c = acoustic wavenumber= 21.25 rad/s /
p= surface wavenumber= 0.63 rad/cm surface wave velocity =40 cm/s
surface frequency= 25.2 rad/s
rms surface height=0.26 cm
grazing angle = 17 ø
surface correlation distance in the y direction
= 12 cm
=
(2 =
(3 = rio= too=distances from transducers to specular point
--304.8 cm. Transducer beam width= 22r x 106/½o
= 22 ø for 0.5-MHz signal
flayleigh parameter = 2kr sine = 3.23
mean square surface slope = 0.04
skew of surface slope pdf = 0.2
2y½'s= 3rd-order surface slope moment=0.0032.
1E. Y. Harper and F. M. Labianca, "Scattering of Sound from a
Point Source by a Rough Surface Progressing over an Iso-
velocity Ocean," J. Acoust. Soc. Am. 58, 249-264 (1975).
2F. M. Labianca and E. Y. Harper, "Asymptotic Theory of
Scattering by a Rough Surface Progressing over an Inhomoge-
neous Ocean," J. Acoust Soc. Am. 59, 799-812 (1976).
aW. A. Kuperman, 'n Scattering from a Moving Rough Surface
without a Kirchhoff or Farfield Approximation," J. Acoust.
Soc. Am. 64, 1113-1116 (1978).
4F. B. Tuteur and H. Tung, "Asymmetric Doppler Amplitudes
in the Surface Scatter Channel," J. Acoust. SOe. Am. Suppl.
59, S89 (1976). Details of the work described in this abstract
are in Tech. Rep. C/ (1 April 1976), Department of Engi-
neering and Applied Science, Yale University, CT.
5H. Tung, "Frequency Spreading in Underwater Acoustic Signal
Transmission," Ph.D. thesis, Yale University, May 1980.
1191 J. Acoust. Soc. Am., Vol. 68, No. 4, October 1980 Tuteur eta/.' Asymmetric Doppler amplitudes 1191
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 155.33.16.124 On: Thu, 27 Nov 2014 05:09:15Also available as Final Report 8001, ONR Contract N00014-
77-C-0237 (April 1980).
6j. G. Zornig and J. F. McDonald, "Direct Measurement of
Surface Scatter Coherence by Impulse Probing," J. Acoust.
Soc. Am. 55, 1205-1211 (1974).
?D. R. Melton and C. W. Horton, St., "Importance of the
Fresnel Correction in Scattering from a Rough Surface; I-
phase and Amplitude Fluctuations," J. Acoust. Soc. Am. 47,
290-298 (1970).
8j. F. McDonald, "Fresnel-Corrected Second Order Interfre-
quency Correlation for a Surface-Scatter Channel," IEEE
Trans. Commun. Theory COMM-22(2), 138-145 (1974).
9B. E. Parkins, "Reflection and Scattering from a Time-Vary-
ing Rough Surface--The Nearly Complete Lloyd's Mirror
Effect," J. Acoust. Soc. Am. 49, 1484-1490 (1971). 1øB. Kinsman, Wind Waves, Their Generation and Propagation
on the Ocean Surface (Prentice-Hall, Englewood Cliffs, NJ,
1965), p. 372.
llH. Medwin and C. S. Clay, "Dependence of Spatial and Tem-
poral Correlation of Forward-Scattered Underwater Sound on
Surface Statistics; II Experiment," J. Acoust. Soc. Am. 47(5)
(part 2), 1419-1429 (1970).
12j. G. Zornig, P.M. Schultheiss, J. Snyder, S. Singhal, and
D. C. Gilbert, "Surface Scattering Studies," Final Report
7905, ONR Contract N00014-75-C-1014 (August 1979).
13j. G. Zornig, "Physical model studies of forward scatter fre-
quency spreading," J. Acoust. Soc. Am. 64(5), 1492-1499
(1978).
laF. B. Tuteur and H. Tung, "Frequency Spreading in Under-
water Acoustic Surface Scatter" (in preparation).
1192 J. Acoust. Soc. Am., Vol. 68, No. 4, October 1980 Tuteur et al.: Asymmetric Doppler amplitudes 1192
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 155.33.16.124 On: Thu, 27 Nov 2014 05:09:15 |
1.3284515.pdf | Ultrafast spin-transfer switching in spin valve nanopillars with perpendicular
anisotropy
D. Bedau, H. Liu, J.-J. Bouzaglou, A. D. Kent, J. Z. Sun et al.
Citation: Appl. Phys. Lett. 96, 022514 (2010); doi: 10.1063/1.3284515
View online: http://dx.doi.org/10.1063/1.3284515
View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v96/i2
Published by the American Institute of Physics.
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Downloaded 08 Jun 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsUltrafast spin-transfer switching in spin valve nanopillars with perpendicular
anisotropy
D. Bedau,1,a/H20850H. Liu,1J.-J. Bouzaglou,1,5A. D. Kent,1J. Z. Sun,2J. A. Katine,3
E. E. Fullerton,4and S. Mangin5
1Department of Physics, New York University, 4 Washington Place, New York, New York 10003, USA
2IBM T. J. Watson Research Center, P .O. Box 218, Yorktown Heights, New York 10598, USA
3San Jose Research Center, Hitachi-GST, San Jose, California 95135, USA
4CMRR, University of California, San Diego, La Jolla, California 92093-0401, USA
5IJL, Nancy-Université, UMR CNRS 7198, F-54042 Vandoeuvre Cedex, France
/H20849Received 30 October 2009; accepted 10 December 2009; published online 14 January 2010 /H20850
Spin-transfer switching with short current pulses has been studied in spin-valve nanopillars with
perpendicularly magnetized free and reference layers. Magnetization switching with current pulsesas short as 300 ps is demonstrated. The pulse amplitude needed to reverse the magnetization isshown to be inversely proportional to the pulse duration, consistent with a macrospin spin-transfermodel. However, the pulse amplitude duration switching boundary depends on the applied fieldmuch more strongly than predicted by the zero temperature macrospin model. The results alsodemonstrate that there is an optimal pulse length that minimizes the energy required to reverse themagnetization. © 2010 American Institute of Physics ./H20851doi:10.1063/1.3284515 /H20852
Spin-transfer devices hold great promise for achieving
fast and high density nonvolatile magnetic random-accessmemory /H20849ST-MRAM /H20850.
1There has been an intense effort
world-wide to reduce the switching current and the switchingtime, while maintaining thermally stable nanometer scalemagnetic elements.
2–6Devices with perpendicular anisotropy
show great promise in this respect, since the high anisotropymay lead to stable elements at room temperature, even atsub 10 nm lateral sizes. In addition, the spin-transfer figureof merit, the ratio of the threshold current for switchingto the energy barrier to thermal reversal, has been shown tobe small in all-perpendicularly magnetized spin-valvenanopillars,
7as expected based on theoretical models.8Re-
cently switching currents on the order of 100 /H9262A have been
observed in quasistatic studies of 50 nm diameter nanopillarsat room temperature.
7
A key issue for memory applications is the possibility of
fast, energy-efficient switching under short current pulses.Several groups have reported magnetization dynamics onshort time scales in nanopillars with in-plane magnetizedelements.
9–13Short-time switching has not been studied thus
far in perpendicularly magnetized layers. However, resultshave been reported for devices with an in-plane free layerand a perpendicular polarizer.
14–16
In this letter, we present studies of ultrafast spin-transfer
switching in all-perpendicularly magnetized spin-valve nano-pillars. We measure the probability of switching for short/H20849100 ps–10 ns /H20850current pulses as a function of pulse ampli-
tude and duration. We also examine how the amplitude-duration boundaries depend on the applied perpendicularmagnetic field and find a simple relation between the pulse
amplitude and duration.
Our experiments were conducted on multilayer spin-
valves fashioned into nanopillars that contain two magneticlayers, a reference and a free layer, both with perpendicularmagnetic anisotropy, separated by a thin Cu spacer layer. Thefree layer is a Co/Ni multilayer while the reference layer is
designed to have higher magnetic anisotropy and coercivitythan the free layer. It consists of a composite of Co/Pt andCo/Ni multilayers. The layer stack is Ta /H208495/H20850/Cu/H2084930/H20850/Pt/H208493/H20850
/H11003/H20851Co/H208490.25 /H20850/Pt/H208490.52 /H20850/H20852/H110034/Co/H208490.25 /H20850//H20851Ni/H208490.6/H20850/Co/H208490.1/H20850/H20852/H110032/
Cu/H208494/H20850//H20851Co/H208490.1/H20850/Ni/H208490.6/H20850/H20852/H110032/Co/H208490.2/H20850/Pt/H208493/H20850/Cu/H2084920/H20850/Ta/H208495/H20850
/H20849layer thickness in nanometer /H20850, similar to samples used in
Ref. 7. The samples were patterned combining e-beam and
optical lithography. The sample sizes studied were 50
/H1100350,
100/H11003100, and 100 /H11003150 nm2. In total 19 samples were
studied. The field was always applied perpendicular to thefilm plane, i.e., along the easy anisotropy axis. Before start-ing an experiment the reference layer was saturated with aperpendicular field of about 0.5 T. During pulse injection theapplied fields do not exceed 200 mT, and do not affect themagnetic state of the reference layer. All measurements wereperformed at room temperature.
The samples have two pairs of contacts to the top and
bottom electrode. On both contact pairs bias tees were usedto split the high and low frequency circuits. The first contactpair was connected to a pulse generator and the second pairto an oscilloscope to record the transmitted waveform.Through the dc port of the bias tee, a low frequency currentof 300
/H9262Armswas injected into the sample to measure the ac
resistance using a standard lock-in technique. While measur-ing the ac resistance the sample could be subjected both toshort time pulses as well as to dc currents. A conventionalcurrent flowing from the bottom to the top layer is defined aspositive.
By measuring the resistance for all possible contact con-
figurations, an equivalent circuit model was generated thatallows relating the pulse amplitude to the current through thespin valve. Replacing the pulse generator with a microwavesource and measuring the amplitude of the transmitted signalwith the oscilloscope, we find that the measured transmissionparameters agree with the circuit model with a few percentaccuracy.
a/H20850Electronic mail: db137@nyu.edu.APPLIED PHYSICS LETTERS 96, 022514 /H208492010 /H20850
0003-6951/2010/96 /H208492/H20850/022514/3/$30.00 © 2010 American Institute of Physics 96, 022514-1
Downloaded 08 Jun 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsTo determine the behavior of the sample in the quasi-
static case, the dc current was swept for a series of fixedfields. To determine the dynamic behavior of the samples,pulses between 100 ps and 10 ns duration and amplitudes ofup to 2 V were injected while maintaining a constant externalmagnetic field. After each pulse injection the ac resistancewas measured to determine whether a switching event hadoccurred. Each pulse injection was repeated 100 to 10 000times to obtain the switching probability.
Figure 1/H20849a/H20850shows the switching phase diagram of a typi-
cal 100 /H11003100 nm
2nanopillar. The color in this diagram is
the resistance obtained during the down sweep subtractedfrom the one from the up sweep. This type of phase diagramillustrates the bistable region for which both parallel /H20849P/H20850and
antiparallel /H20849AP/H20850magnetization configurations coexist. Fig-
ure1/H20849b/H20850shows a zero field current sweep across the hyster-
etic region. The critical current to switch from P to AP is 6.8mA and from AP to P is 3.2 mA, and the magnetoresistanceis 0.3%. We note that for these quasistatic measurements weexpect lower switching fields and switching currents than forshort pulses due to the longer time available for thermallyassisted reversal.
From the width of the AP region obtained for a field
sweep at zero current we can determine the switching field tobe 95 mT. We note that the field-swept hysteresis loop is notcentered at zero field. This is associated with dipolar inter-actions between the reference and free layers. The loop shiftis 16 mT.
For the following analysis we will focus on the AP to P
transition. The P to AP transition has been shown to givesimilar results. The switching probability as a function ofpulse duration and amplitude is shown in Fig. 2/H20849a/H20850. For low
current durations/amplitudes the switching probability iszero /H20849blue /H20850, whereas for longer durations or amplitudes the
switching probability reaches one /H20849red/H20850. The black curve de-
picts a switching probability of 50% and defines the bound-ary between the P and AP states. Magnetization switching forpulse durations down to 300 ps is observed.
We now describe the behavior in the framework of a
zero temperature macrospin model. Starting from theLandau-Lifshitz-Gilbert /H20849LLG /H20850equation with spin torque,
Sun
8linearized the equations around the initial angle to de-
termine the limits of the stability in the small cone anglelimit. Sun then equated the onset of instability with the oc-currence of a switching event, obtaining both the critical cur-rent as well as the switching time. The predicted zero tem-perature critical current for AP-P /H20849I
c0+/H20850and P-AP switching
/H20849Ic0−/H20850has the following form:
Ic0/H11006=/H9251
pg/H110062e
/H6036MsV/H92620/H20849H/H11006HK/H20850, /H208491/H20850
where /H9251is the damping and pis the spin-polarization of the
current. g/H20849/H9258/H20850describes the angular variation in the spin
torque, for AP to P switching g+=g/H20849/H9266/H20850and correspondingly
g−=g/H208490/H20850for P to AP switching.
Alternatively to the linearization,8the full LLG equation
can be integrated up to the time when the critical angle isreached, equating this time with the switching time. Both thelinearization in the small cone angle limit as well as the fullsolution of the LLG equation lead to a switching time of thefollowing form:
1/
/H9270=A/H20849I−Ic0/H20850. /H208492/H20850
The parameter A, which we denote the dynamic parameter,
governs the switching rate. Plotting the 50% switching prob-ability boundary of Fig. 2/H20849a/H20850after a change of coordinates,
we see that the boundary is indeed linear, Fig. 2/H20849b/H20850.
From the slopes and the intercepts of the 50% bound-
aries for different applied magnetic fields the dynamic pa-rameter A, as well as the extrapolated critical currents wereobtained, shown in Fig. 3. The critical current varies linearly
with the applied external field, as is expected from the mac-rospin model. The zero temperature/zero field critical currentextrapolated from Fig. 2/H20849b/H20850has a value of 6.6 mA. This value
is twice as high as the room temperature critical current de-termined for longer timescales /H20851Fig.1/H20849b/H20850/H20852, 3.2 mA. Indeed,
when the pulse duration is short enough, the magnetizationdynamics enter a “dynamic regime” for which thermal fluc-tuations do not perturb the switching trajectory but only setthe initial magnetization configuration.
17
Figure 3shows the dynamic parameter A and the critical
current determined from the curves in Fig. 2/H20849b/H20850. Green disks
correspond to the measured dynamic parameters, while redtriangles correspond to the measured switching currents. Thecurves are computed from the macrospin model. From thecorresponding equations we can extract the coercive field
/H92620HK=245 mT, the damping /H9251=0.011, the spin polarization
P=0.015 and /H92580=1.36 rad, the initial angle of the macrospin
with respect to the zaxis. The ratio /H9251/P=0.77 and /H92620HKare
close to the values determined from the quasistatic measure-ments, but the initial angle is unphysically large and the spinpolarization is unphysically small. For an initial angle of
/H92580=0.05 rad, determined from a Boltzmann distribution, the
field dependence of A /H20849dashed line /H20850cannot explain the strong
FIG. 2. /H20849Color online /H20850/H20849a/H20850Switching probability as a function of the duration
and amplitude of the current pulse for zero applied field. The 50% switchingprobability line is shown in black. /H20849b/H2085050% switching probability for differ-
ent applied fields
/H92620Hfrom 0 to /H1100260 mT.
FIG. 1. /H20849Color online /H20850/H20849a/H20850Phase Diagram of a 100 /H11003100 nm2nanopillar
spin valve. The resistance obtained for decreasing currents was subtractedfrom the resistance obtained for increasing currents to make the metastableregion visible. /H20849b/H20850Zero field current sweep. For both measurements the field
was fixed while the current was swept at a rate of 200
/H9262A/s. The junction
resistance is 6.6 /H9024.022514-2 Bedau et al. Appl. Phys. Lett. 96, 022514 /H208492010 /H20850
Downloaded 08 Jun 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsvariation with field of the measured A. The predicted reversal
rate A depends strongly on the initial angle. For an assumedinitial angle of
/H92580=1.36 rad the macrospin model predicts a
rate 30 times larger than the experimentally determinedvalue, for
/H92580=0.05 rad the predicted rate is lower than the
experimentally obtained value by a factor of 10.
The macrospin model predicts a zero temperature energy
barrier to reversal proportional to the coercive field
UK=MSV/H92620HK/2. /H208493/H20850
From the coercive field /H92620HK=245 mT, determined using
Eq. /H208491/H20850, we obtain a zero temperature energy barrier of
8.7 eV or 338 kT. The spin-transfer figure of merit thereforehas a value of 19
/H9262A/kT. Using the zero temperature coer-
cive field and the energy barrier, we can estimate the roomtemperature coercive field assuming the magnetization relax-ation rate obeys an Arrhenius law.
7We find a room tempera-
ture coercive field of 180 mT, about two times larger than themeasured value.
From the relationship between pulse duration and cur-
rent given in Eq. /H208492/H20850we can determine the energy require-
ment for switching as a function of the pulse duration, E
/H11008I
2/H9270. For short times the energy increases with 1 //H9270, while
for long pulse durations the energy is proportional to /H9270, with
an optimum at 1 //H20849AI c0/H20850/H11015770 ps for /H92620H=0 and Ic0
=6.6 mA. Experimentally, we find an optimal pulse duration
of 660 ps, requiring E=0.9 pJ, five times less than for a
pulse of 10 ns duration. Applying higher amplitudes theswitching probability at 660 ps is 100%. These results showreliable sub-nsec pulse switching with low energy consump-
tion in all perpendicular devices, making them promisingcandidates for ST-MRAM.
In conclusion, we have demonstrated ultrafast switching
in an all-perpendicular spin valve with high efficiency. Wehave analyzed the behavior in the framework of the singledomain model and find a general agreement. The amplitudeneeded for switching is proportional to 1 /
/H9270, as predicted by
the macrospin model. We find a linear dependence of thecritical current and dynamic parameter on the applied easy-axis field as expected for the case of a uniaxial anisotropy.Even though the switching process follows the predictedscaling law, the macrospin model does not account for thematerial parameters. The model cannot explain the strongdamping and predicts a very low spin polarization and aninitial angle which is far too large. Possible explanations arethat the reversal process occurs by domain wall nucleation orpropagation and that there is additional dissipation associatedwith the excitation of spin-waves.
The research at NYU was supported by USARO /H20849Grant
No. W911NF0710643 /H20850.
1J. A. Katine and E. E. Fullerton, J. Magn. Magn. Mater. 320, 1217 /H208492008 /H20850.
2D. Chiba, Y. Sato, T. Kita, F. Matsukura, and H. Ohno, Phys. Rev. Lett.
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3P. M. Braganca, I. N. Krivorotov, O. Ozatay, A. G. F. Garcia, N. C. Emley,
J. C. Sankey, D. C. Ralph, and R. A. Buhrman, Appl. Phys. Lett. 87,
112507 /H208492005 /H20850.
4O. Ozatay, P. G. Gowtham, K. W. Tan, J. C. Read, K. A. Mkhoyan, M. G.
Thomas, and G. D. Fuchs, Nature Mater. 7, 567 /H208492008 /H20850.
5S. Mangin, Y. Henry, D. Ravelosona, J. A. Katine, and E. E. Fullerton,
Nature Mater. 5, 210 /H208492006 /H20850.
6T. Devolder, J. Hayakawa, K. Ito, H. Takahashi, S. Ikeda, P. Crozat, N.
Zerounian, J.-V. Kim, C. Chappert, and H. Ohno, Phys. Rev. Lett. 100,
057206 /H208492008 /H20850.
7S. Mangin, Y. Henry, D. Ravelosona, J. A. Katine, and E. E. Fullerton,
Appl. Phys. Lett. 94, 012502 /H208492009 /H20850.
8J. Z. Sun, Phys. Rev. B 62, 570 /H208492000 /H20850.
9A. A. Tulapurkar, T. Devolder, K. Yagami, P. Crozat, C. Chappert, A.
Fukushima, and Y. Suzuki, Appl. Phys. Lett. 85, 5358 /H208492004 /H20850.
10S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek, J. A.
Katine, and M. Carey, J. Magn. Magn. Mater. 286, 375 /H208492005 /H20850.
11M. L. Schneider, M. R. Pufall, W. H. Rippard, S. E. Russek, and J. A.
Katine, Appl. Phys. Lett. 90, 092504 /H208492007 /H20850.
12T. Devolder, C. Chappert, J. A. Katine, M. J. Carey, and K. Ito, Phys. Rev.
B75, 064402 /H208492007 /H20850.
13S. Garzon, L. Ye, R. A. Webb, T. M. Crawford, M. Covington, and S.
Kaka, Phys. Rev. B 78, 180401 /H208492008 /H20850.
14O. J. Lee, V. S. Pribiag, P. M. Braganca, P. G. Gowtham, D. C. Ralph, and
R. A. Buhrman, Appl. Phys. Lett. 95, 012506 /H208492009 /H20850.
15C. Papusoi, B. Delaet, B. Rodmacq, D. Houssameddine, J.-P. Michel, U.
Ebels, R. C. Sousa, L. Buda-Prejbeanu, and B. Dieny, Appl. Phys. Lett.
95, 072506 /H208492009 /H20850.
16J.-M. L. Beaujour, D. Bedau, H. Liu, M. R. Rogosky, and A. D. Kent,
Proc. SPIE 7398 , 73980D /H208492009 /H20850.
17J. Z. Sun, T. S. Kuan, J. A. Katine, and R. H. Koch, Proc. SPIE 5359 ,4 4 5
/H208492004 /H20850.
FIG. 3. /H20849Color online /H20850Field dependence of the dynamic parameter A and the
critical current on the external field. The dots and triangles are experimentalresults, while the lines are obtained from the macrospin model for differentparameters. The solid lines are a fit corresponding to
/H92620HK=245 mT, /H9251
=0.011, P=0.015, and /H92580=1.36 rad. The experimentally found strong de-
pendence of the dynamic parameter on field manifests itself in a large initialangle. Assuming a Boltzmann distribution, we obtain an average initialangle of 0.05 rad, leading to the dashed curve. The dynamic parameters forthe different curves have been normalized with the following coefficients:A
0,Exp=3.3/H110031011s−1A−1,A0,/H92580=1.36=1013s−1A−1, and A0,/H92580=0.05=5
/H110031010s−1A−1.022514-3 Bedau et al. Appl. Phys. Lett. 96, 022514 /H208492010 /H20850
Downloaded 08 Jun 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions |
1.4730637.pdf | Influence of the winding number on field- and current driven dynamics of magnetic
vortices and antivortices
Michael Martens, Thomas Kamionka, André Drews, Benjamin Krüger, and Guido Meier
Citation: Journal of Applied Physics 112, 013917 (2012); doi: 10.1063/1.4730637
View online: http://dx.doi.org/10.1063/1.4730637
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/112/1?ver=pdfcov
Published by the AIP Publishing
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131.230.73.202 On: Wed, 24 Dec 2014 04:29:04Influence of the winding number on field- and current driven dynamics
of magnetic vortices and antivortices
Michael Martens,1,a)Thomas Kamionka,1,b)Andre ´ Drews,2Benjamin Kru ¨ger,3
and Guido Meier1
1Institut fu ¨r Angewandte Physik und Zentrum fu ¨r Mikrostrukturforschung, Universita ¨t Hamburg,
Jungiusstrasse 11, 20355 Hamburg, Germany
2Arbeitsbereich Technische Informatiksysteme, Universita ¨t Hamburg, Vogt-Ko ¨lln-Str. 30, 22527 Hamburg,
Germany
3I. Institut fu ¨r Theoretische Physik, Universita ¨t Hamburg, Jungiusstrasse 9, 20355 Hamburg, Germany
(Received 12 March 2012; accepted 22 May 2012; published online 11 July 2012)
The excitation of magnetic singularities in ferromagnetic thin films by radio frequency currents and
fields is of high technological interest. Theoretical and experimental work often focuses on the
dynamics of vortices and not on antivortices as their topological counterparts with inverted windingnumber of the domain structure. A comprehensive analytical description is presented for vortices
and antivortices excited by spatial homogeneous two-dimensional in-plane currents and fields. In
particular, the case of rotational excitation is investigated that is known to exhibit an efficient andselective coupling to the intrinsic gyrotropic eigenmode but here shows a crucial dependence on the
winding number. The analytical model is compared with numerical results obtained by micromagnetic
simulations.
VC2012 American Institute of Physics .[http://dx.doi.org/10.1063/1.4730637 ]
I. INTRODUCTION
Manipulation of the magnetization in micron-sized ferro-
magnetic elements is of great interest because of possible
applications for data storage devices. Several concepts for
the representation of single bits have been proposed includ-ing domain-walls
1,2or magnetic vortices3and antivortices.4
In these concepts, the spin-transfer torque, i.e., the influence
of spin-polarized currents on the magnetization, is a drivingforce for magnetization dynamics enabling read and write
processes. While following the direction of magnetization ~M,
the spins of the conduction electrons exert a torque on themagnetization and inhomogenities ~r~Mare moved in the
direction of the electron flow.
5As this effect is rather small
in soft-magnetic materials, high current-densities have to beapplied that generate accompanying Oersted fields.
6Depend-
ing on the experiment, the contribution of the Oersted field to
the magnetization dynamics cannot be neglected and is ingeneral difficult to determine.
7For magnetic vortices, this
requires an experimental technique with high temporal and
spatial resolution to distinguish between these two drivingforces.
8The detailed understanding of driven vortex and anti-
vortex motion is not only of theoretical interest as an ideal
model system for magnetization dynamics but it is also aprerequisite for the realization of future storage devices.
Vortices and antivortices in thin magnetic films are char-
acterized by a curling magnetization around a core, whichpoints out of plane described by its polarization p¼61.
When the core is deflected from its equilibrium position, it
will start a spiral motion which can be described as a quasiparticle in a two-dimensional harmonic oscillator potential
9
with a typical eigenfrequency in the lower gigahertz range.This eigenmode can be excited by magnetic fields or by
spin-polarized currents.10–14Most efficient are circular rota-
tional forces that couple directly into the resonant eigenmode
of gyration and also allow polarization selective excitationand core switching according to the intrinsic sense of
gyration.
15–17As shown for vortices by Lee et al. ,18,19also
the more common case of linear oscillating forces can beexpressed by the superposition of the two circular-rotational
eigenmodes. In this work, we consider the combination of
two-dimensional spin currents and Oersted fields on themotion of magnetic singularities as well as the influence of
the topological winding number on the circular rotational ex-
citation. We show that the in-plane magnetization of antivor-tices causes an asymmetric response to rotating fields and
currents thus enabling an access to the interpretation of ex-
perimental results.
This paper is organized as follows. In Sec. II, we extend
the analytical model for the description of the driven vortex
and antivortex motion
9and find a solution for the special
case of rotational excitation. The results are compared with
micromagnetic simulations in Sec. III, allowing for a test of
the applicability of the analytical approximations. The papercloses with a conclusion in Sec. IV.
II. ANALYTICAL CALCULATIONS
The extended Landau-Lifshitz-equation5
d~M
dt¼/C0c~M/C2~Heffþa
Ms~M/C2d~M
dt/C0bj
M2
s~M/C2½~M/C2ð~j/C1~rÞ~M/C138
/C0nbj
Ms~M/C2ð~j/C1~rÞ~M; (1)
describes the magnetization dynamics of soft ferromagnets
in an effective field ~Heffand in the presence of an electricala)michael.martens@physnet.uni-hamburg.de.
b)tkamionk@physnet.uni-hamburg.de.
0021-8979/2012/112(1)/013917/5/$30.00 VC2012 American Institute of Physics 112, 013917-1JOURNAL OF APPLIED PHYSICS 112, 013917 (2012)
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131.230.73.202 On: Wed, 24 Dec 2014 04:29:04current ~j, where bj¼PlB=eMsð1þn2Þis the coupling con-
stant between the current and the magnetization. Further-more, ais the Gilbert damping factor, nis the degree of non-
adiabaticity, M
sis the saturation magnetization, and Pis the
spin polarization of the current. The angle Uof the in-plane
magnetization of a skyrmion with topological winding num-
bernis given by the relation20
U¼nbþU0;U0¼pc
2; (2)
where bis the polar coordinate with respect to the position
o ft h ec o r e( s e eF i g . 1). For vortices, the winding number is
n¼1 and for antivortices n¼–1. In the case of vortices, the
chirality c¼1(–1) denotes an anticlockwise (clockwise)
curling of the magnetization. For antivortices, there is no
rotational symmetry and the analogously defined c-value
c2ð /C0 2;2/C138(see Ref. 21) depends on the orientation relative
to the coordinate system.
The Thiele equation22describes the motion of the core
and was extended by Thiaville23to include the effect of
spin-polarized currents which yields the velocity of the
core24and reads
ðG2
0þD2
0a2Þ~v¼~G/C2~F/C0D0a~F/C0ðG2
0þD2
0anÞbj~j
þbjD0~G/C2~jðn/C0aÞ; (3)with the non vanishing diagonal element D0of the dissipa-
tion tensor. Regarding singularities in thin films with thick-
ness t, the gyrovector ~G¼G0/C1~ezwith G0¼/C02pMsl0tnp=c
is collinear to the magnetization of the core. To calculate
the force ~Fon the core, we determine the micromagnetic
energies.9For small displacementsffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X2þY2p
of the core, the
stray-field energy ESand the exchange energy EExchcan be
modeled by a two-dimensional harmonic potential
ESþEExch¼1
2mx2
rðX2þY2Þ; (4)
where mx2
ris the curvature of this paraboloid. If we
further assume a homogeneous in-plane magnetic field
~H¼ðHx;Hy;0Þ, the Zeeman energy EZof the skyrmion is
given by
EZ¼þl0MsHxltðcosU0X/C0nsinU0YÞ
þl0MsHyltðsinU0XþncosU0YÞ: (5)
Here the characteristic length ldepends on the geometry of
the ferromagnetic element, in particular l¼Rpfor a disk of
radius R. The spatial derivation of the total energy yields the
force exerted by the strayfield and the external field:
~F¼/C0 ~rðEZþESþEExchÞ
¼/C0l0MsltðHxcosU0þHysinU0Þ/C1~ex
/C0l0Msltð/C0nHxsinU0þnHycosU0Þ/C1~ey
/C0mx2
rX~ex/C0mx2
rY~ey: (6)
By inserting Eq. (6)in the extended Thiele equation (3)and
defining the free frequency xand the damping C(see Refs.
9and24)
x¼/C0npG 0mx2
r
G2
0þD2
0a2;C¼/C0D0amx2
r
G2
0þD2
0a2; (7)
the equation
/C18_X
_Y/C19
¼/C18/C0C/C0npx
npx/C0C/C19/C18X
Y/C19
/C0bj1þn/C0a
aC
x2þC2/C18C npx
/C0npxC/C19 /C20/C21 /C18jx
jy/C19
þcl
2px
x2þC2/C18npx/C0C
C npx/C19/C18n0
01/C19/C18sinU0/C0cosU0
cosU0 sinU0/C19/C18Hx
Hy/C19
(8)
of motion for vortices and antivortices results. Equation (8)
is a system of coupled inhomogeneous first-order differentialequations and its right-hand side consists of three parts: first,
the free motion of the core as a quasi particle in a parabolic
potential, second the spin-current driven dynamics, and thirdthe motion driven by magnetic fields. In the first part, the
gyrotropic force induces a sense of gyration that depends on
the product npof winding number and polarization, result-ing in a (left-)right-hand grip-rule for (anti-)vortices,
25,26
whereas the damping moves the core in the direction towards
the origin. The second part, i.e., the current term is independ-
ent of the actual size land the orientation U0of the structure
and, in the case of n¼a, moves the core exactly in the direc-
tion of the electron flow. In the third part, the coupling to the
in-plane magnetic field is more complicated and involves
rotations and reflections depending on the domain structure.
FIG. 1. Skyrmions with c¼1. The angle of the magnetization vector at polar
coordinate b¼0i si nb o t hc a s e s U0¼p=2. (a) Vortex with winding number
n¼1 and chirality c¼1. (b) Antivortex with n¼–1 and c-value c¼1.013917-2 Martens et al. J. Appl. Phys. 112, 013917 (2012)
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131.230.73.202 On: Wed, 24 Dec 2014 04:29:04Neglecting the damping ðC/C28xÞ, we get the direction of
motion by firstly rotating the field vector by an angle of
p=2/C0U0. Only for antivortices, this is followed by a reflec-
tion across the y-axis which is a crucial influence of the
winding number on the field coupling. Finally an additional
rotation by phas to be applied for a product of np¼–1. Note
that for a vortex with its discrete values of c, this implies an
induced velocity that is always parallel or antiparallel to the
direction of the field. For antivortices all angles are possibledepending on the orientation of the structure. A high damp-
ing only adds a perpendicular component to this velocity.
The presented equation of motion is rather general and
includes the motion of vortices and antivortices under the
influence of in-plane fields and currents of arbitrary time de-
pendence, e.g., radio frequency pulses or harmonic excitationand thus describes a large variety of experiments.
8,14,16,27–30
We now assume the special case of an exciting rota-
tional spin-polarized current ~jwith frequency Xand an Oer-
sted field ~Hthat is perpendicular to the current according to
Biot-Savart’s law in thin films. With the complex ansatz
~jrot;r¼j0/C1eiXt/C1ð~ex/C0ir~eyÞ;
~Hrot;r¼H0/C1eiXt/C1ðir~exþ~eyÞ;(9)
where r¼/C01ð1Þdenotes an (anti-)clockwise sense of rota-
tion of the current and the field, we get the exact steady-statesolution
/C18^X
^Y/C19
¼e
iXtv/C16iXþC irnpx
npx/C0irðiXþCÞ/C17/C18^vjþ^vH
^vjþn^vH/C19
;(10)
with the susceptibility vof the harmonic oscillator
v¼1
x2þðiXþCÞ2(11)
and the complex velocities
^vH¼/C0H0cl
2px
x2þC2
/C1½pxð/C0irsinU0þcosU0ÞþCðircosU0þsinU0Þ/C138
(12)
and
^vj¼j0bj/C01þn/C0a
aC
x2þC2ð/C0CþinprxÞ/C20/C21
: (13)
We would like to point out that for vortices ( n¼1) both
velocities enter only as a sum into Eq. (10), whereas for anti-
vortices ( n¼–1) also their difference contributes. Equation
(10) can be rewritten in the form
/C18^X
^Y/C19
¼eiXtv/C0j0bjnpCn
xa
/C0Cn
xanp0
BB@1
CCA/C18inp ir
rnp 1/C192
664
/C0H0cl
2p/C18ncosU0 sinU0
/C0sinU0ncosU0/C19/C18inp inr
rp 1/C19#
/C2x2X
x2þC2
xþixCX
x2þC20
BB@1
CCA; (14)that gives more insight into the resonant excitation or sup-
pression of gyration. This can be seen as follows. For small
damping and excitation near resonance C/C28x/C25X, the
components of the rightmost vector in Eq. (14) are almost
identical, and it is multiplied with matrices that depend only
on the sense of rotation of the excitation r, the polarization p
and the winding number n. For a resonant current excitation,
the relation r¼npmust hold. Otherwise the resulting vector
almost vanishes and the resonance is suppressed. In contrast,we obtain for resonant field excitation the necessary relation
r¼pindependent of the winding number.
For vortices with n ¼1, small damping and excitation
near resonance C/C28x/C25Xand the reasonable value n¼a
for permalloy
31the solution can be simplified to
/C18^X
^Y/C19
/C25eiXtvðxþXÞ/C18
i
p/C19
ð~vHþ~vjÞforr¼p;(15)
with
~vH¼H0cl
2pðisinU0/C0pcosU0Þ
~vj¼/C0j0bj:(16)
The exact solutions (10) and(14) for vortices as well as this
approximation describes circular trajectories,29independent
of the combination of field and current. For antivortices(n¼–1), we have to distinguish between two cases depend-
ing on the sense of rotation rof the excitation
/C18^X
^Y/C19
/C25e
iXtvðxþXÞ/C18
i
/C0p/C19
/C1~vHforr¼p
~vjforr¼/C0p:8
<
:(17)
Again, the trajectories of this approximation are circular
with a radius that is proportional to either field strength orcurrent amplitude.
28In the general case of Eq. (10), the tra-
jectory of antivortices may become elliptical aside the reso-
nance frequency. This is typically accompanied by adrastical decrease of the amplitude of gyration and so only of
lower relevance for experimental issues where the spatial re-
solution is limited. By comparing Eqs. (15) and(17), we can
see that vortices couple constructively to both, the rotational
spin-torque and the accompanying Oersted field, if the sense
of rotation of the current coincides with the intrinsic sense ofgyration of the vortex core, i.e., r¼p. No resonant behavior
is found for the case r¼–p. In contrast, antivortices can be
excited by a counterrotating magnetic field. Here, the forceson the core due to the spin-transfer torque and the Oersted
fields rotate in a contrary sense due to the inverted winding
number of the domains. Note that this is directly apparentfrom Eq. (8)and the additional reflection for antivortices in
the field term. A resonant excitation of antivortices by spin-
transfer torque is only possible for r¼–pand by Oersted
fields only for r¼p. A summary of the relations necessary
for resonant coupling can be found in Table I.
As the differential equation (8)is linear in the fields ~H
and currents ~j, a linear oscillating excitation can be written as
the superposition of two counterrotating circular excitations013917-3 Martens et al. J. Appl. Phys. 112, 013917 (2012)
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131.230.73.202 On: Wed, 24 Dec 2014 04:29:04~jlin¼j0;lin/C1eiXt/C1~ex¼1
2ð~jrot;/C01þ~jrot;þ1Þ;
~Hlin¼H0;lin/C1eiXt/C1~ey¼1
2ð~Hrot;/C01þ~Hrot;þ1Þ:(18)
By adding the steady-state solutions of Eq. (10) for both,
r¼61, this reproduces the result of Kru ¨ger et al.9
For a static magnetic field ~HðtÞ¼ð Hx;Hy;0Þand the
absence of electrical currents the deflection
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X2þY2p
¼cl
2px
x2þC2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
H2
xþH2
yq
(19)
of the core in its equilibrium state is proportional to the char-
acteristic length land the amplitude of the field.
III. NUMERICAL CALCULATIONS
To verify the results of the analytical calculations we
have performed numerical calculations with a modified OOMMF
code.32The code is extended by the current-dependent terms
in Eq. (1)to implement a rotational current and checked with
the proposed micromagnetic standard problem including
the spin-transfer torque.33Material parameters of Permalloy
ðNi80Fe20Þare used: exchange constant A¼13/C110/C012J=m,
saturation magnetization Ms¼8/C1105A=m, and Gilbert
damping a¼0.01. For the degree of non-adiabaticity we
choose the reasonable value n¼a(see Ref. 31). The sizes of
the cells are chosen to be 2 nm in x- and y-direction, which is
well below the exchange length of Permalloy lex/C255 nm. For
vortices a square with an edge length of l¼200 nm is simu-
lated (see Fig. 2(a)). Antivortices are simulated in a cross-
shaped sample with a stripe width of 200 nm (see Fig. 2(b)).
To stabilize the desired magnetization configuration, the
shape anisotropy at the outer edges is overcome by a high uni-
axial anisotropy. The thickness of both samples is chosen tobet¼20 nm. For the simulation of the gyration amplitude of
the core, first the equilibrium configuration of the (anti-)vor-
tex in the absence of current and field is calculated and thenexposed to the rotational excitation of Eq. (9)for a period of
150 ns. This is long enough to reach a steady-state motion.
The position of the core is determined by interpolating theout-of-plane magnetization giving an accuracy much better
than the cell size. The trajectories are subsequently fitted by acircle. For the analytical model, three element specific param-
eters have to be determined by additional simulations. The
free frequency xand the damping Care determined by fitting
the free gyration of the core by an exponentially damped spi-
ral. For a square-shaped element with a landau pattern and
four homogeneous magnetized domains the characteristiclength lshould equal the edge length of the square. Because
of the finite width of the domain walls, deviations are
expected. For the exact determination of l, the deflection of
the core under an applied static magnetic field is simulated
and then compared with Eq. (19). The following values
are obtained: For the vortex, x¼4:327/C110
9s/C01,
C¼6:364/C1107s/C01,a n d l¼247 nm. For the antivortex,
x¼2:994/C1109s/C01,C¼4:224/C1107s/C01, and l¼165 nm.
The simulations and the analytical solutions are compared inFigs. 2(d)–2(g). The lines represent the analytical approxima-
tion of Eqs. (15) and(17) for the simulated structures. For
both structures, the amplitude of the exciting counterclock-wise rotational current is kept at a constant level, while the
accompanying field is increased in steps of 50 A/m. As for
both structures np¼1, the intrinsic sense of gyration of the
cores is counterclockwise. As expected for vortices, both driv-
ing forces contribute to the achieved stationary gyration am-
plitudeffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
2þY2p
, as it increases with the amplitude of the
field (see Fig. 2(d)). In addition, the phase of the motion, i.e.,
the position of the core with respect to the direction of theTABLE I. Necessary relations for resonant excitation of magnetic vortices
and antivortices with rotational fields and currents. The crucial parametersare the winding number n, the polarization p, and the sense of rotation of the
excitation r. Constructive coupling to the eigenmode of gyration is marked
with a check mark.
Vortex n¼þ1 Antivortex n¼–1
p¼þ1 p¼/C01 p¼þ1 p¼/C01
Currentr¼þ1 H/C2/C2H
r¼/C01/C2HH/C2
Fieldr¼þ1 H/C2H/C2
r¼/C01/C2H/C2H
FIG. 2. Illustration of the simulated microstructures for (a) a simulated vor-
tex with polarization p¼þ1 and chirality c¼þ1 in a 200 nm wide and
20 nm thick square shaped element and (b) a simulated antivortex with
polarization p¼–1 and c-value c¼þ2 in a cross shaped element of 20 nm
thickness. The width of the crossing wires is 200 nm. (c) Illustration of therotational excitation with r¼1. (d)-(g) Comparison of micromagnetic simu-
lations (symbols) with the analytical results (lines) for rotational excitation
with different combinations of currents and fields. (d) Gyration amplitude
for current and field induced vortex gyration. (e) Gyration phase. A spin-
current density of j/C1P¼12/C110
9Am/C02is used. (f) Gyration amplitude for
current and field induced antivortex gyration. (g) Gyration phase. A spin-
current density of j/C1P¼8/C1109Am/C02is used.013917-4 Martens et al. J. Appl. Phys. 112, 013917 (2012)
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131.230.73.202 On: Wed, 24 Dec 2014 04:29:04current flow, shifts by a phase of p/2 from the case of pure
spin torque to the pure field driven case (Fig. 2(e)). For anti-
vortices, Eq. (17) predicts a resonant behavior that is com-
pletely independent of the variable field (see Figs. 2(f)and
2(g)). The numerical results of the simulations are also shown
and clearly confirm the predictions of the analytical model.The comparison also shows that the approximation provides a
precise description of the dynamics over the whole frequency
range of interest.
IV. CONCLUSION
In conclusion, we present an analytical model to
describe the dynamics of magnetic vortices and antivortices
that includes in-plane spin-current and field excitation of ar-
bitrary time dependence. The model was applied to the tech-nological relevant case of rotational excitation showing the
suppression of the effects of either field or current for anti-
vortices, depending on the sense of rotation. This also opensnew possibilities for the study of pure spin-transfer torque
driven magnetization dynamics.
28A precise examination of
spin-torque related parameters, such as the controversiallydiscussed degree of non-adiabaticity and spin polarization
could be based on our theory.
ACKNOWLEDGMENTS
We thank Max Ha ¨nze and Christian Adolff for assis-
tance with the micromagnetic simulations and are grateful tosupport and encouragement by Ulrich Merkt. Financial sup-
port by the Deutsche Forschungsgemeinschaft via the Son-
derforschungsbereich 668 and Graduiertenkolleg 1286 aswell as the Forschungs- und Wissenschaftsstiftung of the
City of Hamburg via the cluster of excellence Nano-
Spintronics is gratefully acknowledged.
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131.230.73.202 On: Wed, 24 Dec 2014 04:29:04 |
1.3662175.pdf | Destabilization of serially connected spin-torque oscillators via non-
Adlerian dynamics
Ezio Iacocca and Johan Åkerman
Citation: J. Appl. Phys. 110, 103910 (2011); doi: 10.1063/1.3662175
View online: http://dx.doi.org/10.1063/1.3662175
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Downloaded 05 Sep 2013 to 139.80.2.185. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsDestabilization of serially connected spin-torque oscillators via non-Adlerian
dynamics
Ezio Iacocca1,a)and Johan A ˚kerman1,2
1Physics Department, University of Gothenburg, 412 96 Go ¨teborg, Sweden
2Department of Microelectronics and Applied Physics, Royal Institute of Technology,
Electrum 229, 164 40 Kista, Sweden
(Received 18 August 2011; accepted 15 October 2011; published online 28 November 2011)
The transient dynamics of phase-locking in serially connected nanopillar spin-torque oscillators
(STOs) is studied both analytically and numerically. A variety of transient behaviors are observed
stemming from the high oscillator nonlinearity and the interplay between the damping to coupling
strength ratio and the phase delay of the coupling. Non-Adlerian (ringing) dynamics is found to bethe main regime of synchronization where the synchronization time depends strongly on the phase
delay. Somewhat nonintuitively, sufficiently strong coupling can also destabilize the system,
destroying the synchronized regime even for identical STOs. This transient behavior is also foundto dominate when the STOs have different frequencies. These results highlight fundamental issues
that must be considered in the design of serially synchronized STOs.
VC2011 American Institute of
Physics . [doi: 10.1063/1.3662175 ]
I. INTRODUCTION
Spin torque oscillators (STOs) are nanosized devices that
rely on spin transfer torque (STT) to generate an oscillatory
signal.1,2STT occurs when spin-polarized electrons impinge
on a magnetic material and can, under the proper conditions,induce a precession of the magnetization direction.
3,4In a
STO, a non-magnetic spacer (metallic1,2,5–15or insulating16–27
separates two magnetic materials: a thick or ‘fixed’ layer that
spin-polarizes the current and a thin or ‘free’ layer where the
precession takes place. The high frequencies achieved,28high
modulation rate,29–33and nanosized dimension are additional
characteristics that nourish the interest to implement STOs in
current silicon technology. However, the limited microwave
output power, and large linewidth,8,24,25,34,35imposes a seri-
ous limitation on its use.2,36
One method to increase the peak power and decrease the
generation linewidth is to coherently couple or mutuallyphase lock a STO array. Mutual phase locking in nanocon-
tacts sharing the same unpatterned free layer, so-called
parallel synchronization, has been achieved for two high-frequency STOs
35,37–39and four low-frequency vortex based
STOs.40Analytically, these observations can be explained
by a set of coupled phase equations, neglecting the powervariations.
41–43However, it has been recently shown that,
while the power variations are indeed negligible in the stable
regime, they are fundamental in the transient regime due tothe STO’s high nonlinearity.
44,45
Serial synchronization, on the other hand, refers to mu-
tual phase locking of electrically connected STO nanopillarsthat couple via a shared microwave current.
9,46Here, a
resistive load in parallel with the STOs provides a fixed ref-
erence for the current thus creating a microwave component
that couples the STOs similarly to an injection lockingexperiment.10,47–50More complex interconnections were
analytically studied considering the STOs as phase oscilla-
tors in the Kuramoto framework, hence neglecting their
power dependencies.51In a different approach, a reactive
element was numerically added to the load, showing a signif-
icant enhancement of the locking region9related to the
intrinsic preferred phase of the STO.52More recently, a reso-
nator was introduced as the load providing means of control-
ling the phase to lag or delay the STO series.53The authors
showed regimes of synchronization and ‘frustration’ depend-ing on the chosen phase delay. The abovementioned studies
focused on the stable regime as a common feature, i.e., dis-
tinguishing between synchronized and non-synchronizedstates. Despite these numerical demonstrations and sugges-
tions, an experimental confirmation of serial synchronization
is still lacking. To further explore possible reasons behindthis issue, we here investigate the transient synchronization
dynamics by also considering the important power variations
of STOs.
This paper describes the synchronization dynamics of
serially connected nanopillar STOs coupled by a purely reac-
tive load [Fig. 1(a)]. Through macrospin simulations
54,55and
analytical calculations, we show that the dynamics are gener-
ally non-Adlerian or oscillatory and, in fact, the only possi-
ble regime when the STOs have different free-runningfrequencies. Moreover, we give conditions for the onset of
phase instability that leads to the synchronized regime’s
destabilization. The paper is organized as follows: in Sec. II
the nonlinear auto-oscillator theory
42is used to analytically
describe the system. Macrospin simulations are performed in
Sec. IIIshowing a good quantitative agreement with the ana-
lytical results. In Sec. IVthe influence of the field-like torque
term and the limited non-linearity of Magnetic Tunnel Junc-
tion (MTJ) based STOs is briefly discussed in the mutualcoupling framework. Finally, the conclusions and implica-
tions of this paper are summarized in Sec. V.
a)Electronic mail: ezio.iacocca@physics.gu.se.
0021-8979/2011/110(10)/103910/6/$30.00 VC2011 American Institute of Physics 110, 103910-1JOURNAL OF APPLIED PHYSICS 110, 103910 (2011)
Downloaded 05 Sep 2013 to 139.80.2.185. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsII. ANALYTICAL FORMULATION
The dynamical regimes of STOs can be analytically
studied within the nonlinear auto-oscillator framework.42.
The complex amplitude equation of an oscillator is
dc
dtþixðpÞcþCþðpÞc/C0C/C0ðpÞc¼Ft;X
ci/C16/C17
;(1)
where pis the oscillation’s power and x(p),Cþ(p) and
C/C0(p) are the power dependent oscillation frequency, damp-
ing and negative damping rate, respectively. The perturba-
tion term F(t,Rci) is any function of time and complex
amplitudes and its form determines the coupling. In the caseof two serially connected STOs [Fig. 1(a)],I
dcis diverted to
the reactive load depending on the combined voltage loss
over both STOs. Due to the resonant bandpass character ofthe load, the dccomponent is suppressed and only an alter-
nating current, I
ac, flows through it. Consequently, Iachas
the same frequency as the STO series and its amplitude andphase depends on the load’s frequency response. By varying
the resonant frequency, it is possible to tune the phase to lag
or delay the STOs.
53We stress that the coupling current, i.e.,
the current flowing through the STO series, is DI¼Idc/C0Iac,
and therefore has an inherent phase delay of 180/C14. Taking
these considerations into account, the coupling term can bewritten as the sum of both the STO contributions and the
phase delay, b, between DIand the STO resistance (the
phase delay is denoted by a positive b). Then,
Ft;X
c
i/C16/C17
¼fjc2je/C0iu2ðtÞþjc1je/C0iu1ðtÞ/C16/C17
eib; (2)
where uis the instantaneous phase of each STO and the sub-
scripts 1 and 2 denote each STO in Fig. 1(a). The coupling fac-
tor for electrical connections, f/C1ct, can be approximated as42
f/C1ct¼nax Htanco
2ffiffiffi
2pl
N: (3)
Here fis the coupling strength, ct¼jc1jþjc2j,n¼Idc/Ithis
the supercriticality parameter, Ithis the threshold current for
oscillations, ais the Gilbert damping, xHis the ferromag-
netic resonance frequency, cois the out-of-plane current
polarization angle, and l¼jDIj/Idcis the modulation depth.
The factor Nrepresents the number of oscillators (two in thiscase) and bounds the magnitude of f/C1ctin the so-called ther-
modynamic limit N!1 .56In this way, the unphysical sce-
nario where f/C1ctincreases as STOs are added to the circuit is
prevented.
The equations of motion are obtained by expanding
Eq.(1)as a set of coupled equations. The complex amplitude
of each oscillator is thus perturbed by Eq. (2)and can be
written as c¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipoþ2dppe/C0iuwhere dprepresents power
variations and po!n/C01 is the free-running power. Assum-
ing similar STOs ( jc1j/C25jc2j) we obtain the power and phase
equations
dw
dt¼Dxþ2/C23CpDp/C02fcosbsinw; (4a)
dDp
dt¼/C02CpDp/C02fsinbsinw; (4b)
where the notations w¼u2/C0u1andDp¼dp2/C0dp1are
used and Dx¼x2/C0x1is the frequency mismatch between
the STOs. The term 2 /C23CpDpis the phase-power coupling,
dependent on the dimensionless nonlinearity parameter /C23
and the total damping (or restoration rate) Cp¼aðn/C01ÞxH.
Note that the assumption of similar STOs eliminates any de-pendence on the complex amplitude, c.
From Eq. (4)it is clear that the coupling strength, f,
plays a fundamental role in the synchronization dynamics.
Based on Eq. (3), we see that fis directly proportional to l,
whose variation is related to the STOs’ MR ratio. The cou-pling strength is also a function of n=c
t/n=ffiffiffiffiffiffiffiffiffiffiffin/C01pwhich
is a decreasing function of n. In other words, fhas a stronger
impact on small-amplitude oscillations since the perturbationbecomes comparable in magnitude. Here, we restrain our-
selves to the low supercriticality regime, where the coupling
is strong and the precession amplitudes are small.
For the case where /C23¼0, Eq. (4a) reduces to Adler’s
equation.
57However, for GMR based STOs, /C23/C25100 and an
approximate solution of Eq. (4)can be obtained by neglect-
ing the power variations. Interestingly, the resulting phase
difference equation (not shown) has the same form of
Eq. (65) a in Ref. 42, derived for nanocontact-based STOs.
Good agreement with experiments has been achieved in this
way by describing the phase locking bandwidth, Dxo.
Taking into account the power variations allows us to
study the synchronization dynamics. Assuming that these
variations are small ( dp/C28po), Eq. (4)can be linearized
about po. The new set of linear differential equations has a
characteristic polynomial
k¼/C0CsðÞ6ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
CsðÞ2/C04Cp~fcoswocosb/C0arctan ð/C23Þ ½/C138q
;(5)
where ~f¼fffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ/C232p
andCs¼Cpþfcosbcoswo. The sta-
tionary phase wo¼arcsin Dx=Dxoand phase locking band-
width Dxo¼2~fjcosðb/C0arctan /C23Þjare obtained from the
steady state solution of Eq. (4). Due to the high nonlinearity
considered, the approximation cos[ b/C0arctan( /C23)]¼sin(b)i s
used in the subsequent discussion.
The influence of bandfon Eq. (5)is shown in Fig. 1(b)
as a pole-zero plot. Two important features are clearly
FIG. 1. (a) Serially connected nanopillar STOs coupled via a shared micro-
wave current. The load is a resonator matching the STO frequency and the
whole system is driven by a direct current source. (b) Pole-zero plot of Eq. (5)
for several phase delays, b, and sweeping the coupling strength, f. Clearly, the
system is potentially unstable for b>90/C14and sufficiently large f.103910-2 E. Iacocca and J. A ˚kerman J. Appl. Phys. 110, 103910 (2011)
Downloaded 05 Sep 2013 to 139.80.2.185. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsvisible: ( i) imaginary components exist for sufficiently strong
f;(ii) real positive components (instability) exist when
b>90/C14. These features can be parametrized from the onset
of complex solutions of Eq. (5)and the sign change of its
real part (or time constant s), respectively,
fring/C21ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C2
pþ4Dx2q
4/C23sinb; (6a)
s/C01¼Cs¼Cpþfcosbcoswo: (6b)
Thus, Eq. (6b) is valid if Eq. (6a)is satisfied.
The set of Eq. (6)is the main result of this paper, provid-
ing important qualitative information that is lost in steady
state analysis. First, satisfying Eq. (6a)leads to an oscillatory
approach to synchronization and hence establishes a lower
bound to the time constant of coupled STOs.44While in the
injection locking case the limit is set by Cpalone, here we
note that bhas a critical influence [Eq. (6b)]. Indeed, the
time constant is reduced by keeping the phase delay in the
range b/C20j90/C14jand its minimum value occurs when b¼0.
Second, in the range b>j90/C14j, the dynamics become slower
and even unstable if the sign of s/C01changes, i.e., if the ratio
Cp=fcosbcoswo<1. Recalling that fis a decreasing func-
tion of n, note that the instability can be avoided by increas-
ingnat the price of loosing coupling strength. Third, the
condition for non-Adlerian solutions [Eq. (6a)] has a mini-
mum value at b¼90/C14whereas it tends to infinity when
b¼0. In the serial circuit considered, the operation regime
lies in the range b>j90/C14jso that b!90/C14is the optimal
choice to reduce the time constant and avoid instability.
Finally, the influence of the device variability can be para-
metrized as the increase of the frequency mismatch. As Dx
grows, Eq. (6a) approaches Dx/C202/C23fringsinb/C25Dxoin the
case of high /C23, so that synchronization is expected to occur
only in the non-Adlerian regime.
We note that Eq. (6)quantitatively addresses previous
observations on the coupling strength impact on synchroni-
zation46and the phase locking enhancement for b/C2590/C14.9
Furthermore, we stress that these circuits inherently lie in the
unstable region and that the maximum coupling strength for
synchronization is limited by the total damping of the freelayer.
III. MACROSPIN SIMULATIONS
To further investigate the described regimes we here per-
form macrospin simulations of the circuit shown in Fig. 1(a).
We first consider two identical GMR based STOs of circularcross-section. The free layer is assumed to be a soft magnetic
material with in-plane, uniaxial anisotropy, such as Permalloy.
The magnetization precession of the free layer follows theLandau-Lifshitz-Gilbert-Slonczewski equation
54,55
d^m
dt¼/C0c^m/C2~Heffþa^m/C2d^m
dtþaJ^m/C2^m/C2^M;(7)
where ^mis the free layer magnetization direction and
c=2p¼28 GHz/T is the gyromagnetic ratio. The effective
field ~Heff¼HappzþMSð^m/C1zÞzincludes the external fieldloHapp¼1:5 T, applied normal to the free layer’s surface,
and the uniaxial anisotropy l0MS¼0:8 T where lois the vac-
uum permeability. The spin torque magnitude is parametrizedbya
J¼/C22hgDI=2l0MSeVwhere /C22his the reduced Planck’s con-
stant, ethe electron charge, V/C253:7/C2103nm3the nanopillar
volume, and g¼0:35 the dimensionless spin torque effi-
ciency. The angle dependence of the spin torque efficiency is
not included due to the high applied field considered which
limits the precession to small orbits, in correspondence withthe auto-oscillator theory. The magnitude and phase of DIis
obtained by solving the differential equations that describe
Fig. 1(a). The resonator is tuned about the oscillation’s
frequency, with L/C251n Ha n dC /C252 pF (exact values depend
on the operation point and desired phase delay). The fixed
layer magnetization unity vector ^Mdefines the spin polariza-
tion direction of the current, here considered tilted at an angle
c
o¼60/C14from the surface normal due to the high Happ.I nt h i s
configuration, the threshold current Ith¼2.82 mA and the
free-running frequency fSTO¼24.266 GHz are found. Finally,
the parallel and antiparallel resistances are R0¼10Xand
Rp¼11X, respectively, and the angular dependence is
defined as R¼[(R0þRp)/C0(Rp/C0R0)^m^M]/2.
The above description of the simulation parameters
takes into account a perpendicularly magnetized GMR nano-pillar. The case of an in-plane magnetized free layer is not
considered due to the limitations of macrospin simulations,
as pointed out by Berkov and Miltat
58while comparing with
full micromagnetic simulations. The analytical results are
nontheless valid for this regime, given that the proper angu-
lar dependencies for the auto-oscillator parameters are takeninto account.
14,59
The stability of the system is governed by the ratio Cp/f,
shown as a function of the supercriticality nin the inset
of Fig. 2. Here the resonator is set at a frequency such that
b/C25180/C14in the supercriticality span and the synchronization
is in-phase ( wo¼0). In this particular configuration, the
unstable region lies near the oscillation threshold, where the
ratio is less than unity. This is a consequence of the high
applied field, which leads to the fast increase of both poand
FIG. 2. Inverse synchronization time, s/C01, as a function of b, determined
from macrospin simulation for the supercriticalities n1¼1.23 (triangles) and
n2¼1.03 (circles), which represent two dynamically different scenarios
(stable and unstable, respectively). The corresponding analytical estimates
are also shown in solid and dashed lines, respectively. The inset shows the
ratioCp/fas a function of supercriticality in common logarithmic scale for
Happ¼1 T (dashed line) and 1.5 T (solid line). Increasing the applied field
reduces the unstable region.103910-3 E. Iacocca and J. A ˚kerman J. Appl. Phys. 110, 103910 (2011)
Downloaded 05 Sep 2013 to 139.80.2.185. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsCpwith current. Consequently, a lower applied field widens
the unstable region, as shown in the inset of Fig. 2.
As discussed in Sec. II,s/C01gives a complete picture of
the synchronization dynamics of the system. The numerical
calculation of s/C01was performed by fitting a decaying expo-
nential to the envelope of the transient phase difference, w.
The STOs are simulated for 20 ns in the free-running state af-
ter which the load is connected and the coupling current DI
flows through the STOs. The parameter s/C01is shown as a
function of bin Fig. 2for the two supercriticality values cho-
sen in its inlay. In both cases, the numerical calculation of
s/C01(triangles and circles) show good agreement with the
analytic estimates of Eq. (6b) (solid and dashed lines). In the
case n1¼1.23, the dynamics are always stable and occur in
the non-Adlerian regime. On the other hand, for n2¼1.03,
the dynamics are slower and ultimately lead to a sign change
ofs/C01, becoming unstable. Close to the transitions to anti-
phase lock ( b>180/C14sowo¼180/C14) and instability at
n1¼1.23 and n2¼1.03, respectively, the dynamics is dis-
torted and the fitting algorithm fails.
The transition from the synchronized to the unstable re-
gime for n2is shown in Fig. 3. Clearly, the time constant
increases as btends toward 180/C14[Figs. 3(a)–3(b)] until it
approaches s/C01/C250 [Fig. 3(c)]. A further increase of bleads
to instability but wremains bounded as shown in Fig. 3(d).
This is qualitatively understood from the phase-amplitude
coupling in STOs. The amplitude is bounded to the unitsphere by Eq. (7)so that the frequency must be also bounded
and, consequently, the phase difference. Interestingly, suchphase variation suggests that the phase of each oscillator
‘beats’ periodically, with a frequency given by jwj(since
each zero crossing represents a beat cycle). Indeed, comput-ing the Power Spectral Density (PSD) of the voltage drop
over the STO series (not to be confused with the phase dif-
ference w) shows two sidebands at /C25644 MHz from the
carrier [Fig. 3(e)], consistent with twice the /C2522 MHz
observed in Fig. 3(d). Although the similarity, this scenario
must not be related to a STO modulation,
29,30,60as discussed
below. Note that the frequency /C2519.805 GHz differs from
the free-running frequency fSTOas a consequence of the cou-
pling (see discussion in Ref. 42, Sec. VI B).
Device variability is included in the simulations as a
change of the nanopillar cross-section. As suggested by
Eq.(6a), the non-Adlerian condition converges to the phase
locking condition as Dxincreases, i.e., only non-Adlerian
dynamics lead to synchronization. Simulations of this sce-
nario are performed by keeping the supercriticality atn¼1.23 and b/C2590
/C14so that the phase locking bandwidth is
Dxo/2p/C25200 MHz. The assumptions of similar STOs and
dp/C28poare not strictly valid in this case but the qualitative
behavior of the dynamics is conserved.
The transient phase difference between the oscillators is
shown for several frequency mismatches in Fig. 4(a).A sDx
diverges from zero, the oscillators approach a stable phase
difference in a non-Adlerian fashion. Above a critical
FIG. 3. Phase difference between two identical STOs for n¼1.03 and (a)
b¼90/C14, (b) b¼120/C14, (c) b¼135/C14, and (d) b¼150/C14.A s bincreases, the
synchronization time increases until the phase difference no longer con-
verges to wo¼0. At instability, wis however bounded in [ /C0180/C14, 180/C14]
(dotted lines). (e) Normalized PSD of the total STO signal in (d). Sidebands
at/C2544 MHz correspond to the beating of the oscillations.
FIG. 4. Phase difference between two STOs with different free-running fre-
quencies. (a) At n¼1.23 and b/C2590/C14, the STOs phase lock up to a critical
frequency mismatch, Dxmax/2p¼115 MHz, outside of which locking is
replaced by frequency pulling. In the locking regime, the approach to syn-
chronization is clearly non-Adlerian with significant ringing. (b) Atn¼1.23, b/C2590
/C14, andDx/2p¼30 MHz the oscillators cannot lock due to
instability. Here, the frequency mismatch creates a period-2 oscillation in the
phase difference. The oscillations have periods T 1/C2550 ns and T 2/C2541 ns. (c)
PSD of the total STO signal in (b). The mean oscillation frequency of each
STO is pointed with arrows while several asymmetric sidebands appear as a
consequence of the non-trivial dynamics of the STO series.103910-4 E. Iacocca and J. A ˚kerman J. Appl. Phys. 110, 103910 (2011)
Downloaded 05 Sep 2013 to 139.80.2.185. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsfrequency mismatch Dx/2p¼115 MHz, the STOs cannot
phase-lock but pull each other in a quasiperiodic motion.56
This motion is a general feature of coupled oscillators and it
has been studied and measured for linear oscillators.61
We note that the breakdown of the approximate solution
is evident from the premature unlocking of the STOs(Dx
o,max/2p¼115 MHz instead of the expected Dxo/2p
¼200 MHz).
Finally, nominally different STOs can also reach an
unstable regime, similarly to Fig. 3(d).H e r e , n¼1.03,
b¼150/C14,a n dDx=2p/C2530 MHz. The phase difference shows
period-2 oscillations due to the mismatch of free-running fre-quencies whose periods are indicated in Fig. 4(b). Such behav-
ior is in clear disagreement with any modulation comparison.
The PSD [Fig. 4(c)] further confirms this observation by the
presence of asymmetric sidebands and two STO mean fre-
quencies (indicated by arrows) indicating nonlinear variations
of the individual frequencies.
IV. MUTUAL COUPLING OF MAGNETIC TUNNEL
JUNCTIONS
STOs, based on nanopillar MTJs, have been extensively
investigated due to their high tunneling magnetoresistance
(TMR) and hence high generation power.18,20,21,26In our
mutual coupling framework, MTJs have three important con-
sequences: ( i) the perpendicular torque in MTJs, bj, introdu-
ces an additional phase shift20so that bcan be pushed
toward the stable regime; ( ii) The high TMR enhances fand
faster synchronization can be achieved. We stress that this
feature can also widen the unstable regime if bis not prop-
erly tuned; ( iii) MTJ based STOs have significantly lower
nonlinearity compared to their GMR counterparts. In fact, /C23
has been estimated to lie between 1 and 3 in recent phasenoise measurements;
24,25(iv) The use of in-plane fields will,
in general, change the parameters used in Eqs. (6)due to
their angle depdencies14,59although dynamics in perpendicu-
larly magnetized MTJs has been recently observed.62
All the abovementioned features can be summarized in
Eq.(6a) considering a weak nonlinearity, v. We remind the
reader that the approximation cos b/C0arctan ð/C23Þ ½/C138 ¼ sinðbÞ
was used in Eq. (6). Thus, the general form is given by
fring/C21ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C2
pþ4Dx2q
4/C23cosb/C0arctan ð/C23Þ ½/C138: (8)
For weak nonlinearities in the range 0.1 </C23< 5, the cosine’s
argument varies between 5/C14and/C2580/C14. In such a case, the
control of bbecomes a fundamental issue in order to set the
critical ringing strength, fring. For serially connected STOs,
however, the limited coupling strength Eq. (3)suggests that
generally f<fringeven for MTJs. Consequently, MTJs are
expected to behave almost linearly, hence, following phase
models such as the Adler’s equation of synchronization57to
a much larger extent, and therefore be less susceptible to theinstabilities discussed above.
On the other hand, recent experimental attempts of serial
synchronization have focused on MTJ based vortexSTOs.
40,49In contrast to MTJ injection locking experiments,50
the MTJ based vortex STO was successfully phase locked tothe fundamental frequency, suggesting a much larger coupling
strength, f.W h i l e vhas yet to be determined for these types of
STOs, their limited frequency tunability also suggests a simi-larly limited non-linearity. The synchronization dynamics for
serially coupled MTJ based vortex STOs is then expected to
be strongly dependent on the onset for non-Adlerian dynam-ics, determined by Eq. (8).
V. CONCLUSION
The synchronization dynamics is generally non-
Adlerian for serially connected GMR nanopillars coupled by
a resonant load. The high nonlinearity of GMR-based STOshas a qualitative impact on the dynamical regimes in this
system. In particular, the synchronization time is strongly de-
pendent on band the system becomes unstable if
C
p=fcosb>1. Instability manifests itself as the impossibil-
ity to achieve phase locking even for identical STOs and,
counterintuitively, is favored by stronger coupling. This in-formation provides boundaries that must be considered to
find the system’s optimal operation point needed for experi-
mental observation of serial synchronization. To this end, therelevant issue of device variability was included in the simu-
lations, and similar transient behavior was obtained for
STOs with different free-running frequencies. Moreover, thefine tuning of the phase delay, b, is shown to be critical in
the serial synchronization of weakly nonlinear devices, such
as MTJ based vortex STOs. We expect similar dynamics inthe case of nanocontact STOs coupled via spin-waves where
both the phase delay and the coupling strength can be
controlled by the separation between the nanocontacts. We
believe that the presented discussions are critical to effi-
ciently design synchronized STO arrays.
Support from the Swedish Research Council (VR) is
gratefully acknowledged. Johan A ˚kerman is a Royal Swed-
ish Academy of Sciences Research Fellow supported by a
grant from the Knut and Alice Wallenberg Foundation.
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1.4982045.pdf | Two-photon absorption spectroscopy of stilbene and phenanthrene: Excited-state
analysis and comparison with ethylene and toluene
Marc de Wergifosse , Christopher G. Elles , and Anna I. Krylov
Citation: The Journal of Chemical Physics 146, 174102 (2017); doi: 10.1063/1.4982045
View online: http://dx.doi.org/10.1063/1.4982045
View Table of Contents: http://aip.scitation.org/toc/jcp/146/17
Published by the American Institute of PhysicsTHE JOURNAL OF CHEMICAL PHYSICS 146, 174102 (2017)
Two-photon absorption spectroscopy of stilbene and phenanthrene:
Excited-state analysis and comparison with ethylene and toluene
Marc de Wergifosse,1Christopher G. Elles,2,a)and Anna I. Krylov1,a)
1Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482, USA
2Department of Chemistry, University of Kansas, Lawrence, Kansas 66045-7582, USA
(Received 13 December 2016; accepted 11 April 2017; published online 1 May 2017)
Two-photon absorption (2PA) spectra of several prototypical molecules (ethylene, toluene, trans - and
cis-stilbene, and phenanthrene) are computed using the equation-of-motion coupled-cluster method
with single and double substitutions. The states giving rise to the largest 2PA cross sections are
analyzed in terms of their orbital character and symmetry-based selection rules. The brightest 2PA tran-
sitions correspond to Rydberg-like states from fully symmetric irreducible representations. Symmetry
selection rules dictate that totally symmetric transitions typically have the largest 2PA cross sections
for an orientationally averaged sample when there is no resonance enhancement via one-photon acces-
sible intermediate states. Transition dipole arguments suggest that the strongest transitions also involve
the most delocalized orbitals, including Rydberg states, for which the relative transition intensities can
be rationalized in terms of atomic selection rules. Analysis of the 2PA transitions provides a founda-
tion for predicting relative 2PA cross sections of conjugated molecules based on simple symmetry and
molecular orbital arguments. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4982045]
I. INTRODUCTION
Non-linear spectroscopies have revolutionized many
areas of materials and life sciences. Techniques based on
two-photon absorption (2PA), for example, provide tem-
poral and spatial control of the optical excitation process,
allowing localized fluorescence excitation and chemical acti-
vation. These 2PA methods enable biological applications
ranging from 3D and super-resolution imaging to photody-
namic therapy, optogenetics, and targeted cell deactivation,1–7
often at wavelengths that are long enough for deep tissue pen-
etration. Two-photon activation has similarly been used for
non-linear optical (NLO) material applications, including
microfabrication, nanolithography, 3D optical data storage,
ultrafast electro-optic switching, and a variety of novel
nanobiophotonics applications.8–14In addition, 2PA spec-
troscopy is a powerful research tool that provides comple-
mentary (and sometimes more detailed) information about
the electronic structure of a molecule relative to one-photon
absorption (1PA) spectroscopy.15–17Thus, the theoretical pre-
diction of 2PA spectra is essential for both the in silico design
of better two-photon chromophores and the interpretation of
experimental spectra.
Designing new two-photon active materials and interpret-
ing experimental 2PA spectra require a fundamental under-
standing of the quantum-mechanical principles behind the
NLO process. As described by Maria G ¨oppert-Mayer in
1931,182PA becomes allowed in the second order of per-
turbation theory and has a quadratic dependence of the
absorption strength on the light intensity. Consequently,
two-photon transitions are governed by different selection
a)Authors to whom correspondence should be addressed. Electronic
addresses: elles@ku.edu and krylov@usc.edu.rules than one-photon transitions, and the magnitude of
2PA cross sections is generally much lower than for 1PA.
Specific details of the 2PA spectrum are more difficult to
obtain because of both the difficulty in measuring exper-
imental 2PA spectra and the inherent challenges in cal-
culating higher-order molecular properties.19If asked to
predict the main features of the 2PA spectrum of a molecule,
little can be said beyond symmetry-imposed selection
rules20,21(e.g., only gerade-gerade andungerade-ungerade
transitions are allowed for centro-symmetric molecules, in
contrast to the 1PA selection rules), possible resonance
enhancement of some transitions, and a handful of struc-
tural predictions derived from simple empirical (“few states”)
models.11,22Moreover, the computational protocols for com-
puting 2PA spectra and the reliability of existing methods have
not yet been fully established. Although several benchmark
studies carefully investigated the effects of various approxi-
mations on the computed cross sections by comparing differ-
ent theoretical methods,23–26no comprehensive comparisons
between calculated and experimental spectra exist, the set of
molecules investigated in previous validation studies is rather
small, and the emphasis was on low-lying electronic states.
The goal of this work is to investigate trends in the 2PA
spectra of several prototypical molecules (ethylene, toluene,
stilbene, and phenanthrene). These compounds represent a
series of conjugated hydrocarbons of increasing size, where
ethylene and toluene can be viewed as building blocks of the
NLO-active stilbene and phenanthrene. The choice of stil-
bene is motivated by its interesting photochemistry, which
includes cis/trans isomerization and electrocyclization path-
ways that represent the key reaction channels in many pho-
tochromic molecular switches.13,27–34The 2PA spectroscopy
of stilbene has been a subject of theoretical studies for sev-
eral decades,15,35,36however, only a handful of studies have
0021-9606/2017/146(17)/174102/14/ $30.00 146, 174102-1 Published by AIP Publishing.
174102-2 de Wergifosse, Elles, and Krylov J. Chem. Phys. 146, 174102 (2017)
employed correlated many-body electronic structure meth-
ods.37–39To treat stilbene and larger conjugated systems,
many studies employed semi-empirical approaches and time-
dependent density functional theory (TDDFT).40–49Several
important general trends have emerged from these studies,
including an increasing 2PA intensity with an increasing
oligomer length,40,44larger response of trans - relative to cis-
polyenes,40and proposed design guidelines for enhanced 2PA
cross sections in dendritic molecules.41–43Other studies exam-
ined steric effects on the 2PA cross sections of substituted
stilbenes,46,47solvent effects on the 2PA of trans -stilbene
and charge-transfer homologues (e.g., D- -D, D--A, and
A--A),50the photoswitching of 2PA properties,45and
enhanced 2PA upon dimerization,51among other NLO prop-
erties of stilbene and stilbene analogues.
In this paper, we examine the calculated 2PA spec-
tra of ethylene, toluene, stilbene, and phenanthrene using
a recently implemented25method for calculating 2PA cross
sections with equation-of-motion coupled-cluster wave func-
tions with single and double substitutions (EOM-CCSD).
Analyzing the electronic structure of the states that give
rise to the dominant features in the 2PA spectra using wave
function analysis tools52–54provides insight into the 2PA
process and allows further calibration of computational pro-
tocols for the 2PA calculations. The structure of the paper
is as follows. In Sec. II, we outline the details of theoreti-
cal methods and computational protocols, including a detailed
description of the 2PA cross section calculation and related
symmetry-based properties. We discuss general trends and
symmetry-imposed selection rules from group theory and for
atomic transitions. We then present the results for the computed
1PA and 2PA spectra and analyze the character of the states
that give rise to the most prominent features in the 2PA spectra.
The comparison of the computed 2PA spectra with the experi-
mental spectra can be found in Ref. 55, where we also analyzed
the performance of the electronic structure methodology.
II. THEORETICAL METHODS
AND COMPUTATIONAL DETAILS
A. Equation-of-motion coupled-cluster method
for excitation energies
EOM-CC methods provide an efficient and robust
computational approach for describing multiple electronic
states, including electronically excited states that have multi-
configurational wave functions.56–62The EOM-CC methods
are similar, or equivalent (in some aspects), to linear response
CC.63–67The EOM-CC wave function has the following form:
j i=ˆReˆTj0i, (1)
where the linear EOM operator ˆRacts on the reference CC
wave function, eˆTj0i. The operator ˆTis an excitation operator
satisfying the reference-state CC equations
hj¯H ECCj0i=0, (2)
where are-tuply excited (with respect to 0) deter-
minants, ¯His the similarity-transformed Hamiltonian ( ¯H
=e ˆTˆHeˆT), and the reference CC energy is
ECC=h0j¯Hj0i. (3)The EOM amplitudes Rand the corresponding ener-
gies are found by diagonalizing ¯Hin the space of target
configurations defined by the choice of operator ˆRand ref-
erence 0(see Ref. 59 for examples of different EOM
models),
¯HRk=EkRk. (4)
In EOM-EE-CCSD (EOM-CCSD for excitation energies), the
CC and EOM operators are truncated as follows:
ˆT=ˆT1+ˆT2and ˆR=R0+ˆR1+ˆR2, (5)
where only the single and double excitation operators (1 h1p
and 2 h2p) are retained in ˆTandˆR. Since ¯His a non-Hermitian
operator, its left and right eigenstates, LkyandRk, are not
Hermitian conjugates of each other. Rather, they form a
biorthogonal set
¯HRk=EkRk, (6)
Lky¯H=LkyEk, (7)
h0LmjRn0i=mn, (8)
where ˆL=ˆL1+ˆL2.
B. Calculation of 2PA cross sections using EOM-CCSD
wave functions
For exact wave functions, the two-photon transition
moments at frequencies !1and!2between statesj0iandjki
are given by the following expressions:
Mk 0
bc= X
n hkjˆcjnihnjˆbj0i
n0 !1+hkjˆbjnihnjˆcj0i
n0 !2!
, (9)
M0 k
bc= X
n h0jˆbjnihnjˆcjki
n0 !1+h0jˆcjnihnjˆbjki
n0 !2!
, (10)
where ˆdenotes the dipole moment operator,
n0is the tran-
sition energy between states j0iandjni, and the sums run over
all electronic states of the system.
In the expectation-value approach to molecular proper-
ties,19this expression serves as a starting point for deriving
two-photon moments for approximate wave functions. This
strategy was used in the implementation of 2PA cross sections
within the EOM-EE-CCSD and ADC (algebraic diagrammatic
construction) frameworks.25,68In this approach, the states and
the transition energies in Eqs. (9) and (10) are replaced by the
EOM-CCSD (or ADC) wave functions and transition energies.
Then the expressions are converted into a computationally
efficient form by using the resolvent expression and by intro-
ducing auxiliary vectors that are solutions of response-like
equations.25Although the programmable expressions have dif-
ferent form from the above sum-over-state equations, they are
identical, both formally and numerically. Thus, for the pur-
poses of this paper, we will use Eqs. (9) and (10) for the analysis
of the computed spectra.
From the two-photon transition moments, the compo-
nents of the transition strength matrix, Sab,cd, are computed
as the products of the left and right two-photon transition
amplitudes
Sab,cd=0.5(M0 k
abMk 0
cd+M0 k
cdMk 0
ab). (11)174102-3 de Wergifosse, Elles, and Krylov J. Chem. Phys. 146, 174102 (2017)
Using atomic units, the rotationally averaged 2PA strength,
h2PAi, reads21,69
D
2PAE
=F
30X
a,bSaa,bb+G
30X
a,bSab,ab+H
30X
a,bSab,ba, (12)
where F,G, and Hare integer constants, which depend on
the polarization of the incident light. F=G=H= 2 for
parallel linearly polarized light, whereas F= 1,G= 4, H
= 1 for perpendicular linearly polarized light. We caution
that these coefficients, which were originally derived by Mon-
son and McClain,69are sometimes reported incorrectly in the
literature (i.e., a sign error for perpendicular polarization70).
The macroscopic 2PA cross section, 2PA, is measured in GM
(G¨oppert-Mayer) units. It is related to the molecular transition
strength in the degenerate case26(!1=!2=Eexc
2) through the
following expression:
2PA=3 a5
0(2!)2
cD
2PAE
S(2!,!0, ), (13)
where is the fine structure constant, a0is the Bohr radius,
!is the photon energy for the excitation of 2 !,cis the speed
of light, and S(2!,!0, ) is the line-shape function describing
spectral broadening effects. In the non-degenerate case, i.e.,
when two different frequencies are used, the macroscopic cross
section reads
2PA=23 a5
0!2
cD
2PAE
S(!,!0, ), (14)
where!=!1+!2. Note that this expression includes an
additional factor of two compared with Eq. (13) to account for
the lower photon dissipation rate when the two photons are
absorbed. The photon dissipation rate of each laser equals the
2PA rate for excitation with two beams, whereas the dissipation
rate is twice the 2PA rate when both photons come from the
same laser beam.26
The line-shape function introduces a phenomenological
broadening for the computed stick spectrum. It includes an
empirical damping parameter , which describes the spec-
tral broadening due to rovibrational excitations and collisional
dynamics. The Lorentzian line-shape function is usually used
for gas-phase calculations
L(!,!0, )=1
(! !0)2+ 2, (15)
where!0is the excitation energy and is the half-width at
half-maximum (HWHM). This function has a maximum at
!=!0,
L(!0, )=1
. (16)
In the condensed phase, a Gaussian line shape is commonly
used to describe inhomogeneous broadening
G(!,!0, )=p
ln 2p exp"
ln 2(! !0
)2#
. (17)
The maximum of the Gaussian is
G(!0, )=p
ln 2p . (18)In this paper, we use a relatively small damping factor
( = 0.1 eV) for the 2PA spectra so that the dominant tran-
sitions do not spill over the entire spectrum. Using a dif-
ferent broadening factor affects not only the width of each
peak but also the maximum amplitude. The uncertainty in the
choice of damping factors can limit the predictive power of the
calculations.
For calibration purposes, we also calculated 1PA spectra.
For ethylene and toluene, 1PA spectra were computed with a
Gaussian broadening of = 0.4 eV applied to each transition
(see Section 1 of the supplementary material). The calculations
presented here also neglect Franck-Condon factors (arising
due to the structural relaxation in the excited states), non-
Condon electronic effects,71and solvent contributions, all of
which can broaden or shift the 2PA bands. Additionally, we
report only the 2PA spectra for degenerate excitation: !1=!2
=Eex/2.
C. Symmetry-imposed selection rules
From group theory, one-photon transitions between states
IandFare allowed when
I
i
F A1g, (19)
where ndenote irreducible representations (irreps; not to
be confused with linewidths from Sec. II B) and irepre-
sents the x,y, orzpolarized component of the incident light.
In centrosymmetric molecules, for example, only transitions
between states of opposite parity ( gtouorutog) are allowed.20
For two-photon transitions, the dipole moment operator acts
twice in the sum-over-states expressions, Eqs. (9) and (10).
Consequently, two-photon transitions between states IandF
are allowed when
I
i
int. A1gand int.
j
F A1g, (20)
where intis the irreducible representation (irrep) of an inter-
mediate state in the sum-over-states expression, and iand
jdescribe the polarizations of the two photons. As the 2PA
transition moments are obtained by summing over all states
of the system, there should exist at least one intermediate
state that is dipole-allowed for both the initial and final states
in order for the transition to be 2PA-allowed. For example,
in the centrosymmetric molecules, the dipole-allowed tran-
sitions are gtou; thus, in 2PA, only gtogandutouare
allowed. Furthermore, one may expect larger cross sections
for 2PA transitions between the states that are connected by a
larger number of intermediate states. Using these arguments,
we illustrate below that the largest 2PA cross sections are
expected for transitions where IandFbelong to the same irrep.
For example, for closed-shell molecules, the excited states
from the fully symmetric irrep will feature the brightest 2PA
transitions.
To illustrate this point, let us consider two specific exam-
ples: the C2vpoint group and the case of atomic transitions.
Figure 1 shows the allowed two-photon transition pathways as
well as the corresponding two-photon transition matrices for
molecules in the C2vpoint group, which has four irreps (A 1,
A2, B1, and B 2). For 1A 1!nA1transitions, the intermediate174102-4 de Wergifosse, Elles, and Krylov J. Chem. Phys. 146, 174102 (2017)
FIG. 1. Two-photon transition pathways and selection rules for the C2vpoint
group.
states can include states from three irreps (B 1, B2, and A 1),
each providing one contribution to the two-photon transition
matrix ( Mxx,Myy, and Mzz, respectively). For example, the B 1
intermediate states only contribute to MnA1 1A1xx ,
MnA1 1A1xx = X
i hnA1jˆxjiB1ihiB1jˆxj1A1i
nA1 1A1 !1
+hnA1jˆxjiB1ihiB1jˆxj1A1i
nA1 1A1 !2!
. (21)
Similarly, the MnA1 1A1yy andMnA1 1A1zz elements include con-
tributions from all B 2and A 1intermediate states, respec-
tively. Note that only the diagonal elements of the matrix
are nonzero in this case due to symmetry requirements for
the totally symmetric transition. For 1A 1!nB1transitions,
on the other hand, only the xzandzxoff-diagonal compo-
nents of the two-photon transition matrix are nonzero (see
Figure 1).
The diagonal matrix elements contribute to all three
terms in the orientationally averaged parallel 2PA cross sec-
tion, Eq. (12), giving a total of 15 terms for 1A 1!nA1
transitions
D
2PAE
nA1 1A1=2
30 3Sxx,xx+Sxx,yy+Sxx,zz+Syy,xx+ 3Syy,yy
+Syy,zz+Szz,xx+Szz,yy+ 3Szz,zz!
.
(22)
In contrast, the two off-diagonal components for the 1A 1
!nB1transitions give only four terms in the orientationally
averaged 2PA cross section
D
2PAE
nB1 1A1=2
30 Sxz,xz+Szx,zx+Sxz,zx+Szx,xz. (23)
Similar results are obtained for transitions to the nA2and
nB2states, for which there are only two off-diagonal ele-
ments in the two-photon transition matrix, leading to four
terms in the 2PA cross section. Thus, the 2PA spectrum ofaC2vmolecule will be dominated by totally symmetric two-
photon transitions based on the larger number of pathways to
the A 1states as well as the larger contribution of the diagonal
elements in calculating the 2PA cross sections (i.e., 15 terms
compared with only four). In general, the diagonal tensor ele-
ments of the two-photon transition matrix belong to the totally
symmetric irreps for all point groups; therefore, the totally
symmetric states will dominate parallel 2PA spectra under non-
resonant conditions (i.e., when there is no resonant one-photon
FIG. 2. Selection rules for 2PA transitions in centrosymmetric systems. Only
l=0,2 transitions are allowed. Each 2PA pathway on the left represents a
contribution to the transition matrices on the right. The totally symmetric s!s
andpz!pztransitions are expected to have the largest 2PA cross sections due
to the larger number of pathways and increased contribution from diagonal
matrix elements. The non-totally symmetric transitions pz!px,y,s!d,
andpz!f(not shown) are also allowed but weaker due to fewer pathways to
each final state and a smaller contribution from off-diagonal elements.174102-5 de Wergifosse, Elles, and Krylov J. Chem. Phys. 146, 174102 (2017)
transition to an intermediate state). We note that a similar anal-
ysis of the structure of the 2PA tensor has been presented in
Ref. 20.
A similar analysis of the atomic selection rules helps to
rationalize the cross sections for transitions involving Rydberg
states of polyatomic molecules. Since only l=1 transitions
are dipole-allowed, only l=0,2 transitions are allowed in
2PA. Figure 2 illustrates this point in more detail; it also shows
intermediate states for selected transitions. Based on the num-
ber of intermediate states available for a particular transition,
one may expect l=0 transitions (e.g., s!sandp!p) fea-
turing, in general, larger 2PA cross sections than the l=2
transitions (e.g., s!dandp!f). For example, the top
row of the figure shows that there are three possible pathways
fors!stransitions, each contributing a diagonal element to
the two-photon transition matrix, M(n+1)s ns
aa . In contrast, the
s!dtransitions have only two off-diagonal pathways to each
of the dxy,dxz, and dyzorbitals and one diagonal pathway to
dz2. Given larger contributions from the diagonal elements to
Eq. (12), the s!stransition will dominate the 2PA spectrum
relative to the s!dxy,dxz,dyz, ordz2transitions. Two-photon
transitions to dx2 y2are not symmetry-allowed.
The last two rows of Figure 2 illustrate a similar result
for transitions originating from the npzorbital, for which there
are two off-diagonal pathways to each of the pxandpyfinal
orbitals (i.e., via the sanddxzordyzintermediate states),
compared with four diagonal pathways to pz(one passing
through s, one through dz2, one through dxz, and one through
dyz). The key result is that the totally symmetric s!sand
pz!pzatomic transitions have larger cross sections than the
non-totally symmetric transitions based on simple symmetry
arguments.
D. Transitions via a dominant intermediate state
The symmetry arguments in Sec. II C are most appro-
priate for excitation conditions in which none of the
intermediate one-photon accessible states dominate the sum-
over-states expressions in Eqs. (9) and (10). This condition
(i.e., no resonance enhancement) is achieved when all one-
photon transition energies
n0are very different from the
photon energies, !1and!2. However, it is also instructive
to consider a special case in which a single intermediate state
dominates the 2PA transition. For example, intermediate state
jiiwill dominate the sum-over-states expressions if it is low-
lying and is strongly dipole-coupled to both the initial and the
final states. In this case, the 2PA cross section for degenerate
excitation (!1=!2=
k0
2) and parallel polarization takes the
form
0k
k24
30h0jˆnjii2hijˆnjki2
(
i0
k0
2)2, (24)
or, in the case of the non-hermitian EOM theory
0k
k24
30h0jˆnjiihijˆnjkihkjˆnjiihijˆnj0i
(
i0
k0
2)2. (25)
Such a three-state model has been used to rationalize trends
in the non-linear properties of a variety of conjugated chro-
mophores.47,72–74Within the three-state model, we can limitour attention to the ground, intermediate, and target excited
states. As follows from Eq. (24), large 2PA cross sections can
be attained by maximizing the brightness of the 1PA transi-
tions from the ground to the intermediate state and from the
intermediate state to the target state.
The one-photon transition dipole moment between two
states is
IF=h Ij~rj Fi=Tr[
IFr], (26)
where
IFis the one-particle transition density matrix con-
necting two states, Iand F. The transition density matrix
is defined as
IF
pqh Ijˆpyˆqj Fi, (27)
where ˆ pyand ˆqare the creation and annihilation operators
corresponding to orbitals pand q. As one can see from
Eqs. (26) and (27) (and as discussed in Ref. 47), the transition
dipole moment is large when the corresponding one-particle
transition density is delocalized. This can be achieved when
the electronic transition involves significant charge-resonance
character over a large area. One can further simplify the
analysis by considering a pair of states that differ by the
state of one electron, such as HOMO !LUMO excitation
from the closed-shell ground state. Then the expression for
the transition dipole moment can be written in terms of a
single matrix element between the orbitals involved in the
transition
( i! a)=p
2h ij~rj ai. (28)
As described below, the wave function analysis based on the
one-particle transition density matrix allows one to describe
electronic transitions between correlated many-electron states
in terms of natural transition orbitals, so Eq. (28) can be
used to rationalize the brightness of transitions between
states described by general multi-configurational wave
functions.
Eq. (28) shows that large one-photon transition dipoles
can be obtained for transitions between bonding and anti-
bonding orbitals, such as !transitions in ethylene. The
delocalization of the two orbitals due to increased conjugation
in larger molecules will result in even larger dipole moments.
A large matrix element can also be obtained when one orbital is
a delocalized valence orbital (such as an extended orbital in a
conjugated molecule) and the second is a delocalized Rydberg
orbital. In order to maximize both M0iandMik, the target state
should be a one-electron transition from the intermediate state
and the respective molecular orbitals should be delocalized.
Assuming that the intermediate bright state is accessed through
a bonding-antibonding transition, a large value of Mikcan be
achieved for two different types of target states: (i) a dou-
bly excited valence state, such as a hypothetical state of
ethylene (note that such doubly excited configuration belongs
to the fully symmetric irrep) or (ii) a singly excited state of
Rydberg character, such as the !Ry(3p) state of ethylene.
We note that diffuse character of the target state can result in
larger 2PA cross sections even beyond the three-state model
because a delocalized orbital can lead to larger transition dipole
moments for many intermediate states in the sum-over-states
expression.174102-6 de Wergifosse, Elles, and Krylov J. Chem. Phys. 146, 174102 (2017)
E. Wave function analysis
Let us now discuss the molecular orbital character of
electronic states giving rise to large 2PA cross sections. The
character of an excited state can be assigned by analyzing
the leading electronic configuration of the underlying wave
function, that is, by identifying the leading amplitude and
the respective MOs. A more rigorous (and orbital-invariant)
way52–54,75–77to analyze excited states is based on the one-
particle transition density matrix, Eq. (27).
IFcontains the
information needed to compute one-electron transition prop-
erties such as transition dipole moments (i.e., Eq. (26)). It also
contains a compact description of the changes in electronic
structure between Iand F. The norm of
IFmeasures the
degree of one-electron character in the I! Ftransition.
For example,jj
jj= 1 for those transitions which are of purely
one-electron character,78e.g., for the transitions between the
Hartree-Fock ground state and target configuration interaction
single (CIS) states. The norms of the EOM amplitudes R1and
R2from Eq. (5) arejjR1jj1 andjjR2jj0 when the transition
has one-electron character.
The one-particle transition density matrix can be decom-
posed using the singular value decomposition procedure giv-
ing rise to the most compact description of the electronic
transition
IF=UVT, (29)
where UandVare the unitary matrices and is a diagonal
matrix containing non-negative numbers
diag(p
1,p
2,:::). (30)
Usually, only a small number of iare significant. For exam-
ple, for the CIS transition of pure HOMO-LUMO character,
there is only one non-zero iand it equals one (for spinless
represented in the basis of spatial MOs). For each significant
i, one can define a pair of natural transition orbitals (NTOs);
they can be interpreted as hole and particle orbitals for the
transition
hole
i(r)=X
pUpi p(r), (31)
particle
i(r)=X
pVpi p(r). (32)
To quantify the number of significant contributions to the
specific transition, one can define the NTO participation ratio
PRNTO= P
ii!2
P
i2
i. (33)
For the CIS transition of pure HOMO !LUMO character,
PRNTO= 1, whereas for the state which has equal weights of the
HOMO!LUMO and HOMO 1!LUMO+1 configurations,
PRNTO= 2 (for example, PR NTOequals 1.11 for the !
transition of ethylene and 2.06 for toluene).
Using the NTO basis, one can define the exciton wave
function as follows:
(~rh,~re)=X
ip
i hole
i(~rh) particle
i(~re). (34)
The square of the exciton wave function gives the probability
that the hole is located at the position rhand the electron at theposition re. One can define the average positions of the hole
and the electron as the centroids of the respective NTOs
~rh =1
jj
jjX
iiD
hole
ij~rj hole
iE
,
~re =1
jj
jjX
iiD
particle
ij~rj particle
iE
.(35)
The exciton can be characterized by the absolute value of
the mean separation vector ~dh!e =
~re ~rh that quan-
tifies the charge separation between the hole and the elec-
tron; the hole size h=qD
~r2
hE
~rh 2; the electron size
e=qD
~r2eE
~re 2; and the electron-hole pair correlation
coefficient Reh=h(~rh h~rhi)(~re h~rei)i
he. In the analysis below,
we provide these quantities as well as the jj
IFjjand || R1||2val-
ues, which quantify the degree of single excitation involved in
a transition.
Additional insight can be derived from the properties of
the electronic states, such as permanent dipole moments and
the spatial extent of the wave functions.79The size of the
wave function can be characterized by the expectation value
ofR2,
h jR2j i=h jX2j i+h jY2j i+h jZ2j i. (36)
Since the size of the electronic wave function increases with
the system size, we consider the difference between hR2iof
the target EOM-EE-CCSD state and of the CCSD reference
state79
hR2ih EOMjR2j EOMi h CCSDjR2j CCSDi. (37)
For a valence state, theD
R2E
value is similar to the ground-
state value, but for Rydberg states, the value is larger. Although
the magnitude of the individual Cartesian components of R2
depends on the choice of the coordinate system (or molec-
ular orientation), the absolute value ( hR2i) is invariant with
respect to the molecular orientation, owing to the properties of
the trace. In the following, we use this quantity to distinguish
between Rydberg and valence states.
F. Computational details
The structures of ethylene ( D2h), toluene ( Cs),trans -
stilbene ( C2h),cis-stilbene ( C2), and phenanthrene ( C2v) were
optimized at the B3LYP/cc-pVTZ level of theory. All rele-
vant geometries are given in the supplementary material. The
effect of the basis set and of using Cholesky decomposition
on the computed properties was investigated using ethylene
and toluene. We employed singly and doubly augmented Dun-
ning’s double- and triple- basis sets. We also considered “-df”
versions of the bases derived from the parent naug-cc-pV XZ
bases by removing dandffunctions from the diffuse augment-
ing sets. The errors due to freezing core electrons and due to
using Cholesky decomposition with a threshold of 10 2have
been also quantified. The results of this analysis are summa-
rized in the supplementary material. We found that (i) at least
a doubly augmented Dunning basis set is needed, (ii) when
considering low energy states (below 9 eV , for ethylene), we
can use a double- basis set and remove the ddiffuse func-
tions from both augmenting sets, and (iii) using the Cholesky174102-7 de Wergifosse, Elles, and Krylov J. Chem. Phys. 146, 174102 (2017)
FIG. 3. 1PA and 2PA spectra of ethylene computed at the EOM-EE-CCSD/d-aug-cc-pVTZ level of theory. 2PA spectrum is computed for parallel polarization
( 1PA= 0.4 eV and 2PA= 0.1 eV).
decomposition and freezing the core electrons have a negli-
gible effect on 2PA cross sections (less than 3%). Thus, the
d-aug(-d)-cc-pVDZ is the minimal basis set for computing the
2PA cross sections of low energy states. In the production cal-
culations, we employed d-aug-cc-pVTZ basis set for ethylene,
d-aug(-df)-cc-pVTZ basis set for toluene, and d-aug(-d)-cc-
pVDZ basis set for larger molecules. Except for ethylene, we
freeze core electrons and use the Cholesky decomposition with
a threshold of 10 2.
All calculations were performed using the Q-Chem elec-
tronic structure program.80,81The reported symmetry labels
of the electronic states and MOs correspond to the standard
molecular orientation used in Q-Chem, which differs from the
Mulliken convention.82,83
III. RESULTS AND DISCUSSION
A. Ethylene
The ethylene molecule belongs to the D2hpoint group
which has 8 irreps, four with ungerade symmetry and four
with gerade symmetry. One-photon transitions from the X1Ag
ground state can access only ungerade states with B 1u, B2u,
and B 3usymmetry, but two-photon transitions to all of the
gerade states are symmetry-allowed. The totally symmetric
transitions to A gstates have three diagonal components of
the two-photon transition matrix, whereas each of the other
three irreps (B 1g, B2g, and B 3g) has only two off-diagonal
components that are nonzero.84Therefore, based on symmetry
considerations alone, transitions to nAgstates are expected to
dominate the 2PA spectrum.
Figure 3 shows the computed 1PA and 2PA spectra of ethy-
lene. As expected from the symmetry-imposed selection rules,
the two spectra are markedly different. Here, we computed ten
EOM-EE-CCSD singlet states per irrep. The lowest ionization
energy of ethylene is 10.75 eV (EOM-IP-CCSD/d-aug-cc-
pVTZ) and corresponds to ionization from the b 1uHOMO.Thus, the bound part of each spectrum ends at 10.75 eV and the
spectral features beyond this value correspond to autoionizing
resonances (the description of these states in our calculations
is very approximate).
Applying a phenomenological broadening of = 0.4 eV
to each of the transitions in the 1PA spectrum results in sev-
eral distinct absorption bands, including a single strong band
below 9 eV . This low-energy band is dominated by two bright
transitions, 11B1uand 11B3u. Similarly, the broadened 2PA
spectrum ( = 0.1 eV) has a single, strong band below 9 eV and
several additional bands at higher energy. The lowest-energy
2PA band contains a dominant transition to the 21Agstate at
8.51 eV , with weaker transitions to 11B2gand 11B3ggiving
rise to a shoulder on the low-energy side of the band.
To illustrate the orbital character of the dominant 1PA and
2PA transitions below 9 eV , Figure 4 shows the natural transi-
tion orbitals (NTOs) associated with the 11B1u, 11B3u, 21Ag,
11B2g, and 11B3gstates of ethylene. The relevant properties
derived from the wave function analysis are summarized in
Table I. All transitions have participation ratios around 1; thus,
each transition can be described by a single pair of hole-particle
FIG. 4. NTOs corresponding to the 11B1u, 11B3u, 21Ag, 11B2g, and 11B3g
excited states in ethylene. Left: hole NTO. Right: particle NTOs. Isovalue ofp
310 6is used.174102-8 de Wergifosse, Elles, and Krylov J. Chem. Phys. 146, 174102 (2017)
TABLE I. Properties of the 11B1u, 11B3u, 21Ag, 11B2g, and 11B3gexcited states of ethylene.
State E(eV) || R1||2jj
jjPRNTOj~dh!ej(Å)h(Å)e(Å) Rehe h(Å)hR2i(Å2)
11B1u 7.46 0.951 0.939 1.00 0.0 1.24 3.58 0.01 2.35 10.0
11B3u 8.08 0.956 0.961 1.11 0.0 1.24 2.42 0.00 1.18 3.7
11B3g 8.10 0.952 0.941 1.05 0.0 1.24 3.95 0.01 2.71 12.8
11B2g 8.17 0.949 0.937 1.00 0.0 1.24 4.24 0.01 3.00 15.0
21Ag 8.51 0.953 0.942 1.02 0.0 1.24 4.70 0.01 3.46 19.1
NTOs. The hole NTO is very similar for all five states and can
be described as a valence orbital nearly identical to the b1u
HOMO. The particle NTO for the 11B3ustate hascharacter
and is accessible via !valence excitation, whereas the
11B1ustate has more diffuse Ry(s)-like character. The valence
and Rydberg-like character of the two 1PA-accessible states is
evident from the very different spatial extent of the wave func-
tions, as measured by hR2i, which equals 3.7 and 10.0 Å2for
11B3uand 11B1u, respectively. The other three particles’ NTOs
in Fig. 4 represent the excited states responsible for the lowest-
energy 2PA band. Their shapes suggest that these states can
be described as Rydberg-type states of 2 pcharacter. As pre-
dicted on the basis of the atomic selection rules (Fig. 2), the
two-photon transition to the 21Agstate is brighter than the tran-
sitions to 11B2gand 11B3g. The transition to 21Agresembles
apz!pzexcitation, which can be accessed via the interme-
diate s,dz2,dxz, and dyzstates. Each of these four pathways
contributes to a diagonal element to the two-photon transition
matrix. The 11B2gand 11B3gstates resemble pxandpyRyd-
berg orbitals that are accessible through the intermediate sand
dxzordyzstates, respectively, and therefore contribute only to
two off-diagonal elements for each transition (see Fig. 2). As
discussed above, the additional pathways and larger contribu-
tions from diagonal elements explain the larger relative 2PA
cross section for the 21Agstate.Alternatively, one can rationalize the relative 2PA cross
sections in the context of the three-state model in Eq. (24).
The weight of the leading amplitude in the EOM-CCSD wave
function of the 21Agstate is 0.95; thus, this is a singly excited
state. Singly excited character of this state is further con-
firmed by the square norm of transition density matrix,
,
which is equal to 0.942. In a three-state model, excitation
to the 21Agstate can be described as !!pztransi-
tion via the brightest 1PA intermediate state, 11B3u. Both the
valence!and Rydberg !pztransitions have large
transition dipoles according to Eq. (28), therefore, the over-
all transition has a large 2PA cross section. In addition, the
one-photon accessible 11B1uintermediate state facilitates an
alternative!s!pzpathway to the 21Agstate, as well as
the!s!px,pytransitions to the 11B3gand 11B2gstates,
respectively.
The diffuse character of the 2PA active states is con-
firmed by the quantitative wave function analysis, which
gives hR2ivalues in the range 13-19 Å2. The size dif-
ference between the electron and hole NTOs, e h,
provides an additional measure of the relative diffuseness
of the states, with values ranging from 2.7 to 3.5 Å. We
note that for these three states, a more diffuse charac-
ter correlates with larger 2PA cross sections, as predicted
above.
FIG. 5. 1PA and 2PA spectra of toluene computed at the EOM-EE-CCSD/d-aug(-df)-cc-pVTZ level of theory. 2PA spectrum is computed for parallel polarization
( 1PA= 0.4 eV and 2PA= 0.1 eV).174102-9 de Wergifosse, Elles, and Krylov J. Chem. Phys. 146, 174102 (2017)
TABLE II. Properties of the 71A0, 21A00, 51A0, and 131A0excited states of toluene.
State E(eV) || R1||2jj
jjPRNTOj~dh!ej(Å)h(Å)e(Å) Rehe h(Å)hR2i(Å2)
21A006.48 0.935 0.916 1.02 0.2 1.77 4.43 0.06 2.66 14.1
51A06.94 0.937 0.920 1.03 0.1 1.89 5.22 0.03 3.33 20.9
41A006.98 0.932 0.917 1.63 0.5 1.81 3.93 0.06 2.12 10.3
71A07.21 0.931 0.919 2.06 0.4 1.87 3.86 0.06 1.99 9.8
131A07.80 0.937 0.914 1.03 0.2 1.78 6.31 0.04 4.53 33.4
B. Toluene
Toluene belongs to the C ssymmetry group, which has
only two irreps (A0and A00both of which are active in both 1PA
and 2PA. We calculated 15 states per irrep to obtain the 1PA and
2PA spectra in Fig. 5. The lowest ionization energy of toluene
is 9.21 eV (EOM-IP-CCSD/d-aug-cc-pVDZ), well above the
energies of the states calculated here. The brightest transitions
in the low-energy region of the 1PA spectrum correspond to
the 41A00and 71A0states. In the 2PA spectrum, the three states
with the largest cross sections are 21A00, 51A0, and the much
stronger 131A0. The key parameters characterizing these states
are collected in Table II, and the respective NTOs are shown in
Fig. 6. The || R1||2andjj
jjvalues show that all of these states
are singly excited.
The two leading NTOs of the 71A0state, which have the
largest 1PA cross section, contribute with weights of 0.49 and
0.33. The first pair of NTOs can be described as a !dz2
Rydberg transition, whereas the second pair of NTOs corre-
sponds to a!transition. The partial valence character of
this transition is evident from the moderately diffuse value of
hR2i= 9.8 Å2. The second brightest 1PA state, 41A00, also has
mixed Rydberg/valence NTOs ( !pand!), although
the Rydberg-like transition carries more weight (0.64 and 0.19,
respectively) and therefore gives a final state with slightly more
diffuse character of hR2i= 10.3 Å2.
Each of the three brightest 2PA states has only one dom-
inant pair of NTOs ( 10.85). The particle NTOs for all
three states have significant Rydberg character and become
FIG. 6. NTOs corresponding to the 71A0, 21A00, 41A00, 51A0, and 131A0
excited states in toluene. Isovalue ofp
310 6is used.increasingly diffuse for higher-lying states. The transition to
the 21A00state has!sRydberg character ( hR2i= 14.1 Å2),
the transition to 51A0has!pRydberg character ( hR2i
= 20.9 Å2), and the transition to 131A0has the largest 2PA cross
section and!dRydberg character ( hR2i= 33.4 Å2). As for
ethylene, the values of e hcorrelate perfectly with hR2i
due to the similarity of the hole orbital for all three transitions.
Because of the lower symmetry of toluene relative to ethylene,
the states in toluene have moderate charge-transfer character,
as indicated by the non-zero j~dh!ejvalues in Table II. We
observe the largest j~dh!ejvalues for the 41A00and 71A0states
(the two brightest 1PA states), which have the least diffuse
character of the five states discussed here.
Judging from the shapes of the NTOs, the 131A0
state resembles a totally symmetric dyz!dyz-type transition,
whereas 21A00and 51A0resemble d!sandf!ptransi-
tions. Thus, on the basis of the atomic selection rules (Fig. 2),
one would expect the following trend in the respective 2PA
cross sections: 131A0>51A0>21A00because of the number of
pathways between the respective (atomic orbital-like) NTOs,
which is consistent with the computed values. Here we again
observe that more diffuse character correlates with larger cross
sections.
Within the few-state model, the dominant one-photon
transitions to 71A0and 41A00provide a glimpse of the
molecular orbital pathways. The leading Ry(dz2)-like NTO
of 71A0provides a likely intermediate for the f!dz2!p
pathway to the 51A0state, and the Ry(pz)-like NTO for 41A00
provides possible pathways for the d!sandd!dtransitions
to 21A00and 131A0, respectively. Although other pathways are
also likely to contribute to the sum-over-states 2PA expres-
sion, the strong one-photon transitions and partial Rydberg-
like character suggest that the 71A0and 41A00states play an
important role as intermediates.
C.Trans -stilbene, cis-stilbene, and phenanthrene
The compounds trans -stilbene, cis-stilbene, and phenan-
threne have similar chemical structures but belong to different
symmetry point groups. trans -stilbene belongs to the C2hpoint
group, which has two irreps of gerade symmetry that are active
in 2PA and two of ungerade symmetry that are active in 1PA.
The A gtransitions should dominate the 2PA spectrum because
the two-photon transition matrix has five nonzero components
including the three diagonal elements, compared with two off-
diagonal elements for B gtransitions.69cis-stilbene is in the C2
point group, with two irreps (A and B) that are active in both
1PA and 2PA. Totally symmetric transitions to the A states
should dominate the 2PA spectrum in this point group as well,174102-10 de Wergifosse, Elles, and Krylov J. Chem. Phys. 146, 174102 (2017)
FIG. 7. 2PA spectra of trans -stilbene, cis-stilbene, and phenanthrene computed at the EOM-EE-CCSD/d-aug(-d)-cc-pVDZ level of theory (parallel polarization,
!1=!2, = 0.1 eV).
due to five non-zero matrix elements, compared with only two
off-diagonal elements for states with B symmetry. Phenan-
threne belongs to the C2vpoint group, which has four total
irreps that are all active in 2PA but only three of which are
active in 1PA. As illustrated in Fig. 1, the states belonging to
the totally symmetric irrep of C2valso should have the largest
cross sections because the diagonal elements of the two-photon
transition matrix are nonzero for A 1transitions, whereas the
other irreps each have only two off-diagonal elements that are
nonzero.
Figure 7 shows the computed 2PA spectra of trans -
stilbene, cis-stilbene, and phenanthrene. The spectra were
computed using 8 states per irrep. The vertical ionization
energies are 7.58 eV , 7.84 eV , and 7.79 eV (EOM-IP-CCSD/d-
aug(-d)-cc-pVDZ) for trans -stilbene, cis-stilbene, and phenan-
threne, respectively. In the analysis below, we focus on the
states with the largest 2PA cross sections, which generally are
not the lowest-energy transitions. Table III collects relevant
properties of the selected excited states. The || R1||2andjj
jjof
these states reveal singly excited character, and the respective
NTOs are shown in Figs. 8–10.
The 2PA spectra of all three compounds are dominated
by the fully symmetric (A g, A, and A 1) transitions, as pre-
dicted above. The spectrum of trans -stilbene has a very strong
transition to the 51Agstate near 6.3 eV . The cross section
of that transition is 10 times larger than the next most
intense transition in the molecule (to the 41Agstate) and the
strongest of all the 2PA transitions calculated in this paper. Thespectrum of cis-stilbene has a strong 2PA band near 6.3 eV that
includes transitions to the 61A and 91A states, as well as con-
tributions from several weaker transitions. Finally, the 2PA
spectrum of phenanthrene has two main bands due to the 51A1
and 91A1states, respectively, with some weaker underlying
contributions.
The absolute 2PA cross sections for trans -stilbene, cis-
stilbene, and phenanthrene are larger than for ethylene and
toluene, which can be explained by the extended delocalization
of the respective states. Most notable is the very strong transi-
tion to the 51Agstate of trans -stilbene, which has two dominant
NTO pairs (Fig. 8). The first pair of NTOs resembles a !
transition with some diffuse Ry(p)-like character, but the sec-
ond pair has more compact !character. The two pairs
of NTOs resemble linear combinations of the 51A0state of
toluene, having !pRydberg character. The leading pair of
NTOs can be described as the symmetric combination of NTOs
from toluene, and the second pair as the antisymmetric com-
bination with additional character of the ethylene bridge
(resembling the 11B3uparticle NTO of ethylene). With hR2i
= 10.2 Å2, the 51Agstate of trans -stilbene is more diffuse than
a purevalence state but not as diffuse as the Rydberg states
of the other compounds due to its mixed Rydberg/valence
character.
The strongest 2PA transitions of cis-stilbene (61A and
91A) also resemble combinations of the toluene NTOs
but have more complicated character due to the reduced
conjugation of the nonplanar structure. The twisted structure
TABLE III. Properties of the brightest 2PA states in trans -stilbene, cis-stilbene, and phenanthrene.
State E(eV) || R1||2jj
jjPRNTOj~dh!ej(Å)h(Å)e(Å) Rehe h(Å)hR2i(Å2)
trans -stilbene
51Ag 6.28 0.879 0.856 1.75 0.0 3.08 4.80 0.05 1.72 10.2
cis-stilbene
61A 6.27 0.910 0.888 1.60 1.1 2.87 5.55 0.15 2.68 18.4
91A 6.49 0.921 0.902 1.28 0.6 2.84 5.86 0.13 3.02 22.9
Phenanthrene
51A1 6.11 0.910 0.889 2.01 0.1 2.70 4.93 0.04 2.23 13.9
91A1 6.81 0.895 0.869 2.75 0.3 2.67 4.61 0.06 1.94 11.2174102-11 de Wergifosse, Elles, and Krylov J. Chem. Phys. 146, 174102 (2017)
FIG. 8. NTOs corresponding to the 51Agexcited state in trans -stilbene.
Isovalue ofp
310 6is used.
limits the interaction between the two phenyl rings and reduces
the electronic coupling across the ethylene bridge. This is evi-
dent from the leading NTO of the 61A state, which resembles
an anti-symmetric combination of the 51A0state (!p)
of toluene for each side of the molecule. The deviation from
planarity also reduces the intensity of the 1PA transitions and
explains the reduced 2PA cross sections in cis-stilbene relative
totrans -stilbene.
The lower cross section for cis-stilbene compared with
trans -stilbene can be rationalized based on the molecular
orbital characteristics and Eqs. (24)–(27). The delocalization
effect is quantified in Table IV using a three-state model to
compare the calculated 2PA cross sections for a single interme-
diate state. Despite the very similar 1PA intensities for initial
to intermediate states, and for intermediate to target states, the
FIG. 9. NTOs corresponding to the 61A and 91A excited states in cis-stilbene.
Isovalue ofp
310 6is used.
FIG. 10. NTOs corresponding to the 51A1and 91A1excited states in
phenanthrene. Isovalue ofp
310 6is used.
overall 2PA cross section of trans -stilbene via the low-energy
11Bustate is an order of magnitude higher than any of the other
transitions in the table.
Finally, for phenanthrene, the 51A1and 91A1states have
the largest 2PA cross sections although they are less intense
than the two strongest transitions in cis-stilbene. The 51A1
state is related to the 51Agstate of trans -stilbene, with the
leading pair of NTOs resembling a !p-like Rydberg
transition. The transition to the higher-energy 91A1state has
TABLE IV . Parallel 2PA cross sections for the brightest 2PA transitions in
trans -stilbene, cis-stilbene, and phenanthrene.
Three-state transition !0!k!0!i
0!i!k (hartree) (hartree) 0k
k(a.u.)a0k
k(a.u.)b
trans -stilbene
11Ag!11Bu!51Ag 0.167 0.231 24 469 40 786
11Ag!21Bu!51Ag 0.178 0.231 1 448 40 786
cis-stilbene
11A!21B!61A 0.180 0.230 3 832 3 768
11A!21B!91A 0.180 0.239 2 340 3 126
Phenanthrene
11A1!31B1!51A1 0.207 0.225 1 244 2 084
11A1!31B1!91A1 0.207 0.251 1 590 1 785
aEstimated using the three-state model.
bFull calculation.174102-12 de Wergifosse, Elles, and Krylov J. Chem. Phys. 146, 174102 (2017)
three main pairs of NTOs: the first and third pairs resem-
bling two different !d-like Rydberg transitions and the
second pair corresponding to !-like transition. The
mixed valence/Rydberg character of the two phenanthrene
states is evident from the moderately diffuse orbitals ( hR2i
= 11-14 Å2), compared with predominantly Rydberg transi-
tions in cis-stilbene ( hR2i>18 Å2). The reduced conjugation
ofcis-stilbene may be partially compensated by the more
delocalized character of the dominant states in that spectrum
compared with the other two compounds (see Table III). The
NTOs of trans -stilbene and phenanthrene are not as diffuse
(5 Å), possibly due to the partial valence character of the
excited states. We note that the relationship between 2PA cross
section and electronic delocalization, as well as the reduced
cross sections for twisted structures, has been documented
in previous studies.40,47,51A more detailed discussion of the
valence and Rydberg character of 2PA transitions is provided
in Ref. 55. The distinction is especially important for com-
parison with experimental spectra in solution, where Rydberg
states might be pushed up to higher energies.
IV. CONCLUSION
In this contribution, we investigated 2PA spectra of
several prototypical molecules (ethylene, toluene, cis- and
trans -stilbene, and phenanthrene) and analyzed the electronic
structure of the states that give rise to the dominant spec-
tral features. By employing the best available methodology25
for calculating 2PA cross sections for molecules of this size,
which is based on the EOM-CCSD formalism, our calculations
provide an important high-quality reference data for future
theoretical and experimental studies. We also use an orbital-
invariant density-matrix based approach53for analyzing the
characters of the states in question. This analysis, which is
more rigorous than qualitative description of the characters
of MOs involved in the transition, provides a valuable con-
tribution towards understanding photophysics of stilbene and
phenanthrene.
We reviewed the symmetry-imposed selection rules and
showed that, since the diagonal tensor elements of the two-
photon transition matrix belong to the totally symmetric irreps,
those transitions typically dominate the 2PA spectra for par-
allel polarization. Similar symmetry arguments based on the
atomic selection rules predict that l= 0 transitions have
the largest 2PA cross sections among Rydberg-like transi-
tions, which further helps to rationalize the calculated 2PA
spectra. We also note that for the same molecule, a more dif-
fuse character of the target state corresponds to a larger 2PA
cross section, which can be explained by considering transi-
tion dipole moments between the molecular orbitals involved
in the brightest valence transitions and the diffuse Rydberg
orbitals.
trans- stilbene features the largest 2PA cross sections
among the molecules investigated here owing to the most
extensive delocalization of the relevant states. The 51Agstate
has the largest cross section. The first pair of the correspond-
ing NTOs can be described as !p-like Rydberg transi-
tion, where the diffuse NTO of the electron resembles the
bonding linear combination of p-like Rydberg MOs on eachphenyl ring. In cis-stilbene, due to the nonplanar geometry,
the exciton delocalization is reduced, resulting in lower 2PA
cross sections ( !61A and 91A) relative to the 51Agstate of
trans -stilbene. In phenanthrene, the largest 2PA cross sections
correspond to pandd-like Rydberg target states.
The general trends that affect the 2PA transition strengths
derived on the basis of simple symmetry arguments and
atomic selection rules provide an important foundation for
predicting and rationalizing 2PA spectra of increasingly com-
plex chromophores. Such fundamental insight is essential for
the development of new two-photon active compounds and
NLO materials, as well as interpreting the spectra of exist-
ing compounds. For example, the larger 2PA cross sections
for Rydberg states compared with valence states may play
an important role in determining the most efficient interme-
diate states for resonance-enhanced multi-photon ionization
(REMPI) schemes involving a two-photon transition.
SUPPLEMENTARY MATERIAL
See supplementary material for additional details and
relevant Cartesian geometries.
ACKNOWLEDGMENTS
A.I.K. acknowledges support by the U.S. National Sci-
ence Foundation (Grant No. CHE-1264018). C.G.E. acknowl-
edges support from the U.S. National Science Foundation
through a CAREER Award (No. CHE-1151555). We are grate-
ful to Dr. Kaushik Nanda for valuable discussions and his
help with the calculations. We thank Dr. Thomas Jagau for
his feedback about the manuscript.
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1.4746381.pdf | Fast domain wall dynamics in MnAs/GaAs films
M. Tortarolo, L. Thevenard, H. J. von Bardeleben, M. Cubukcu, V. Etgens, M. Eddrief, and C. Gourdon
Citation: Applied Physics Letters 101, 072408 (2012); doi: 10.1063/1.4746381
View online: http://dx.doi.org/10.1063/1.4746381
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202.28.191.34 On: Mon, 22 Dec 2014 06:46:00Fast domain wall dynamics in MnAs/GaAs films
M. Tortarolo,1,2L. Thevenard,3H. J. von Bardeleben,3M. Cubukcu,3,4V. Etgens,2,3
M. Eddrief,2,3and C. Gourdon3
1Laboratoire de Chimie Physique, Universit /C19e Pierre et Marie Curie, CNRS UMR 7614, 11 rue Pierre et Marie
Curie, F-75005 Paris, France
2International French Argentinian Nanoscience Laboratory LIFAN, CNEA, CONICET, Argentina, CNRS,
France
3Institut des Nanosciences de Paris, Universit /C19e Pierre et Marie Curie, CNRS, UMR7588, 4 place Jussieu,
F-75005 Paris, France
4Laboratoire Nanostructures et Magn /C19etisme, INAC, CEA-Grenoble, France
(Received 5 June 2012; accepted 1 August 2012; published online 16 August 2012)
Field-induced domain wall (DW) dynamics was investigated in MnAs/GaAs(100) films by means
of longitudinal Kerr microscopy and a field pulse technique. Saw-tooth type domains wereobserved in the studied temperature range. DW velocities up to 950 m s
/C01were measured at 200 K
and up to 540 m s/C01at 290 K, at the beginning of the nucleation of the non-ferromagnetic bphase
within the ferromagnetic aphase. Different propagation regimes were observed depending on the
magnitude of the Walker field compared to the depinning field, which depends on the progressive
nucleation of an unordered bphase. The results are interpreted in the framework of the one-
dimensional model (1D) for DW propagation. VC2012 American Institute of Physics .
[http://dx.doi.org/10.1063/1.4746381 ]
The possibility of storing information in magnetic domains
separated by DWs moving along domain tracks1,2has stimu-
l a t e dar e n e w e di n t e r e s ti nD Wp ropagation in in-plane magne-
tized materials for spintronics. Most studies are performed on
permalloy (Py) where high DW velocities (up to 1000 m s/C01)
can be achieved.3,4However, it is well known that MnAs is a
promising candidate for spintronics applications5–7as it can be
grown on different semiconductors.8,9Even though DW dy-
namics in ferromagnetic films has b een intensively investigated
during the last decade,4,10,11D Wp r o p a g a t i o ni nM n A sw a s
studied only recently. Up to now, studies were limited to thelow velocity regime (a few tens of lms
/C01) observed under
application of a magnetic field cl ose to the coercive field where
DW propagation is governed by pinning defects.12–14In parti-
cular, a magnetization reversa l study reported on the tempera-
ture dependent critical scaling behavior of Barkhausen
avalanches.14,15This dependency was ascribed to the decrease
of the saturation magnetizatio n with temperature due to the
magneto-structural phase coex istence of the ferromagnetic a
MnAs phase and the non-ferromagnetic bMnAs phase in the
280–330 K temperature interval.
Field-induced DW dynamics in the intrinsic regime
(above the depinning regime) has not been investigated inMnAs yet. The effect of the a=bphase coexistence on DW
dynamics in intrinsic regimes therefore remains an open ques-
tion and particularly relevant to potential applications ofMnAs in data storage. In this letter, we report on the DW
dynamics in MnAs/GaAs over a wide range of magnetic fields
and temperature, in the pure aphase, and at the onset of the
bphase (T ¼290 K). Very high DW velocities, up to
940 m s
/C01, i.e., comparable to those found for Py (Refs. 3and
4) were measured. Strikingly, the maximum velocity at 290 K
is still as high as 540 ms/C01. Moreover, the analysis of the ve-
locity curves leads to the identification of different dynamical
regimes, depending on the relative values of the field at theend of the depinning regime and the Walker field H
W.A sdepinning critically depends on temperature for MnAs/GaAs
due to its characteristic phase coexistence, the DW velocitycurve shows different shapes depending on temperature.
The sample is a 300 nm thick MnAs/GaAs (100) film
grown as described elsewhere,
9,16in the epitaxial condition
MnAs ð/C01100ÞkGaAs ð100Þwith MnAs ½0001/C138kGaAs [1–10].
This determines a well-defined uniaxial magnetic anisotropy
in the sample plane with the easy axis along the [11–20] direc-tion, an intermediate anisotropy axis along [ /C01100] and a hard
one along [0001]. High resolution magnetic force microscopy
measurements at remanence showed that in MnAs/GaAs(100), the DWs are 180
/C14Bloch type.17The sample was placed
in a helium-flow cryostat. The hysteresis cycles were meas-
ured by magneto-optical longitudinal Kerr effect for severaltemperatures from 200 K to 290 K, showing square loops in
this range of temperature. The temperature dependence of the
coercive fields H
cobtained from these loops (Fig. 1, inset)
shows a characteristic increase similar to that of Ref. 18.A t
200 K, the MnAs film exhibits a single a-phase structure. The
FIG. 1. DW velocity as a function of applied field. The lines are guides for
the eyes. (inset) Dependence of the coercive field on temperature.
0003-6951/2012/101(7)/072408/4/$30.00 VC2012 American Institute of Physics 101, 072408-1APPLIED PHYSICS LETTERS 101, 072408 (2012)
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202.28.191.34 On: Mon, 22 Dec 2014 06:46:00progressive nucleation of the bphase is initiated above 200 K.
It remains an unordered phase up to 280–290 K, beyond whichit adopts a self-organized stripe pattern, leading to a strong
increase of the coercive field.
18,27,28
The DW dynamics depends critically on the micromag-
netic parameters: magnetization, magnetic anisotropy con-
stants, and exchange constant A. The magnetization was
measured by a superconducting quantum interference devicemagnetometer. The magnetic anisotropy constants and the
exchange constant were obtained from ferromagnetic reso-
nance (FMR) experiments using a Q-band setup. The rangeof available static fields allows the determination of the mag-
netic anisotropy constants only at 295 K. Taking into account
the fraction of the aphase and the demagnetization factor of
thea-phase stripes,
19we obtain K2?¼3:2/C2105Jm/C03and
K2k¼4/C2105Jm/C03for the a-phase using the same defini-
tions as in Ref. 19. We determine Afrom the linear fit of the
successive spin-wave resonances as a function of the square
of the mode number (Fig. 2), assuming Kittel’s boundary
conditions. We obtain A¼6:6p J m/C01at 285 K.20A reliable
determination of Aat higher temperature is not possible
because of additional lines in the spectra.
We then observed the magnetic domains by longitudinal
Kerr microscopy, using a light emitting diode source
(k¼635 nm), a high numerical aperture (0.4) objective, and
an off-centered aperture diaphragm to probe the horizontalcomponent of the magnetization. The contrast was enough to
accurately image the magnetic domains up to 290 K. The
field was aligned along the easy axis of the sample. In thisconfiguration, we observed wedge shaped domains in the full
range of studied temperatures (Fig. 3). The domains
expanded along the easy axis direction, much like the saw-tooth domains observed in Refs. 13and14. This kind of pat-
tern is known to develop in thin ferromagnetic films with
strong uniaxial in-plane anisotropy in order reduce the mag-
netic charge density when the magnetization of two adjacentdomains meets head-on.
21The DW velocity was then meas-
ured using a pulsed magnetic field technique. To be able to
track high velocities, pulses were generated by micro-coilsplaced inside the cryostat, and DW motion was driven by a
sequence of field pulses of varying duration. Details can be
found in Ref. 22. The DW velocity was systematically meas-
ured at the site of the largest DW displacement, i.e., at the
tip of the wedge domains, as indicated by the white line inFig.3(b). DW displacements were measured in areas with no
visible pinning defects.
Figure 1shows the velocity measurements for several
temperatures: 200, 270, and 290 K. The DW velocity becomes
measurable above the depinning field H
dep.Hdepis of the
order of the coercive field, which explains the unusualincrease of the depinning field with the temperature because
of the progressive growth of the bphase fraction (inset). At
high field (9 mT) and 270 K, thermally activated domainnucleation prevents further measurement of the DW displace-
ments. At 290 K however, nucleation is statistically less likely
as the volume fraction of the ferromagnetic phase (90%–95%)is decreased compared to the alpha-phase regime. Points
can then be taken up to 14 mT without problems. Above H
dep,
the velocity rises up to a velocity peak of 900 m s/C01at 200 K,
480 m s/C01at 270 K, and 540 m s/C01at 290 K and then
decreases. Note that the maximum velocities are comparable
to the velocities measured in Py magnetic tracks.4The peak
velocity of 540 m s/C01observed at T ¼290 K is over a million
fold higher than the velocities that can be deduced from
images of magnetization reversal in the coexistence regimepreviously reported in Refs. 13and14. Several issues can
now be addressed: the shape and temperature dependence of
the velocity curves (Fig. 1) but also the possible mechanisms
at play in the DW propagation in the coexistence regime.
The 1D DW propagation model is a good starting point
to explain these curves. Initially developed to describe thepropagation of Bloch DWs in perpendicularly magnetized
layers,
23,24it was later adapted to head-to-head or tail-to-tail
charged walls in in-plane magnetized tracks.25This is the sit-
uation encountered at the apex of the domain tips in MnAs.
When compared to micromagnetic simulation results, the 1D
model was found to provide the correct behavior of velocity-versus-field curves although the maximum velocity could
differ by a factor of two. DW propagation exhibits features
similar to the perpendicular case: a stationary regime at lowfield, followed by a precessional regime beyond the Walker
field H
W. The model assumes effective anisotropy constantsFIG. 2. Dependence of the resonance field of the spin wave modes on the
square of the mode number (inset: FMR spectrum at 285 K). The dashed lineis the linear fit at 285 K used to determine the exchange constant A.
FIG. 3. Kerr microscopy images of magnetic domains
taken at T ¼270 K after 3 successive propagation
pulses of amplitude 4.1 mT and duration 0 :6ls.072408-2 Tortarolo et al. Appl. Phys. Lett. 101, 072408 (2012)
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202.28.191.34 On: Mon, 22 Dec 2014 06:46:00with an anisotropy energy of the form: Eanis¼K0sin2h
þP1
n¼1Knsin2nhsin2ðnUþUnÞ, where his the polar angle
of the magnetization vector with respect to the in-plane easy
axis ( x) along which the DW propagates and Uthe azimuthal
angle. Neglecting the long-range contribution of the charged
wall to the effective magnetic anisotropy and taking into
account the symmetry of the magnetic anisotropy in MnAs,we keep only the K
1term in the summation and use the fol-
lowing relations: K0¼K2k/C0K2?þl0M2
s=2 and K1¼K2?
/C0l0M2
s=2, where Msis the saturation magnetization. Assum-
ing that the DW position can be fully described by the sole
time derivative of the angle h(1D approximation), and using
the propagation equations thus derived from the Landau-
Lifshitz-Gilbert equation, the DW velocities in the stationary
and precessional regime are given by
vstat¼cD0l0H
a1þj
21/C0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1/C0H
HW/C18/C192s0
@1
A0
@1
A/C01=2
;
hvpreci¼1
TðT
0DðtÞadU
dtþcl0HW
asinð2UÞ/C20/C21
dt; (1)
with the DW width D¼D0=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1þjsin2UÞq
;D0¼ffiffiffiffiffiffiffiffiffiffiffi
A=K0p
and the time-dependence of the azimuthal angle given by
tanU¼HW
Hþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1/C0HW
H/C0/C1 2q
tancl0HW
1þa2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
HW
H/C0/C1 2/C01q
t/C20/C21
.ais the
Gilbert damping factor, j¼K1=K0and HW¼aK1=l0Ms.
Figure 4shows the velocity curve calculated from Eq. (1)
with the micromagnetic parameters ( K1;K0,A, and Ms)
obtained experimentally (full curve). The velocity rises witha sublinear behavior (stationary regime) up to a field veryclose to the Walker field. After that, it decreases since theDW enters the precessional regime where it moves back andforth, hence with a lower average velocity. Actually theshape of the velocity curve, especially in the stationary re-
gime, depends mostly on the value of j,
22,25equal to 1.1
here. The maximum velocity is proportional to D0.
The experimental curves can then be qualitatively
understood as shown in Fig. 4for the T ¼290 K curve. The
depinning regime starts at Hdep. The velocity rises but with
an average value lower than in the intrinsic stationary regime
since the wall is pinned part of the time. The end of thedepinning regime occurs at the field Hflwhere the curve
meets the intrinsic flow regime. If Hfl/C21HW, the velocity
curve shows a peak at Hfl. This scenario qualitatively
explains the experimental velocity curve at 290 K providedthe Gilbert damping factor is taken equal to 0.028. This
value is larger than the one we determined by FMR (homo-
geneous damping a¼0:002 for a 15 nm thick layer) as usu-
ally encountered in DW propagation experiments (see Ref.
11and references therein) but compatible with the one
extracted from the pump-probe experiments in Ref. 26
(a¼0:017). The K
0constant is also in principle an adjusta-
ble parameter since it should include phenomenologicallythe contribution of the demagnetization energy from the
charged wall (demagnetization field along the easy axis, op-
posite to the x-component of the magnetization). The dashed
curve in Fig. 4, which is in very good agreement with the ex-
perimental one, is calculated using K
0¼0:8/C2105Jm/C03.
Simulations for the 200 K and 270 K curves using the anisot-ropy constants of Refs. 18and19and estimated exchange
constants show that the succession of dynamical regimes is
likely to be the same, i.e., first a depinning regime up to themaximum velocity and then the precessional flow regime.
Let us note that H
flis larger by a factor of 2 at 290 K as com-
pared to 200–270 K. This might indicate that the Walkerfield is larger at 290 K presumably because of the decrease
ofK
1=Mswith temperature.
At 290 K, the sample is in the early stages of the trans-
formation of the aphase into a mixed phase, which raises the
question of the effect of the bphase on the expansion of the
domains. The magnetic domain pattern observed at this tem-perature is unchanged from the one shown in Fig. 3, in par-
ticular the angle of the wedged domains has not varied:
the displacement of the side walls of domains appears notto have been much disturbed. This is consistent with the
T¼16
/C14(T¼289 K) images in Ref. 27showing a very
dilute bphase, and the persistence of the pointed domains,
unhindered by the nucleating stripes. The main effect of the
coexistence of the a=bphase on the DW velocity at
T¼290 K is found in a depinning regime that extends from
6 to 13.5 mT, i.e., over a larger range than at lower temper-
atures. In the flow regime however, the DWs easily go past
the defects formed by the nucleating bphase, and the veloc-
ity reached at H flis high, in agreement with the simple 1D
model. A more elaborate model taking into account the full
structure of the DW would likely refine the predicted veloc-ities. Above T ¼290 K, the loss of Kerr contrast is very
likely due to the domains being broken up by the non-
ferromagnetic stripes whose volume fraction increases rap-idly with temperature.
As a conclusion, we have studied the field induced DW
dynamics in a MnAs/GaAs (100) film and identified distinctdepinning and precessional flow propagation regimes. We
highlight here that the early stage of the a-btransformation
in MnAs offers the unique opportunity to study the effect ofa temperature-dependent and totally reversible well-defined
DW pinning potential landscape. The high velocity reported
near room temperature makes MnAs a promising candidatefor magneto-logic devices, which would require nano-
patterning. From this study, we can anticipate that nanostruc-
turing these films into wires parallel to the easy axis would
FIG. 4. DW velocity calculated from Eq. (1)with A¼6:6p J m/C01;K1¼2:1
/C2105Jm/C03;K0¼1:9/C2105Jm/C03(full curve) and K0¼0:8/C2105Jm/C03
(dashed curve), a¼0:028. Symbols: experimental curve at T ¼290 K.072408-3 Tortarolo et al. Appl. Phys. Lett. 101, 072408 (2012)
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202.28.191.34 On: Mon, 22 Dec 2014 06:46:00give similar results, i.e., high DW velocities relatively unhin-
dered by the bphase at 290 K. In this geometry, the DW will
be charged and wide, reduced to the apex part of the domains
in this study. Patterning wires along the hard axis has al-
ready shown that the organization of the mixed phase nota-
bly changes.16Under applied field, one further expects the
propagation of pure, uncharged Bloch walls propagatingalong the structure. As these are narrower, DW velocities in
these structures can be expected to be smaller than when pat-
terned along the easy axis.
We acknowledge valuable technical assistance from
M. Bernard and S. Majrab at INSP.
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1.1588734.pdf | Variation of magnetization and the Landé g factor with thickness in Ni–Fe films
J. P. Nibarger, R. Lopusnik, Z. Celinski, and T. J. Silva
Citation: Applied Physics Letters 83, 93 (2003); doi: 10.1063/1.1588734
View online: http://dx.doi.org/10.1063/1.1588734
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/83/1?ver=pdfcov
Published by the AIP Publishing
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69.166.47.134 On: Sat, 13 Dec 2014 23:03:34Variation of magnetization and the Lande ´gfactor with thickness
in Ni–Fe films
J. P. Nibarger,a)R. Lopusnik, Z. Celinski,b)and T. J. Silva
National Institute of Standards and Technology, Boulder, Colorado 80305
~Received 17 February 2003; accepted 1 May 2003 !
WehavemeasuredtheLande ´gfactor,theeffectivemagnetization Meff,theuniaxialanisotropy Hk,
and the Gilbert damping parameter a, as a function of Permalloy film thickness from 2.5 to 50 nm.
We used a pulsed inductive microwave magnetometer capable of generating dc bias fields of 35.2kA/m ~440 Oe !. A significant decrease in gis observed with decreasing thickness below 10 nm.
Also,M
effdecreases with decreasing thickness consistent with a surface anisotropy constant of
0.196 60.025 mJ/m2. The decrease in gcan arise from the orbital motion of the electrons at the
interface not being quenched by the crystal field. We also compare our data to a model of aneffective gfactor suggesting that the decrease in gfactor might also stem from the Ni–Fe interface
with a Ta underlayer. @DOI: 10.1063/1.1588734 #
As the magnetic-data-storage industry develops disk
drives with data transfer rates approaching 1 Gbit/s, under-standing the underlying dynamics of the soft magnetic com-ponents used in recording heads becomes increasingly im-portant. Two important material parameters that govern the
response and precessional frequency of a magnetic film arethe effective magnetization M
effand the Lande ´gfactor.Meff
affects the dynamics by generating internal demagnetizing
fields during the switching process that greatly accelerate theprecessional motion. The Lande ´gfactor sets the proportion-
ality of angular momentum and magnetic moment for theindividual spins that results in precessional motion. For state-of-the-art heads with exceedingly small magnetic layer thick-nesses, interfaces play a large role, and understanding theeffect of interfaces on M
effandgis crucial for the engineer-
ing of high-performance recording systems. The thicknessdependence of M
effandgin the case of thin Permalloy films
was first measured by ferromagnetic resonance.1
We demonstrate the ability of a pulsed inductive micro-
wave magnetometer ~PIMM !to measure simultaneously the
effective magnetization Meff, the uniaxial anisotropy Hk,
and the spectroscopic Lande ´gfactor, at high dc bias fields.
This is done for a thickness series of Permalloy films(Ni
81Fe19) ranging from 2.5 to 50 nm. By applying large dc
fields @35.2 kA/m ~440 Oe !#along the easy axis of the
sample during measurements, we are able to extract Meff,g,
andHksimultaneously using a nonlinear, three-parameter fit.
This is in contrast to most permeameters, which require aseparate measurement of M
eff. In addition, the Gilbert damp-
ing parameter a, was extracted as a function of thickness.
Polycrystalline Permalloy films were deposited on 1
cm31c m 3100mm~0001!oriented sapphire coupons. The
sapphire substrates were cleaned using ion milling in Ar/O 2
and Ar atmospheres to remove contaminants. Then, a dcmagnetron operating in an Ar atmosphere at 0.533 Pa ~4
mTorr !was used to sputte ra5n mT a adhesion layer. Per-
malloy films of 2.5, 5, 7.5, 10, 15, 25, or 50 nm thicknesseswere then deposited followed b ya5n m capping layer of Cu
to protect the Permalloy against oxidation. Samples weregrown in a 20 kA/m ~250 Oe !external magnetic field to
induce uniaxial anisotropy. Photolithography and a nitricacid etch was used to patter na3m m 33 mm square in the
center of the Permalloy coupon. The reduced sample areawas required to guarantee high uniformity of the dc bias fieldacross the area of the sample during measurements. Figure 1upper inset shows typical hard- and easy-axis hysteresisloops of the unpatterned 50-nm-thick sample characterizedusing an induction-field looper to verify their quality.
Samples were measured by use of a PIMM.
2A coplanar
waveguide of 50 Vimpedance and 100 mm center conductor
was used.The easy axis of the sample was aligned parallel tothe center conductor, as shown in the lower inset of Fig. 1.Acommercial pulse generator provided 10Vpulses, with 50 ps
a!Electronic mail: nibarger@boulder.nist.gov
b!Present address: Department of Physics, University of Colorado at Colo-
rado Springs, Colorado Springs, CO 80918.
FIG. 1. Frequency squared as a function of bias field fo ra5n ms a m p l e ;t h e
error bars are the size of the circles. Data for bias field 0.8–7.16 kA/m~10–90 Oe !are shown with filled-in circles and fitted linearly with a solid
line to demonstrate the deviation from linearity of the data at high biasfields. All of the data were fitted with Eq. ~1!~dashed line !. Lower inset
shows the measurement geometry used for pulsed inductive microwavemagnetometer measurements, with the easy axis of the sample parallel to theapplied dc bias field. Upper inset shows induction field looper measurementsof the unpatterned 50 nm thick sample showing the easy- and hard-axishysteresis loops with easy-axis squareness of 0.99.APPLIED PHYSICS LETTERS VOLUME 83, NUMBER 1 7 JULY 2003
93 0003-6951/2003/83(1)/93/3/$20.00
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69.166.47.134 On: Sat, 13 Dec 2014 23:03:34rise times and 10 ns durations. The pulsed field Hp, was
oriented along the hard axis of the sample. The nominal fieldpulse amplitudes were found by the use of the Karlquistequation for fields from a current strip
3to be 800 A/m ~10
Oe!. The Permalloy films were placed facing the waveguide.
To prevent shorting of the coplanar waveguide a thin layer ofphotoresist ~,1
mm!was spin coated onto the sample.
Static longitudinal bias fields ( Hbin Fig. 1 lower inset !,
ranging from 0.8 to 35.2 kA/m ~10 to 440 Oe !, were gener-
ated by an electromagnet with soft iron pole pieces and acircular yoke.
4Field calibration was performed to avoid any
effects of remanence and allowed the fields to be set with anuncertainty of 1%. Field uniformity along the waveguide wasbetter than 1% over a length of 4 mm. Coil resistance wasmonitored to determine if any heating had occurred thatcould lead to field drift. If the resistance was more than 2.5%above room-temperature resistance, then data acquisitionwas temporarily stopped until the coils cooled.
Precessional response was measured with a 20 GHz-
bandwidth digital sampling oscilloscope. The measured pre-cession frequencies ranged from 1 to 6.5 GHz and were wellwithin the bandwidth of the detection system.
2Abackground
response was obtained with an applied saturation field of 2.4kA/m ~30 Oe !along the hard axis and zero field along the
easy axis. The precessional dynamics was extracted by sub-tracting the measured and background signals.
The induced voltage of the precessional response mea-
sured in the time domain was converted into frequency spec-tra by fast Fourier transform for further analysis. The Gilbertdamping parameter
a, was extracted from the full width at
half maximum of the imaginary part of the spectrum Dv,
such that: a’Dv/(gm0Meff).5The resonance of the signal
was extracted from the zero crossing of the real part of thespectrum.The resonance frequency as a function of bias fieldcan be described by the Kittel formula for a thin film
6
v025SgmBm0
\D2
~Meff1Hk1Hb!~Hk1Hb!, ~1!
where mBis the Bohr magneton, \is Planck’s constant di-
vided by 2 p, and m0is the permeability of free space. A
simultaneous three-parameter fit of v02vsHbcan be used to
extractMeff,g, andHk. We emphasize that a three-
parameter fit is possible only when a sufficiently large fieldrange is used such that terms in Eq. ~1!quadratic in bias field
are no longer negligible. Fortunately, the applied dc fields
need not be as large as M
efffor the nonlinearity in v02vsHb
to be measurable. Since surface anisotropies may exist for
very thin magnetic films, the demagnetizating fields inducedby out-of-plane motion of the magnetization vector differsfrom the saturation magnetization by the usual surface an-isotropy term
7
m0Meff5m0Ms22Ks
Msd, ~2!
where dis the film thickness and Ksis the average anisot-
ropy, consisting of the sum of the Cu/NiFe and Ta/NiFe in-terface surface anisotropies.
Figure 1 is a plot of frequency squared, f
25(v/2p)2as
a function of longitudinal bias field for a 5-nm-thick film.The uncertainty in f
2is found to vary from 5% at 1 GHz2to0.8% at 40 GHz2. The data can be fit using Eq. ~1!~dashed
line!to yield values of Meff,g, andHk. To highlight the
deviation from linearity, data for 0.8–7.2 kA/m ~10–90 Oe !
bias fields ~shown with filled-in circles !were fitted to a lin-
ear function of Hb, with the fit extrapolated to high fields.
The data are as much as 8% greater than the linear extrapo-lation from low field data, showing the magnitude of thenonlinearity to be fitted in the extraction of M
eff,g, andHk.
For each thickness, multiple measurements were made todetermine statistics for repeatability and to decrease noisethrough averaging. Due to the 1% uncertainty in bias fields,a systematic error of 2.5% for g, 4% forM
eff, and 12% error
inHkare presumed. Results for Meffandgas a function of
thickness d, are shown in Fig. 2. Hkandaas a function of
thickness is shown in Fig. 2 inset. The avalues plotted are
from data with an 8 kA/m longitudinal bias field.
The Gilbert damping parameter, a, increased with de-
creasing film thickness, consistent with previous measure-ments in Permalloy.
8Hkappears to vary randomly with an
average value of 408 640 A/m ~5.160.5 Oe !for 2.5 ,d,15
nm, with no observable trend within the error bars for themeasurement. However, both M
effandgdecrease signifi-
cantly with decreasing film thickness below 10–20 nm. Inpractical terms, the reduction in M
effandgis a decrease in
the intrinsic ferromagnetic resonance frequency for the thin-nest Permalloy by 27% relative to the thickest films, equiva-lent to a shift in the precessional frequency of 230 MHz.
Values obtained for
m0Meffwith the PIMM are consis-
tent with values obtained from an alternating gradient mag-netometer ~AGM !. For 50 and 25 nm sample thicknesses the
AGM measured values of
m0Meffwere 1.0630 and 1.0180 T,
respectively, compared to 1.0462 and 1.0398 T, respectively,from the PIMM measurements. The observed decrease in
m0Meffwith decreasing thickness is consistent with a surface
anisotropy contribution given by Eq. ~2!.Afi tt oE q . ~2!is
shown in Fig. 2 as a dashed line. Ksis 0.196 60.025 mJ/m2
FIG. 2. m0Meffandgas a function of thickness, d. Inset shows Hkandaas
a function of d.T h e avalues plotted are with 8 kA/m longitudinal bias field
applied. m0Meffwas fitted to Eq. ~2!~dashed line !, yielding m0Ms51.0553
T andKs50.196 mJ/m2. The measured gfactor is compared to Eq. ~7!
~solid line !whereTTa/NiFe 50.6 nm ~see Ref. 12 !,TCu/NiFe 50.4 nm,
gNiFe52.1~see Refs. 1, and 13 !gTa/NiFe 51.58 ~see Ref. 14 !,gCu/NiFe 52.05
~see Ref. 15 !.94 Appl. Phys. Lett., Vol. 83, No. 1, 7 July 2003 Nibargeret al.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
69.166.47.134 On: Sat, 13 Dec 2014 23:03:34andm0Msis 1.0553 60.046 T. The error in Ksandm0Ms
accounts for both random error and the uncertainty of 1% for
the bias field.
In general, both the orbital and spin angular momentum
contribute to the total angular momentum of an electron. Assuch, the gfactor may be written as
g52m
e
emS1mL
^S&1^L&, ~3!
where mSandmLare the contributions to the electron mag-
netic moment due to the spin and orbital components, re-spectively. For a symmetric crystal lattice, the orbital motionof the electron during gyromagnetic precession is quenchedby the crystal field, i.e.,
^L&50. Thus, the orbital contribu-
tion to the electron angular momentum is zero even thoughthe orbital contribution to the magnetic moment is nonzeroresulting in a gfactor that is always greater than two:
9
g52me
emS1mL
^S&52S11mL
mSD. ~4!
However, the orbital motion is not entirely quenched at
surfaces and interfaces where the crystal field is no longersymmetric since the interface breaks inversion symmetry.Under such circumstances, the orbital motion can still con-tribute to the gyromagnetic motion. Equation ~3!can then be
written as
g52m
e
emS
^S&S11mL
mSD
S11^L&
^S&D’2S12mL
mSD, ~5!
since ^S&5mSme/e,^L&52mLme/eand expanding the Tay-
lor’s series to first order. Thus, surfaces and interfaces allowfor the possibility that the gfactor is less than 2. Two physi-
cal mechanisms are plausible sources for this interface effect.First, the orbital motion is not quenched by the crystal field,i.e.,
^L&Þ0. In addition, material mixing at the interface
could alter the gfactor. We can model the later hypothesis of
interface mixing by relying on the concept of an effective g
factor,geff, first proposed by Wangness10,11
rNiFeVNiFe1rTa/NiFeVTa/NiFe 1rCu/NiFeVCu/NiFe
geff
5rNiFeVNiFe
gNiFe1rTa/NiFeVTa/NiFe
gTa/NiFe1rCu/NiFeVCu/NiFe
gCu/NiFe, ~6!
where ri,Vi, andgiare the spin density, volume, and g
factor for each of the respective layers or interfaces(i5NiFe, Ta/NiFe, Cu/NiFe !. The volume V
imay be set
equal to the thickness tisince the interface area is the same
for each layer. Equation ~6!can be rewritten as a function of
the thickness of the Permalloy film, d:
geff~d!5d1tTa/NiFe 1tCu/NiFe
d
gNiFe1tTa/NiFe
gTa/NiFe1tCu/NiFe
gCu/NiFe. ~7!
We assume that the spin density riis invariant through the
film thickness and that reasonable assumptions for the Cu/NiFe and Ta/NiFe interface thickness and the g-factors for
the films and interfaces can be made.The mixing at Ta/NiFe interfaces has been well studied
for magnetic random access memory ~MRAM !and giant
magnetoresistance applications. Kowalewski et al.
12found
the interface thickness for unannealed samples to be 0.6 nm.The thickness of the Cu/NiFe interfaces is approximatelytwo monolayers ~0.4 nm !. The measured gfactor for NiFe
from this experiment for the thickest films is 2.1, which isconsistent with other published values ~2.08,
12.08,13and
2.17!.13For thegfactor at the Ta/NiFe interface, we make a
very coarse approximation and use the Ta bulk value of1.58.
14Likewise, the gfactor for Cu/NiFe is simply that of
bulk Cu, 2.05,15a value not too different from that of bulk
Permalloy. A plot of Eq. ~7!with the earlier assumption is
shown in Fig. 2 with no adjustable parameters. The calcu-lated reduction in gwith decreasing NiFe thickness is in
large part the result of the Ta interface, which has a largeorbital contribution to the moment.
This model works surprisingly well as an explanation for
the thickness variation of g, in spite of the particularly crude
assumptions made of uniform spin density and bulk valuesfor thegfactors at the various interfaces. These two assump-
tions stem from the presumption that ferromagnetism persistseven in an intermixed atomic environment, though the orbitalmomentum contribution to the total angular momentum ofthe ferromagnetic spins is dominated by the electronic struc-ture of the nonmagnetic constituent. We conclude that eitherthe orbital motion at the interface is not quenched by thecrystal field, i.e.,
^L&Þ0, or that interfacial mixing of ferrous
and nonferrous materials, or some combination of these twoeffects can explain the significant deviations of the preces-sional dynamics in thin Permalloy films from that predictedfrom bulk values of the gfactor.
The authors would like to thank T. Kos for technical
assistance. Z.C. acknowledges the financial support fromARO ~Grant No. DAAG19-00-1-0146 !.
1M. H. Seavey and P. E. Tannenwald, J. Appl. Phys. 29,2 9 2 ~1958!.
2T. J. Silva, C. S. Lee,T. M. Crawford, and C.T. Rogers, J.Appl. Phys. 85,
7849 ~1999!, A. B. Kos, T. J. Silva, and P. Kabos, Rev. Sci. Instrum. 73,
3563 ~2002!.
3O. Karlquist, Trans. R. Inst. Tech. Stockholm 86,3~1954!.
4A. B. Kos, J. P. Nibarger, R. Lopusnik, T. J. Silva, and Z. Celinski, J.
Appl. Phys. 93, 7068 ~2003!.
5J. P. Nibarger, R. Lopusnik, and T. J. Silva, Appl. Phys. Lett. 82,2 1 1 2
~2003!.
6C. Kittel, Introduction to Solid State Physics , 7th ed. ~Wiley, New York,
1995!.
7L. Ne´el, Compt. Rend. 238, 3051468 ~1953!.
8S. Ingvarsson, L. Ritchie, X. Y. Liu, Gang Xiao, J. C. Slonczewski, P. L.
Trouilloud, and R. H. Koch, Phys. Rev. B 66, 214416 ~2002!.
9S. Chikazumi, Physics of Magnetism ~Krieger, Malabar, FL, 1964 !,p .5 2 .
10R. K. Wangsness, Phys. Rev. 91, 1085 ~1953!.
11M. Farle, A. N. Anisimov, K. Beberschke, J. Langer, and H. Maletta,
Europhys. Lett. 49,6 5 8 ~2000!.
12M. Kowalewski, W. H. Butler, N. Moghadam, G. M. Stocks, T. C.
Schulthess, K. J. Song, J. R.Thompson,A. S.Arrott,T. Zhu, J. Drewes, R.R. Katti, M. T. McClure, and O. Escorcia, J.Appl. Phys. 87, 5732 ~2000!.
13R. M. Bozorth, Ferromagnetism ~IEEE, New York, 1978 !.
14C. Schober and V. N. Antonov, Phys. Status Solidi B 143, K31 ~1987!.
15G. E. Grechnev, N. V. Savchenko, I. V. Svechkarev, M. J. G. Lee, and J.
M. Perz, Phys. Rev. B 39, 9865 ~1989!.95 Appl. Phys. Lett., Vol. 83, No. 1, 7 July 2003 Nibargeret al.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
69.166.47.134 On: Sat, 13 Dec 2014 23:03:34 |
1.5122753.pdf | J. Appl. Phys. 126, 183902 (2019); https://doi.org/10.1063/1.5122753 126, 183902
© 2019 Author(s).Tunable magnetic domain walls for
therapeutic neuromodulation at cellular
level: Stimulating neurons through magnetic
domain walls
Cite as: J. Appl. Phys. 126, 183902 (2019); https://doi.org/10.1063/1.5122753
Submitted: 02 August 2019 . Accepted: 17 October 2019 . Published Online: 08 November 2019
Diqing Su
, Kai Wu
, Renata Saha
, and Jian-Ping Wang
COLLECTIONS
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neuromodulation at cellular level: Stimulating
neurons through magnetic domain walls
Cite as: J. Appl. Phys. 126, 183902 (2019); doi: 10.1063/1.5122753
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Submitted: 2 August 2019 · Accepted: 17 October 2019 ·
Published Online: 8 November 2019
Diqing Su,1,a)
Kai Wu,2,a)
Renata Saha,2
and Jian-Ping Wang2,b)
AFFILIATIONS
1Department of Chemical Engineering and Material Science, University of Minnesota, Minneapolis, Minnesota 55455, USA
2Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, Minnesota 55455, USA
a)Contributions: D. Su and K. Wu contributed equally to this work.
b)Author to whom correspondence should be addressed: jpwang@umn.edu
ABSTRACT
Cellular-level neuron stimulation has attracted much attention in the areas of prevention, diagnosis, and treatment of neurological disorders.
Herein, we propose a spintronic neurostimulator based on the domain wall movement inside stationary magnetic nanostructures driven by
the spin transfer torques. The electromotive forces generated by the domain wall motion can serve as highly localized stimulation signals for
neuron cells. Our simulation results show that the induced electric field from the domain wall motion in permalloy nanostructures can
reach up to 14 V =m, which is well above the reported threshold stimulation signal for clinical applications. The proposed device operates on
a current range of several microamperes that is 103times lower than the current needed for the magnetic stimulation by microcoils. The
duration and amplitude of the stimulating signal can be controlled by adjusting the applied current density, the geometry of the nanostruc-
ture, and the magnetic properties of the material.
Published under license by AIP Publishing. https://doi.org/10.1063/1.5122753
I. INTRODUCTION
In 2013, the National Institutes of Health (NIH), the Defense
Advanced Research Projects Agency (DARPA), and the National
Science Foundation (NSF) launched a project named the BRAINInitiative
1,2to accomplish the prevention, diagnosis, and treatment
of brain disorders such as Alzheimer ’s disease, attention de ficit
hyperactivity disorder (ADHD), Parkinson ’s disease, migraines,
and traumatic brain injury (TBI). The primary challenge in thisprocess is the lack of understanding of the pathogenesis, which
makes it necessary to investigate the interactions within the brain
from the cellular level to the complex neural circuits throughbrain stimulation. Since the early work of Wise et al. ,
3research
studies in neuroscience and neural engineering have experienced
rapid growth, especially in exploring new probe materials and new
fabrication technologies to produce miniaturized, customized, andhigh-density electrode arrays for the stimulation of neurons.Despite their great potential, the electrode arrays employed in
most of the current brain stimulation technologies are constantlyaffected by the migration of cells (such as astrocytes) around the
devices, which leads to increas ed impedance and alterations of
the electric field in the stimulation processes. One way to avoid
the in fluence of surrounding neuron cells on the stimulation
signal is magnetic stimulation, where a magnetic field is gener-
ated and is not a ffected by the encapsulation of astrocytes or any
other cells. Transcranial magnetic stimulation (TMS) is a com-monly used noninvasive brain sti mulation technique that utilizes
a strong alternating magnetic field (1.5 T to 3 T) to modulate the
neuron activities.
4–6However, due to the bulky and noninvasive
nature of this setup, it is impossible to generate a highly focusedmagnetic field. Moreover, as the magnetic field decays exponen-
tially over distance, this technique cannot stimulate neurons
located deep inside the brain. As a complementation of TMS,
deep brain stimulation (DBS) implants electrodes in certainregions of the brain permanently to activate deeply located neurons.
7–9
Nevertheless, heating e ffects and large power consumption due to
the constant application of relatively large-amplitude current are
major drawbacks of DBS. Consequently, the development of anJournal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 126, 183902 (2019); doi: 10.1063/1.5122753 126, 183902-1
Published under license by AIP Publishing.implantable magnetic neurostimulator with the ability of generat-
ing a highly localized magnetic field through a low power input
is essential for both the study of neuron activities and the treat-ment of neuron disorders.
The domain walls (DWs) from magnetic nanostructures are
promising candidates for the m agnetic stimulation of neuron
cells.
10The displacement of magnetic domain walls through the
current has been widely studied to switch local magnetization inhigh-density magnetic recording applications.
11–13The velocity
of the domain walls can be controlled by the applied currentdensity. Pulses of highly spin polarized current can move the
entire pattern of DWs coherently along the nanostructure. A
neuron cell placed at a certain location on top of the nanostruc-ture will be stimulated with an electromotive force (EMF) that isinduced by the change of the magnetic stray field generated by
the domain wall movement within the nanostructure. Since the
width of the domain wall ranges from the nanometer scale to the
submicrometer scale, the stimul ation is highly localized and is
free from the in fluence of the surrounding environment, which
facilitates neuron stimulation at the cellular level. Like the electri-cal stimulators, special attention needs to be paid to the biotoxic-
ity of the magnetic stimulators since most of the magnetic
materials are also highly toxic. As a result, passivation techniquesare required to isolate the magnetic materials from the neuroncells. It has been shown that neuron cells can be successfully
grown on the surface of the magnetic tunnel junctions passivated
with SiO
2/Si3N4/SiO 2.14In addition, since the magnetic stimulator
arrays are fully compatible with most of the fabrication processesfor electrical stimulators, various passivation materials employed inthe electrical stimulators such as polydimethylsiloxane (PDMS),
15
poly(3,4-ethylenedioxythiophene) (PEDOT),16and SiC17can also
be applied to the magnetic stimulators. In this paper, we have theo-retically demonstrated the feasibility of stimulating an individualneuron under adjustable magnetic field strength and frequency with
nanofabricated magnetic nanostructure arrays ( Fig. 1 ).II. METHODS
The magnetic dynamics including spin transfer torque (STT)
terms with an extended Landau-Lifshitz-Gilbert (LLG) equation isused in the simulation, which can be expressed as
dm
dt¼γ0Heff/C2mþαm/C2dm
dt/C0(u/C1∇)mþβm/C2(u/C1∇)m, (1)
u¼JcPgμB
2eM s, (2)
where m¼M
Msis the unit magnetization vector, Msis the saturation
magnetization, Heffis the e ffective magnetic field,γ0is the absolute
value of the gyromagnetic ratio, and αis the Gilbert damping
parameter. The last two terms on the right side of Eq. (1)are adia-
batic and nonadiabatic torque terms. The dimensionless quantity β
represents the degree of nonadiabaticity, u(inm=s)i st h ee ffective
drift velocity of the conduction electron spins, JCis the charge
current density, Pis the spin polarization of the current, gis the
Landé factor, μBis Bohr magneton, and eis the electron charge.
Here, we consider the permalloy (Ni 80Fe20) nanostructure with
the current applied along the wire axis, and the material parametersare assumed as follows: α¼0:02,β¼0:04,M
s¼8/C2105A=m,
P¼0:6, and exchange constant A¼20 pJ =m that gives an exchange
length of 5 nm. The permalloy nanostructure has dimensions of
10μm × 8 nm × 8 nm, and the simulation cell size is set to be
2×2×2n m3(Table I ). We used the object oriented micromagnetic
framework (OOMMF)18code for simulations that solve the LLG
equation incorporating the STT terms.
Based on the Maxwell-Faraday law, alternating magnetic flux
density can induce an electromotive force (EMF),
þ
E/C1dl¼/C0ðð@B
@t/C1dS, (3)
FIG. 1. Schematic illustration of the on-chip magnetic stimulation and sensing of a neuron cell. The device is designed to be implanted in vivo with passivation materials
coated on the surface. The region highlighted in blue consists of a magnetic stimulator (left) and a magnetic nanosensor (right). There are multiple d omain walls in the
stimulation device, which can generate the electromotive forces. Different parts of the neuron cell can be stimulated by multiple magnetic stimulat ors, and the resulting
magnetic field generated within the neuron after the stimulation will be sensed by the magnetic nanosensor adjacent to the magnetic stimulator. The red arrow ind icates
the direction of the applied current.Journal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 126, 183902 (2019); doi: 10.1063/1.5122753 126, 183902-2
Published under license by AIP Publishing.where Bis the magnetic flux density, Eis the electric field,lis the
contour, and Sis the surface area. It is reported that the neuron cells
c a nb es t i m u l a t e db ya ne l e c t r i c field higher than 10 V =mw i t hd u r a -
tion longer than 50 μs19–22This can be determined by the domain wall
velocity and can, thus, be controlled by the applied current density.III. RESULTS AND DISCUSSION
A. Generation of highly localized magnetic field
As shown in Fig. 2(a) , due to large shape anisotropy, the
magnetizations in the domains of a magnetic nanostructure with
very high aspect ratio (10 μm:8 nm) tend to lie along the long axis
(x axis), resulting in negligible short axis (y or z axis) componentsof the stray field. To minimize the total energy, there are usually
multiple domains in the nanostructure, which are separated by the
domain walls. As the magnetizations rotate either toward y or zdirection within the domain wall, the y or z components of thestray field become nonzero. During the current-driven domain
wall motion, the stray field experienced by a neuron cell located at
afixed location in proximity to the nanostructure surface will
either increase or decrease, generating an electromotive force, i.e.,a stimulation signal. Without further notation, the stray field in
the following content refers to the stray field perpendicular to the
nanostructure surface ( H
z) that is used to generate the stimulation
signal. Since the structure of the nanomaterial is symmetrical in
the y and z direction, the analysis of the stray field in the y direc-
tion should be similar to that in the z direction and, thus, will notbe discussed again in this paper.TABLE I. Simulation parameters of the magnetic domain wall movement in permal-
loy nanostructures.
Parameter Description Value
Dimensions Length × Width × Thickness 10 μm×8n m×8n m
Cell size Length × Width × Thickness 2 × 2 × 2 nm3
α Gilbert damping factor 0.02
β Nonadiabatic spin transfer
torque factor0.04
A Exchange constant 20 × 10−12J/m
P Polarization factor 0.6
Ms Saturation magnetization 0.8 × 106A/m
JC Charge current density 1011−2.4 × 1013A/m2
I Charge current 6.4 μA–1.5 mA
FIG. 2. (a) Schematic view of different magnetic domain walls in a magnetic nanostructure: (i) perpendicular transverse wall and (ii) transverse wall. Red a rrow represents
the magnetization in that domain. (b) Out-of-plane magnetic stray field component Hzattenuates from 320 kA/m at the surface of the nanostructure to below 1 A/m at
150 nm from the surface. (c) The z component of the stray fieldHzacross the nanostructure as a function of the distance from the xy surface.Journal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 126, 183902 (2019); doi: 10.1063/1.5122753 126, 183902-3
Published under license by AIP Publishing.To realize cellular-level neuron stimulation, the out-of-plane
stray field should be highly localized around the domain wall,
which is determined by the fast decay of the stray field and the
width of the domain wall. The amplitude of the stray field along
the z direction ( Hz) is plotted against the distance from the nano-
structure surface in Fig. 2(b) . The maximum stray field at the
surface of the nanostructure is 3 :2/C2105A=m, which is in the same
order of the saturation magnetization (8 /C2105A=m) and decays to
less than 1 A =m at a distance of 150 nm. Since the size of the
neuron cells is in the micrometer range, while the thickness of thecell membranes is in the order of several nanometers, the distribu-
tion of the domain wall stray field can penetrate the membrane of
the cell on top of its surface without further in fluence on other sur-
rounding cells, which makes it possible to realize single-cell stimu-lation. Based on this argument, the calculation of the electricalstimulation signal in the subsequent section will be focused only in
the region within the width of the domain wall. Furthermore, mag-
netic nanostructures in the array can be controlled by separate elec-trodes with only one nanostructure activated each time so thatother neurons located in proximity to other nanostructures will notbe stimulated. The distribution of the stray field in the transverse
direction is determined by the domain wall width. The width of the
domain wall is around 40 nm for a permalloy nanostructure with athickness of 8 nm as observed in our simulation, which is consis-tent with the previously reported value and can be adjusted by
simply altering the geometry of the nanostructure.
23This facilitatesthe highly localized stimulation at a speci fic spot of the neuron
cells. As shown in Fig. 2(c) ,Hzis nonzero within each domain wall
in the nanostructure and decays to zero at the edges of the domainwall. The amplitude of H
zis different at each domain wall due to
the di fference in the magnetization orientation within the wall,
which will be discussed in Sec. III B . Due to the nanoscale domain
structure of the permalloy nanostructure and the fast decay of the
magnetic field in nonmagnetic space, a highly localized stimulation
signal can be generated, whose viable stimulation region can bemodified by multiple parameters, including the exchange constant
and the anisotropy constant of the nanostructure material as well
as the geometry of the nanostructures.
B. Current-driven domain wall motion
To induce a change in the stray field, a spin polarized current
is applied to the nanostructure. Driven by the spin transfer torque,
the magnetic domain wall moves along the +x direction, which canbe characterized by the change in the distribution of the demagnet-ization field pro file along the nanostructure at di fferent timepoints.
The system is stabilized for 25 ns after the application of the
current (see Fig. S1 in the supplementary material ). As shown in
Fig. 3(a) , the demagnetization field distributions are plotted at
different locations along the long axis of the nanostructure every
0:15 ns. The applied current Jis 1 :2/C210
13A=m2, which corre-
sponds to an e ffective drift velocity uof 600 m =s. Each peak in the
FIG. 3. (a) Distribution of the demagnetization field along the center line of the nanostructure ( y¼4 nm, z¼4n m ) a t t¼58:22 ns ,58:37 ns ,58:52 ns ,
58:67 ns ,58:82 ns ,and 58 :97 ns under an applied current of 1 :2/C21013A=m2. The red, orange, and yellow lines connect the locations of one domain wall at different time-
points, indicating linear displacements of the domain wall. The domain wall velocity is calculated to be 590 m =s. (b) Magnetization distribution in the nanostructure at
t¼58:22 ns ,58:37 ns, and 58 :52 ns under an applied current of 1 :2/C21013A=m2. The green arrow indicates the movement of the domain walls. (c) Domain wall position
at different timepoints for u¼200 m =s, 400 m =s, and 600 m =s. (d) Magnetization in the y and z direction at the cross section of each domain wall along the x axis under an
applied current density of 1 :2/C21013A=m2. The blue line indicates the oscillation of the domain wall magnetization for domain walls located at x¼4:019μm–4:845μm.
The red line indicates the oscillation of the domain wall magnetization for domain walls located at x¼2:565μm–3:201μm.Journal of
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J. Appl. Phys. 126, 183902 (2019); doi: 10.1063/1.5122753 126, 183902-4
Published under license by AIP Publishing.profile represents one domain wall, and all the domain walls move
toward +x direction consistently after the application of the spin
polarized current, which can also be con firmed by the magnetiza-
tion distribution presented in Fig. 3(b) . The domain wall pro files
foru¼400 m =s and u¼200 m =s, which correspond to current
densities of 8 /C21012A=m2and 4 /C21012A=m2, also exhibit a
similar trend as shown in Fig. S2 in the supplementary material .B y
plotting the locations of one domain wall in the nanostructure atdifferent timepoints, the domain wall velocities vunder di fferent
applied currents can be calculated, which turn out to be 200 m =s,
400 m =s, and 590 m =s under current densities of 4 /C210
12A=m2,
8/C21012A=m2, and 1 :2/C21013A=m2, respectively [ Fig. 3(c) ]. It is
observed that v/C25ufor the utilized current densities, which is in
accordance with the results from other literatures when the currentis above the threshold current, also known as Walker limit.
24–26In
this case, the Walker limit is estimated to be 2 /C21010A=m2.24
For transverse domain walls, when the applied current is
below the Walker limit, the spin transfer torque is counteracted bydamping, which cants the spin out of the plane.
27Once the current
is above the Walker limit, the spin transfer torque will be muchlarger than the torque due to damping, driving the domain walls to
move continuously while precessing around the x axis. The rotation
of domain wall planes can be con firmed by the inconsistent peak
values of B
zinFig. 3(a) . As shown in Fig. 3(d) , the magnetizations
at the center of the domain wall oscillate around the z axis. Since
the domain walls move at a constant velocity, this e ffect can also be
viewed as the oscillation of the domain wall planes for a given loca-tion along the nanostructure at di fferent timepoints, which is
confirmed by the projection of magnetizations in the M
y/C0Mzplane in Fig. 4(a) . The reason behind this oscillation pattern can be
explained qualitatively as follows. Due to dipolar interaction, the
magnetizations at the end of the nanostructure tend to tilt awayfrom the x axis. According to the first term of the LLG equation,
γ
0Heff/C2m, where Heffmainly consists of the demagnetization
field, the magnetizations at the end of the nanostructure precess
around the x axis. When the damping term is taken into consider-
ation, the magnetizations will be pushed toward the x axis with adecreasing precession angle. The spin transfer torque is either inthe same or in the opposite direction of the damping term depend-ing on the direction of the applied current, which will either move
the magnetizations toward or against the x axis.
28Upon the appli-
cation of the spin polarized current, the nonzero y and z compo-nents of the magnetizations can be transferred into thenanostructure. Consequently, the end of the nanostructure can beviewed as a domain wall generator. Since the domain walls are con-
stantly injected into the nanostructure by the current, the equilib-
rium domain wall structure will depend on the possibleconfigurations of adjacent domain walls upon collisions. According
to Kunz,
29two domain walls can either annihilate with each other
or form a stabilized structure depending on the types of topological
defects. It was shown that only domain walls with the same topo-
logical charges can be preserved during the collision, resulting in a360° domain wall, while the domain walls with opposite topologicalcharges will annihilate, forming a single domain. In our cases, only
adjacent domain walls with opposite signs of m
zcan survive, which
explains the oscillation of domain wall planes around the z axis. Asa result of the domain wall rotation, the demagnetization field gen-
erated by the domain wall along the y and z direction exhibits a
FIG. 4. (a) Three-dimensional plot of the magnetizations at the center of the nanostructure under an applied current of 1 :2/C21013A=m2. The projection in the x/C0My,
x/C0Mz, and My/C0Mzplanes are also shown. The distributions of the demagnetization fieldBy(b) and Bz(c) along the x axis are calculated from (a).Journal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 126, 183902 (2019); doi: 10.1063/1.5122753 126, 183902-5
Published under license by AIP Publishing.wavelike form, as shown in Figs. 4(b) and4(c), where the frequency
of the oscillation is determined by the distance between adjacent
domain walls and the velocity of the domain walls.
C. Electromotive force for neuron stimulation
According to Faraday ’s law, the alternating stray fields from
the nanostructure can generate electromotive forces. Assuming that
the contour of Efield is a circle with a diameter comparable to the
width of the domain wall, the Faraday ’s law can be rewritten as
2πrE¼/C0ΔB
Δtπr2, (4)
which gives
E¼/C0r
2ΔB
Δt, (5)
with v¼r
Δt,
E¼/C0v
2ΔB, (6)
where Eis the electrical field,Bis the stray field from the domain
wall, ris half of the width of the domain wall, Δtis the time di ffer-
ence between two adjacent data points, and vis the velocity of the
domain wall motion. The instantaneous electrical field is propor-
tional to the change of the stray field and the domain wall velocity
and, thus, can be adjusted by tuning the magnetic properties of the
nanostructure material or, more conveniently, the applied currentdensity. The calculated electromotive force at the surface of a per-malloy nanostructure under an applied current of 1 :2/C210
13A=m2
is shown in Fig. 5 . Since the domain walls are moving at a constant
velocity, the spatial distribution of the electrical field can be con-
verted to the time domain. Like the patterns of the stray field, both
electromotive forces along the y and z axis exhibit wavelike behav-iors with a maximum electrical field of 14 V =m, which is larger
than the minimum requirement for neuron modulation (5 V =m).
20
It is worth noting that the threshold electrical field may vary with
the pulse width of the stimulation signal.30Here, the frequency
of the stimulation signal is 4.76 GHz. When the applied currentdensities are decreased to 8 /C210
12A=m2and 4 /C21012A=m2, the
amplitudes of the stimulation signal are decreased to 7 :8V=m and
3:8V=m, respectively, and the frequencies are decreased to
2:86 GHz and 2 :78 GHz, respectively. The magnitude of the stimu-
lation signal needs to be adjusted according to di fferent clinical
applications by changing the applied current density, the geometry
of the nanostructure, and/or the magnetic properties of the mate-
rial. Since the domain wall motion can be continuously driven bythe spin polarized current, the duration of the signal can be easilycontrolled by turning the current on and o ff. Based on the applica-
tion, an alternating current with high frequency can also be
employed to obtain the required pattern of the stimulation signal.
The commonly used neuron stimulation technologies such as elec-trical current-based DBS and microcoil require electrical currentsin the milliampere range,
31,32which is not only power consuming
but can also lead to heating e ffects. Domain wall-based spintronic
neuron stimulation, however, only requires current densities of1012-1013A=m2, which corresponds to currents in the microampere
range. The corresponding power consumptions within the nano-structure are calculated to be 27 :7 mW, 12 :3 mW, and 3 :08 mW
under the applied currents of 1 :2/C210
13A=m2,8/C21012A=m2, and
4/C21012A=m2, respectively.
IV. CONCLUSIONS
Current neuron modulation technologies su ffer from multiple
problems such as encapsulation of cells around the devices, bulky
equipment, large power consumption, and heating e ffects.
Magnetic domain wall-based stimulation was studied in this letteras a potential candidate with the capability of overcoming thesedifficulties. When the applied electrical current is above the Walker
limit, the magnetic domain walls within the nanostructure can
move at a constant velocity comparable to the drift velocity of theconduction electron spins. Due to domain wall rotation, the strayfields generated by the domain walls exhibit wavelike patterns
along the nanostructure. The resulting electromotive force from the
domain wall motion has a maximum amplitude of 14 V =m with a
frequency of 4.76 GHz. With the application of domain wall-basedspintronic nanodevices, the required electrical current can bereduced from tens of milliamperes to several microamperes, which
significantly reduces the power consumption and the heating
effects. Based on the speci fic requirements of clinical applications,
FIG. 5. The electrical field generated by the alternating stray field during the
domain wall motion along the (a) y axis and (b) z axis under an applied currentdensity of 1 :2/C210
13A=m2.Journal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 126, 183902 (2019); doi: 10.1063/1.5122753 126, 183902-6
Published under license by AIP Publishing.the frequency and amplitude of the stimulation signal can be
adjusted by altering the frequency and amplitude of the applied
current, the geometry of the nanostructure, and the magnetic prop-erties of the nanostructure material. Since these neuron stimulatorscan be fabricated with microfabrication techniques, they can beeasily integrated with other functional devices such as magnetic
nanosensors on flexible substrates to realize stimulation and detec-
tion on a single chip.
SUPPLEMENTARY MATERIAL
See the supplementary material for the simulation results
under other electrical current densities.
ACKNOWLEDGMENTS
This study was financially supported by the Institute of
Engineering in Medicine of the University of Minnesotathrough FY18 IEM Seed Grant Funding Program, the NationalScience Foundation MRSEC Facility Program, the Centennial
Chair Professorship, and Robert F. Hartmann Endowed Chair
from the University of Minnesota. The authors declare no con flict
of interest.
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1.4907696.pdf | Large amplitude oscillation of magnetization in spin-torque oscillator
stabilized by field-like torque
Tomohiro Taniguchi,1,a)Sumito Tsunegi,2Hitoshi Kubota,1and Hiroshi Imamura1
1National Institute of Advanced Industrial Science and Technology (AIST), Spintronics Research Center,
Tsukuba 305-8568, Japan
2Unit/C19e Mixte de Physique CNRS/Thales and Universit /C19e Paris Sud 11, 1 Ave. A. Fresnel, Palaiseau, France
(Presented 4 November 2014; received 24 August 2014; accepted 18 October 2014; published
online 9 February 2015)
Oscillation frequency of spin torque oscillator with a perpendicularly magnetized free layer and an
in-plane magnetized pinned layer is theoretically investigated by taking into account the field-like
torque. It is shown that the field-like torque plays an important role in finding the balance between
the energy supplied by the spin torque and the dissipation due to the damping, which results in asteady precession. The validity of the developed theory is confirmed by performing numerical sim-
ulations based on the Landau-Lifshitz-Gilbert equation.
VC2015 AIP Publishing LLC .
[http://dx.doi.org/10.1063/1.4907696 ]
Spin torque oscillator (STO) has attracted much atten-
tion as a future nanocommunication device because it can
produce a large emission power ( >1lW), a high quality fac-
tor (>103), a high oscillation frequency ( >1 GHz), a wide
frequency tunability ( >3 GHz), and a narrow linewidth
(<102kHz).1–9In particular, STO with a perpendicularly
magnetized free layer and an in-plane magnetized pinned
layer has been developed after the discovery of an enhance-
ment of perpendicular anisotropy of CoFeB free layer byattaching MgO capping layer.
10–12In the following, we focus
on this type of STO. We have investigated the oscillation
properties of this STO both experimentally6,13and theoreti-
cally.14,15An important conclusion derived in these studies
was that field-like torque is necessary to excite the self-oscillation in the absence of an external field, nevertheless
the field-like torque is typically one to two orders of magni-
tude smaller than the spin torque.
16–18We showed this con-
clusion by performing numerical simulations based on the
Landau-Lifshitz-Gilbert (LLG) equation.15
This paper theoretically proves the reason why the field-
like torque is necessary to excite the oscillation by using the
energy balance equation.19–27An effective energy including
the effect of the field-like torque is introduced. It is shownthat introducing field-like torque is crucial in finding the
energy balance between the spin torque and the damping,
and as a result to stabilize a steady precession. A good agree-
ment with the LLG simulation on the current dependence of
the oscillation frequency shows the validity of the presentedtheory.
The system under consideration is schematically shown
in Fig. 1(a). The unit vectors pointing in the magnetization
directions of the free and pinned layers are denoted as mand
p, respectively. The z-axis is normal to the film-plane,
whereas the x-axis is parallel to the pinned layer magnetiza-
tion. The current Iis positive when electrons flow from thefree layer to the pinned layer. The LLG equation of the free
layer magnetization mis
dm
dt¼/C0cm/C2H/C0cHsm/C2p/C2m ðÞ
/C0cbHsm/C2pþam/C2dm
dt; (1)
FIG. 1. (a) Schematic view of the system. (b) Schematic views of the con-
tour plot of the effective energy map (dotted), Eq. (2), and precession trajec-
tory in a steady state with I¼1.6 mA (solid).a)Author to whom correspondence should be addressed. Electronic mail:
tomohiro-taniguchi@aist.go.jp.
0021-8979/2015/117(17)/17C504/3/$30.00 VC2015 AIP Publishing LLC 117, 17C504-1JOURNAL OF APPLIED PHYSICS 117, 17C504 (2015)
where cis the gyromagnetic ratio. Since the external field is
assumed to be zero throughout this paper, the magneticfield H¼ðH
K/C04pMÞmzezconsists of the perpendicular
anisotropy field only, where HKand 4 pMare the crystalline
and shape anisotropy fields, respectively. Since we areinterested in the perpendicularly magnetized free layer, H
K
should be larger than 4 pM. The second and third terms on
the right-hand-side of Eq. (1)are the spin torque and field-
like torque, respectively. The spin torque strength,H
s¼/C22hgI=½2eð1þkm/C1pÞMV/C138, includes the saturation mag-
netization Mand volume Vof the free layer. The spin polar-
ization of the current and the dependence of the spin torquestrength on the relative angle of the magnetizations arecharacterized in respective by gandk.
14According to Ref.
15,bshould be negative to stabilize the self-oscillation.
The values of the parameters used in the following calcula-tions are M¼1448 emu/cc, H
K¼20.0 kOe, V¼p/C260/C260
/C22n m3,g¼0:54,k¼g2,b¼/C00.2, c¼1.732 /C2107rad/
(Oe/C1s), and a¼0.005, respectively.6,15The critical current
of the magnetization dynamics for b¼0i s Ic¼½4aeMV
=ð/C22hgkÞ/C138ðHK/C04pMÞ’1:2 mA, where Ref. 15shows that
the effect of bon the critical current is negligible. When
the current magnitude is below the critical current, the mag-netization is stabilized at m
z¼1.
In the oscillation state, the energy supplied by the spin
torque balances the dissipation due to the damping. Usually,the energy is the magnetic energy density defined as
E¼/C0MÐ
dm/C1H,
28which includes the perpendicular ani-
sotropy energy only, /C0MðHK/C04pMÞm2
z=2, in the present
model. The first term on the right-hand-side of Eq. (1)can be
expressed as /C0cm/C2½ /C0 @E=@ðMmÞ/C138. However, Eq. (1)indi-
cates that an effective energy density
Eeff¼/C0MH K/C04pM ðÞ
2m2
z/C0b/C22hgI
2ekVlog 1 þkm/C1p ðÞ (2)
should be introduced because the first and third terms on the
right-hand-side of Eq. (1)can be summarized as /C0cm
/C2½/C0@Eeff=@ðMmÞ/C138. Here, we introduce an effective magnetic
fieldH¼/C0@Eeff=@ðMmÞ¼ð b/C22hgI=½2eð1þkmxÞMV/C138;0;
ðHK/C04pMÞmzÞ. Dotted line in Fig. 1(b)schematically shows
the contour plot of the effective energy density Eeffprojected
to the xy-plane, where the constant energy curves slightly shift
along the x-axis because the second term in Eq. (2)breaks the
axial symmetry of E. Solid line in Fig. 1(b)shows the preces-
sion trajectory of the magnetization in a steady state withI¼1.6 mA obtained from the LLG equation. As shown, the
magnetization steadily precesse s practically on a constant
energy curve of E
eff. Under a given current I, the effective
energy density Eeffdetermining the constant energy curve of
the stable precession is obtained by the energy balance
equation27
aMaðEeffÞ/C0MsðEeffÞ¼0: (3)
In this equation, MaandMs, which are proportional to the
dissipation due to the damping and energy supplied by the
spin torque during a precession on the constant energy curve,are defined as
14,25–27Ma¼c2þ
dt½H2/C0ðm/C1HÞ2/C138; (4)
Ms¼c2þ
dtH s½p/C1H/C0ðm/C1pÞðm/C1HÞ/C0ap/C1ðm/C2HÞ/C138:
(5)
The oscillation frequency on the constant energy curve deter-
mined by Eq. (3)is given by
f¼1=þ
dt: (6)
Since we are interested in zero-field oscillation and from the
fact that the cross section of STO in experiment6is circle, we
neglect external field Hextor with in-plane anisotropy field
Hin/C0plane
K mxex. However, the above formula can be expanded
to system with such effects by adding these fields to Hand
terms /C0MHext/C1m/C0MHin/C0plane
K m2
x=2 to the effective energy.
In the absence of the field-like torque ( b¼0), i.e.,
Eeff¼E, there is one-to-one correspondence between the
energy density Eandmz. Because an experimentally measura-
ble quantity is the magnetoresistance proportional to
ðRAP/C0RPÞmax½m/C1p/C138/max½mx/C138¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1/C0m2
zp
,i ti ss u i t a b l e
to calculate Eq. (3)as a function of mz, instead of E,w h e r e
RPðAPÞis the resistance of STO in the (anti)parallel alignment
of the magnetizations. Figure 2(a) shows dependences of
Ms;/C0aMaand their difference Ms/C0aMaon mz
(0/C20mz<1) for b¼0, whereMsandMaare normalized by
cðHK/C04pMÞ. The current is set as I¼1.6 mA ( >Ic). We also
showMs;/C0aMaand their difference Ms/C0aMafor
b¼/C00.2 in Fig. 2(b),w h e r e mxis set as mx¼/C0ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1/C0m2
zp
.
Because /C0aMais proportional to the dissipation due to theFIG. 2. Dependences of Ms;/C0aMaand their difference Ms/C0Manormal-
ized by cðHK/C04pMÞonmz(0/C20mz<1) for (a) b¼0 and (b) b¼/C00.2,
where I¼1.6 mA.17C504-2 Taniguchi et al. J. Appl. Phys. 117, 17C504 (2015)damping, /C0aMais always /C0aMa/C200. The implications of
Figs. 2(a)and2(b) are as follows. In Fig. 2(a),Ms/C0aMais
always positive. This means that the energy supplied by thespin torque is always larger than the dissipation due to the
damping, and thus, the net energy absorbed in the free layer is
positive. Then, starting from the initial equilibrium state(m
z¼1), the free layer magnetization moves to the in-plane
mz¼0, as shown in Ref. 14. On the other hand, in Fig. 2(b),
Ms/C0aMais positive from mz¼1 to a certain m0
z, whereas it
is negative from m0
ztomz¼0(m0
z’0:4 in the case of Fig.
2(b)). This means that starting from mz¼1, the magnetization
can move to a point m0
zbecause the net energy absorbed by
the free layer is positive, which drives the magnetization dy-
namics. However, the magnetization cannot move to the film
plane ( mz¼0) because the dissipation overcomes the energy
supplied by the spin torque from mz¼m0
ztomz¼0. Then, a
stable and large amplitude precession is realized on a constant
energy curve.
We confirm the accuracy of the above formula by com-
paring the oscillation frequency estimated by Eq. (6)with
the numerical solution of the LLG equation, Eq. (1). In Fig.
3, we summarize the peak frequency of jmx(f)jfor
I¼1.2–2.0 mA (solid line), where mx(f) is the Fourier trans-
formation of mx(t). We also show the oscillation frequency
estimated from Eq. (6)by the dots. A quantitatively good
agreement is obtained, guaranteeing the validity of Eq. (6).
In conclusion, we developed a theoretical formula to
evaluate the zero-field oscillation frequency of STO in the
presence of the field-like torque. Our approach was based on
the energy balance equation between the energy supplied bythe spin torque and the dissipation due to the damping. An
effective energy density was introduced to take into account
the effect of the field-like torque. We discussed that intro-ducing field-like torque is necessary to find the energy bal-
ance between the spin torque and the damping, which as a
result stabilizes a steady precession. The validity of thedeveloped theory was confirmed by performing the numeri-
cal simulation, showing a good agreement with the present
theory.The authors would like to acknowledge T. Yorozu, H.
Maehara, H. Tomita, T. Nozaki, K. Yakushiji, A.Fukushima, K. Ando, and S. Yuasa. This work wassupported by JSPS KAKENHI No. 23226001.
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1.4772071.pdf | Dynamics of the vortex core in magnetic nanodisks with a ring of magnetic
impurities
D. Toscano, S. A. Leonel, P. Z. Coura, F. Sato, R. A. Dias et al.
Citation: Appl. Phys. Lett. 101, 252402 (2012); doi: 10.1063/1.4772071
View online: http://dx.doi.org/10.1063/1.4772071
View Table of Contents: http://apl.aip.org/resource/1/APPLAB/v101/i25
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Downloaded 26 Dec 2012 to 152.14.136.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionsDynamics of the vortex core in magnetic nanodisks with a ring of magnetic
impurities
D. Toscano,1S. A. Leonel,1,a)P . Z. Coura,1F . Sato,1R. A. Dias,1and B. V. Costa2
1Departamento de F /C19ısica, Laborat /C19orio de Simulac ¸~ao Computacional, Universidade Federal de Juiz de Fora,
Juiz de Fora, Minas Gerais 36036-330, Brazil
2Departamento de F /C19ısica, Laborat /C19orio de Simulac ¸~ao, Universidade Federal de Minas Gerais, Belo Horizonte,
Minas Gerais 30123-970, Brazil
(Received 25 September 2012; accepted 29 November 2012; published online 17 December 2012)
In this work, we used numerical simulations to study the effect of a ring of magnetic impurities on
the vortex core dynamics in nanodisks of Permalloy. The presence of the ring not only allowed usto modulate the gyrotropic frequency but also provided us a way to confine the vortex core. We
observed that the gyrotropic frequency depends on the ring parameters. Moreover, we have noticed
that the switching of the vortex core polarity can be obtained from the vortex core-impurityinteraction under peculiar conditions, in particular, when the ring works for pinning the vortex
core.
VC2012 American Institute of Physics .[http://dx.doi.org/10.1063/1.4772071 ]
Nowadays, nanomagnets can be fabricated to present
different magnetization states, depending on its geometric
characteristics and the material of which it is made of. In
particular, a soft nanomagnet in the shape of a disk can ex-hibit a magnetic vortex in the remanent state.
1–5The compe-
tition between the exchange and dipolar energies is
responsible for the curl configuration of the magneticmoments that arises as an intermediate state between mono
and multidomain.
6At the core of the magnetic vortex a
magnetization perpendicular to the plane of disk can bedeveloped; the vortex polarity determines the sense of this
out-of-plane magnetization (up or down). From the techno-
logical point of view, these discrete states can be useful tostore information, so that, a nanodisk containing a vortex
configuration could store up to 2 bits of data.
7
It is well known that the simplest effect induced by an in-
plane external magnetic field is the vortex core displacement.
Since the vortex core is polarized , it describes an elliptical tra-
jectory around some fixed point. This motion is known as thegyrotropic mode. The sense of rotation depends only on the vor-
tex core polarity. The frequency of the gyrotropic mode is of
order of 10
2MHz being determined by the disk aspect ratio.8,9
In the last few years, much effort has been dedicated to
control the polarity switching. It has been experimentally
observed that applying a small-amplitude field pulse10or a
spin polarized current11can switch the polarity.
Some works reported that defects can influence the vortex
core dynamics. One can distingu ish two classes of defects,
magnetic and nonmagnetic. The vortex core pinning by non-
magnetic defects (vacancy or cavity) was predicted theoreti-
cally12and observed experimentally.13Such defects can attract
and capture the vortex core. Corresponding analytical and
micromagnetic calculations, mod elling the vacancy defects, are
in good agreement with reported experiments.14–16Further-
more, it was observed that the gyrotropic frequency at low
excitation amplitudes was signifi cantly influenced by intrinsic
defects.17,18If the vortex core is captured by artificial defects inthe form of cavities, its out-of-pl ane component is substantially
reduced or even vanishes. In thi s case, the commonly observed
gyrotropic mode is suppressed.19,20The basic physics behind
the mechanism of the vortex core pinning by nonmagneticdefects is widely discussed in Ref. 21. Nevertheless, for some
special conditions, it has been ob served in simulations that the
vortex core switching can occur due to the interaction between
the vortex core and the nonmagnetic defects.
22A model for
structural defects in nanomagnets was proposed in Ref. 23;t h e
authors discussed two possible t ypes of pointlike defects acting
as pinning or scattering sites for the vortex core. There, the fol-
lowing question is pointed out: It is known that nonmagnetic
impurities act as pinning sites, but what could act as scattering
sites? In a recent work,24the answer to this question was
addressed. Numerical results indicate that a possible origin of
the pinning or scattering defects in nanomagnets could be thelocal reduction or increase in the exchange constant, respec-
tively. In another recent work,
25it was shown that the gyro-
tropic frequency is affected by the magnetic impurities. From
the experimental point of view, magnetic impurities can be
lithographically inserted in nanomagnets.
The main goal in this paper is to investigate how the dy-
namics of the vortex core is influenced by an arrangement of
magnetic impurities in nanodisks. In particular, we study a
circular distribution of the impurities as shown in Fig. 1.
In an early work, we have presented a Hamiltonian
model describing two types of pointlike magnetic impurities
that can behave as pinning and scattering sites for the vortexcore.
24We have considered a classical ferromagnet model
described by the following Hamiltonian:
H¼J(
/C01
2X
<i6¼i0;j>^mi/C1^mj/C0J0
2JX
<i0;j>^mi0/C1^mj
þD
2JX
i;j^mi/C1^mj/C03ð^mi/C1^rijÞð^mj/C1^rijÞ
ðrij=aÞ3"#
/C0Z
JX
i^mi/C1~bext
i)
; (1)a)Author to whom correspondence should be addressed. Electronic mail:
sidiney@fisica.ufjf.br.
0003-6951/2012/101(25)/252402/4/$30.00 VC2012 American Institute of Physics 101, 252402-1APPLIED PHYSICS LETTERS 101, 252402 (2012)
Downloaded 26 Dec 2012 to 152.14.136.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionswhere ^mk/C17ðmx
k;my
k;mz
kÞis a dimensionless vector with
j^mkj¼1 representing the magnetic moment located at the
sitekof the lattice. The first term in Eq. (1)represents the
ferromagnetic coupling only for sites without impurities,whereas the second take into account the exchange interac-
tion between sites with and without impurities. The exchange
interactions between magnetic sites and that one containingthe impurity were modelled by ferromagnetic coupling with
the exchange constant strength J
0differing of its value for
sites without defects J. In this way, we describe two possible
types of magnetic impurities, acting as pinning ( J0<J)o r
scattering ( J0>J) sites for the vortex core. The following
terms are dimensionless versions of dipolar and Zeemaninteractions, respectively. The Hamiltonian (1) can be rewrit-
ten as H¼JH, where His the dimensionless term in curly
brackets. The system energy is measured in unities of J.
The dynamics of the system is followed by solving
numerically the discrete version of the Landau-Lifshitz-Gilbert
equation given by
d^mi
ds¼/C01
1þa2½^mi/C2~biþa^mi/C2ð^mi/C2~biÞ/C138; (2)
where ~bi¼/C0@H
@^miis the dimensionless effective field at site i,
containing individual contributions from the exchange, dipo-
lar, and Zeeman fields.
In the micromagnetics approach, the interaction constants
depend on the material parameters and also the manner in
which the system is partitioned into cells. As in Refs. 24and
25, we have chosen to use cubic cells of edge length a.I nt h i s
case, the interaction constants between the cells are given by
J¼2AaandD
J¼1
4pða
kÞ2. If there is an external applied mag-
netic field, the coefficient of Zeeman interaction isZ
J¼ðakÞ2.
We have used the typical parameters for Permalloy-79: the
saturation magnetization MS¼8:6/C2105A=m, the exchange
stiffness constant APy¼1:3/C210/C011J=m, and the damping
constant a¼0:01. Using these parameters, we have estimated
the exchange length as k¼ffiffiffiffiffiffiffiffi
2A
l0M2
Sq
/C255:3 nm and the unit cellsize was chosen as 5 /C25/C25n m3.T h et i m e tis obtained by
t¼s=x0,w h e r e sis the dimensionless simulation time and
x0¼ðk
aÞ2l0cMs.F o rP e r m a l l o y - 7 9 , x0/C252:13/C21011s/C01.
The equation of motion (2) was integrated forward by using afourth-order predictor-corrector scheme with time step
Ds¼0:01. In our simulations, we have used the nanodisks
with diameter d¼170 nm and thickness L¼10 nm, differing
one to another only in the ring parameters: local variation of
the exchange constant J
0=Jalong the ring and ring diameter
d0. The corresponding disk without ring was taken as a
reference.
We have chosen as initial condition the disk with a vortex
configuration with upward polarity and counter-clockwise
chirality. The integration of the equations of motion (2) at
external magnetic field ~bext
i¼~0 leads the system to a local
energy minimum configuration and we assumed that the
nanodisk remanent state was reached. The states obtained in
this way were saved to be used as initial configurations in thestudies of the gyrotropic mode and switching of vortex
polarity.
Over a wide range of the ring diameter d
0and exchange
constant ratio J0=J, we numerically calculated the dynamic
response of the remanent state to a homogeneous in-plane
magnetic field pulse
~Bext
i¼^yBsinð2p/C23tÞ: (3)
All simulations were done using /C23¼0:5 GHz. The relation
between the applied magnetic field and its dimensionless
corresponding is ~Bext
i¼l0MS~bext
i.
To excite the gyrotropic mode, we used a low excitation
amplitude B¼3 mT. We observed that the vortex core
describes a circular trajectory within the ring. The gyrotropicfrequency dependence with the ring parameters is shown in
Fig. 2. As a reference, we plot the gyrotropic frequency of
the nanodisk without the ring ðJ
0=J¼1Þ, being shown as the
dashed line. A local reduction of the exchange constant
(J0=J<1) always lowers the gyrotropic frequency of the
nanodisk. On the other hand, if ( J0=J>1) the gyrotropic fre-
quency increases. The variation in gyrotropic frequency is
more pronounced for smaller ring diameters. That might be
expected, since the interaction between the vortex core andthe magnetic impurity is short-ranged. As expected, for
FIG. 1. Schematic of the modified nanodisks. The blue circles represent small
clusters containing impurities; they define the ring of magnetic impurities.
FIG. 2. Gyrotropic frequency depending on the local variation of theexchange constant J
0=Jand ring diameter d0. The dashed line, in black, cor-
responds to the gyrotropic frequency of the nanodisk without the ring of
magnetic impurities, in this case 0.543 GHz. The effect of an attractive ring(J
0=J<1) is to lower the gyrofrequency, whereas the effect of a repulsive
ring ( J0=J>1) is to increase gyrofrequency.252402-2 Toscano et al. Appl. Phys. Lett. 101, 252402 (2012)
Downloaded 26 Dec 2012 to 152.14.136.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionslarger values of the ring diameter, the gyrotropic frequency
tends to the frequency of the reference disk ( J0=J¼1).
The reason why the magnetic impurities insertion modi-
fies the gyrotropic frequency can be understood using aThiele approach
26and is discussed in Ref. 25.
The dynamics of the vortex core is substantially affected
if the ring diameter approaches the gyrotropic trajectory di-ameter (30 nm for the disk without the ring). When
(J
0=J>1), we observed the confinement of the vortex core,
namely, the gyrotropic trajectory diameter is reduced. In
extreme cases ( J0=J/C291), due to the radial symmetric distri-
bution of repulsive impurities, the vortex core can get stuckin its equilibrium position and the gyrotropic mode is not
excited. Above a critical amplitude B, the vortex core
escapes from inside the ring.
Augmenting the excitation amplitude, we have noticed a
phenomenon not yet observed: the switching of polarity can
be obtained from the interaction between the vortex core andmagnetic impurity under peculiar conditions. The mecha-
nism of switching involves necessarily a pinning site
(J
0=J<1), the relative position of the attractive magnetic
impurity in relation to the disk center and an external agent.
A set of 30 nanodisks with different ring of magnetic
impurities was submitted to an in-plane magnetic field forvaried excitation amplitudes B, see Fig. 3. The chosen values
ofBare not strong enough to reverse the vortex polarity in
the nanodisk without the ring. For B¼20 mT, the vortex
core expulsion was observed in our reference nanodisk butnot if the ring is present. Thus, using the ring, we can expect
an increase in the saturation field. For a strong excitation am-
plitude B, the dominant event is a random multiple switch,
such that the polarity cannot be reversed in a controllablyway. For moderate amplitudes, B¼10, 12, and 15 mT, the
multiple switches region is mainly concentrated in the region
of smaller ring diameters. Whenever the pinning effects areweak, ( J
0=J!1), no reversal is observed. In the case of
strong pinning effects, ( J0=J!0), the polarity cannot be
controlled because the vortex core is pinned in some point of
the ring. As shown in Fig. 3, the control of the polarity
occurs in a very well defined range of parameters. In short,the polarity switching mechanism must involve a single
interaction between the vortex core and the ring. The exter-
nal field must be strong enough to provide the kinetic energynecessary, so that, the vortex core does not get stuck. Besides
attracting the vortex core, a magnetic impurity of the pinning
site reduces its out-of-plane component; as a result of theinteraction, the vortex core magnetization can be reversed.
In summary, we have shown how the dynamics of the
vortex core in a nanodisk can be managed introducing a dis-tribution of impurities in the system; with this we can control
the gyrotropic mode and the vortex core magnetization. A
fine tuning of the gyrotropic frequency can be obtainedthrough the control of the exchange constant strength and the
impurity distribution. When the gyrotropic trajectory is
described inside the ring, the effect of an attractive ring(J
0=J<1) is to reduce the gyrotropic frequency, whereas the
FIG. 3. Polarity controllability diagram
for the magnetic impurities ring parame-
ters in Py nanodisks with diameter
d¼170 nm and thickness L¼10 nm.
Red triangles correspond to a combina-tion of parameters when a single switch-
ing occurs and black circles include the
following events: dynamics of vortex
core without switching, multiple
switches, pinning, or expulsion of the
vortex core.252402-3 Toscano et al. Appl. Phys. Lett. 101, 252402 (2012)
Downloaded 26 Dec 2012 to 152.14.136.96. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissionseffect of a repulsive ring ( J0=J>1) is to increase the gyro-
tropic frequency. Moreover, we showed an alternative mech-
anism to switch the vortex polarity, mediated by the
interaction between the vortex core and attractive magneticimpurities. The control of the polarity occurs in a very well
defined range of parameters. In our simulations, the polarity
switching can be obtained for an amplitude of /C2410 mT with
the switching time about 1.5 ns. The great differential of this
polarity switching process is that it does not require a high
excitation amplitude. A nonmagnetic defect, such as a cav-
ity, has already been intentionally incorporated in Permalloy
disks by using an image reversal electron beam lithographyprocess.
13,14We believe that a magnetic impurity can
be lithographically inserted in nanodisks by depositing a
ferromagnetic material into a cavity previously created. Inorder to verify our predictions, a ring of cluster of Ni or
Fe (A
Ni¼0:86/C210/C011J=m;AFe¼1:98/C210/C011J=m),27for
example, could be inserted in Permalloy nanodisks to act asattractive or repulsive ring for the vortex core, respectively.
AsA
Ni<APy, we expect that JNi<J0<JPyfor attracting,
and as AFe>APy, we expect that JPy<J0<JFefor scatter-
ing the vortex core. We consider here only one possible real-
ization of the vortex core dynamics controllability with a
very single ring. We believe that the usage of others mag-netic impurity distributions lithographically inserted will be
promising for different applications of nanomagnets.
This work was partially supported by CNPq and FAPE-
MIG (Brazilian Agencies). Numerical works were done at theLaborat /C19orio de Simulac ¸~ao Computacional do Departamento
de F/C19ısica da UFJF.
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1.4914496.pdf | Microwave-induced dynamic switching of magnetic skyrmion cores in nanodots
Bin Zhang, Weiwei Wang, Marijan Beg, Hans Fangohr, and Wolfgang Kuch
Citation: Applied Physics Letters 106, 102401 (2015); doi: 10.1063/1.4914496
View online: http://dx.doi.org/10.1063/1.4914496
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/106/10?ver=pdfcov
Published by the AIP Publishing
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132.203.227.62 On: Mon, 29 Jun 2015 03:19:44Microwave-induced dynamic switching of magnetic skyrmion cores
in nanodots
BinZhang,1Weiwei Wang,2Marijan Beg,2Hans Fangohr,2and Wolfgang Kuch1,a)
1Institut f €ur Experimentalphysik, Freie Universit €at Berlin, Arnimallee 14, 14195 Berlin, Germany
2Faculty of Engineering and the Environment, University of Southampton, SO17 1BJ Southampton,
United Kingdom
(Received 10 December 2014; accepted 27 February 2015; published online 9 March 2015)
The nonlinear dynamic behavior of a magnetic skyrmion in circular nanodots was studied
numerically by solving the Landau-Lifshitz-Gilbert equation with a classical spin model. Weshow that a skyrmion core reversal can be achieved within nanoseconds using a perpendicular
oscillating magnetic field. Two symmetric switc hing processes that correspond to excitations of
the breathing mode and the mixed mode (combination of the breathing mode and a radial spin-wave mode) are identified. For excitation of the breathing mode, the skyrmion core switches
through nucleation of a new core from a transient u niform state. In the mixed mode, the skyrmion
core reverses with the help of spins excited both at the edge and core regions. Unlike the mag-netic vortex core reversal, the excitation of radial spin waves does not dominate the skyrmion
core reversal process.
VC2015 AIP Publishing LLC .[http://dx.doi.org/10.1063/1.4914496 ]
A rich variety of magnetic configurations can emerge in
nanoscale magnetic materials. For instance, in a circularnanodisk, both vortex and skyrmion states may occur.
1–4As a
topological excitation, a skyrmion has a topological chargeQ¼61, whereas the vortex topological charge
5is61/2. The
reversal of a vortex core has been intensively studied in thepast few years
2,4,6–15because of its potential use in data stor-
age devices and the importance in fundamental physics.Recently, the creation and annihilation of individualskyrmions was demonstrated,
16–20which illustrates the poten-
tial application of topological charge in future information-storage devices. Moreover, the spin-wave modes of askyrmion have been investigated both experimentally
21,22
and using simulations,3,23and three k¼0 skyrmion “optical”
modes have been identified. It was found that anticlockwiseand clockwise rotation modes are excited when an alternatingcurrent (AC) magnetic field is applied in the skyrmion plane,while the breathing mode is found if the AC field is appliedperpendicular to the skyrmion plane.
23Very recently, a hyste-
retic behavior and reversal of an isolated skyrmion formed ina nanodisk by a static magnetic field have been demon-strated.
24However, the influence of a time-dependent mag-
netic field, such as a microwave field, on the skyrmion corereversal is still unexplored. This is the focus of this letter, inwhich we study the reversal of an individual skyrmion in acircular magnetic dot by applying an AC magnetic field per-pendicular to the skyrmion plane.
The Dzyaloshinskii-Moriya interaction (DMI)
25,26is an
antisymmetric interaction which arises if inversion symmetryin magnetic systems is absent,
27,28either because of a non-
centrosymmetric crystal structure or due to the presence ofinterfaces.
29Consequently, these chiral interactions can be
classified as bulk or interfacial, depending on the type ofinversion symmetry breaking.
29In magnetic thin films, a
Bloch-type chiral skyrmion can be formed by bulk DMI,while a N /C19eel-type radial (“hedgehog”) skyrmion requires the
presence of interfacial DMI.30In this study, we consider the
bulk DMI, which exists in non-centrosymmetric magnets
such as MnSi31,32and FeGe.33
We consider a classical Heisenberg model on a two-
dimensional regular square lattice with ferromagnetic
exchange interaction (represented by J) and uniaxial anisot-
ropy ( Ku) along the zaxis.23,34,35Apart from that, a time-
dependent magnetic field h(t) is applied to the system in the
positive zdirection. Therefore, the total Hamiltonian of the
system is given by
H¼/C0 JX
hi;jimi/C1mjþX
hi;jiDij/C1½mi/C2mj/C138
/C0X
iKuðez/C1miÞ2/C0X
ijlijh/C1mi; (1)
where miis the unit vector of a magnetic moment li¼/C0/C22hcS
withSbeing the atomic spin and c(>0) the gyromagnetic ra-
tio. The DMI vector Dijcan be written as Dij¼D^rijin the
case of bulk DMI, where Dis the DMI constant and ^rijis the
unit vector between liandlj. In this study, dipolar interac-
tions are not included, because in systems with small sizes of
10–100 nm, dipolar interactions are comparably weak.23,36
The spin dynamics at lattice site iis governed by the
Landau-Lifshitz-Gilbert (LLG) equation
@mi
@t¼/C0cmi/C2Heffþami/C2@mi
@t; (2)
where adenotes the Gilbert damping and the effective field
Heffis computed as Heff¼/C01
jlij@H
@mi. The Hamiltonian (1)
associated with the LLG equation (2)can be understood as a
finite-difference micromagnetic model. In this study, we
have chosen J¼1 as the energy unit.23All simulations are
performed in a circular dot with a diameter of 121 lattice
sites. We have chosen J¼/C22h¼c¼S¼1 as simulation pa-
rameters, and therefore the coefficients for conversion of thea)Electronic mail: kuch@physik.fu-berlin.de
0003-6951/2015/106(10)/102401/4/$30.00 VC2015 AIP Publishing LLC 106, 102401-1APPLIED PHYSICS LETTERS 106, 102401 (2015)
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132.203.227.62 On: Mon, 29 Jun 2015 03:19:44external field h, time t, and frequency xto SI units are
J=ð/C22hcSÞ;/C22hS=J, and J=ð/C22hSÞ, respectively, as shown in Table
I. The Gilbert damping a¼0.02 is selected for most of the
simulations. A DMI constant with a value D¼0.08 is used,
except for Fig. 1, where Dis varied. This value of Dresults
in the spiral period k/C252pJa/D/C2539 nm and the skyrmion di-
ameter23of approximately 55 nm for the lattice parameter
a¼0.5 nm.
There are various stable states for the system described
by Hamiltonian (1). The phase diagram presented in Fig. 1(a)
shows the ground state as a function of DandKufor the mag-
netic nanodot. The phase diagram is obtained by comparing
several possible states such as skyrmion, uniform, and dis-
torted states, as shown in Figs. 1(b)–1(e) . Figure 1(b) shows
an isolated skyrmion with topological charge (skyrmion num-ber) Q¼Ðqdxdy/C25/C00:86, where q¼m/C1ð@
xm/C2@ymÞis
the topological charge density. The isolated skyrmion state
emerges in the brown region with skyrmion number /H11351/C00.6,
while the distorted states are shown in the purple area, wherethe skyrmion number is chosen to be positive to distinguish
from the skyrmion state. The uniform state with skyrmion
number /C250 is found in the remaining area. We fix parameters
D¼0.08 and K
u¼0.004 in the remaining of this work, as
indicated in Fig. 1(a).
To extract the dynamic response of the skyrmion to the
microwaves, we calculate the magnetic absorption spectrumof the skyrmion
3,37by applying a magnetic field pulse in the
vertical ( z) direction with a temporal shape given by a sinc-
function field hz¼h0sinc(x0t)¼h0sin(x0t)/(x0t), where
h0¼8/C210/C06andx0¼0.1. The magnetic spectrum of a sin-
gle skyrmion is shown in Fig. 2with a¼0.04. There are two
main resonance lines at x¼3.6/C210/C03andx¼1.4/C210/C02,
corresponding to f¼0.87 GHz and f¼3.39 GHz, respectively,
ifJ¼1 meV and S¼1 are chosen. The insets in Fig. 2show
the corresponding eigenmode amplitude plots, which are
obtained by computing the fast Fourier transform (FFT) to the
spatial mzoscillations of the sample. These patterns indicate
the spatially resolved fluctuation amplitude. The lower-frequency resonance peak corresponds to the breathing modeof the skyrmion,
22,23where the skyrmion radially expands and
shrinks as a function of time with significant fluctuation
around the skyrmion core region. The higher-frequency reso-nance peak with weaker intensity can be viewed as a mixedmode that combines the breathing mode and the radial spin-wave mode,
3since the spin excitation arises both around the
core area as well as at the dot edges.
After determining the eigenfrequencies of the skyrmion
formed in a nanodot, we study the switching driven by a si-nusoidal magnetic field along the zdirection, i.e.,
h
x
z¼hxsinðxtÞ,w h e r e hxandxare the field amplitude
and frequency, respectively. Two cases of microwave-induced dynamics of a single skyrmion at h
x¼2.4/C210/C03
(/C2520 mT for J¼1 meV and S¼1) with AC fields of
x¼3.6/C210/C03and 1.4 /C210/C02corresponding to the breath-
ing and mixed resonant modes, respectively, are shown in
Fig.3. Figures 3(a) and3(b) present serial snapshots, while
the corresponding mz(t)a n d hx(t)a r es h o w ni nF i g . 3(c) by
black and red curves for the two frequencies. The initialskyrmion state ( t¼0) is obtained by relaxing the system
with maximum jdm=dtj<10
/C06. In the initial state, the sky-
rmion number /C00.86 and the zcomponent of the magnetiza-
tion mzof the skyrmion core are negative, while mzat the
edge of the nanodot is positive. In the first scenario, shownTABLE I. Unit conversion table for J¼1 meV and S¼1.
Magnetic field hJ =ð/C22hcSÞ/C25 8.63 T
Time t /C22hS=J /C250.66 ps
Frequency x J=ð/C22hSÞ/C25 1.52/C2103GHz
FIG. 1. (a) The phase diagram with
topological charge of the ground state
as a function of Dand Ku. Possible
ground states are (b) skyrmion (SkX),(c) uniform, and (d)–(e) distorted
states. The diameter of the dot is 121
sites.
FIG. 2. Imaginary part of the susceptibility spectrum ( vzz) obtained after
applying a field pulse hz¼h0sinc ( x0t) along the zaxis to the system with
h0¼8/C210/C06andx0¼0.1. The insets show the spatial distribution of the
FFT power at eigenfrequencies x¼3.6/C210/C03and 1.4 /C210/C02, respectively.
The lines are profiles of the mzcomponent across the core center.102401-2 Zhang et al. Appl. Phys. Lett. 106, 102401 (2015)
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132.203.227.62 On: Mon, 29 Jun 2015 03:19:44in Fig. 3(a) [see also Multimedia view], the breathing mode
is excited by applying an AC field with xa¼3.6/C210/C03,
which is switched off at ta
off¼1750, as indicated by the
black dotted line in Fig. 3(c).A sar e s p o n s et ot h eA Cm a g -
netic field, the skyrmion starts to breathe, i.e., the skyrmioncore expands when the field points to the negative z-direc-
tion and shrinks if the direction of applied magnetic field is
positive. The core thus becomes smaller than in the initialstate ( t¼0) after the first positive half period ( t¼780).
After one full period ( t¼1760), the skyrmion core expands
over the entire dot, the edges become negative, and a tran-sient uniform state is reached. Afterwards, the tilted spins at
the edge of the disk circulate to the inside of the dot and
propagate to the center as shown in Fig. 3(a) at times
1920–2120. At the same time, m
zo ft h ec o r eb e g i n st o
increase and an upward core is nucleated at t¼2340 [Fig.
3(a)]. Subsequently, high-frequency radial spin waves are
excited from the core center. The switching of the skyrmion
from a downward to an upward core orientation is finished.
At last, the system enters into a damped oscillation with thefrequency of the breathing mode up to t/C2525 000, as shown
by the black line in Fig. 3(c).
Now we turn to the simulation results of the mixed mode
case, x
b¼1.4/C210/C02, with tb
off¼5600, shown in Fig. 3(b)
[see also Multimedia view] and by the red curves in (c). By
applying the AC magnetic field, both the spins at the edge ofthe structure and around the skyrmion core are excited in the
mixed mode, as shown at t¼2720. After t¼3000, the lower-
frequency breathing mode starts and then dominates thedynamic properties. The most expanded core is obtained at
t¼3920 with a ring of tilted spins around the middle, similar
to the case of t¼1920 and t¼2120 in Fig. 3(a), however, no
reversed core is formed for these field parameters. After the
core has shrunk to a smaller size at t¼4760, it does not
recover to the previous size because of a compressionresulting from the excitation of the edge area ( t¼5520).
Instead, the core shrinks even further, and immediately there-after, the down-core vanishes and is replaced by an up-core,while spin waves are emitted from the core area (seet¼5720). After switching off the AC field at t¼5600, the
reversed skyrmion starts to dissipate the energy via both the
breathing and mixed modes.
Similar to the vortex core reversal with out-of-plane
fields,
13,15,38,39the reversal processes for the two frequencies
are symmetric, and the switching mechanisms are similar to
the radial-spin-wave-mode-assisted switching for a vortex
core.13At the lower frequency, which corresponds to the
breathing mode, the skyrmion first switches to a transientuniform state via axial core expansion to the edge, and thena new core is formed from the center. For the mixed mode
excitation at the higher frequency, the reversed skyrmion
results from the shrinking of the core with the help of spinexcitations from the edge region. High-frequency spin wavesare emitted during the core reversal, however, a spin-waveassisted re-reversal, i.e., a switching back to the original stateby a spin wave, as it has been observed in vortex core rever-
sal,
13,15,38has not been found here. Our simulation results
also show that the switching time and switching field for askyrmion-core reversal are similar to those needed for avortex-core reversal.
Besides the duration time, the skyrmion reversal process
also depends on the frequency xand the AC field amplitude
h
x. Figure 4shows the switching phase diagram as a func-
tion of xand hxwith a maximum temporal duration
t¼10 000 ( /C246.6 ns for J¼1 meV and S¼1). The diagram
can be divided into three different regions. The first is theweak-susceptibility region I (Gray symbols), in which a
skyrmion core never switches for AC field duration up to
t¼10 000, mainly because the field is not strong enough
with respect to the dynamic susceptibility of the skyrmion.
FIG. 3. Spin dynamics of the skyrmion
e x c i t e db ya nA Cm a g n e t i cfi e l d hx
z
¼hxsinðxtÞwith hx¼2:4/C210/C03up
to time 25 000 ( /C2416.5 ns for J¼1m e V
andS¼1). Selected time evolution of
the magnetization distribution with two
examples: (a) xa¼3.6/C210/C03,ta
off
¼1750 and (b) xb¼1.4/C210/C02,
tb
off¼5600. The color-coded mz(out-
of-plane) and mx(in-plane) components
of the magnetization are displayed in
the first and third line, respectively,while the second line displays a per-
spective view (topography) with the
color code of m
z. (c) Time trace of the
magnetization component mz(top
graph), the AC magnetic field (dashed
lines), and the skyrmion number (both
in bottom graph). Black and red linesrefer to the data of the two different
examples, while the black circles and
red prisms indicate the times corre-
sponding to the snapshots shown in (a)
and (b), respectively. (Multimedia
view) [URL: http://dx.doi.org/10.1063/
1.4914496.1 ] [URL: http://dx.doi.org/
10.1063/1.4914496.2 ]102401-3 Zhang et al. Appl. Phys. Lett. 106, 102401 (2015)
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132.203.227.62 On: Mon, 29 Jun 2015 03:19:44In the second region (II), a skyrmion converts to either the
uniform state (up-pointing triangles) or the distorted state(down-pointing triangles). The distorted state is formed
directly from the skyrmion due to the instability of the core
under the AC field with intermediate intensities, similar tothe case of magnetic vortices.
14The boundary between
regions I and II reflects the susceptibility spectrum of the
skyrmion, cf. Fig. 1, in such a way that at around the eigen-
frequencies of the system switching to the uniform or dis-
torted states occurs at lower field amplitudes than at other
frequencies. In the third region (III, red), a skyrmion corecan be reversed by the AC field, as had been shown for two
instances with t
a
off¼1750 and tb
off¼5600 in Figs. 3(a) and
3(b), respectively. Comparing to the results for a static mag-
netic field ( x¼0), where in this amplitude regime the final
state is either unchanged or uniform depending on whether
hxis below or above about 3.6 /C210/C03, it is obvious that an
AC magnetic field within a certain frequency range substan-
tially helps to reverse the skyrmion core.
In summary, we have studied the nonlinear skyrmion
dynamics in a circular magnetic nanodot induced by a verti-
cal oscillating magnetic field using micromagnetic simula-
tions. We found that in a certain frequency window a fastskyrmion switching can be achieved for field amplitudes that
do not lead to a reversed skyrmion under static conditions.
We presented two examples of skyrmion switching in detail,in which the excitation frequencies corresponded to the
breathing and mixed modes of the system. Under excitation
with the breathing-mode frequency, the skyrmion reversesvia a transient uniform state, while exciting it with the
mixed-mode frequency spin waves from the edge area assist
the switching. Our results show that a skyrmion core can bereversed within nanoseconds by means of microwaves per-
pendicular to the skyrmion plane.
B.Z. gratefully acknowledges funding by the China
Scholarship Council. W.W. and M.B. acknowledge financial
support from EPSRC’s DTC Grant No. EP/G03690X/1. TheHigh Performance Computing Center at the Freie Universit €at
Berlin (ZEDAT) is acknowledged for computational time.
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FIG. 4. Phase diagram of the skyrmion switching induced by a field hx
z
¼hxsinðxtÞas a function of hxandx. Red dots indicate skyrmion reversal,
gray dots mean no switching, up-pointing and down-pointing triangles
skyrmion switching to uniform and distorted states, respectively.102401-4 Zhang et al. Appl. Phys. Lett. 106, 102401 (2015)
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132.203.227.62 On: Mon, 29 Jun 2015 03:19:44 |
1.3365220.pdf | Vocal fold and ventricular fold vibration in period-doubling
phonation: Physiological description and aerodynamicmodeling
a)
Lucie Bailly,b/H20850Nathalie Henrich, and Xavier Pelorson
Département Parole et Cognition, Grenoble Images Parole Signal Automatique (GIPSA-lab), UMR5216
CNRS, Grenoble INP , UJF , Université Stendhal, 961 Rue de la Houille Blanche, DomaineUniversitaire, BP 46, 38402 Saint Martin d’Hères Cedex, France
/H20849Received 10 July 2009; revised 20 January 2010; accepted 21 January 2010 /H20850
Occurrences of period-doubling are found in human phonation, in particular for pathological and
some singing phonations such as Sardinian A Tenore Bassu vocal performance. The combined
vibration of the vocal folds and the ventricular folds has been observed during the production ofsuch low pitch bass-type sound. The present study aims to characterize the physiological correlatesof this acoustical production and to provide a better understanding of the physical interactionbetween ventricular fold vibration and vocal fold self-sustained oscillation. The vibratory propertiesof the vocal folds and the ventricular folds during phonation produced by a professional singer areanalyzed by means of acoustical and electroglottographic signals and by synchronized glottalimages obtained by high-speed cinematography. The periodic variation in glottal cycle duration andthe effect of ventricular fold closing on glottal closing time are demonstrated. Using the detectedglottal and ventricular areas, the aerodynamic behavior of the laryngeal system is simulated usinga simplified physical modeling previously validated in vitro using a larynx replica. An estimate of
the ventricular aperture extracted from the in vivo data allows a theoretical prediction of the glottal
aperture. The in vivo measurements of the glottal aperture are then compared to the simulated
estimations. © 2010 Acoustical Society of America. /H20851DOI: 10.1121/1.3365220 /H20852
PACS number /H20849s/H20850: 43.75.Rs, 43.70.Gr, 43.70.Bk, 43.70.Jt /H20851AH /H20852 Pages: 3212–3222
I. INTRODUCTION
The ventricular folds, also called false vocal folds or
ventricular bands , are two laryngeal structures located above
the vocal folds, superior to the laryngeal /H20849or Morgagni /H20850ven-
tricle /H20849see Fig. 1/H20850. These laryngeal structures are not com-
monly involved as a vibrating structure during normal pho-nation. Their physical properties /H20849high viscosity and low
stiffness /H20850are different from those of biomechanical oscilla-
tors such as the vocal folds /H20849Haji et al. , 1992 /H20850. Yet, their
vibration has been observed during specific vocal gestures:Asian throat singing /H20849Fuks et al. , 1998 ;Lindestad et al. ,
2001 ;Sakakibara et al. , 2001 ,2004 /H20850, Mediterranean tradi-
tional polyphony /H20849Henrich et al. , 2006 /H20850, rock singing /H20849Zang-
ger Borch et al. , 2004 /H20850, pathological phonation /H20849Lindestad
et al. , 2004 ;Nasri et al. , 1996 ;V on Doersten et al. , 1992 /H20850.
Several vibratory gestures can be distinguished: periodic oraperiodic, in phase or not with the vocal fold vibration, withor without ventricular contact. In this study, we focus on a
particular type of ventricular fold vibratory movement, re-ferred to as vocal-ventricular phonation mode byFuks et al.
/H208491998 /H20850. In this phonatory gesture, the ventricular fold vibra-
tory movement is periodic, occurring every two glottalcycles, in antiphase with the glottal vibration /H20849Fuks et al. ,
1998 ;Lindestad et al. , 2001 ;Sakakibara et al. , 2001 ,2002
,2004 ;Henrich et al. , 2006 /H20850. It goes with a period-doubling
phenomenon, i.e., a perceived octave jump below the origi-nal tone.
The physiological and physical nature of this complex
laryngeal vibratory gesture is poorly understood. As both thevocal folds and ventricular folds are vibrating, is there anyphysical interaction between these two laryngeal structures?If so, what is the nature of this interaction? In light of recentstudies, it seems interesting to explore the hypothesis of apossible aerodynamic interaction. Previous experimental in-vestigations dealing with in vitro set-ups have provided an
initial insight into the influence of a supra-glottal constrictionon the glottal airflow /H20849Shadle et al. , 1991 ;Pelorson et al. ,
1995 ;Agarwal, 2004 ;Kucinschi et al. , 2006 ;Finnegan and
Alipour, 2009 ;Bailly et al. , 2008 ;Bailly, 2009 /H20850. Using a
rigid non-oscillating replica combining vocal folds and ven-tricular folds, Agarwal /H208492004 /H20850observed an influence of the
laryngeal geometry on the translaryngeal airflow resistance.An increase in translaryngeal airflow resistance has also beenevidenced on excised canine larynges by Alipour et al.
/H208492007 /H20850and using the same experimental set-up, a median or
antero-posterior ventricular compression has resulted in amean subglottal-pressure increase and an airflow decrease/H20849Finnegan and Alipour, 2009 /H20850. Experiments made on in vitr o
static replicas have shown that the presence of a supra-glottalconstriction results in a decreased glottal-jet curvature
/H20849Shadle et al. , 1991 ;Kucinschi et al. , 2006 /H20850, and a down-
stream shift of the position of glottal separation point, induc-ing a conservation of the flow laminar properties over a
a/H20850This paper is based partially on a talk presented at the 6th ICVPB, Tam-
pere, Finland, 6–9 August 2008.
b/H20850Present address: Laboratoire Sols Solides Structures Risques /H208493S-R /H20850,D o -
maine Universitaire, BP53, 38041 Grenoble Cedex 9, France.
3212 J. Acoust. Soc. Am. 127 /H208495/H20850, May 2010 © 2010 Acoustical Society of America 0001-4966/2010/127 /H208495/H20850/3212/11/$25.00longer distance /H20849Kucinschi et al. , 2006 /H20850. A significant pres-
sure recovery associated with a reattachment of the jet-flowto the constriction has been observed in Pelorson et al.
/H208491995 /H20850and also measured and theoretically predicted in
Bailly et al. /H208492008 /H20850. The geometry of the constriction affects
the phonation threshold pressure and fundamental frequencyof a self-oscillating vocal fold replica /H20849Bailly et al. , 2008 ;
Bailly, 2009 /H20850. In complement to these in vitro observations,
explorations of throat singing have shown that ventricularfold closure coincides with a decrease in glottal-flow ampli-tude every two glottal cycles /H20849Fuks et al. , 1998 ;Lindestad
et al. , 2001 /H20850. Higher oesophageal pressures have been mea-
sured by Fuks et al. /H208491998 /H20850during a switch from modal
phonation to vocal-ventricular mode.
This study explores the physiological correlates of
vocal-ventricular periodic vibrations, and the aerodynamicimpact of ventricular vibration on the vocal fold self-sustained oscillation. The vocal gesture of a professionalsinger is analyzed by the use of high-speed cinematographycombined with acoustic and electroglottographic /H20849EGG /H20850re-
cordings, detailed in Part II /H20849II A and II B /H20850. A simplified
physical modeling of phonation is presented in Sec. II C,which was previously validated in vitro using a vocal fold/
ventricular fold replica. Part III provides a quantitativephysiological description of the co-oscillations, deducedfrom the detection of glottal and ventricular areas, the kymo-graphic processing of the laryngeal images and the analysisof EGG signals /H20849Sec. III A /H20850. The glottal aperture and laryn-
geal pressure distribution are theoretically predicted from themeasured ventricular area as a function of the subglottal-pressure /H20849Sec. III B /H20850.
II. MATERIAL AND METHOD
In the followings, if Xis a function of time t, Xnrefers to
the corresponding normalized quantity, such as: Xn=X/max t/H20849X/H20850. Each geometrical variable Xintroduced in the
theoretical description refers to a real quantity measurable at
human scale, noted X˜.
A. Data recordings
The experiment was conducted at the University Medi-
cal Center Hamburg-Eppendorf in the Department of V oice,Speech and Hearing Disorders. A professional male singer/H20849MW, age 41 /H20850was recorded while performing different pho-
nations, among which two particular utterances are selectedand compared within the scope of this study: a case of nor-mal phonation, and an example of specific growly phonation,perceptually similar to Asian throat singing. This latter caseis characterized by vocal-ventricular periodic vibrations.
High-speed cinematographic recordings of the laryngeal
movement were made by inserting a rigid endoscope into theoral cavity /H20849Wolf 90° E 60491 /H20850with a continuous light
source /H20849Wolf 5131 /H20850and a digital black-and-white CCD cam-
era /H20849Richard WOLF, HS-Endocam 5560 /H20850. The recording se-
quence duration was approximately 4s, with a camera framerate of 2000 frames/s and an image resolution of 256/H11003256 pixels. Audio signal was recorded simultaneously
with a microphone placed at the end of the endoscope /H20849Wolf
5052.801 /H20850. The electroglottographic signal was recorded si-
multaneously with a dual-channel electroglottograph /H20851EG2,
Rothenberg /H208491992 /H20850/H20852; two electrodes were placed either side
of the larynx.
B. Data processing
1. EGG signal processing
Glottal closing instants /H20849GCIs /H20850and glottal opening in-
stants /H20849GOIs /H20850are detected on the time derivative of the EGG
signal /H20849DEGG /H20850, using a threshold-based peak detection
method /H20849Henrich, 2001 ;Henrich et al. , 2004 /H20850. As illustrated
in Fig. 2, the GCI peaks are numbered in order of appear-
ance, a parameter, which completes time and amplitude in-formation for each peak. A distinction is made between the
FIG. 1. /H20849Color online /H20850/H20849a/H20850Rear view coronal section of the larynx, /H20849b/H20850/H20851resp.
/H20849c/H20850/H20852description of the minimal aperture width at the constriction between
the ventricular folds, h˜venf/H20849z,t/H20850/H20851resp. the vocal folds, h˜vof/H20849z,t/H20850/H20852on a frontal
view of the human larynx during phonation /H20849in vivo high-speed recordings;
venf: ventricular fold; vof: vocal fold /H20850.
FIG. 2. /H20849Color online /H20850A typical example of eight-period normalized DEGG
/H20849DEGGn/H20850signal, extracted from the database during the specific growly
phonation and processed with a peak detection method /H20849GCI: glottal closing
instants; GOI: glottal opening instants /H20850.
J. Acoust. Soc. Am., Vol. 127, No. 5, May 2010 Bailly et al. : Vocal fold/ventricular fold periodic vibrations 3213two cycles inside a sequence of two glottal cycles. The du-
ration of each glottal cycle /H20849resp. t1andt2/H20850is calculated as
the duration between two successive GCIs /H20849resp. odd-even
GCIs and even-odd GCIs /H20850. Glottal open time /H20849resp. top1and
top2/H20850is measured as the duration between a GOI and the
following GCI. Open quotient /H20849Oq/H20850is calculated as the ratio
between glottal open time and cycle duration.
2. High-speed image processing
a. Glottal and ventricular aperture area extraction.
Using the MATLAB Image Processing Toolbox, high-speed
images are resized to focus on the area of interest. They aresmoothed using a bicubic scaling filter /H20849radius 4 /H20850. Glottal and
ventricular contours are manually detected by shaping Beziercurves on each high-speed image, following an algorithmdescribed in Serrurier and Badin /H208492008 /H20850/H20849see Fig. 3/H20850. The
glottal and ventricular space areas /H20849A
˜vofandA˜venf/H20850are com-
puted from the detected contours by generic surface triangu-lar meshes /H20849Serrurier and Badin, 2008 /H20850. For each selected
time sequence, the detected areas A˜vof/H20849t/H20850andA˜venf/H20849t/H20850are nor-
malized by their maximum values, and resampled to the
sampling frequency of the synchronized EGG and DEGGsignals /H2084944 170 Hz /H20850with cubic interpolation. Therefore, the
measured parameters are A˜vofn/H20849t/H20850andA˜venfn/H20849t/H20850.
During the growly phonation, the investigated phonatory
gesture includes a narrowing of the supraglottic airway. Thislaryngeal configuration makes it difficult to detect the effec-tive values of ventricular fold aperture and glottal areas. In
such a case, both areas A
˜vofandA˜venfare obviously underes-
timated. Nevertheless, the aryepiglottic constriction beyondthe ventricular folds does not vary much during the se-quence. Therefore, we can assume that the method providesa good estimate of the dynamical evolution of the measuredareas.
Due to the demanding video-laryngoscopic procedure,
magnitudes of recorded vocal and ventricular motions havenot been calibrated. As an alternative, we chose to graduatetheir values with the physiological data available in the lit-erature /H20849Hollien and Colton, 1969 ;Wilson, 1976 ;Kitzing
and Sonesson, 1967 ;Hirano et al. , 1983 ;Agarwal et al. ,
2003 ;Agarwal, 2004 /H20850. Assuming that h
˜vof/H20849z,t/H20850andh˜venf/H20849z,t/H20850
represent the opening width observed at the constriction be-
tween the vocal and the ventricular folds respectively /H20849seeFig. 1/H20850, the maximal glottal and ventricular metric magni-
tudes observed during the phonatory gesture are imposed in
accordance with the bibliographical study /H20849h˜vofrefandh˜venfref/H20850
and defined as follows:
h˜vofref= max t,z/H20849h˜vof/H20849z,t/H20850/H20850=1 m m
/H208491/H20850
h˜venfref= max t,z/H20849h˜venf/H20849z,t/H20850/H20850= 2.5 mm.
Note that in Eq. /H208491/H20850, the ventricular magnitude h˜venfrefis ar-
bitrarily adjusted smaller than the mean value measured in
previous physiological studies during normal phonation/H20851around 5mm in Agarwal et al. /H208492003 /H20850andAgarwal /H208492004 /H20850
for instance /H20852, in order to account for the initial ventricular
constriction observed to switch into the specific studiedgrowly phonation.
The depth of the section along the zdirection, noted W,i s
assumed identical and x-invariant across the vocal fold and
the ventricular fold constrictions. Similarly, its value is cho-sen in agreement with physiological studies mentioned above/H20849W=15 mm /H20850. Two calibration factors are defined such as:
C
vof=h˜vofref·Wand Cvenf=h˜venfref·W. /H208492/H20850
Two additional geometrical parameters, the glottal and ven-
tricular mean apertures measured along the zdirection, are
deduced from the detected areas A˜vofn/H20849t/H20850andA˜venfn/H20849t/H20850. Those
are defined under the approximation of a rectangular mean
section area at the glottal and ventricular levels, such as:
/H20855h˜vof/H20856/H20849t/H20850=Cvof·A˜vofn/H20849t/H20850/W
/H208493/H20850
/H20855h˜venf/H20856/H20849t/H20850=Cvenf·A˜venfn/H20849t/H20850/W.
In the end, the conversion of the measured areas A˜vof/H20849t/H20850and
A˜venf/H20849t/H20850from pixels into m2is achieved under the approxima-
tion:
A˜vof/H20849t/H20850=Cvof·A˜vofn/H20849t/H20850=/H20855h˜vof/H20856/H20849t/H20850·W
/H208494/H20850
A˜venf/H20849t/H20850=Cvenf·A˜venfn/H20849t/H20850=/H20855h˜venf/H20856/H20849t/H20850·W.
Note that the perspective effect occurring within the visual
field of the camera yields to the underestimation of the ratio
FIG. 3. Illustration of a typical high-speed laryngeal image and the detected contours of the glottal /H20849a/H20850and ventricular fold aperture /H20849b/H20850areas.
3214 J. Acoust. Soc. Am., Vol. 127, No. 5, May 2010 Bailly et al. : Vocal fold/ventricular fold periodic vibrations/H20855h˜venf/H20856//H20855h˜vof/H20856, as compared to the physiological reality. Yet,
this optical phenomenon is left aside from the high-speed
image processing, because the axial width of the larynx/H20849along the xdirection /H20850cannot be graphically measured, and
because the discrepancy thus committed is negligible to afirst approximation /H20849Bailly, 2009 /H20850.
b. Kymographic analysis. High-speed images are also
analyzed by means of a kymographic method inspired fromŠvec and Schutte, 1996 . It consists of a visualization method
for high-speed investigation of laryngeal vibrations. A line isselected on the image, perpendicular to the median glottalaxis. This line is plotted as a function of time, along with thesynchronized EGG and DEGG signals. This method does notallow for visualization of glottal vibrations along the wholeglottal length, but it provides a detailed representation of thevocal dynamics at the selected position on the glottis.
C. Theoretical modeling
First, a simplified theoretical description of the laryngeal
airflow dynamics is proposed. Then, corresponding aerody-namic forces are combined with mechanical forces related toa two-mass model of the vocal folds.
1. Flow theory through the larynx
The basic aerodynamic impact implied by the presence
of the ventricular folds in the larynx on the pressure distri-bution and the vocal fold self-oscillations is extensively de-scribed in Bailly et al. , 2008 andBailly, 2009 . A schematic
representation of the laryngeal geometry considered in thisinvestigation is given in Fig. 4along with all relevant param-
eters of the aerodynamic study.
The larynx is assumed to be symmetric with respect to
thexandzaxes. In the following, indices icorrespond to
specific positions along the xaxis, as indicated in the figure.
h
i=hi/H20849xi,t/H20850refers to the height of the channel flow at the
position xi. The parameters hvof /H20849resp. hvenf,hventricle /H20850corre-
sponds to the minimal aperture of the vocal folds /H20849resp. the
ventricular folds, the ventricle /H20850. Note that in this study, hvof
always equals h1, whereas hvenfmay differ from h3, for spe-
cific geometric laryngeal configurations where the glottal jet-flow does not interact with the ventricular bands and de-
taches from the ventricular walls /H20849Bailly, 2009 /H20850.Avof /H20849resp.
Avenf/H20850refers to the glottal area /H20849resp. the ventricular area /H20850in
the axial plane x1=constant /H20849resp. x3=constant /H20850. In our the-
oretical approximation, these areas are rectangular: Avof=W
/H11003hvofandAvenf=W/H11003hvenf.Pi=Pi/H20849xi,t/H20850represents the rela-
tive pressure predicted at xi, as compared to the ambient
atmospheric pressure.
Three coupled subsystems are considered for modeling
the airflow dynamics through the larynx:
• the pressure drop across the glottis: /H9004Pvof=P0−Ps1;
• the emerging jet evolving in the ventricle, with a dissipa-
tion of kinetic energy: /H9004Pjet=Ps1−P2;
• the pressure drop across the ventricular folds: /H9004Pvenf=P2
−Ps3.
All theoretical aspects used in the followings directly
refer to this simplified model of phonation, applied under theassumptions of a semi-empirical Liljencrant’s flow separa-tion model, a “turbulent” jet-flow geometrical expansion inthe ventricle, dissipation /H9004P
jetbeing neglected, and a quasi-
steady Bernoulli flow dynamics description /H20851seeBailly et al.
/H208492008 /H20850andBailly /H208492009 /H20850for more details /H20852.
2. Simulation of the vocal fold dynamics in
interaction with the ventricular fold constriction
A distributed two-mass model /H20849M2M /H20850combining me-
chanical and airflow theoretical descriptions is used to simu-late the glottal behavior in time, through the predictions ofthe mass apertures h
vof1 /H20849t/H20850andhvof2 /H20849t/H20850, as illustrated in Fig.
4. For the sake of simplicity, the acoustical propagation in
the resonators downstream to the glottal source is not imple-mented in this study.
a. Geometrical parameters. For all simulations dis-
cussed below, we chose a laryngeal configuration consistentwith physiological measurements /H20849Hollien and Colton, 1969 ;
Wilson, 1976 ;Kitzing and Sonesson, 1967 ;Hirano et al. ,
1983 ;Agarwal et al. , 2003 ;Agarwal, 2004 /H20850:
FIG. 4. /H20849a/H20850Geometrical sketch of the larynx and relevant quantities of the aerodynamic study. /H20849b/H20850Sketch of the two-mass model of the vocal folds /H20849M2M /H20850
combined with the theoretical flow description of the ventricular fold influence and the ventricular geometry used in this study. Ps3=0.
J. Acoust. Soc. Am., Vol. 127, No. 5, May 2010 Bailly et al. : Vocal fold/ventricular fold periodic vibrations 3215h0= 20 mm, dvof= 4 mm, hventricle =2 3 m m ,
/H208495/H20850
Lventricle = 4.7 mm, dvenf= 5.5 mm.
The following two input values of the ventricular aperture
hvenf/H20849t/H20850are imposed, depending on the phonation investi-
gated.
•hvenf/H20849t/H20850=hventricle is considered for the study of normal pho-
nation with no implication of the ventricular folds in the
phonatory gesture.
•hvenf/H20849t/H20850=/H20855h˜venf/H20856/H20849t/H20850/H20851see Eq. /H208493/H20850/H20852is considered in the case of
vocal-ventricular periodic vibrations.
The initial glottal aperture hvof/H20849t=0/H20850=min /H20849hvof1/H20849t
=0/H20850,hvof2/H20849t=0/H20850/H20850is fixed to 0.2mm, so that the two-mass
model could simulate stable self-sustained glottal oscillations
with complete closure of the vocal folds in the absence ofventricular folds /H20851forh
venf/H20849t/H20850=hventricle /H20852.
b. Mechanical model The applied reduced mechanical
model is a variation in the symmetrical two-mass model pro-posed by Lous et al. /H208491998 /H20850, and further detailed in Ruty
/H208492007 /H20850andRuty et al. /H208492007 /H20850. It is controlled by a set of
mechanical parameters: the mass /H20849m
vof/H20850, spring stiffness
/H20849kvof,kcvof/H20850and damping /H20849rvof/H20850.
In the following, the simulated sequences correspond to
four ventricular cycles selected during vocal-ventricular pe-riodic vibrations. The value of m
vofis chosen in agreement
with previous studies on voice modeling using a two-massmodel of the vocal folds /H20849Miller et al. , 1988 ;Vilain, 2002 ;
Ruty, 2007 /H20850. The stiffness /H20849k
vof/H20850and damping /H20849rvof/H20850input
parameters are fitted so that the frequency of two consecutive
glottal cycles 1 //H20849t1+t2/H20850is identical to the measured one
/H20849mean value of 75 Hz /H20850with a relative error below 1%, when
the ventricular folds are vibrating. The coupling constant kcvofis arbitrarily set equal to 0.5. kvof. The mechanical parameters
are thus summarized below:
mvof= 0.05 g, kvof= 42.65 N m−1,
/H208496/H20850
rvof=1 0−3Nsm−1.
c. Airflow model. The laryngeal airflow is simulated as
previously described, with viscous losses considered in addi-tion along the glottal channel. The pressure P
0upstream of
the two-mass model is parametrically controlled as inputdata. For all simulations presented below, it is set to 900Pa,within the range of subglottal pressures commonly measuredin the trachea.
The pressure at glottal separation P
s1, usually set equal to
the atmospheric pressure /H20849Vilain, 2002 ;Ruty, 2007 ;Ruty
et al. , 2007 /H20850, is equal to P2, the pressure estimated in the
ventricle, by means of the fluid-flow theory before-mentioned. Thus, the simulated laryngeal flow accounts forthe pressure recovery owed to the ventricular folds.
The temperature is fixed at 37.2° and the atmospheric
pressure is set to 101.1kPa. Under such thermodynamic con-ditions,/H9267= 1.13 kg m−3,/H9262= 1.85 /H1100310−5kg m−1s−1,
c= 353.3 m s−1. /H208497/H20850
where /H9267is the air density, /H9262the dynamic viscosity, and cthe
sound celerity.
III. RESULTS
A. Physiological description of the co-oscillations
1. Description of ventricular fold vibration
Figure 5presents an illustrative kymographic compari-
son between the normal phonation and the phonation withvocal-ventricular periodic vibrations. Regarding this lattercase, different features can be highlighted, in line with pre-vious studies /H20849Fuks et al. , 1998 ;Lindestad et al. , 2001 ;
Sakakibara et al. , 2001 ,2002 ,2004 ;Henrich et al. , 2006 /H20850.
The ventricular folds are moving closer each glottal cycle,out of phase with glottal closing. A periodic contact of theventricular folds is observed for every two glottal cycles. Theperceived low pitch corresponds therefore to the fundamentalperiod of the ventricular vibration. It is interesting to notethat the singer does not feel the ventricular contact, whereashe mentions proprioceptive feelings during another growlyphonation characterized by a strong aryepiglottic constric-tion.
a. Contact recorded by the EGG measurements during
vocal-ventricular vibrations. Figure 6zooms on five glottal
cycles of vocal-ventricular phonation selected from thegrowly voice recordings. It shows both a kymographic plotof the laryngeal vibratory movement and the EGG andDEGG signals. Figure 7displays corresponding glottal and
ventricular aperture areas together with EGG and DEGG sig-
nals. Note that no detection of A
˜vof/H20849t/H20850is possible when the
ventricular fold motion hides the glottal aperture, which ex-
plains the incomplete signal plotted in Fig. 7. In this case, the
FIG. 5. Comparison between two kymographic views of the selected line
AB on the high-speed image during normal phonation and vocal-ventricularperiodic vibrations. Time scaling is different for both analyses. During nor-mal phonation /H20849up/H20850, the periodic contact of the vocal folds is illustrated each
glottal cycle. During the growly phonation /H20849down /H20850, the ventricular fold pe-
riodic motion is observed in the foreground, and superimposed to the vocalfold periodic vibration backward.
3216 J. Acoust. Soc. Am., Vol. 127, No. 5, May 2010 Bailly et al. : Vocal fold/ventricular fold periodic vibrationsventricular contact occurs all along the ventricular fold
length in the zdirection, so that A˜venfn/H20849t/H20850=0 at the end of
their closing phase.
A periodic alteration of both the EGG and DEGG signals
is observed during vocal-ventricular periodic vibrations /H20849see
Figs. 6and 7/H20850, illustrating a period-doubling pattern, in
agreement with previous observations /H20849Fuks et al. , 1998 ;
Lindestad et al. , 2001 ;Sakakibara et al. , 2001 ,2002 ,2004 ;
Henrich et al. , 2006 /H20850. What does this alteration reflect? Se-
lected instants in Figs. 6and7provide further insights into
this matter.
• Each time a DEGG positive peak of highest amplitude
occurs /H20851see instant A in Fig. 6, instants /H20849a/H20850and /H20849e/H20850in Fig.
7/H20852, the ventricular folds are coming closer to each other butnot touching, while the vocal folds are closing. In this case,
the amplitude variations on EGG and DEGG signals aredue to glottal motions only.
• Each time a DEGG positive peak of smallest amplitude
occurs /H20851see instant E in Fig. 6and instant /H20849c/H20850in Fig. 7/H20852, the
ventricular folds are already in contact. Data on vocal foldbehavior is thus not available. Yet, the observed suddenDEGG amplitude peak is not correlated with any suddenchanges in ventricular motion; in fact, the closing of theventricular folds precedes this instant and is not reflectedby a DEGG amplitude peak, although concurrent to amodification of the EGG signal. Therefore, this suggeststhat the DEGG smallest amplitude peak corresponds onlyto the closure of the vocal folds. These observations sup-port the thesis that though the EGG may combine informa-tion about ventricular and glottal contact areas, the DEGG
positive peaks exclusively reflect glottal closing states.Consequently, times t
1andt2/H20849see Fig. 7/H20850define two con-
secutive glottal cycle durations.
• When a DEGG negative peak arises /H20851see instants B and F
in Fig. 6, instants /H20849b/H20850and /H20849d/H20850in Fig. 7/H20852, the ventricular
folds are already apart from each other, whereas the vocalfolds are starting their opening phase. Thus, the DEGGnegative peaks occur at GOI, in the same way as duringnormal phonation. Consequently, times t
op1and top2 /H20849see
Fig. 7/H20850define two consecutive glottal open phase
durations.
b. Correlation between ventricular and glottal motions.
The ventricular fold contact follows the glottal opening /H20851time
B in Fig. 6and /H20849b/H20850in Fig. 7/H20852. The ventricular contact starts
during the glottal open phase /H20851time B to E in Fig. 6, time /H20849b/H20850
to/H20849c/H20850in Fig. 7/H20852. It stops during the closing phase of the
following glottal cycle /H20851time E to F in Fig. 6, time /H20849c/H20850to/H20849d/H20850
in Fig. 7/H20852. The ventricular opening is initiated at the time of
glottal closure. These observations do not depend on the spe-cific kymographic line considered.
FIG. 6. Zoom on five glottal cycles of a kymographic view, detailing three ventricular fold contacts, along with the synchronized and normalized EGG an d
DEGG signals. The selected kymographic line is represented on the high-speed image, perpendicular to the median axis. Shot instants are plotted as ve rtical
lines, and the corresponding laryngeal images are given on panels below /H20849A–F /H20850.
FIG. 7. /H20849Color online /H20850Normalized EGG /H20849EGGn/H20850and DEGG /H20849DEGGn/H20850sig-
nals as a function of time measured during growly phonation, along with
synchronized and normalized areas A˜venfn/H20849t/H20850/H20849up/H20850andA˜vofn/H20849t/H20850/H20849down /H20850derived
from the high-speed images. Shot instants /H20849a–e /H20850are plotted as dashed verti-
cal lines.
J. Acoust. Soc. Am., Vol. 127, No. 5, May 2010 Bailly et al. : Vocal fold/ventricular fold periodic vibrations 32172. Description of vocal fold vibration
In this part, the impact of the behavior of the ventricular
fold on the glottal motion during vocal-ventricular periodicvibrations is quantified.
a. Ventricular impact on glottal speed of contact
area. Figure 8displays the amplitude of the DEGG glottal
closing peaks detected on the growl sequence as a functionof time. The peaks are numbered in order of appearance: thecrosses correspond to even numbers, and the dots to oddnumbers /H20849see Fig. 2/H20850. Once the period-doubling phenomenon
is established, a constant alteration of the amplitude betweentwo consecutive peaks is periodically observed.
As illustrated in Figs. 6and7, the peaks of lower ampli-
tude on the DEGG signal /H20849even numbers /H20850occur during a
ventricular contact whereas the DEGG peaks of higher am-plitude occur at a moment when the ventricular folds havemoved apart. As the amplitude of the DEGG signal reflectsthe speed of vocal fold contact area, the observed periodicalteration of the glottal cycle suggests that the vocal foldspeed of contact is noticeably reduced under the influence ofthe downstream ventricular fold contact.
Note that at the very beginning of the sequence /H20849t
/H110210.5 s /H20850, the singer did not succeed in producing the inves-
tigated growl but performed a creaky-like sound instead. In
this case, the variations of the DEGG signal do not lead toany obvious period-doubling pattern.
b. Ventricular impact on the glottal cycle frequencies.
Not only is the speed of glottal contact modified by thedownstream ventricular vibration, but also the duration ofglottal cycle. Figure 9presents the evolution of glottal cycle
frequencies as a function of time. A distinction is made be-tween the glottal cycles, depending on whether the ventricu-lar folds close during the cycle /H20849f
1=1 /t1/H20850or open /H20849f2
=1 /t2/H20850. The frequency 2 //H20849t1+t2/H20850represents twice the fre-
quency of two consecutive glottal cycles. It corresponds to
the glottal fundamental frequency, which would have beenobserved if the ventricular folds were not vibrating. It alsocorresponds to an octave above the perceived pitch/H208492.f
0,f0=1 /t0being the acoustical fundamental frequency /H20850.
In normal phonation, f1=f2=f0.
During the creaky voice performed at the beginning of the
sequence /H20849t/H110210.5 s /H20850, the fundamental frequency f0is mea-
sured at 155 Hz /H20849D#3 /H20850on average. Once the periodic vocal-
ventricular vibrations have established, it decreases from 80Hz /H20849D#2 /H20850to 65 Hz /H20849C2/H20850/H20849mean value 72 Hz /H20850. During vocal-
ventricular periodic vibrations, two following glottal cyclesdo not have the same duration /H20849t
1/HS11005t2/H20850: a glottal cycle with
ventricular folds in closing phase is longer than a glottal
cycle with a ventricular aperture /H20849t1/H11022t2/H20850, implying a lower
glottal cycle frequency /H20849f1/H11021f2/H20850.f1goes from 149 Hz to 121
Hz with a mean value of 133 Hz /H20849C3/H20850;f2goes from 186 Hz
to 133 Hz with a mean value of 158 Hz /H20849D#3 /H20850. The maximal
difference between two consecutive glottal cycle frequenciesrises to 51 Hz and fluctuates around a mean value of 25 Hzin that case. It is interesting to note that the frequencies f
0,f1
and f2are superimposed at the beginning of the musical
sentence, which does not exhibit any period-doubling pat-tern.
Alteration of the glottal cycle duration during vocal-
ventricular periodic vibrations is confirmed in Fig. 10, which
shows the variation in glottal opening durations /H20849t
op1and
top2/H20850for two successive glottal cycles. It demonstrates that
top1is consistently longer that top2during growl; both dura-
tions also vary according to fundamental period t0fluctua-
tions. In the studied period-doubling sequence, this lengthen-ing may reach a maximal delay of 2.6 ms /H20849average value 1.5
ms/H20850. The duration relative to fundamental period, i.e., the
open quotient O
q, is also much higher in the case of ventricu-
lar fold closing than in case of ventricular fold opening /H20849rela-
tive gap of 11% in average, 23% at maximum /H20850. These fea-
tures are not observed during normal phonation, nor duringthe creaky sound produced at the start of the sentence.
FIG. 8. /H20849Color online /H20850Amplitude of DEGG signal at GCI as a function of
time.
FIG. 9. /H20849Color online /H20850Glottal-cycle frequencies as a function of time.
FIG. 10. /H20849Color online /H20850Open phase duration with /H20849top1/H20850and without /H20849top2/H20850a
ventricular fold contact, as a function of time.
3218 J. Acoust. Soc. Am., Vol. 127, No. 5, May 2010 Bailly et al. : Vocal fold/ventricular fold periodic vibrationsWhat could explain these variations in glottal opening du-
rations? The ventricular folds are closing during phase 1,whereas they are opening and remain completely open dur-ing phase 2, as displayed in Fig. 6. This suggests that the
duration of the glottal cycle could be extended with a ven-tricular contact. In Bailly et al. /H208492008 /H20850, it is shown that a
decreasing ratio h
˜venf /h˜vofcan imply a pressure recovery
downstream of the glottis, thus reducing the pressure dropand consequently the Bernoulli effect involved in the vocalfold adduction. In light of these previous results, we proposeto further explore the hypothesis of a possible aerodynamicinteraction between the glottal-flow and the ventricular foldvibration, which could explain the lengthening of the glottalopening duration.
B. Theoretical prediction of the vocal-ventricular
interaction
In this part, the aerodynamic influence of the ventricular
fold motion /H20849measured during period-doubling phonation /H20850on
the vocal fold self-sustained oscillations is predicted. Figures11and12present numerical simulations of the glottal behav-
ior obtained using a two-mass model and accounting for theaerodynamic interaction between the vocal folds and theventricular folds, as described in Sec. II C.
1. Configuration of reference with an input static
ventricular geometry
Glottal behavior during normal phonation is simulated
considering a standard geometry in the input parameters ofthe two-mass model, characterized by a static ventricularfold aperture such as h
venf /H20849t/H20850=hventricle .
Figure 11illustrates this latter configuration. No pres-
sure recovery is predicted in such a case /H20849P2/H20849t/H20850=0/H20850and the
two-mass model of the vocal folds oscillates periodically
with a single fundamental frequency 1 /t0ref /H20849mean value 173
Hz/H20850.2. Simulation of a period-doubling phenomenon at
the glottis
This part presents the glottal behavior and the transla-
ryngeal pressure predicted during vocal-ventricular periodicvibrations, in contrast to the situation shown in Fig. 11.
In this case, the input value of the ventricular aperture
h
venf/H20849t/H20850is estimated in agreement with the measurements re-
corded from explorations of this type of period-doubling
phonation in humans, detailed in Sec. II B. Thus, the param-
eter hvenf/H20849t/H20850follows the variations of the area A˜venfn/H20849t/H20850de-
tected on the high-speed images and plotted in Fig. 7, and is
approximated by /H20855h˜venf/H20856/H20849t/H20850. If coupled to Eqs. /H208492/H20850and /H208493/H20850, this
approximation yields to:
hvenf/H20849t/H20850=h˜venfref·A˜venfn/H20849t/H20850. /H208498/H20850
Finally, the input ventricular aperture hvenf/H20849t/H20850estimated by
Eq. /H208498/H20850has been dephased in concordance with the chosen
initial glottal aperture hvof/H20849t=0/H20850=0.2 mm. Figure 12pre-
sents the predicted glottal aperture min t/H20849hvof1,hvof2 /H20850and pres-
sure P2/H20849t/H20850, with the ventricular motion hvenfmeasured on
high-speed images as an input parameter.
Similarly to Eq. /H208498/H20850, the glottal aperture extracted from
the measurements is deduced from the equation below andcompared to the simulated M2M oscillations /H20849see Fig. 12/H20850:
/H20855h˜vof/H20856/H20849t/H20850=h˜vofref·A˜vofn/H20849t/H20850. /H208499/H20850
Under such conditions, the following three main features can
be deduced from the two-mass simulations.
• A large pressure recovery P2is predicted, in contrast to the
normal phonation simulation /H20849see bottom panels in Figs.
11and12/H20850. It even reaches the driving pressure P0each
time the ventricular folds are in contact. Therefore, thepressure drop across the vocal folds, /H9004P
vofis affected and
the vocal fold self-oscillation is altered.
• An alteration of the glottal aperture amplitude between two
consecutive vocal fold oscillations is theoretically pre-
FIG. 11. /H20849Color online /H20850Two-mass simulations of mint/H20849hvof1/H20849t/H20850,hvof2/H20849t/H20850/H20850and
P2/H20849t/H20850as a function of time. Configuration of reference with hvenf/H20849t/H20850
=hventricle ,P0=900 Pa. 1 /t0ref=173 Hz /H20849mean value /H20850. Input /H20849resp. output /H20850
data are displayed in thin solid /H20849resp. thick dotted /H20850line.
FIG. 12. /H20849Color online /H20850Two-mass simulations of mint/H20849hvof1/H20849t/H20850,hvof2/H20849t/H20850/H20850and
P2/H20849t/H20850as a function of time. hvenf/H20849t/H20850=/H20855h˜venf/H20856/H20849t/H20850,P0=900 Pa. 1 /t0=74 Hz
/H20849mean value /H20850. Input /H20849resp. output /H20850data are displayed in thin solid /H20849resp. thick
dotted /H20850line.
J. Acoust. Soc. Am., Vol. 127, No. 5, May 2010 Bailly et al. : Vocal fold/ventricular fold periodic vibrations 3219dicted. The rise of pressure P2/H20849t/H20850up to cancellation of the
pressure drop /H9004Pvofoccurs during the opening phase of the
glottal cycle characterized by a maximal amplitudemin
t/H20849hvof1,hvof2 /H20850. The ventricular fold contact thus gener-
ates an increase in the glottal aperture, according to the
simulations. Therefore, the modeling of the aerodynamic
vocal-ventricular interaction leads to the simulation of aperiod-doubling phenomenon at the glottis: a second fre-quency appears in the vocal fold vibratory pattern, whichequals to the fundamental frequency of the ventricular foldoscillation and of the resulting sound, 1 /t
0=74 Hz /H20849see
Fig.12/H20850.
• An anti-phase shift between the glottal and the ventricular
vibrations is also predicted. Figure 13compares the delay
between the simulated glottal apertures and the measureddata during vocal-ventricular periodic vibrations. To thisend, DEGG signal variations are compared to the time de-rivative of the simulated glottal apertures for the standardconfiguration /H20849as displayed in Fig. 11/H20850, and the case with
the ventricular motion extracted from the in vivo measure-
ments /H20849as displayed in Fig. 12/H20850. The peak detection method
allows to define the amplitude maxima of the signal−dmin
t/H20849hvof1/H20849t/H20850,hvof2/H20849t/H20850/H20850/dt, thus reached at simulated
GCI. The delay between two consecutive simulated GCI
defines the theoretical duration of the corresponding glottalcycle. Two main results are illustrated in Fig. 13regarding
the period-doubling phonation /H20849bottom panel /H20850.
• The simulation predicts an alteration of the
−dmin
t/H20849hvof1 /H20849t/H20850,hvof2 /H20849t/H20850/H20850/dtpositive peaks amplitude ev-
ery two glottal cycles; in other words, a noticeable de-
crease in speed of glottal closing is predicted, in agreementwith the measured data.
• The simulation predicts a difference of duration between
two consecutive glottal cycles. In the studied case, the dis-crepancy between two consecutive glottal cycle frequen-cies varies from 10 Hz to 46 Hz according to the simula-tion /H20849mean value 31 Hz during the four selected ventricular
cycles, if calculated from a simulated GCI peak of highestamplitude /H20850, while it varies from 39 Hz to 42 Hz according
to the DEGG measurements /H20849mean value 41 Hz /H20850.
These results suggest that the aerodynamic interaction
between the vocal folds and the ventricular folds does alterthe glottal vibrations in terms of frequency and amplitude inthe simulations, which may play an important role in theperiod-doubling phenomenon. Yet, it is shown in Fig. 13that
an inconsistent phase shift remains between the simulatedglottal motions and the measured DEGG signal, and thecycle duration lengthened by the ventricular closure, as ob-served in phonation /H20849see Fig. 9/H20850, is not reproduced under
such modeling assumptions.
IV. CONCLUSION
In this study, the laryngeal dynamics of a specific
growly phonation produced by a professional singer usingperiod-doubling as a musical performance is explored. Thisphonation is perceptually similar to Mongolian Kargyraa ,
Tibetan voice or Sardinian Bassu singing and involves the
vibrations of the ventricular folds. In vivo investigation of
this phonation has been carried out using acoustic and elec-troglottographic measurements, together with high-speedcinematography. A quantification of the laryngeal dynamicsis proposed, extracting glottal and ventricular areas from thehigh-speed images. These data are used as input parametersto a simplified model of phonation accounting for the aero-dynamic interaction between the vocal and ventricular folds.
The conclusions that can be drawn from this study are as
follows.
• It is observed that, although the EGG signal may combine
information about ventricular and glottal contact areas dur-ing this specific phonation, the DEGG signal exclusivelyreflects glottal vibratory behavior.
• A correlation between the vocal fold vibration and the ven-
tricular fold motion is demonstrated.
• As shown in high-speed images, the ventricular folds are
moving closer each glottal cycle, out of phase with glottalclosing. A contact between the ventricular folds is ob-served every two glottal cycles. The perceived low pitchcorresponds therefore to the fundamental period of theventricular vibration. The ventricular fold contact startsduring the glottal open phase and stops during the closingphase of the following glottal cycle. The ventricular open-ing corresponds to a glottal closure.
• From the processed in vivo data, it is shown that the ven-
tricular fold closing affects vocal fold movements. Twoconsecutive glottal cycles do not have the same durationand the duration of the glottal cycle is extended with aventricular contact. In particular, glottal opening durationis increased when the ventricular folds are closed.
• From the theoretical simulations, it is shown that the alter-
ation of the vocal fold vibration amplitude between twoconsecutive glottal cycles can be explained by the aerody-namic impact of the ventricular folds. Thus, the simulationexhibits a period-doubling phenomenon at the glottis. Inother words, extraction of the ventricular behavior ob-served during vocal-ventricular periodic vibrations, if com-
FIG. 13. /H20849Color online /H20850Comparison between the normalized DEGG signal
DEGGn/H20849top /H20850, the opposite of the time derivative signal of the glottal aper-
ture simulated by the M2M model dmint/H20849hvof1/H20849t/H20850,hvof2/H20849t/H20850/H20850/dtforhvenf/H20849t/H20850
=hventricle /H20849middle /H20850, and for hvenf/H20849t/H20850=/H20855h˜venf/H20856/H20849t/H20850/H20849bottom /H20850.P0=900 Pa.
3220 J. Acoust. Soc. Am., Vol. 127, No. 5, May 2010 Bailly et al. : Vocal fold/ventricular fold periodic vibrationsbined to a simplified aerodynamic modeling of the vocal-
ventricular interaction, suffices to predict the period-doubling phenomenon characteristic of this specificphonation.
• Finally, this work provides quantified in vivo information
on vocal fold and ventricular fold interaction recorded dur-ing period-doubling phonation on a professional singer,which is of much help for investigating the physical impactof the ventricular folds on the glottal oscillations. There-fore, further study is ongoing to characterize these physi-ological correlates on a greater number of subjects, and toovercome the main limitations of this study. In particular:
• A better understanding of the delay between the measure-
ments and the predictions of the glottal behavior may bealso provided if including an accurate and validated de-scription of the impact stresses occurring during vocalfolds collision in the modeling.
• A driven model of the ventricular band variation has been
chosen as a first step to determine their aerodynamic im-pact on the translaryngeal pressure distribution and theglottal vibrations. Yet, exploration of the ventricular mo-tion physical origins will need further modeling, such as atheoretical description of the glottal vibrations mechanicaltransmission along the laryngeal mucosa.
ACKNOWLEDGMENTS
This research has been partly supported by a Ph.D.
Grant from the French Ministry of Research and Education.The authors gratefully acknowledge Pierre Badin for his veryhelpful contribution to the contour detection method used toprocess the data. They also acknowledge Frank Müller, AnnaKatharina Licht, Markus Hess and Mal Webb for their pre-cious help during the experimental procedure. They wouldlike also to thank Peter Murphy and Joël Gilbert for theirsuggestions and participation on this work.
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5.0019024.pdf | Appl. Phys. Lett. 117, 152403 (2020); https://doi.org/10.1063/5.0019024 117, 152403
© 2020 Author(s).Current-controlled magnon propagation in
Pt/Y3Fe5O12 heterostructure
Cite as: Appl. Phys. Lett. 117, 152403 (2020); https://doi.org/10.1063/5.0019024
Submitted: 21 June 2020 . Accepted: 27 September 2020 . Published Online: 13 October 2020
Md Shamim Sarker ,
Hiroyasu Yamahara , and
Hitoshi Tabata
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Cite as: Appl. Phys. Lett. 117, 152403 (2020); doi: 10.1063/5.0019024
Submitted: 21 June 2020 .Accepted: 27 September 2020 .
Published Online: 13 October 2020
Md Shamim Sarker,
Hiroyasu Yamahara,a)
and Hitoshi Tabata
AFFILIATIONS
Department of Electrical Engineering and Information Systems, Graduate School of Engineering, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
a)Author to whom correspondence should be addressed: yamahara@bioxide.t.u-tokyo.ac.jp .Tel.:þ81358418846. Fax: 81358418846
ABSTRACT
We present a dynamic spin wave (SW) modulation technique using direct current (DC) to manipulate the magnetic properties of an
ultralow-damping Y 3Fe5O12thin film. The microwave excitation and detection technique with two coplanar waveguide antenna arrange-
ments on the Y 3Fe5O12(YIG) surface is used to characterize the SW. An additional platinum (Pt) stripe connected to a current source is inte-
grated between the coplanar waveguide pair to demonstrate the SW resonant frequency and amplitude modulation by current induction. Weselected a Pt stripe due to its significantly lower spin wave absorption property. The application of current through the Pt stripe generateslocal joule heating that modifies the magnetic properties of the YIG film. Temperature variation through local heating modifies the saturationmagnetization of the YIG film, which, in turn, modulates the SW frequency. Moreover, the amplitude of the SW spectra is found to be tuned
by the current amplitude. This phenomenon is mainly described by magnon–magnon scattering induced by the spin Seebeck effect in the
case of local heating. Furthermore, the group velocity of the proposed device is also found to be responsive to the current, which has beenexplained by both magnon–magnon and magnon-phonon scattering.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0019024
S p i nw a v e s( S W s )o f f e rd a t at r a n s m i s s i o na sw e l la sd a t ae n c o d i n g
capabilities in both amplitude and phase, providing an additional degree
of freedom
1–3compared to their conventional electronic counterparts.
Recently, they have attracted considerable interest due to their interest-
ing fundamental physics and potential technological applications such
as spin wave FETs,4,5magnonic logic gates,6,7spin wave magneto-
meters,8and reservoir computing.9The application of SWs in data proc-
essing requires active and robust control of the frequency and/or
amplitude, externally. There are several techniques to control a spin
wave, including strain,10electro-magnon,4voltage-controlled magnetic
anisotropy (VCMA),5spin–orbit torque (SOT),11and temperature12-
based techniques. This study deals with the temperature control of SW
propagation using current. Although current-induced Oersted fields
have been studied for controlling the propagating SW,13,14they can
affect the neighboring devices increasing their bit error rates.15Our pur-
pose is to use current for SW control by manipulating the magnetic
property of magnonic media through temperature redoing. SW modu-
lation by temperature control using a Peltier device has been previously
investigated,16but the centimeter-order device size limits on chip appli-
cation. For local temperature control, laser-induced heating to generate
a thermal landscape or localized heating area has been reported;12,17however, a bulky laser arrangement and difficulty in confining the laserlight restrict its efficacy for on chip-device application.
In this study, we investigate the control of SW propagation in
Y
3Fe5O12(YIG) thin films using current flow through a platinum (Pt)
film deposited on the YIG surface. We apply current-induced local heat-ing to control the SW behaviors associated with the magnetic propertiesof the YIG film. We observe the frequency shifts and amplitude reduc-tions of the transmitted SW depending on the magnitude of the current.In addition, the current-dependent group velocity is investigated.Compared to previous reports dealing with temperature control of
SW,
12,16this study presents a more convenient technique using simple
electric current, which can be extended to thermal mapping for morecomplex SW devices. Moreover, the differences in the physical mecha-nisms governing the case of local and uniform heating have beenexplored in this article, which are lacking in the existing literature.
An 80-nm-thick homogenous YIG film was grown on a single
crystalline (100)-oriented gadolinium gallium garnet (Gd
3Ga5O12,
GGG) substrate through pulsed laser deposition (PLD). The growthtemperature, oxygen pressure, and pulse rate were maintained at750
/C14C, 0.1 Pa, and 5 Hz, respectively, as an optimum environment.
An ArF-excimer laser at k¼193 nm was applied. The sample was
Appl. Phys. Lett. 117, 152403 (2020); doi: 10.1063/5.0019024 117, 152403-1
Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apladditionally annealed at 800/C14C for 3 h in air to improve the crystalline
quality. Subsequently, two 100 nm thick gold (Au) coplanar wave-guides (CPWs) were fabricated on the YIG film surface using electron
beam lithography and magnetron sputtering, and approximately 3 nm
thick chromium (Cr) was used as the adhesion layer. The width Wof
the individual lines (signal and ground) and their separations were
maintained at 10 lm. These dimensions were selected to match the
total impedance of the antenna with the input impedance of the vectornetwork analyzer (VNA). The separation lbetween the closest edges of
the signal lines was maintained at 145 lm. Two different types of devi-
ces were fabricated, as depicted in Figs. 1(a) and1(b). The first type
was a simple CPW SW device to serve as the reference. In the second
type, a 90 nm thick Pt stripe with a width of 85 lm was fabricated in
the SW propagation path between the CPWs. The two edges of the Ptstripe were connected to a current source through Au electrodes for
injecting current through the Pt stripe [ Fig. 1(c) ]. Pt was selected
because it has lesser SW suppression than the other generally usedmetals (e.g., Au, Al, and Ag where suppression is significant due to
their diamagnetism) when placed on the YIG surface.
18
For device characterization, we employed the microwave tech-
nique using the VNA. A DC-biased magnetic field ( l0Hext)w a s
applied in the in-plane direction and perpendicular to the SW propa-gation path, which aligns all the magnetic moments of the YIG film
along the magnetic field direction. On applying RF current to the
CPWs, an Oersted field is created around the signal wire, which exertsa torque on the magnetic moments beneath the CPWs on the YIG sur-
face. The torque is transferred to the nearest neighboring spin. To
investigate SW propagation, we followed a similar approach as in Ref.19. Initially, we obtained the FMR spectra for l
0Hext¼40 mT by
measuring the scattering parameters ( S11,S21). The plotted spectra
were obtained by subtracting the background signal ( S11andS21mea-
sured at l0Hext¼0m T )f r o m S11andS21measured at a bias magnetic
field of l0Hext¼40 mT. Here, S11andS21represent the reflection and
propagation of SW signals, respectively.
The real part of S11(Re(S11)) for l0Hext¼40 mT is shown in
Fig. 2(a) . Here, the solid black curve denotes the reference device FMR
spectra. The clear dip in Re( S11) indicates ferromagnetic resonance.
The solid blue line (SW device with a Pt stripe) displays a subtle shiftin the resonant frequency toward a higher-frequency region. This can
be attributed to the enhancement of the effective magnetization due to
the positive susceptibility of the paramagnetic Pt in the Pt/YIG bilayerdevice.
18The red line (FMR spectra upon application of 90-mA DC,
corresponding to a current density ( Jdc)o f1 2 /C2109Am/C02that is anorder of magnitude smaller than those of previous reports20)i n d i c a t e s
that current application reduces the resonant frequency. The transmis-
sion spectra as the real part of S21(Re(S21)) shown in Fig. 2(b) demon-
strate clear SW signal propagation between the CPW pair. The blackdotted line representing the real part of S
12for the reference device is
shown in Fig. 2(b) , which indicates higher insertion loss due to the
nonreciprocity of the surface SW. The center frequency of the S21
spectra exhibits a similar shift toward higher frequency in the Pt-
striped device, compared to the reference device. Moreover, an obviousshift in frequency between Re( S
11)a n dR e ( S21) spectra is observed.
This is because S11represents coherent excitation similar to the stan-
dard FMR and corresponding to wavevector k¼0, while propagating
spin waves correspond to k>0. According the dispersion relation
reported previously,21S21should always be at higher frequency than
S11. Moreover, after the connection of long Au electrodes for DC probe
connection, multiple peaks are observed in S21due to standing electro-
magnetic wave generation in the Au electrode, which worked as theantenna stub. According to previous reports, the stub antenna gener-ates multiple resonance
22in the electromagnetic wave, which is super-
imposed on the propagating SW in our device.
For detailed characterization, the DC density through the Pt stipe
was tuned from 0 to 25 /C2109Am/C02(in steps of 0, 4 /C2109,8/C2109,
12/C2109,1 6/C2109,2 1/C2109;and 25 /C2109Am/C02)a t l0Hext
¼30 mT; the corresponding response of the S11spectra is displayed in
Fig. 2(c) . It can be observed that the FMR frequency gradually shifts
lower (from 1.96 GHz to 1.75 GHz) with the increase in current. Thisbehavior cannot be attributed to the current-induced magnetic fieldbecause the response does not depend on the sign of J
dc.Moreover, the
magnitude of the Jdc-induced magnetic field for a similar structure was
predicted to be 60:03 mT (extracted in the middle of the YIG section)
forI¼650 mA, which is very small compared to the applied field.23
However, this phenomenon can be attributed to the current-induced
heating of the magnonic crystal. The DC flowing through the Pt layer
generates joule heating, increasing the YIG film temperature. Due to
this current-induced heating, the saturation magnetization ( Ms)o ff e r -
rimagnetic YIG reduces gradually and disappears at the N /C19eel tempera-
ture (559 K for YIG). However, the relation between the temperature
(T)a n d Msis almost linear from the cryogenic temperature to approx-
imately 450 K,24which can be modeled as12
MsTðÞ/C25Ms;RT/C0gT/C0TRT ðÞ ; (1)
where Ms;RTis the room-temperature saturation magnetization (140
kAm/C01)a n d gis the fitting parameter (313 A K/C01m/C01). In support of
FIG. 1. Optical images of the (a) reference SW device and (b) SW device with a dynamic control arrangement. (c) SW schematic diagram. Microwave signals from th e VNA
excite the SW, while the DC manipulates the magnetic properties of YIG by increasing its temperature and tuning the SW frequency.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 117, 152403 (2020); doi: 10.1063/5.0019024 117, 152403-2
Published under license by AIP Publishingthis linear assumption, we have presented the temperature-dependent
saturation magnetization of the YIG film by the superconducting
quantum interference measurement (SQUID) shown insupplementary material Fig. S1. The saturation magnetization is
related to the effective magnetization ( M
eff) and anisotropic field
(Hani)a sf o l l o w s : Meff¼Ms/C0Hani.A s Msand Meffdecrease with
increasing temperature, the FMR frequency reduces according to the
Kittle equation: fFMR¼c
2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
HextHextþMeff ðÞp
.Figure 2(d) depicts the
shift in the resonant frequency with the current; here, the filled circles
denoting the experimental data and the dotted line denoting the
theoretical prediction are in good agreement. To model the theoreticalprediction, we integrated the temperature-dependent saturation mag-
netization into the Kittel equation,
f
FMR¼c
2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
HextHextþMs;RT/C0gT/C0TRT ðÞ /C0Hani/C0/C1q
:(2)
Using this equation, we can express the temperature-dependent
fFMRshift as DfFMR T/C0TRT ðÞ /C25DfFMRDTðÞ¼fFMR TðÞ/C0fFMR T¼TRT ðÞ .
To quantify the current-dependent temperature change, we measured
the current-dependent resistance ( RIðÞ)o ft h eP tl a y e rs h o w ni n Fig.
2(e)and found that it quadratically increases with the applied current,
defined by RIðÞ¼R0þR2I2,w h e r e R2¼520XA/C02is the fitting parame-
ter and R0¼50Xis the total resistance of the Pt layer at room temper-
ature. Considering the thermal conductivity of YIG to be 5W/m/K
and calculating it for Pt using the Wiedemann–Franz relation, N.Thiery et al. analytically defined the current-induced temperature
change as DT¼j
PtðRIðÞ/C0R0Þ=R0,w h e r e jPt¼254K is specific to Pt.20
Figure 2(f) represents the model extracted DTas a function of I2:The
temperature increase in the Pt layer is estimated to be 38 60:4K for acurrent density of 16 /C2109A/m2, which according to Eq. (2)reduces
fFMRby 82MHz, which is in very good agreement with the experimen-
tal observation (79MHz). The demonstrated dynamic frequency tun-ing of the microwave with the current can be used for filteringapplications in the microwave range.
Furthermore, in Fig. 2(b) , the reference device displays a trans-
mitted signal with a higher amplitude and broader frequency rangecompared to the Pt-striped device. This is because spin pumping fromthe ferrimagnetic YIG to the Pt layer causes signal absorption, result-ing in smaller amplitude and narrower transmission in the Pt/YIGhybrid device.
25,26Spin pumping from YIG to Pt introduces additional
damping in YIG, reducing the transmission spectra amplitude.
Moreover, the introduction of current through the Pt stripe results in
additional damping. For rigorous analysis, the current-dependenttransmitted spin wave amplitude is plotted in Figs. 3(a)–3(d) .H e r e ,
theS
21spectra for current densities Jdc¼0, 12/C2109,1 6/C2109;and
21/C2109Am/C02are displayed in Figs. 3(a)–3(d) , respectively. A con-
tinuous shift in the resonant frequency with the decrease in the ampli-tude of the SW spectra can be observed with the increasing current.The relationship between the transmitted SW amplitude and current
density is depicted in Fig. 3(e) , which implies the enhancement of the
Gilbert damping constant with the increase in current, similar to a pre-vious report for the Pt/YIG structure.
23This damping enhancement
can be attributed to the nonlinear magnon–magnon scatteringbetween the transmitted SW and spin Seebeck effect (SSE)-inducedSW modes. Local surface heating may cause space distribution of thetemperature, which may result in an additional spin wave mode due to
the spin Seebeck effect (SSE).
27,28These SW modes will interact with
transmitted SW and will reduce the amplitude of the transmitted SW
FIG. 2. Real part of the (a) S11and (b) S21spectra of the reference device (black) and the Pt/YIG bilayer device without (blue) and with (red) a current density of
12/C2109Am/C02through the Pt layer at a bias magnetic field of 40 mT. The black dotted curve in (b) represents S12spectra of the reference device. (c) S11spectra for different
levels of Jdcthrough the Pt layer. Data are recorded at a bias magnetic field of 30 mT. (d) Jdcdependence of the SW resonant frequency shift. Injected current I2-dependent
(e) Pt stripe resistance and (f) local temperature increase.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 117, 152403 (2020); doi: 10.1063/5.0019024 117, 152403-3
Published under license by AIP Publishingspectra. To clarify the contribution of the SSE, we performed an exper-
iment wherein the SW was characterized under uniform heating
instead of local heating. The Pt/YIG sample was placed on a Peltier
device that was 1 cm2in size (much larger than the SW device), and
the temperature was raised from 24/C14Ct o7 9/C14C. The variation of
transmission spectra under this uniform heating can be seen in supple-
mentary material Fig. S2. It has been observed that uniform heating
has shifted the transmission spectra toward the lower frequency band
in a similar fashion like the local heating case. However, no significant
reduction in the SW amplitude was detected in the case of uniform
heating. Less than 1% reduction in SW amplitude was noted for a 55 K
temperature variation in the case of uniform heating, while a similar
amount of temperature variation resulted in a 33% intensity reduction
in the case of local heating. This result clearly suggests that the reduc-
tion in transmitted SW amplitude mainly originates from the local
heating-mediated SSE-induced magnon scattering with the transmit-
ted SW mode. Moreover, minor (less than 1%) reduction in the case
of global heating may arise from the magnon–phonon scattering at
elevated temperatures. Furthermore, magnetization distribution due tothe current through the Pt layer may generate multiple SW modes
29in
the YIG film. Interaction among these SW modes has the potential to
introduce magnon–magnon scattering, which can enhance damping in
the YIG film. Another possibility is the magnon annihilation due to the
reverse polarized spin current into YIG from the Pt layer by the spin
Hall effect (SHE). This may introduce additional damping torque that
can contribute to the amplitude reduction of the transmitted SW sig-
nal.30However, this argument requires the counter effect. When the
sign of applied current is reversed, it should add anti-damping torque,
increasing the transmitted SW amplitude. However, in the conducted
current reversal experiment, such a damping switching phenomenon
was not observed. Hence, we can overlook the magnon injection phe-
nomenon and can conclude that the S21amplitude modulation mostly
results from the magnon–magnon scattering due to the SSE.Furthermore, we calculated the current-dependent group velocity
vg¼Df/C2lfollowing a well-established procedure.21Here,Dfis the
frequency difference between two oscillating maxima located at the
center in the highest intensity mode of Im( S21)s h o w ni n Fig. 3(g) .
Figures 3(g)–3(k) show that Dfreduces gradually upon increasing Jdc
from 0 to 25 /C2109Am/C02. As a result, the SW group velocity also
exhibits a gradual reduction from almost 5.2 km/s at Jdc¼0A m/C02to
3.0 km/s at Jdc¼25/C2109Am/C02, which is shown by the solid black
circle in Fig. 3(f) forl0Hext¼30 mT. To confirm the magnetic field
dependency of this behavior, the current-dependent group velocity of
as i m i l a rd e v i c ef o r l0Hext¼50 mT was calculated [shown in the solid
red rectangle of Fig. 3(f) ]. A similar trend of group velocity reduction
was found in our observation [(Im( S21)f o rab i a sfi e l do f
l0Hext¼50 mT, as shown in supplementary material Fig. S3].
However, the reduction of group velocity in a higher magnetic field is
consistent with previous reports.19,21To identify the origin of vgreduc-
tion, Im( S21) was investigated for a uniformly heated SW device by a
Peltier device. The experimental data ( supplementary material Fig. S4)
revealed a reduction of Dfby 4.5 MHz with the temperature change in
the range of 24/C14C–70/C14C corresponding to the group velocity reduc-
tion by 0.65 km/s. Since it is expected that the SSE should be very small
in the case of uniform heating, this contribution is a result of the
magnon-phonon interaction. Hence, in the case of local heating, both
magnon–magnon and magnon–phonon scattering play crucial roles
in group velocity reduction. Further observation shows that the vg
reduction rate is slower for smaller current ( Jdc<12/C2109Am/C02)
and faster for higher current density ( Jdc>12/C2109Am/C02).
Joule heating modulation of SW propagation has been reported
in a previous work; however, the contribution of the SSE has not beendiscussed thus far.
23Our experiments demonstrated that the SSE is the
major player for the amplitude reduction of propagating SWs. As an
additional feature, we also investigated the current-dependent groupvelocity, which can be used to design dynamic microwave delay lines.
FIG. 3. Re ( S21) spectra of the Pt/YIG bilayer device for (a) Jdc¼0, (b) Jdc¼12/C2109, (c) Jdc¼16/C2109;and (d) Jdc¼21/C2109Am/C02through the Pt layer. (e) SW
intensity vs Jdcatl0Hext¼30 mT. (f) Jdc-dependent group velocity vg, where the black circle represents vgfor a bias field of 30 mT, whereas the red square is for a bias field
of 50 mT. Imaginary part of S21(Im(S21)) for (g) Jdc¼0 A/m2, (h) Jdc¼12/C2109A/m2, (i)Jdc¼16/C2109A/m2, (j)Jdc¼21/C2109A/m2, and (k) Jdc¼25/C2109A/m2.Df
is observed to gradually reduce with increasing Jdc.Applied Physics Letters ARTICLE scitation.org/journal/apl
Appl. Phys. Lett. 117, 152403 (2020); doi: 10.1063/5.0019024 117, 152403-4
Published under license by AIP PublishingComparing the other methods (such as strain,10electro-magnon,4
VCMA,5and SOT11-based methods), this report proposes a diverse
range of parameter (frequency, intensity, and group velocity) modula-
tion using current. Furthermore, the use of CPWs instead of themicrostrip line antenna for SW excitation and detection is another dis-tinguishable feature compared to other reports on thermal modula-
tion.
12,23By tuning the CPW dimensions, one can control the
wavelength of propagating SWs.21Another prospect of this study can
be the implementation of magnonic transistors by controlling the SHEphenomenon, where spin polarization reversal by reversing the currentdirection can be used to generate two opposing effects (enhancement
and depletion) on SW propagation.
In summary, this study investigated the current-dependent reso-
nant frequency shift of the transmitted SW spectra in an ultralow-damped ferrimagnetic YIG film. A CPW-based reference SW devicewas fabricated for comparison. In the experiments, a systematic shift
in the resonant frequency from 1.96 GHz to 1.75 GHz was observed,
which can be attributed to the current-induced temperature changeand the corresponding change in the saturation magnetization of theYIG film. Moreover, a reduction in the transmitted SW intensity wasobserved with the increase in current, which can be attributed to mag-
non–magnon interactions between transmitted SW and SSE-induced
SW modes. In addition, we calculated the current-dependent SWgroup velocity, which was also explained in terms of magnon–phononand magnon–magnon scattering.
See the supplementary material for the temperature-dependent
saturation magnetization change and the difference in scattering
parameters due to local and uniform heating.
A portion of this work was conducted at the Advanced
Characterization Nanotechnology Platform, The University of Tokyo,
supported by the “Nanotechnology Platform” of the Ministry ofEducation, Culture, Sports, Science, and Technology (MEXT), Japan.This work was supported by JSPS KAKENHI Grant Nos. “16K21001”
and “19K15022.” Md Shamim Sarker expresses his heartfelt gratitude
to the “Yoshida Scholarship Foundation” and the “NakataniFoundation for advancement of measuring technology in biomedicalengineering” for supporting him.
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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Published under license by AIP Publishing |
1.3027187.pdf | RF Electrodynamics in Small Particles of
Oxides—^A Review
V.V. Srinivasu
National Centre for Nanostructured Materials, Council for Scientific and Industrial Research (CSIR)
PO Box 395, Pretoria 0001, South Africa
Abstract. RF electrodynamics, particularly, the low field rf absorption in small
superconducting and manganite particles is reviewed and compared with their respective bulk
counterparts. Experimental and theoretical aspects of the small particle electrodynamical
response which is qualitatively different as compared to its bulk form, atleast in the high-Tc
superconducting YBa2Cu307.x (YBCO) cuprate superconductor and in the CMR manganite
family members are discussed. We focus attention on fascinating new phenomena like rf power
induced 'anomalous rf absorption' in the small particle system of YBCO and room temperature
'Colossal Magneto Impedance' (CMI) in the micron size particles of manganites occurs,
illustrating rich physics and paving way for new device applications.
Keywords: Small Particle Electrodynamics, RF properties. Colossal Magneto Impedance, CMR
manganites, High-Tc Cuprate Superconductors.
PACS: 74.78.Na, 74.25.Nf 75.50.Tt.
INTRODUCTION
Functional Oxide materials such as High-Tc cuprates and CMR manganites have
interesting dc electrical and magnetic properties which paved way for the emergence
of the exciting field now known as 'Oxide Electronics'. However these materials also
have very interesting 'bulk' and 'small particle' zero field and low field rf properties,
which can be used in novel RF devices. While a few reviews and review-like long original
articles are available for the rf properties of the bulk form of these materials [1-5], we believe
that no review is available in the area of the small particle rf range electrodynamics in these
materials and this article shall fill the gap to some extent.
(a) High-Tc Cuprate Superconducting Materials
High-Tc cuprate superconducting materials (HTSC) in the bulk form (thin films,
single crystals and polycrystalline pellets) show an intense low field dependent RF
absorption signals, generally known as 'Non-resonant RF and microwave absorption
(NRMA) [6-13]. This absortion can originate from fraction of free fluxons and weak
links [ 14-16]. Further many novel features are observed in the bulk form of these
cuprates, such as 'anomalous hysteresis' [17], 'Temperature dependent phase reversal'
[18,19], 'Oscillations in the line spectra' [20], 'periodic fine structure' particularly in
single crystals [21], 'Paramagnetic Meissner effect' [22] and microwave power
induced evolution of the second peak in Bi-2212 single crystals [23], to name the
CPl 063, Me505cop/c, Nanoscopic, and Macroscopic Materials, edited by S. M. Bose, S. N. Behera, andB. K. Roul
© 2008 American Institute of Physics 978-0-7354-0593-6/08/$23.00
411 most important features here. However, in this article we focus on the microwave
power induced 'anomalous rf absorption' in the micron size YBCO powder samples,
which is not observed in the bulk form [24].
(b)Manganites
Manganites have become famous in the scientific community because of their
Colossal Magneto Resistance (CMR) property, which is a dc property. However,
manganites in their bulk form, have shown impressive rf magneto impedance (Ml)
properties too [4,5,25-34]. For example, the best thin films of La2/3Bai/3Mn03 by laser
ablation technique and carefully grown ceramic samples have shown impressive
magneto impedance properties at microwave frequencies. In this case a figure of merit
(defined as f = [P(H)-P(0)]/P(0), where P(H) is the power absorption at field H as
high as 0.15 (corresponds to 15% Ml) in a field as low as 200 Oe was achieved at
room temperature [26]. Further in good quality single crystals of Lao vSro sMnOs, a
figure of merit as large as f~0.5 (or 50% Ml) in an apphed field of just 300 Oe was
achieved by exploiting the so called ferromagnetic antiresonance (FMAR) [27,28] .
However these single crystals and thin films needs stringent preparation conditions. In
ceramic samples the magneto Impedance (Ml) effect is significantly smaller,
especially at room temperature [5]. In the micron size powders of manganites, a real
'Colossal Magneto Impedance' (CMl) effect upto 80% (f = 0.8) in the applied fields
of a few hundred Oe and at room temperature was reported [35,36]. In this review, we
bring into focus spectacular magneto impedance properties at microwave frequencies
shown by the micron size manganite powders and which are not observable in the bulk
form.
Thus after a brief introduction of the bulk rf properties of HTSC and manganites as
given above, we shall now discuss in detail the rf properties of the micron size YBCO
and manganite powders, which is our main focus in the following sections.
RF Power Dependent 'Anomalous' NRMA in Micron Size
YBCO Powders
Before we start to discuss the special features that have been observed in the micron
size YBCO powders rf absorption studies, we have to clarify the two types of non-
resonant (low field) rf absorption, namely, 'normal' and 'anomalous' Non-resonant rf
Absorption in High-Tc Superconductors (NRMA) that is observed in the HTSC. In the
case of normal NRMA, the rf absorption is always a minimum at zero applied field
and it increases as the field is increased and saturates at some characteristic high field
[14, 19]. This characteristic field at which the rf absorption saturates depends on either
the weak hnk/Josephson junction dimension or the viscosity coefficient of the fluxons
that are driven by the rf currents [14]. In the case of anomalous NRMA, the rf
absorption has a maximum at zero applied field and shall decrease with increasing
magnetic field [18,19]. In fact one would observe a cross over from 'normal' to
'anomalous' NRMA rf absorption as the temperature is increased from a temperature
well below Tc of the sample to a temperature very close to Tc in HTSC bulk. This
phenomenon is well explained in terms of meissner fraction dependence on the
412 temperature and field and treating the bulk granular HTSC sample as an 'effective
medium' with effective conductivity and permeability expressed in terms of the
meissner fraction. The cross over from 'normal' to 'anomalous' rf absorption also
depends on the normal state conductivity of the sample and the rf frequency too. A
model developed by Bhat-Srinivasu-Kumar [19] explains all these features naturally.
In the case of micron size YBCO powders, a new feature as compared to bulk form
is observed, namely, rf power dependent cross over from 'normal' to 'anomalous'
NRMA signals. Such a behavior is totally new and only has been observed in the
micron size powders of YBCO [24]. In Fig.I(a) (reproduced from ref [24]) one can
see that as the rf field value is increased from about 4 milh gauss to 16 milli gauss, the
phase of the NRMA signal reversed as compared to that of the 'H NMR signal. Here
the in phase and out of phase signals corresponds to the 'anomalous' and the 'normal'
NRMA signals respectively. One can also see the 'transition' signals with structure, at
intermediate rf power levels. Full experimental details can be found in ref [24].
In the model developed by Srinivasu-Bhat-Kumar [24] the 'normal' and
'anomalous' NRMA rf absorption and the cross over can be explained in the following
way:
'Normal' NRMA absorption
The micron size YBCO powder consists of many weakly connected loops. For the
situation where the parameter (3 = 27i LgJc/cpo > 1 (where cpo is the flux quantum, Lg is
the geometric loop self inductance, Jc is the junction critical current density.) the
energy of the loop becomes a multi valued function, leading to an irreversible
hysteretic Josephson-Junction characteristic.
Thus, for the rf amplitude beyond a threshold value, the YBCO powder system
makes transitions between flux states, as the dc flux is ramped. The associated phase-
slippage induces impulsive emf causing ohmic dissipation, through the junction
normal resistance [24, 40]. This response when averaged over a distribution of loop
area and orientation shall give an rf absorption that increases with increasing dc field
(Hdc) [3].
'Anomolous' NRMA absorption
The rf field induced anomalous NRMA absorption occurs only in micron size
powder samples of YBCO and is experimentally absent in well sintered bulk samples.
This means the weak links involved are expected to be mostly the Josephson junctions
between the micron-sized irregularly shaped particles of the powder. These weakly-
hnked superconducting loops can be non-hysteretic, i.e. (3 = 27i LgJc/cpo <1 as the
critical current in these inter particle contact weak links is expected to be very small.
Further as the Hd (junction) « Hd (bulk), these junctions are well suited for
nucleating fluxons and which are weakly pinned.
413 FIGURE 1. (a) Derivative non-resonant rf absorption signals at various Rf fields recorded from a
micron size YBCO powder at 77K. The inset shows 'H NMR signal recorded using the
samespectrometerNote that the non-resonant signal at high rf field levels has the same phase as the
Resonant 'H NMR signal, indicating an 'anomalous' rf absorption as discussed in text, (b) Plot of
simulated power absorbed as a function of Hjc according to the equation (1) and its derivative . This
corresponds to the experimentally observed anomalous response at high rf fields. Reprinted with
permission from Solid State Communications, ref [24], copy right [1994], Solid State Communications.
And note that the volume fraction represented by these junctions is small and the
junction penetration depth is large, one would have a situation where each junction
acquiring a single fluxon, independent of field scan Hdc> 0. The damped motion of
these junction bound fluxons which are driven by the rf currents shall cause
dissipation in the sample. By considering the loop super current and the usual Bessel
function expansion, Srinivasu-Bhat-Kumar [24] have obtained an expression for the
power dissipation (P) as
P = -P„cos(—^//,J JS—AHJ-J\{^AHJ (1)
Where Po is a scale factor, Hdc and Ho, are the dc and rf fields respectively. A is the
loop area and other symbols having their usual meaning. It shows an initial
cosinusoidal decrease of absorption with Hdc for ATT n An
JS—AHJ<J\{—AHJ
This is true for the entire range of Ho, used. Now averaging P over the distribution (=
Ae"^^°) of loop area A and as Hdc » Ho, , simply averaging the rapidly varying
414 cosine factor and replace AHm by its average value in the argument of Bessel function,
Eq. 1 takes the form:
l-16;r'?7' ^ ^ ,4;E4/f^
l + 16;r'77' l-( /.)'^.(—T^) l + 16;r'77' (^„ (2)
/f A Where r\ = —^^^ Figure 1(b) shows the simulation of Eq. 2 (reproduced from
(^»
ref [24]). It can be seen that the power dissipation <P> is a monotonically decreasing
function of Hdc, which is the 'anomalous' NRMA. A cross over to the 'normal'
NRMA at lower rf levels occurs because of the hysteretic-rf-squid behavior as
discussed above.
There is another possible explanation, which is based entirely on the conventional
rf SQUID response. The reduction of Jc with field can make a fraction of the junctions
non-hysteretic (non-dissipative). This hysteretic to non-hysteretic conversion of the
junctions can also lead to the 'anomalous' NRMA. However if this mechanism is
playing a role, one expects this to happen in the bulk granular HTSC samples too. But
in the bulk granular YBCO the rf power induced anomalous NRMA is experimentally
proved to be absent [24]. We have to note that in an rf SQUID response, only
hysteretic junction loops are multivalued in their energies and flux states, leading to
phase slippage and dissipation. Non-hysteretic junction loops can not give any
dissipation in this picture. Thus with a strong experimental proof that the anomalous
NRMA is absent in bulk YBCO and with the above argument, it can be ruled out that
the 'anomalous' NRMA to be originating from the rf-SQUID response. So the only
dissipative mechanism is due to the above model of Srinivasu-Bhat-Kumar, where
weakly pinned single fluxons in the non hysteretic junctions giving rise to the
'anomalous' low field rf absorption (NRMA), from Eq.2. The non-hysteretic junction
nature is ensured because of the very weak inter particle contacts in the powder.
Microwave Absorption and Colossal Magneto Impedance (CMI) in
the Micron size Manganite powders
Infact there are only a few reports on the rf impedance measurements in manganite
small particle powder samples [35-39] as compared to that of bulk. In the nanometric
manganites, Nath et al [ 39] have shown that the MI percent increases with nanometric
grain size. In the micron size Lai-x SrxMnOs powders, Li et al [37,38] found that
microwave loss peak corresponds to the maximum dielectric loss tangent tan 5e near
10.5 GHz, while the measurements being carried out by network analyzers. In this
type of measurements it is difficult to separate the loss contributions from Crf and hrf
fields.
415 Table.1 Powder characteristics and Half-point fields (Ho) "Reprinted with permission from AIP,
ref.[36]. Copyright [1999], American Institute of Physics."
Sampk (dejigna.tirn)
LfluTUiiti.jMiiOj (LBMOJ
Ij^,.,Sr(| iMmO, (LSMOl
Ui,Ca,vtMrtO, LLCMO (j-aj)]
LaspCsiftiMnO, [LCMO U=0,2}]
LOfl jCaojMnOj [LCHO (jr=0,5))
LauCa.^MiiiCh [LCMO ID)]
NdiKTSrr^^flnO.i (NSMO^
Ni-25^01
CiJCr.Sej
Ic
Ni Powder
^'iis
f/iin)
J
3
3
3
3
J
3
3
3
50
3 TciK)
335*J
360±3
260*3
1«5±3
333:1:3
255 ±3
215*3
372
\zm 1043
«33 Tempcrsitun:
(K)
305
295
TS250
395
240
J60
IS5
150
24«
711
205
395
305 W<,
(Oe)
6U0
250
23O0
SOO
900
^()tl
2300
3[(»
1350
ISOO
iSOO
Z5O0
2000
However, in the cavity perturbation technique, as adopted by Srinivasu et al [35,
36] it is possible to completely separate the microwave loss contributions from Crf and
hrf fields. These measurements showed a qualitative and as well as quantitative
discriminative behavior in the rf-electrodynamics, for situations when the sample is
subjected to Crf and hrf fields separately.
The microwave absorption in the micron size manganite powders whose details are
given in Table.1 (reproduced from ref [36]) shows pecuhar temperature and field
dependences, which shall be discussed in two separate sections below.
(a) Temperature dependence:
The microwave absorption in the micron size manganite powders was found to be
qualitatively different from that of their bulk form of the material [35, 36]. Most
importantly the micron size manganites discriminate between the Crf and the hrf fields.
For example when the powder sample is placed in the location at maximal hrf in the
cavity, the microwave absorption data as function of temperature (reproduced from
ref [36]) is shown in Fig.2(a). One can see a clear monotonic rise of absorption as the
sample is cooled down below a characteristic temperature T , which is observed to be
slightly different from that of Tc. Almost all of the manganite family members show
this behavior and to date no exemption is found. This characteristic behavior should be
416 compared with the situation when the sample is located in the Crf. Data (reproduced
from ref [36] is shown in Fig. 2(b). Here in this case the absorption seems to follow
the resistivity. Thus the micron size powders of the manganite family shows
qualitatively different behavior as compared to their bulk form. The pecuhar hrf
behavior, namely a rise of microwave loss as the sample is cooled down, was
explained by Srinivasu et al [35] as in the following.
As the low-T rise in the microwave loss occurs only when the powder sample is in
hrf field and not when it is in Crf field, the microwave loss is purely a magnetic effect.
In as much as the loss is essentially attributed to the joule heating arising due to the
rapidly varying rf flux, the loss is then proportional to ||i^|, where \i is the dynamic
permeability.
ju-l = ATTX •• 47M[H + 47M+iT]
{H + iT\H + 47iM + iT]-(3)
This is the Gilbert's form of dynamical permeability for the spin system. M is the
magnetization and T the resonance width parameter. Other symbols have their usual
meanings. These powders are found to be magnetically inhomogeneous as their FMR
hue widths are very large (~ kOe). Therefore it is possible that a vestige of resonance
absorption remains even at zero field. Assuming that hue width increases rapidly as M
increases when the temperature is decreased, which is the case actually observed
JO
20
10 +
+
+
+ ^, ,<^ „
•
• V
9.9 CHt
i LCMCT
+ LCMQvOI
Obww p2
150 200 25(1
Temperamre (K) 300 'I 'if
nLa(0
OLCMO(JfeO,3)
^ \^^J0^
100 200 3()0 400
r(K)
FIGURE 2. Temperature dependence of microwave absorption in the micron size manganite powders
when placed in (a) hrf field and (b) Crf field. The behavior in the two cases is qualitatively different as
discussed in the text. "Reprinted with permission from AIP, ref. [36] Copyright [1999], American
Institute of Physics."
417 experimentally, and with a simple relation T = [200 Oe + 0.3 coM(T)], Eq.3, with
credible values for T and the magnetization M(T), can qualitatively simulate the
temperature dependence of the microwave loss [35]. It is interesting to note that the
above explanation with the aid of Eq.3 has resonance effects inbuilt in it. It would be
very interesting to perform an experiment at MHz range of frequencies, far away from
any kind of ferromagnetic resonance (that occurs at microwave frequencies mentioned
in ref [35]) and to see how the temperature dependence of rf loss would look like, i.e.,
to see whether we still see the low-T rise of rf loss as shown in Fig.2(a).
(b) Colossal Magneto Impedance:
Once well below T even small applied field of a few hundred Oe was able to
quench the microwave absorption up to ~ 80% , leading to Colossal Magneto
Impedance in these micron size manganite powders. Here a few crucial features have
to be noted which are shown in Fig. 3 (a) (reproduced from ref[36]). (i). Compared to
single crystals, polycrystalhne samples show much larger magneto impedance (MI)
effect, (ii) Most importantly this impressive magneto-impedance effect occurs only
when the sample is placed in the maximal hrf field position in the cavity and no
measurable magneto-impedance effect was observed when the same sample is placed
in the Crf field of the cavity. Which means this MI has to do something with the spin
system rather than the charge system, (iii) and this is significant because the MI effect
is almost 80% in just a few hundred Oe of applied field and at room temperature.
Since the Mi effect is really huge and in very small fields, one can call it 'Colossal
Magneto Impedance' (CMI) effect. This effect in manganites is much larger than the
conventional small particles of Fe and Ni. As a typical example, we show the data of
micron size LBMO, Fe and Ni at ambient temperature in Fig.3 (b) (reproduced from
Ref [36]). One can see the MI effect is so spectacular in the case of micron size
LBMO as compared to the conventional Fe and Ni particles.
It was demonstrated that the figure of merit data collected for almost all the family
members of the manganites at various temperature follows an empirical function G(x)
= x^/(I+x^) and shows a good universality. Here x = H/Ho, Ho being the half point
field or the field at which figure of merit f = 0.5. Experimental data of figure of merit
for various samples plotted as a function of G(x) shows clear universality as shown in
Fig. 4 (reproduced from ref [36]).
Introducing an anisotropy field Hi, and with respective orientations of
magnetization and the anisotropy fields with the applied field being defined as 9i,v|/i, an
expression for figure of merit was derived by minimizing the free energy involved and
averaging over v|/i.
418 19
U •
a(^ •
^ nL30
• \3/0
DLBMQ in e^
S.9(Mz
29SK
U-J
0.2
0.0 ll.K
06
0.2
0(1 [[b) /-
/' M , . , r
9,9GHz •
295 K
I Ni
• Fe
* IBMQ
-
4O0 600 2000 4™ WOO 6O00 lOOOt
(a) (b)
FIGURE 3. Figure of merit vs applied field for (a) LSMO, LBMO and Single crystal piece of LSMO
placed in h^ field. Note that polycrystalline powder samples show the largest effect as compared to that
of bulk single crystal piece. Also almost no effect when and LBMO placed in e^ field, (b) A comparison
of figure of merit between LBMO powder and the conventional Ni and Fe powders. "Reprinted with
permission from AIP, ref [36]. Copyright [1999], American Institute of Physics."
•^ l,U
o.t
0-6
a4
0 2
(lOl 9.9 GHz
• LCMOCjto.SjfZ-iOK)
• 0 NSMOEOSKJ
• LCMO(;(*0,5Ki«5K} rL
• UMD(aiOK> g K
i LaM0(29&Kj ^/
A LaUQ29SK> ^^
VT^B
vlff
tefffi 1 d • - • 0^
^^ A
•
•
•
..I 1 i
0.0 0.2 0,4 0.6
G ii.a 10
FIGURE 4. For most of the manganite family members, the observed figure of merit follows an
empirical function G(x) = x^/(l+x^), as explained in the text. "Reprinted with permission from AIP, ref.
[36]. Copyright [1999], American Institute of Physics."
419 2.,, \ /„;„2^ ^ . . -. 2 ,x2
Sin i//,;-\sin u,^ _ 5 3 3{jf_-l)
sinV,) ~'8 877' I677 •nH)=^—TTV^T—" =-o-^- .^3 ^" (I-77
1 + 77 (4)
Where T] = H/Hj. This explains the magneto impedance 'qualitatively'. Note that
the observed colossal magneto impedance in these micron size manganite powders has
assumed significance because of its quantitative nature. A mere qualitative explanation
as put forward in ref [36] is not certainly sufficient to understand the phenomenon
and calls for a thorough theoretical investigation to explain this spectacular effect
quantitatively.
The above discussed special effects namely a pecuhar rise of microwave absorption
below the characteristic temperature T and its spectacular quenching by small fields
at room temperature in the micron size powders of manganites makes them very
attractive for several apphcations, such as magnetically tunable microwave absorbers,
acoustic modulators, field sensors etc.
SUMMARY
In summary, rf-electro dynamical properties in the small particles of Cuprate
Superconductors and CMR manganites are reviewed. We focused attention on 1. the
theoretical and experimental aspects of RF power induced 'anomalous' NRMA
reported in the micron size YBCO powders, which is absent in the bulk ceramic
samples of the same and 2. (a) Spectacular low field magneto impedance at room
temperature in the micron size CMR manganite powders, (b) discriminative
temperature dependence of microwave loss for to Crf and hrf fields, which was found in
almost all the family members of the manganite powders but not in the bulk form.
Thus at least in these materials, we can conclude that the rf electrodynamics are
qualitatively different in the bulk and small particle forms. In particular as the CMI in
the manganite powders has been experimentally proved to be originating from the spin
system, one can say that these micron size manganite powders are good rf
'spintronics' materials.
ACKNOWLEDGMENTS
We acknowledge useful discussions with B.K. Roul, Alan Portis, N. Kumar, and A.
Andreone.
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421 |
5.0050289.pdf | J. Chem. Phys. 154, 174105 (2021); https://doi.org/10.1063/5.0050289 154, 174105
© 2021 Author(s).Three-state harmonic models for
photoinduced charge transfer
Cite as: J. Chem. Phys. 154, 174105 (2021); https://doi.org/10.1063/5.0050289
Submitted: 13 March 2021 . Accepted: 18 April 2021 . Published Online: 04 May 2021
Dominikus Brian ,
Zengkui Liu ,
Barry D. Dunietz ,
Eitan Geva , and
Xiang Sun
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Three-state harmonic models for photoinduced
charge transfer
Cite as: J. Chem. Phys. 154, 174105 (2021); doi: 10.1063/5.0050289
Submitted: 13 March 2021 •Accepted: 18 April 2021 •
Published Online: 4 May 2021
Dominikus Brian,1,2,3
Zengkui Liu,1,2,3
Barry D. Dunietz,4
Eitan Geva,5
and Xiang Sun1,2,3,a)
AFFILIATIONS
1Division of Arts and Sciences, NYU Shanghai, 1555 Century Avenue, Shanghai 200122, China
2NYU-ECNU Center for Computational Chemistry at NYU Shanghai, 3663 Zhongshan Road North, Shanghai 200062, China
3Department of Chemistry, New York University, New York, New York 10003, USA
4Department of Chemistry and Biochemistry, Kent State University, Kent, Ohio 44242, USA
5Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109, USA
a)Author to whom correspondence should be addressed: xiang.sun@nyu.edu
ABSTRACT
A widely used strategy for simulating the charge transfer between donor and acceptor electronic states in an all-atom anharmonic condensed-
phase system is based on invoking linear response theory to describe the system in terms of an effective spin-boson model Hamiltonian.
Extending this strategy to photoinduced charge transfer processes requires also taking into consideration the ground electronic state in addi-
tion to the excited donor and acceptor electronic states. In this paper, we revisit the problem of describing such nonequilibrium processes
in terms of an effective three-state harmonic model. We do so within the framework of nonequilibrium Fermi’s golden rule (NE-FGR) in
the context of photoinduced charge transfer in the carotenoid–porphyrin–C 60(CPC 60) molecular triad dissolved in explicit tetrahydrofuran
(THF). To this end, we consider different ways for obtaining a three-state harmonic model from the equilibrium autocorrelation functions
of the donor–acceptor, donor–ground, and acceptor–ground energy gaps, as obtained from all-atom molecular dynamics simulations of
the CPC 60/THF system. The quantum-mechanically exact time-dependent NE-FGR rate coefficients for two different charge transfer pro-
cesses in two different triad conformations are then calculated using the effective three-state model Hamiltonians as well as a hierarchy of
more approximate expressions that lead to the instantaneous Marcus theory limit. Our results show that the photoinduced charge transfer
in CPC 60/THF can be described accurately by the effective harmonic three-state models and that nuclear quantum effects are small in this
system.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0050289 .,s
I. INTRODUCTION
Photoinduced charge transfer (CT) plays an important role in
many systems of practical interest, for example, solar energy conver-
sion in photosynthesis and organic photovoltaic (OPV) devices.1–11
The ability to accurately simulate photoinduced CT dynamics could
therefore offer invaluable insights toward the discovery of more effi-
cient artificial energy-conversion materials. Many quantum dynam-
ical methods have been proposed to study electronic transitions in
the condensed phase, such as the semiclassical initial value represen-
tation,12–14mean-field Ehrenfest,15–18fewest switches surface hop-
ping,19–21mixed quantum-classical Liouville,22–25generalized quan-
tum master equation,26–35and hierarchical equations of motion,36–41to name a few. Despite these advances, simulating CT dynamics in
realistic complex systems rigorously and accurately remains highly
challenging.42–49
Importantly, Marcus theory50–54cannot account for the
nonequilibrium nature of photoinduced CT, which is due to the
fact that the initial state after photoexcitation from the ground
state to the excited donor state corresponds to equilibrium on
the ground state potential energy surface (PES) rather than on
the donor state PES.55–60The nonequilibrium nature of the ini-
tial state can have a significant effect on the CT dynamics
when the timescale of nuclear relaxation to thermal equilibrium
on the donor PES is comparable to or longer than the timescale of
the CT.
J. Chem. Phys. 154, 174105 (2021); doi: 10.1063/5.0050289 154, 174105-1
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Accounting for the above-mentioned nonequilibrium effects
is made possible by the nonequilibrium Fermi’s golden rule
(NE-FGR).61–65Combined with the linearized semiclassical (LSC)
approximation, the NE-FGR can also be applied to complex
condensed-phase systems described by all-atom anharmonic Hamil-
tonians.66Furthermore, starting from the LSC NE-FGR, a hierar-
chy of more approximate methods can be derived, which ultimately
leads to the instantaneous Marcus theory (IMT), where the time-
dependent CT rate coefficient is given by a Marcus-like expres-
sion with explicitly time-dependent reorganization energy and reac-
tion free energy.66The time-dependent IMT rate coefficient is
expected to be accurate when (1) the nuclear degrees of freedom
(DOF) can be treated as classical, (2) the nuclear motion occurs
on a timescale slower than the electronic dephasing time, and (3)
the time-dependent distribution of the donor–acceptor energy gap
maintains its Gaussian nature.
An alternative and widely used approach relies on mapping
the all-atom anharmonic Hamiltonian onto an effective harmonic
model Hamiltonian, such as the popular spin-boson model for two-
state systems.44–47,67–70The mapping onto the spin-boson model
typically relies on invoking linear response theory and is based on
the donor–acceptor energy gap time correlation function (TCF) as
obtained from a classical molecular dynamics (MD) simulation of
the all-atom anharmonic system.71,72Such mapping onto an effec-
tive spin-boson model Hamiltonian72has been recently found to
lead to remarkably accurate equilibrium FGR (E-FGR) CT rate
constants in the case of the carotenoid–porphyrin–C 60(CPC 60)
molecular triad11,57,73–76dissolved in an explicit tetrahydrofuran
(THF) solvent.77–81The main advantage of using the effective har-
monic model is that once the mapping is established, the quantum-
mechanically exact expression for both E-FGR and NE-FGR and
the corresponding hierarchy of approximations are known in closed
form.42,64
In this paper, we go beyond the two-state spin-boson
Hamiltonian to construct effective three-state harmonic
Hamiltonians for simulating NE-FGR CT rate coefficients as well as
a progression of increasingly more approximate expressions, which
ultimately lead to IMT. We benchmark the results obtained for the
harmonic model against the all-atom predictions in the context of
the following two photoinduced CT processes in the bent and linear
CPC 60triad conformations in explicit THF (see Fig. 1). The triad
is initially equilibrated on the ground (G) state, CPC 60, before it
is photoexcited impulsively to the P-localized ππ∗state, CP∗C60.
Following the impulsive photoexcitation at time 0, there could be
a nonradiative transition to the excited P-to-C 60CT state, CP+C−
60,
which is denoted as CT1,
CPC 60(G)hν/leftr⫯g⊸tl⫯ne →CP∗C60(ππ∗)→CP+C−
60(CT1), (1)
or to the excited C-to-C 60charge separated state, C+PC−
60, which is
denoted as CT2,
CPC 60(G)hν/leftr⫯g⊸tl⫯ne →CP∗C60(ππ∗)→C+PC−
60(CT2). (2)
In what follows, we construct unique three-state harmonic
Hamiltonians for each of the two conformations and photoinduced
CT processes. It should be noted that mapping a three-state all-atom
anharmonic system onto an effective three-state harmonic system
FIG. 1 . Two characteristic conformations of the carotenoid–porphyrin–C 60(CPC 60)
molecular triad in explicit THF, (a) the bent triad conformation and (b) the linear
triad conformation, and schematic representations of the potential energy surfaces
involved in the photoinduced charge transfer processes in (c) the bent triad and (d)
the linear triad, where vertical photoexcitation brings it from the ground state to the
ππ∗state (black arrow), and then, during the nuclear relaxation on the ππ∗state
(orange arrow), electronic transition to CT1 or CT2 states occurs. The molecular
structure visualization was generated using the Visual Molecular Dynamics (VMD)
package.82
is not as straightforward as the two-state case.83,84This is because,
as pointed out by Cho and Silbey in Ref. 62, the ground, donor,
and acceptor PESs are not independent, and therefore, one needs to
consider the relations between each pair of the PESs consistently. In
particular, three different energy gap TCFs (donor–acceptor, donor–
ground, and acceptor–ground) are needed in order to construct the
harmonic model. In this paper, we propose and test several routes
for achieving the consistency requirement by satisfying constraints
imposed by various reorganization energies. Our previous LSC NE-
FGR and IMT calculations showed a significant nonequilibrium
transient effect in the bent triad conformation that enhanced the
porphyrin-to-C 60[Eq. (1)] CT rate coefficient by a factor of ∼40.66
Therefore, testing the ability of the harmonic three-state Hamil-
tonians to accurately reproduce this effect serves as an important
benchmark. The fact that the effective three-state harmonic model
makes it easy to evaluate the fully quantum mechanical NE-FGR also
allows us to assess the importance of nuclear quantum effects in this
system.72
The remainder of this paper is organized as follows. Section II
summarizes the NE-FGR CT rate coefficient and the correspond-
ing hierarchy of LSC-based approximations that leads to the IMT.
Section III presents three different ways to construct three-state
harmonic models for the nonequilibrium photoinduced CT pro-
cess (models 1–3) as well as a strategy for calculating the IMT CT
rate coefficients directly from the three spectral densities associated
with the three above-mentioned TCFs. The all-atom anharmonic
model for the CPC 60molecular triad dissolved in liquid THF and
MD simulation techniques are described in Sec. IV. The results and
discussion are reported in Sec. V. The conclusions and outlook are
J. Chem. Phys. 154, 174105 (2021); doi: 10.1063/5.0050289 154, 174105-2
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provided in Sec. VI. The derivations of the IMT rate coefficient for
the three-state model and additional supporting tables and figures
are provided in the supplementary material.
II. THEORY
A. Nonequilibrium Fermi’s golden rule (NE-FGR)
Consider a charge transfer system with the Hamiltonian
ˆH=ˆHD∣D⟩⟨D∣+ˆHA∣A⟩⟨A∣+ΓDA(∣D⟩⟨A∣+∣A⟩⟨D∣). (3)
Here, | D⟩and | A⟩represent the diabatic donor and acceptor elec-
tronic states, respectively, and ˆHD/Aare the corresponding nuclear
Hamiltonians, ˆHD/A=ˆP2/2 +VD/A(ˆR), where ˆR=(ˆR1,. . .,ˆRN)
and ˆP=(ˆP1,. . .,ˆPN)are the mass-weighted nuclear coordinates
and momenta, VD/A(ˆR)are the donor/acceptor PESs, and ΓDAis the
electronic coupling coefficient that is assumed to be constant within
the Condon approximation.
We assume that the system starts out at the donor electronic
state with the initial nuclear DOF described by the ground state
equilibrium nuclear density operator ˆρG=e−βˆHG/Trn[e−βˆHG]. Here,
ˆHG=ˆP2/2 + VG(ˆR)is the ground state nuclear Hamiltonian,
VG(ˆR)is the ground state PES, β= 1/ kBTis the inverse temper-
ature, Tr n(⋅) denotes the trace over the nuclear Hilbert space, and
TrnTredenotes the trace over both the nuclear and electronic
Hilbert spaces. The donor state population, PD(t), at a later time tis
given by
PD(t)=TrnTre[e−iˆHt/̵hˆρGeiˆHt/̵h∣D⟩⟨D∣]. (4)
Applying second-order perturbation theory then leads to the fol-
lowing approximate NE-FGR expression for the donor state pop-
ulation:61
PD(t)≈exp[−∫t
0dt′k(t′)], (5)
where the time-dependent rate coefficient is defined as
k(t)=2
̵h2Re∫t
0dτC(t,τ). (6)
Here,
C(t,τ)=∣ΓDA∣2Trn[e−iˆHDt/̵hˆρGeiˆHDt/̵he−iˆHAτ/̵heiˆHDτ/̵h]. (7)
It should be noted that the NE-FGR expression in Eq. (7) is fully
quantum-mechanical, which can still be difficult to evaluate in
complex systems.
We also note in passing that assuming that the initial state of
the nuclear DOF corresponds to equilibrium on the donor state PES,
ˆρD=e−βˆHD/Trn[e−βˆHD],C(t,τ) reduces to
C(τ)=∣ΓDA∣2Trn[ˆρDe−iˆHAτ/̵heiˆHDτ/̵h]. (8)
Also assuming that the CT happens on a timescale longer than the
electronic dephasing time, which is defined by the lifetime of C(τ),then leads to the time-dependent rate coefficient k(t) turning into
the time-independent E-FGR CT rate constant,42
kD→A=2
̵h2Re∫∞
0dτC(τ). (9)
Under those conditions, the donor state population could be esti-
mated by the exponential decay PD(t) = exp(−kD→At). It should be
noted that the rate constant kD→Ais the long-time asymptotic limit
of the time-dependent rate coefficient, k(t).64,66Accounting for the
nuclear initial preparation effect is important when k(t) and kD→A
are significantly different and the timescale for reaching thermal
equilibrium on the donor PES is comparable to or longer than the
timescale of CT, ∼k−1
D→A.
B. The hierarchy of LSC-based approximations
The linearized semiclassical (LSC) approximation provides a
computationally feasible route for calculating NE-FGR rates. The
main idea behind LSC is to express C(t,τ)in terms of the real-
time path integral and then apply the linearization approximation
to the differences between the forward and backward paths, which
filters out the highly oscillatory contributions and leads us to a
classical-like procedure for calculating the time-dependent CT rate
coefficient. The hierarchy of LSC-based approximations for C(t,τ)
is summarized as follows:64
CW-AV(t,τ)=∣ΓDA∣2∫dR0dP0ρG,W(R0,P0)
×exp[−i
̵h∫t−τ
tdt′U(Rav
t′)], (10)
CW-0(t,τ)=∣ΓDA∣2∫dR0dP0ρG,W(R0,P0)exp[i
̵hU(Rt)τ], (11)
CC-AV(t,τ)=∣ΓDA∣2∫dR0dP0ρG,Cl(R0,P0)
×exp[−i
̵h∫t−τ
tdt′U(Rav
t′)], (12)
CC-D(t,τ)=∣ΓDA∣2∫dR0dP0ρG,Cl(R0,P0)
×exp[−i
̵h∫t−τ
tdt′U(RD
t′)], (13)
CC-0(t,τ)=∣ΓDA∣2∫dR0dP0ρG,Cl(R0,P0)exp[i
̵hU(Rt)τ]. (14)
Here, “W” denotes the semiclassical Wigner distribution for initial
sampling of the nuclear DOF,
ρG,W(R,P)=(1
2π̵h)N
∫dZexp(i
̵hZ⋅P)⟨R−Z
2∣ˆρG∣R+Z
2⟩,
(15)
and “C” denotes the classical nuclear sampling using the corre-
sponding classical distribution ρG,Cl(R,P). Following the initial sam-
pling of ( R0,P0) via Wigner or classical nuclear densities, the system
J. Chem. Phys. 154, 174105 (2021); doi: 10.1063/5.0050289 154, 174105-3
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is propagated on the donor PES forward in time over time interval
t, arriving at the configuration Rt, and then backward in time from
time ttot−τ(τ= 0→t), integrating the donor–acceptor energy gap
U(R) =UDA(R) =VD(R)−VA(R) in the phase factor for each ( t,τ).
“AV” denotes that the dynamics over τis propagated on the aver-
age PES, Vav(R) = (VD(R) +VA(R))/2, leading to trajectories of Rav
τ,
“D” denotes that the τ-dynamics is on the donor surface, leading
to trajectories of RD
τ, and “0” denotes no τ-dynamics beyond the t-
relaxation on the donor PES until Rt. The W-AV level is the original
LSC approximation for the NE-FGR.
From the C-0 approximation for the NE-FGR, one can derive
the instantaneous Marcus theory (IMT) expression of k(t),66
kIMT(t)=∣ΓDA∣2
̵h√
2π
σ2
texp⎡⎢⎢⎢⎢⎣−(Ut)2
2σ2
t⎤⎥⎥⎥⎥⎦, (16)
where we assume that the instantaneous distribution of the energy
gap Utis Gaussian with a time-dependent mean Utand the corre-
sponding variance σ2
t=U2
t−(Ut)2at time tafter the photoexcitation
so that Prob (Ut)=1/√
2πσ2
texp[−(Ut−Ut)2/(2σ2
t)]. Here, Utand
σ2
tcan be calculated using energies obtained from all-atom molecu-
lar dynamics simulations.66The IMT CT rate coefficient kIMT(t) can
also be expressed equivalently in terms of time-dependent reaction
free energy ΔE(t) and reorganization energy Er(t),
Er(t)=σ2
t
2kBT=−ΔE(t)−Ut. (17)
In addition, the hierarchy of LSC-based approximations for the
E-FGR rate constants corresponds to replacing the t-relaxation on
the donor PES with equilibrium sampling on the donor PES, so
C(t,τ) in Eqs. (10)–(14) would reduce to W-AV, W-0, C-AV, C-D,
and C-0 levels of approximation for E-FGR C(τ) in Eq. (8), respec-
tively. Assuming the second-order cumulant approximation for Uat
the C-0 level of approximation, the classical Marcus rate constant is
thus given by42
kM
D→A=∣ΓDA∣2
̵h√
π
kBTErexp[−(ΔE+Er)2
4kBTEr]. (18)
The corresponding reorganization energy and reaction free energy
satisfy the relation Er=σ2
eq/(2kBT)=−ΔE−⟨U⟩and the activation
energy Ea=kBT⟨U⟩2/(2σ2
eq), where ⟨⋅⟩is the equilibrium average
on the donor state PES and the equilibrium variance of the energy
gapσ2
eq=⟨U2⟩−⟨U⟩2.
III. CONSTRUCTING THREE-STATE HARMONIC
MODEL HAMILTONIANS
A. Mapping an anharmonic all-atom Hamiltonian
onto the spin-boson model
The spin-boson model has been widely used for simulating CT
dynamics in the condensed phase.44–47,67–70The prototypical spin-
boson Hamiltonian is given by47ˆH=ΓDAˆσx+̵hωDA
2ˆσz+N
∑
j=1⎛
⎝ˆP2
j
2+1
2ω2
jˆR2
j−cjˆRjˆσz⎞
⎠. (19)
Here, ˆσx=∣D⟩⟨A∣+∣A⟩⟨D∣and ˆσz=∣D⟩⟨D∣−∣A⟩⟨A∣are
the Pauli operators; ΓDAis the electronic coupling coefficient;
ΔE=−̵hωDA is the donor-to-acceptor reaction free energy;
{ˆRj,ˆPj,ωj∣j=1,. . .,N}are the mass-weighted coordinates,
momenta, and frequencies associated with the nuclear normal
modes, respectively; and { cj} are the coupling coefficients between
the electronic DOF and nuclear normal modes. The frequencies and
the coupling coefficients, { ωj,cj}, are often given in terms of spectral
density, which is defined by
J(ω)=π
2N
∑
j=1c2
j
ωjδ(ω−ωj). (20)
Next, we outline the procedure typically followed for mapping
an all-atom anharmonic Hamiltonian such as that of the triad in
liquid THF solution onto the spin-boson Hamiltonian. It starts out
by obtaining the donor–acceptor energy gap time correlation func-
tion, CUU(t), from an equilibrium MD simulation on the donor
PES,72
CUU(t)=⟨U(t)U(0)⟩−⟨U⟩2, (21)
and using the following relation to obtain the spectral density (note
that in some literature,71the TCF of half energy gap is used, so we
see a factor of 1/4 difference):
J(ω)=βω
4∫∞
0dtCUU(t)cos(ωt). (22)
It is noted that within this framework, the reorganization energy can
be identified as being given in terms of the donor–acceptor energy
gap variance, CUU(0)=σ2
eq,
Er=CUU(0)
2kBT=4
π∫∞
0dωJ(ω)
ω. (23)
Obtaining a discrete representation of the nuclear DOF in
terms of Nnormal modes ( N≫1) from the continuous spectral den-
sity function in Eq. (22) is based on solving the following equation
for the frequency ωjof the jth mode ( j= 1, 2, . . .,N):71
2Nωj
πCUU(0)∫∞
0dtCUU(t)
ωjtsin(ωjt)=j−1
2. (24)
It is worth noting that following this procedure, the obtained discrete
frequency intervals correspond to equal fractions of the reorgani-
zation energy. Once the frequency of the jth mode is determined,
the corresponding coupling coefficients, cj, are obtained via the
relation
cj=√
Er
2Nωj(j=1, 2, . . .,N). (25)
J. Chem. Phys. 154, 174105 (2021); doi: 10.1063/5.0050289 154, 174105-4
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B. Three-state harmonic models for photoinduced
charge transfer
Next, we consider generalizing this procedure to the case
of a nonequilibrium CT process that takes place in a three-state
system. To this end, we define the following effective harmonic
Hamiltonian:
ˆH=ˆHG∣G⟩⟨G∣+ˆHD∣D⟩⟨D∣+ˆHA∣A⟩⟨A∣+ΓDA(∣D⟩⟨A∣+∣A⟩⟨D∣), (26)
where the nuclear Hamiltonians for the donor (D), acceptor (A), and
ground (G) electronic states are given by
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ˆHD=N
∑
j=1ˆP2
j
2+N
∑
j=11
2ω2
jˆR2
j+̵hωDA,
ˆHA=N
∑
j=1ˆP2
j
2+N
∑
j=11
2ω2
j(ˆRj−Req
j)2
,
ˆHG=N
∑
j=1ˆP2
j
2+N
∑
j=11
2ω2
j(ˆRj+Sj)2+εG.(27)
Here, the energy difference between the donor and acceptor states,
ΔE=−̵hωDA, corresponds to the CT reaction free energy, while
the ground state energy, εG, is defined relative to the energy of the
acceptor state at its equilibrium geometry. It should be noted that
the photoinduced CT rate is independent of the value of εGsince
we assume that the photoexcitation corresponds to a vertical excita-
tion from the ground to the donor state. The displacement between
the donor and acceptor equilibrium geometries along the jth mode
(j= 1, 2, . . .,N) is given by [see Eqs. (25) and (19) for the definition
ofcj]
Req
j=2cj
ω2
j=√
2Er
N1
ωj. (28)
Thus, the reorganization energy between donor and acceptor states
is given by
Er=N
∑
j=12c2
j
ω2
j=N
∑
j=11
2ω2
j(Req
j)2. (29)
Expressed in terms of {Req
j}instead of { cj}, the spectral density can
be written in the following form:
J(ω)=π
8N
∑
j=1ω3
j(Req
j)2δ(ω−ωj). (30)
Finally, { Sj} correspond to the displacements between the ground
and donor equilibrium geometries along the normal mode coordi-
nates, which is responsible for the nonequilibrium initial prepara-
tion of the CT process (see below).
For the three-state harmonic model defined in Eqs. (26) and
(27), we can express the NE-FGR hierarchy of approximations in
the closed form [it should be noted that the LSC approximation
(W-AV) coincides with the exact quantum-mechanical expression
in this case]64Cexact/W-AV(t,τ)=∣ΓDA∣2exp⎧⎪⎪⎨⎪⎪⎩iωDAτ−N
∑
j=1ωj(Req
j)2
2̵h
×[coth(β̵hωj
2)(1−cos(ωjτ))+isin(ωjτ)]
−N
∑
j=1iωjReq
jSj
̵h[sin(ωjt)+ sin(ωjτ−ωjt)]⎫⎪⎪⎬⎪⎪⎭, (31)
CW-0(t,τ)=∣ΓDA∣2exp⎧⎪⎪⎨⎪⎪⎩iωDAτ−N
∑
j=1ωj(Req
j)2
2̵h
×⎡⎢⎢⎢⎣coth(β̵hωj
2)ω2
jτ2
2+iωjτ⎤⎥⎥⎥⎦−N
∑
j=1iωjReq
jSj
̵hcos(ωjt)ωjτ⎫⎪⎪⎬⎪⎪⎭,
(32)
CC-AV(t,τ)=∣ΓDA∣2exp⎧⎪⎪⎨⎪⎪⎩iωDAτ−N
∑
j=1ωj(Req
j)2
2̵h
×[2
β̵hωj(1−cos(ωjτ))+isin(ωjτ)]
−N
∑
j=1iωjReq
jSj
̵h[sin(ωjt)+ sin(ωjτ−ωjt)]⎫⎪⎪⎬⎪⎪⎭, (33)
CC-D(t,τ)=∣ΓDA∣2exp⎧⎪⎪⎨⎪⎪⎩iωDAτ−N
∑
j=1ωj(Req
j)2
2̵h
×[2
β̵hωj(1−cos(ωjτ))+iωjτ]
−N
∑
j=1iωjReq
jSj
̵h[sin(ωjt)+ sin(ωjτ−ωjt)]⎫⎪⎪⎬⎪⎪⎭, (34)
CC-0(t,τ)=∣ΓDA∣2exp⎧⎪⎪⎨⎪⎪⎩iωDAτ−N
∑
j=1ωj(Req
j)2
2̵h
×[ωjτ2
β̵h+iωjτ]−N
∑
j=1iωjReq
jSj
̵hcos(ωjt)ωjτ⎫⎪⎪⎬⎪⎪⎭. (35)
The time-dependent IMT parameters Utandσ2
tin Eq. (16) for
the three-state harmonic model can be expressed in the closed form
(the derivation can be found in the supplementary material)
Ut=̵hωDA−N
∑
j=11
2ω2
j[(Req
j)2+ 2Req
jSjcos(ωjt)]
=⟨U⟩+βCL
1(t), (36)
σ2
t=σ2
0=1
βN
∑
j=1ω2
j(Req
j)2. (37)
Here, CL
1(t)is the cross correlation function of the donor–acceptor
and donor–ground energy gaps,
J. Chem. Phys. 154, 174105 (2021); doi: 10.1063/5.0050289 154, 174105-5
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CL
1(t)=⟨UDA(t)UDG(0)⟩−⟨UDA⟩⟨UDG⟩
=−1
βN
∑
j=1ω2
jReq
jSjcos(ωjt). (38)
It should be noted that σ2
t=σ2
0for the harmonic model, which
implies that any significant time-dependence of σ2
tis a signature
for deviations from this model. It should also be noted that the
same expression can be derived from linear response theory,85which
is exact in the case of the three-state harmonic model (see the
supplementary material).
C. Mapping an anharmonic all-atom Hamiltonian
onto a three-state harmonic model
We now consider the construction of three-state harmonic
models of the form of Eqs. (26) and (27) from inputs obtained from
MD simulations. To this end, and in analogy to the spin-boson case,
we use as inputs the correlation functions of the three energy gaps
(donor–acceptor, donor–ground, and acceptor–ground), which are
calculated from classical equilibrium MD simulations on the donor
PES,
CDA
UU(t)=⟨UDA(t)UDA(0)⟩−⟨UDA⟩2≡CUU(t),
CDG
UU(t)=⟨UDG(t)UDG(0)⟩−⟨UDG⟩2,
CAG
UU(t)=⟨UAG(t)UAG(0)⟩−⟨UAG⟩2.(39)
Here, UXY=VX−VY(X,Y=G,D,A) is the energy gap between
Xand YPESs. Correspondingly, we can define three different
reorganization energies ( XY=DA,DG,AG),
EXY
r=CXY
UU(0)
2kBT. (40)
Following the two-state case, we chose to obtain the frequen-
cies {ωj} of the nuclear modes from CDA
UU(t)based on Eq. (24) and
assume them to be the same for all three states. Once the frequencies
of the nuclear modes are determined, we can turn to assigning val-
ues to the equilibrium geometry displacements, {Req
j}and { Sj}. The
equilibrium geometry displacements between the donor and accep-
tor states, {Req
j}, can be unambiguously determined from EDA
r=Er
as in Eq. (28),
Req
j=RDA
j≡√
2EDAr
N1
ωj(j=1,. . .,N). (41)
Similarly, we also define the equilibrium shifts between DG and
between AG,
RDG
j≡√
2EDGr
N1
ωj(j=1,. . .,N), (42)
RAG
j≡√
2EAGr
N1
ωj(j=1,. . .,N). (43)
One can also show thatEDA
r=CDA
UU(0)
2kBT=∑
j1
2ω2
j(RDA
j)2, (44)
EDG
r=CDG
UU(0)
2kBT=∑
j1
2ω2
j(RDG
j)2, (45)
EAG
r=CAG
UU(0)
2kBT=∑
j1
2ω2
j(RAG
j)2. (46)
Now, we need to determine the initial shifts between the donor
and ground states, { Sj}. From the nuclear Hamiltonians in Eq. (27),
one would expect
EDA
r=∑
j1
2ω2
j(Req
j)2, (47)
EDG
r=∑
j1
2ω2
jS2
j, (48)
EAG
r=∑
j1
2ω2
j(Req
j+Sj)2. (49)
Equations (47) and (48) can be satisfied by using Eq. (41) and by
choosing
∣Sj∣=RDG
j(j=1,. . .,N). (50)
It should be noted that only the absolute values of { Sj} are deter-
mined by Eq. (48). If we choose the nonequilibrium shifts to be
positive and Sj=RDG
j(for all j= 1, . . .,N), the effective AG
reorganization energy will be larger than the actual value EAG,eff
r
=∑j1
2ω2
j(Req
j+RDG
j)2>EAG
r=CAG
UU(0)/(2kBT), which means
that the displacements between the ground and acceptor minima
are too large. It is important to note that the three reorganization
energies are not entirely independent (e.g., UDG=UDA+UAG), so
forcing the nonequilibrium shifts to be colinear with UDA, for exam-
ple, will overestimate the nonequilibrium initial shifts. Therefore,
the relative signs of {Req
j,Sj}need to be chosen in a way that satisfies
Eq. (49).
Figure 2 illustrates the effect of flipping the sign of Sjfor a given
Req
j. The relative signs of Sjand Req
jcan clearly lead to either a posi-
tive or a negative contribution of the jth mode to EAG
rby the amount
ofΔEAG
r(j)=±2ω2
jReq
jRDG
j(see Table S3 of the supplementary
material). Since EAG
r=∑j1
2ω2
j(Req
j+Sj)2can be satisfied for different
selections of relative signs of Sjand Req
j, satisfying Eq. (49) does not
dictate a unique choice of the relative signs of {Req
j,Sj}.
In practice, we choose Req
j=RDA
j≥0(j=1,. . .,N)and
randomly flip the signs of { Sj} such that the deviations between
EAG,eff
r=∑j1
2ω2
j(Req
j+Sj)2and EAG
r=CAG
UU(0)/(2kBT)are minimal.
Alternatively, one can flip the signs of { Sj} relative to {Req
j}evenly,
namely, every few modes, until EAG,eff
r≈EAG
r[the two implementa-
tions turn out to give similar results for the system under considera-
tion in this paper (see the supplementary material)]. In what follows,
we will refer to this approach as model 1 .
J. Chem. Phys. 154, 174105 (2021); doi: 10.1063/5.0050289 154, 174105-6
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FIG. 2 . Flipping the sign of the minimum position of the jth mode on the ground
state (G) with a positive Sj(black) to a negative Sj(gray) would decrease the effec-
tive reorganization energy between the acceptor state and the ground state EAG
r,
whereas the minimum positions of the donor state (D, blue) and the acceptor state
(A, red) do not change. Here, ⟨U⟩,Er,ΔE, andεGare the energetic parameters
of the jth mode contribution (model 1).
An alternative approach for determining { Sj} was inspired by
the work of Cho and Silbey.62The relations between the donor,
acceptor, and ground PESs can be depicted in terms of the two-
dimensional illustration in Fig. 3. Since the shifts between the equi-
librium geometry of different pairs of PESs are proportional to
the square root of the corresponding reorganization energy [see
Eqs. (41)–(43)], i.e., RXY
j∝√
EXYr, we expect the three reorgani-
zation energies to form a triangle with the three edges proportional
to√
EDAr,√
EDGr,√
EAGr, and the angle θbetween DAand DGedges
is given by
FIG. 3 . A schematic representation of the three-state model in two-dimensional
space where qCTis the CT axis and qoris the axis orthogonal to the CT axis. The
donor (D), acceptor (A), and ground (G) potential energy surfaces (PESs) fall onto
the different locations forming a triangle, where θis the angle between DAandDG
PES minima. The effective nonequilibrium shifts {SCT
j}are the projections of the
donor–ground minimum displacements {RDG
j}onto the donor–acceptor CT axis
shown as the horizontal dashed line (models 2 and 3).cosθ=EDA
r+EDG
r−EAG
r
2√
EDArEDGr. (51)
As shown in Fig. 3, the DG edge or {RDG
j}can be decom-
posed into a component along the CT axis, qCT(the horizontal
dashed line along the donor–acceptor displacement direction), and
an orthogonal component qor, which is perpendicular to it,
SCT
j=−RDG
jcosθ(j=1,. . .,N), (52)
Sor
j=RDG
jsinθ(j=1,. . .,N). (53)
In this triangular case, the effective AG reorganization energy is
given by
EAG,eff
r=N
∑
j=11
2ω2
j[(Req
j+SCT
j)2
+(Sor
j)2]
=N
∑
j=11
2ω2
j[(Req
j)2
+(RDG
j)2−2Req
jRDG
jcosθ]
=EDA
r+EDG
r−2 cosθN
∑
j=11
2ω2
jReq
jRDG
j
=EDA
r+EDG
r−2 cosθN
∑
j=11
2ω2
j√
2EDAr
N1
ωj⋅√
2EDGr
N1
ωj
=EDA
r+EDG
r−2 cosθ√
EDArEDGr
=EAG
r, (54)
which agrees with EAG
r=CAG
UU(0)/(2kBT)from MD simulation.
Since the initial shifts along the orthogonal axis are irrelevant to the
CT process, it is reasonable to assume that the effective nonequilib-
rium shifts along the CT axis { Sj} in Eq. (27) are {RDG
j}projected
onto the CT axis SCT
j, i.e.,
Sj=SCT
j=−RDG
jcosθ(j=1,. . .,N). (55)
In what follows, we will refer to this approach as model 2 .
Alternatively, we can extend the number of harmonic modes
in model 2 from Nto 2Nand dividing them into two subgroups
ofNmodes each based on the displacements in their equilibrium
geometries [see Eq. (56)]. To this end, the nuclear Hamiltonians for
the donor, acceptor, and ground electronic states are redefined as
follows:
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ˆHD=2N
∑
j=1ˆP2
j
2+N
∑
j=11
2ω2
j(ˆR2
j+ˆR2
N+j)+̵hωDA,
ˆHA=2N
∑
j=1ˆP2
j
2+N
∑
j=11
2ω2
j[(ˆRj−Req
j)2
+ˆR2
N+j],
ˆHG=2N
∑
j=1ˆP2
j
2+N
∑
j=11
2ω2
j[(ˆRj+SCT
j)2
+(ˆRN+j+Sor
j)2]+εG,
where{SCT
j,Sor
j}are given by Eqs. (52) and (53), respectively. In
what follows, we will refer to this approach as model 3 .
J. Chem. Phys. 154, 174105 (2021); doi: 10.1063/5.0050289 154, 174105-7
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Finally, we note that the time-dependent IMT parameters Ut
andσt[see Eqs. (36) and (37)], which are necessary for calculat-
ing the IMT rate [see Eq. (16)], can be calculated directly from the
spectral densities defined as follows:
JDA(ω)=βω
4∫∞
0dt CDA
UU(t)cos(ωt)
=π
8N
∑
j=1ω3
j(Req
j)2
δ(ω−ωj),
JDG(ω)=βω
4∫∞
0dt CDG
UU(t)cos(ωt)
=π
8N
∑
j=1ω3
j(Sj)2δ(ω−ωj),
JAG(ω)=βω
4∫∞
0dt CAG
UU(t)cos(ωt)
=π
8N
∑
j=1ω3
j(Req
j+Sj)2
δ(ω−ωj).(56)
We thereby bypass the need to assign values to the equilibrium
geometry displacements, {Req
j,Sj}. Using the spectral densities, we
express the TCF in Eq. (38) as follows:
CL
1(t)=−N
∑
j=11
βω2
jReq
jSjcos(ωjt)
=−8
πβ∫∞
0dω
ω⎡⎢⎢⎢⎢⎣π
8N
∑
j=1ω3SjReq
jδ(ω−ωj)⎤⎥⎥⎥⎥⎦cos(ωt)
=4
πβ∫∞
0dω
ω[JDA(ω)+JDG(ω)−JAG(ω)]cos(ωt). (57)
The IMT parameters for Eq. (16) are thus given by
Ut=4
π∫∞
0dω
ω[JDA(ω)+JDG(ω)−JAG(ω)]cos(ωt)+⟨U⟩, (58)
σ2
t=σ2
0=8
πβ∫∞
0dω
ωJDA(ω). (59)
IV. SIMULATION TECHNIQUES
In this section, we describe the all-atom simulations that were
used to provide classical MD inputs for constructing the three-state
harmonic models. The all-atom model of the CPC 60triad dissolved
in explicit THF was adopted from Ref. 81. For the bent and linear
triad conformations, four electronic-state-dependent force fields
were constructed for the ground, ππ∗, CT1, and CT2 states. The
overall PESs of the four electronic states of the triad differ with
respect to the excitation energies and the triad’s atomic charges.
The non-electrostatic interactions are assumed to be the same for
different electronic states. The excitation energies, atomic charges,
and electronic coupling coefficients for this system were computed
using time-dependent density functional theory (TDDFT) with
the range-separated hybrid (RSH) Baer–Neuhauser–Livshits (BNL)
functional86–88using the Q-Chem 4 program package.89The elec-
tronic coupling coefficients between the electronically excited stateswere calculated via the fragment charge difference (FCD) method.90
The overall PES in the αth excited state is given by
Vα(R)=VMD
α(R)+Eα(rTriad)−Vα,T(rTriad)
=VMD
α(R)+Wα. (60)
Here, Eα(rTriad) represents the αth excited state energy with respect
to the ground state for the gas-phase triad, which were obtained from
electronic structure calculations in the characteristic bent and lin-
ear triad geometries, i.e., rTriad. It is crucial to include the excitation
energy in the total potential energy because classical force fields can
only describe the interactions or forces between all atoms but not the
relative energy between states (e.g., from ground →ππ∗, the atomic
charges do not change too much, but still there is a sizable energy dif-
ference due to the electronic excitation). However, simply adding up
the MD potential energy of the entire system VMD
α(R)and the exci-
tation energy Eα(rTriad) of the triad would double count the energy
of the triad intramolecular interactions, so we need to subtract this
double-counted contribution Vα,T(rTriad) evaluated with the force
fields where only the triad is present. Finally, we arrive at the energy
correction Wα=Eα(rTriad)−Vα,T(rTriad) to the MD potential energy
of theαth excited state [Eq. (60)]. Table S1 shows the values of Eα,
Vα,T, and Wαfor theππ∗, CT1, and CT2 states in the bent and linear
conformations. For details of force field parameters and its agree-
ment with experiment on CT rate constants, we refer the reader to
Ref. 81.
MD simulations were performed for a system containing one
triad molecule and 6741 THF molecules in a 100 Å ×100 Å×100 Å
periodic cubic box, performed using the AMBER 18 package.91
The van der Waals cutoff radius was set to be 9 Å. The SHAKE
algorithm92was used to constrain all covalent bonds involving
hydrogen atoms. Particle mesh Ewald summation was used to cal-
culate the electrostatic interactions.93To maintain the linear triad
conformation, the end-to-end distance was constrained at 49.6 Å
using a steep harmonic potential with a force constant of 100
kcal mol−1Å−2. No constraint was used in the simulation of a flex-
ible bent triad in flexible THF. The MD time step was chosen as
δt= 1.0 fs. The system was equilibrated on the ππ∗state at 300 K
using a Langevin thermostat with a collision frequency of 1.0 ps−1.
Two sets of all-atom equilibrium MD simulations comprising
of 200 independent trajectories of length 500 ps each were gener-
ated under the NVE ensemble after equilibration under the NVT
ensemble, one set propagated on the ground state PES and another
on the donor state PES. The initial positions and velocities for the
NVE production runs were sampled every 5 ps from an NVT ensem-
ble at a temperature of 300 K equilibrated on either the ground
state PES or the excited ππ∗state PES, followed by a 50 ps re-
equilibration under constant NVE conditions. Configurations were
sampled every 5 fs in these 200 NVE trajectories, which means a
total sum of 2 ×107configurations were sampled from equilib-
rium NVE trajectories amounting to a total propagation duration
of 100 ns in each case of different triad conformations. The potential
energies on every electronic-state PES were recalculated from these
trajectories.
In the all-atom NEMD simulations employed as benchmark,
a total of 20 000 nonequilibrium NVE trajectories were initially
J. Chem. Phys. 154, 174105 (2021); doi: 10.1063/5.0050289 154, 174105-8
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sampled from an equilibrated ground state in the NVT ensemble
at 300 K and propagated for 4 ps under constant NVE conditions
with configuration recorded every time step, i.e., 1.0 fs. Thus, a total
sum of 8×107nonequilibrium configurations were averaged over
to obtain the IMT rate coefficient. The error bars for k(t) of the
all-atom NEMD simulations were obtained by averaging five sets of
4000 NVE trajectories.66
In the construction of model 1, we selected the number
of flipped modes, Nf, such that the deviation of the effec-
tive reorganization energy EAG,eff
r from the accurate value EAG
ris
minimized,
argmin
Nf∣EAG,eff
r(Nf)−EAG
r∣. (61)
V. RESULTS AND DISCUSSION
In this section, we present the NE-FGR CT rate coefficients
obtained using the three three-state harmonic models (models 1–3)
as well as IMT via J(ω). The results obtained based on the three-
state harmonic models are compared to the IMT rate coefficients
calculated via the LSC method from all-atom NEMD simulations.
We construct the three-state models for two separate photoinduced
CT pathways in the CPC 60triad dissolved in THF. More specif-
ically, we start with a vertical photoexcitation from the equilib-
rium ground state to the donor state, which is then followed by
an electronic transition to one of the two acceptor states [CT1 or
CT2, see Eqs. (1) and (2)]. In what follows, we refer to those path-
ways asππ∗→CT1 andππ∗→CT2. Different sets of results are
shown for the two characteristic triad conformations, i.e., bent and
linear.
A. All-atom simulation of CPC 60in THF
and its harmonic models
We start out with Table I that summarizes the reorganization
energies EDA
r,EDG
r, and EAG
rfor theππ∗→CT1 andππ∗→CT2
transitions in the bent and linear triad conformations, as obtained
from the all-atom MD simulations. Importantly, any reasonable
three-state harmonic model needs to be consistent with those three
reorganization energies, EDA
r,EDG
r, and EAG
r. The inputs from MD
simulations for constructing the three-state harmonic models are
the time autocorrelation functions of the energy gaps between dif-
ferent PESs in the bent and linear triad conformations, which are
shown in Fig. 4. It is seen that most of the normalized CXY
UU(t)
have similar exponential decay profiles with lifetime τ1/e∼1.0 ps
TABLE I . Reorganization energies between different pairs of potential energy sur-
faces ( EXY
rin eV) forππ∗→CT1 andππ∗→CT2 transitions in the bent and linear
triad conformations.
Transition Conf. EDA
r EDG
r EAG
r
ππ∗→CT1 Bent 0.533 0.0914 0.924
Linear 0.687 0.0109 0.666
ππ∗→CT2 Bent 1.46 0.0914 1.75
Linear 1.71 0.0109 1.73
FIG. 4 . Un-normalized (top) and normalized (bottom) time correlation functions
of the energy gap between different potential energy surfaces, CXY
UU(t)for
ππ∗−G (black), CT1 −G (cyan), CT2 −G (orange), ππ∗−CT1 (red), and
ππ∗−CT2 (blue) in the bent (solid line) and linear (dashed line) triad conforma-
tions. The inset of the top panel is CXY
UU(t)for theππ∗−G energy gap in the
linear triad conformation, and the inset of the bottom panel is for the ππ∗−G
energy gap in the bent and linear triad conformation.
except for CG,ππ∗
UU(t)in the linear triad case that exhibits a fast and
non-monotonic decay but with a very small magnitude. The initial
values CXY
UU(0)are proportional to the corresponding reorganization
energies, i.e., CXY
UU(0)=2kBTEXY
r[see Eqs. (47)–(49) and reorga-
nization energies in Table I]. The fact that the charge distribution
on the ground state is similar to that on the ππ∗state results in
the smaller amplitude of CG,ππ∗
UU(t)(black lines). In contrast, since
the difference in charge distributions between the ground/ ππ∗state
and the CT2 state is the largest; this leads to the largest amplitude
ofCG/ππ∗,CT2
UU(t)(orange and blue lines). As expected, the amplitude
ofCG/ππ∗,CT1
UU(t)is intermediate (cyan and red lines). The noticeable
difference between the bent and linear triad conformations reflects
the fact that the solute–solvent interactions depend on the triad
conformation.
Next, Figs. 5 and 6 show the three spectral densities JDA(ω),
JAG(ω), and JDG(ω), as obtained via Eq. (56) using energy gap TCFs
CDA
UU(t),CAG
UU(t), and CDG
UU(t), in the bent and linear triad confor-
mations (see Fig. 4). Due to the fact that the decay profiles of the
normalized CXY
UU(t)/CXY
UU(0)in the bent triad are relatively insensi-
tive to the PESs, the spectral densities that correspond to different
pairs of PESs have a similar shape and only differ by a scaling fac-
tor that is proportional to CXY
UU(0)=σ2
eq=2kBTEXY
r. This justifies
J. Chem. Phys. 154, 174105 (2021); doi: 10.1063/5.0050289 154, 174105-9
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FIG. 5 . Three spectral densities JDA(ω),JAG(ω), and JDG(ω), calculated via
Eq. (56) using energy gap time correlation functions CDA
UU(t),CAG
UU(t), and
CDG
UU(t), respectively. The spectral densities are for the bent triad conformation
undergoing donor-to-acceptor transition ππ∗→CT1 (left) and ππ∗→CT2 (right).
The cyan vertical lines indicate the N= 500 mode frequencies extracted from
JDA(ω).
using the frequencies obtained from JDA(ω) to construct the three-
state models (shown as cyan vertical lines in Fig. 5). However, in the
linear triad conformation, the spectral densities JDA(ω) and JAG(ω)
have the same profile for the same transition except for a scaling fac-
tor, while the spectral density JDG(ω) shows different low frequency
FIG. 6 . Three spectral densities JDA(ω),JAG(ω), and JDG(ω), calculated via
Eq. (56) using energy gap time correlation functions CDA
UU(t),CAG
UU(t), and
CDG
UU(t), respectively. The spectral densities are for the linear triad conforma-
tion undergoing donor-to-acceptor transition ππ∗→CT1 (left) and ππ∗→CT2
(right). The cyan vertical lines indicate the N= 500 mode frequencies extracted
from JDA(ω).distributions, but it is two orders of magnitude smaller than the
other spectral densities (bottom of Fig. 6). Likewise, we choose to
use the frequencies from discretizing JDA(ω) to construct the three-
state models of the linear triad (shown as cyan vertical lines in Fig. 6).
From our previous all-atom simulations, the nonequilibrium effect
in the linear triad is negligible, which is reflected by the very small
magnitude in JDG(ω) that would in turn give rise to rather small equi-
librium geometry shifts between the donor and the ground PESs in
the three-state models.
B. Photoinduced CT dynamics of CPC 60in THF via
three-state harmonic models
We present our main results in Fig. 7 for the bent triad and
Fig. 8 for the linear triad, which show the time-dependent CT rate
coefficients for both ππ∗→CT1 andππ∗→CT2 transitions com-
puted with NE-FGR and the hierarchy of semiclassical approxima-
tions including the IMT using models 1–3 and the IMT via spectral
densities [Eqs. (58) and (59)], compared with the IMT rate coeffi-
cient obtained with all-atom NEMD simulations. The general obser-
vation is that all the three models and IMT via J(ω) can generate
accurate time-dependent CT rate coefficients for all the transitions
and triad conformations under consideration. We also note that
the nuclear quantum effects are small in that the fully quantum-
mechanical NE-FGR rate (equivalent to W-AV) is only slightly faster
than the IMT values (within a factor of 2). The initial CT rate coef-
ficients ( t= 0) agree with the corresponding E-FGR CT rate con-
stants obtained with the equilibrium ground state parameters, and
the long-time CT rate coefficients ( t= 4 ps) approach the corre-
sponding E-FGR CT rate constants obtained with the equilibrium
donor (ππ∗) state simulations.
The temporal profiles of k(t) show a significant nonequilib-
rium effect, caused by the initial nuclear sampling, in the case of the
bent triad (Fig. 7). All three three-state harmonic models capture
the nonequilibrium effect remarkably well and are seen to accu-
rately reproduce the results from all-atom NEMD simulations. The
ππ∗→CT1 process exhibits a significant enhancement in the tran-
sient CT rate coefficient, which can be traced back to the fact that the
vertical photoexcitation brings the triad to a region that is close to
the crossing point between the donor and acceptor PESs (see Fig. 1).
In contrast, the ππ∗→CT2 process exhibits an initial slowdown
in the CT rate coefficient, which is due to the fact that the vertical
photoexcitation puts the triad further away from the crossing point,
compared to the equilibrium state.66
For the bent triad, model 1 shows larger fluctuations than
models 2 and 3, which can be traced back to the fact that model
1 is based on flipping the sign of the displacements along ran-
dom modes, whereas models 2 and 3 are based on scaling all
the displacements uniformly. However, the fact that model 1
works in the first place suggests that the NE-FGR nonequilib-
rium CT rate coefficient is largely shaped by the global parameters,
such as EDA
r,EDG
r,EAG
r, rather than the displacements along specific
modes.
The fact that models 2 and 3 produce the same k(t) profiles
confirms our hypothesis that only the nuclear motion along the CT
axis determines the photoinduced CT dynamics, while the relaxation
along the orthogonal axis (included in model 3) does not influence
the CT rate coefficient.
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FIG. 7 . Time-dependent charge transfer rate coefficients k(t) forππ∗→CT1 (top) and ππ∗→CT2 (bottom) transitions, in the bent triad conformation, calculated with
NE-FGR and different levels of approximation using models 1–3 ( N= 500) as well as IMT via spectral densities [Eqs. (58) and (59)], compared with the IMT result obtained
via NEMD (black lines). In model 1, the total number of flips, Nf, are 80 and 180 for ππ∗→CT1 andππ∗→CT2 transitions, respectively. The corresponding E-FGR
charge transfer rate constants obtained with different levels of approximation using equilibrium ground state parameters ( t= 0) are shown on the left of each panel, and those
obtained using equilibrium ππ∗state parameters ( t=∞) are shown on the right of each panel. The insets show the comparison of the long-time CT rate coefficients and the
E-FGR CT rate constants.
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FIG. 8 . Time-dependent charge transfer rate coefficients k(t) forππ∗→CT1 (top) and ππ∗→CT2 (bottom) transitions in the linear triad conformation, calculated with
NE-FGR and different levels of approximation using models 1–3 ( N= 500) as well as IMT via spectral densities [Eqs. (58) and (59)], compared with the IMT result obtained
via NEMD (black lines). In model 1, the total number of flips, Nf, is 295 and 245 for ππ∗→CT1 andππ∗→CT2 transitions, respectively. Other settings of E-FGR rate
constants and the insets are the same as Fig. 7 except for using the parameters of the linear triad.
J. Chem. Phys. 154, 174105 (2021); doi: 10.1063/5.0050289 154, 174105-12
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In the insets of models 1–3, the orders of the NE-FGR long-
time rate coefficients and E-FGR rate constants calculated at dif-
ferent levels of approximation are similar with the W-AV result
corresponding to the fastest CT rate and the Marcus-level result
corresponding to the slowest CT rate. This suggests that nuclear
quantum delocalization can enhance the CT rate in this system
somewhat. The IMT expression based on the three continuous spec-
tral densities JDA(ω),JAG(ω), and JDG(ω) [Eqs. (58) and (59)] is
also seen to reproduce the all-atom IMT CT rate coefficient rather
well.
In contrast to the bent conformation, the linear triad does
not show significant nonequilibrium effects in both ππ∗→CT1
andππ∗→CT2 transitions (see Fig. 8). This result is consis-
tent with the all-atom IMT result. All the three-state harmonic
models and the IMT via J(ω) predict that the photoinduced CT
dynamics follows Marcusian kinetics except for the very small
initial relaxation in the CT rate coefficient for the ππ∗→CT1 pro-
cess since the nonvanishing EDG
rgives rise to slight DG shifts in
our three-state harmonic models, which were obscured by the fluc-
tuations in the all-atom NEMD simulations. In the ππ∗→CT1
process, the IMT via spectral densities is seen to overestimate the
initial CT rate coefficient although it approaches the equilibrium
Marcus rate constant in 10 ps, while models 1–3 show better
agreement with the NEMD benchmark result. In the ππ∗→CT2
transition, model 1 shows larger fluctuations compared with
models 2 and 3, and the fluctuation is comparable to the error bar
size in the NEMD result. We conclude that all four approaches
generate accurate photoinduced CT dynamics in the linear
triad.
The donor state population can be computed from the time-
dependent CT rate coefficient using Eq. (5), and we show the result
of model 2 for all the CT processes in Fig. 9. The deviation between
FIG. 9 . Donor state population decay for ππ∗→CT1 (left) and ππ∗→CT2 (right)
transitions in the bent (top) and linear (bottom) triad conformations, calculated with
NE-FGR and different levels of approximation using model 2 ( N= 500), compared
with the IMT results obtained via NEMD (black lines) and the Marcus rate constant
predictions (blue dashed lines).the NE-FGR predictions and that of the Marcus CT rate constants
are significant for the ππ∗→CT1 in the bent triad conforma-
tion, which displays the largest nonequilibrium effects in the pho-
toinduced CT process, whereas for the rest of the cases, the dif-
ferences between NE-FGR population dynamics and Marcus the-
ory are small. Figures S1–S3 of the supplementary material show
similar behaviors for the donor state population calculated with
model 1, model 3, and the IMT via the spectral density approach,
respectively.
Figure 10 shows the IMT parameters including the time-
dependent donor–acceptor energy gap Utand its variance σ2
tas well
as the IMT rate coefficients for the bent triad calculated using models
1–3, the IMT expression via spectral densities [Eqs. (58) and (59)],
and the benchmark IMT result from NEMD. Model 1 exhibits more
fluctuations in the energy gap and therefore in k(t). Note that here
the variance of the energy gap is time-independent and exactly the
same for those obtained using models 1–3, and it differs from the
one obtained using IMT expression via spectral densities by only 1%.
The IMT parameters for the linear triad also give rise to a reasonably
accurate prediction of the time-dependent CT rate coefficient and is
shown in Fig. S4. All the Marcus parameters (e.g., ΓDA, |ΔE|, and Er)
as well as the Marcus CT rate constants kMfor all cases are provided
in Table S2.
The generally excellent agreement of the IMT result with the
NE-FGR rate coefficient in all models suggests that the nuclear
FIG. 10 . Comparison of Ut(top),σ2
t(middle), and IMT rate coefficient k(t) (bottom)
forππ∗→CT1 (left) and ππ∗→CT2 (right) transitions in the bent triad confor-
mation calculated using models 1–3 ( N= 500), IMT via spectral densities (orange
lines), and IMT results obtained via NEMD (black lines).
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FIG. 11 . The effect of nuclear degrees of freedom Non the time-dependent charge transfer rate coefficients k(t) forππ∗→CT1 transition in the bent triad conformation
calculated with NE-FGR and different levels of approximation using model 1 (top panels, N= 100, 500, 1000 with random flips Nf= 16, 80, 160, respectively, as well as
N= 500 with evenly flips every six modes, Nf= 80) and model 2 (bottom panels, N= 100, 200, 500, 1000). Other settings of E-FGR rate constants and the insets are the
same as Fig. 7.
J. Chem. Phys. 154, 174105 (2021); doi: 10.1063/5.0050289 154, 174105-14
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quantum effects on the CT rate coefficient are small in this triad
system, which was also observed in our previous study, where map-
ping to a spin-boson model was used for calculating E-FGR CT
rate constants.72Moreover, when dealing with large complex sys-
tems, whenever IMT is valid, using the IMT level of approxima-
tion would be remarkably advantageous since the computational
cost is many orders of magnitude lower compared with the other
LSC-based approximations of NE-FGR.66
Finally, we discuss the dependence of the results on the number
of nuclear modes in the harmonic model, N. Figure 11 depicts the
NE-FGR time-dependent CT rate coefficients for the ππ∗→CT1
transition in the bent triad conformation obtained using model 1
(top panels, N= 100, 500, and 1000 with random flipped modes,
andN= 500 with evenly flipped modes) and model 2 (bottom panels,
N= 100, 200, 500, and 1000 with scaled modes). In general, incorpo-
rating more nuclear modes in both models 1 and 2 damps the fluc-
tuations in k(t) and yields a smoother relaxation profile. However,
the general trend of the predicted nonequilibrium rate coefficient
virtually remains unchanged using different N, and all models with
different Nseem to be able to capture both the transient behavior
and the long-time limit in the nonequilibrium CT rate coefficients
remarkably well.
As discussed above, the fact that model 1 works suggests a
collective solvent effect, where the contributions from individual
modes are relatively small. For example, comparing the N= 500
results with random flipped modes and the N= 500 results with
evenly flipped modes (see the top right panels of Fig. 11), we barely
observe any significant difference in the k(t) evaluated on all levels
of approximation. In addition, Fig. S5 displays the statistical averages
ofk(t) as obtained from ten independent parameter sets of model 1
constructed using different sets of { Sj} whose signs were flipped
randomly, where the error bars are notably small. It is noted that
the number of flipped modes Nfis the same for both stochastic
and deterministic approaches since the AG reorganization energy
change in flipping any mode is the same (see Table S3), as a result
of our spectral density discretization procedure that equally parti-
tions the reorganization energy. The fact that model 1 with different
random sets of flipped modes produces very similar k(t) indicates
that its stochastic uncertainties are generally negligible, and thus,
model 1 is robust in the simulation of nonequilibrium CT dynamics.
Similar figures showing the influence of changing nuclear degrees
of freedom for other combinations of transitions and conforma-
tions are given in Figs. S6–S8 for model 1 and Figs. S9–S12 for
model 2.
VI. CONCLUDING REMARKS
In this paper, we proposed and demonstrated three protocols
for constructing effective three-state harmonic model Hamiltonians
and an alternative strategy to compute the IMT rate coefficient
via the spectral densities in the context of simulating photoin-
duced CT rates, where the nonequilibrium effects may be sig-
nificant due to the initial preparation of the nuclear DOF. Our
proposed three-state models (models 1–3) are based on map-
ping of the all-atom anharmonic Hamiltonian onto a three-state
harmonic Hamiltonian. The accuracy and reliability of the three
models and the IMT via spectral density approach were assessed
by using this strategy to calculate the quantum-mechanical NE-FGRtime-dependent rate coefficient and the hierarchy of semiclassical
approximations leading to the classical IMT in the CPC 60molec-
ular triad dissolved in explicit THF, which were compared with
the benchmark time-dependent CT rate coefficient obtained from
all-atom NEMD simulations.
The model construction process starts out with taking inputs
from MD simulations in the form of time correlation functions of
the energy gaps between different PESs, which are then mapped
onto three spectral densities. JDA(ω) is further discretized to N
nuclear modes constituting the CT axis parameters shared by
models 1–3. For model 1, we employ a one-dimensional CT axis
and randomly flip the shifts Sjbetween the donor and ground state
PESs along this CT axis so as to obtain consistently the value of
EAG
robtained from MD simulations. Models 2 and 3 are based
on a two-dimensional PES scaling approach, where in model 2 we
project the nonequilibrium shift on the CT axis and in model 3
we extend the nuclear DOF of model 2 from Nto 2Nby includ-
ing the projection of the nonequilibrium shift along the orthogonal
axis.
We also showed that IMT can be formulated using only the
spectral densities, which can be obtained directly from the MD
simulations. Our results show that all three models and the IMT
via spectral densities produce remarkably accurate NE-FGR rate
coefficients for all the transitions and triad conformations inves-
tigated here. The results obtained at different levels of LSC-based
approximations are all in agreement with each other, which shows
that nuclear quantum effects are minimal for the system under
consideration. For the IMT level of approximation, the IMT
parameters and the time-dependent rate coefficients obtained via
different strategies are seen to coincide with each other. To provide
in depth coverage on the characteristic behavior of our models, we
also provided an assessment concerning the influence of the number
of modes on the model performance and accuracy.
In summary, we have demonstrated the feasibility of three
strategies to construct three-state harmonic models that could
be used to investigate the significance of the nonequilibrium
effects due to initial nuclear preparation and the nuclear quantum
dynamical effects in the photoinduced charge transfer dynam-
ics in the condensed phase. As a natural extension of this work,
it would be desirable to have a comprehensive study for apply-
ing the models proposed here to more complicated multi-level
molecular systems and combining with other condensed-phase
quantum dynamical methods such as tensor-train split-operator
Fourier transform,94mixed quantum-classical Liouville,95quasiclas-
sical mapping Hamiltonian,96,97and generalized quantum master
equation,34,35as well as hierarchical equations of motion39–41and
stochastic equations of motion.98,99It is also desirable to extend the
current treatment of NE-FGR to more than three states, such as
incorporating CT1 →CT2 transition. Work toward those goals is
currently under way and will be reported in future publications.
SUPPLEMENTARY MATERIAL
See the supplementary material for the derivation of the IMT
expression for the three-state harmonic model, the tabulated data
of energy corrections for the MD potential energy of different elec-
tronic states, the Marcus charge transfer rate constants evaluated
J. Chem. Phys. 154, 174105 (2021); doi: 10.1063/5.0050289 154, 174105-15
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using both ground and ππ∗electronic-state MD simulations, the
plateau values of time-dependent NE-FGR rate coefficients cal-
culated using different models for all transitions and conforma-
tions, and other auxiliary information for determining the num-
ber of flips for model 1. We also provide the figures for donor
state population of all cases evaluated using models 1 and 3, IMT
parameters and rate coefficients for linear triad conformations
obtained with different models, the stability test of the stochastic
effect and the effect of changing nuclear DOF for model 1 in all
cases, and the effect of changing nuclear DOF for model 2 in all
cases.
ACKNOWLEDGMENTS
X.S. acknowledges support from NYU Shanghai, the National
Natural Science Foundation of China (Grant No. 21903054), the
Hefei National Laboratory for Physical Sciences at the Microscale
(Grant No. KF2020008), and the Program for Eastern Young Scholar
at Shanghai Institutions of Higher Learning. E.G. and B.D.D.
acknowledge support from the Department of Energy (DOE),
Basic Energy Sciences through the Chemical Sciences, Geosciences
and Biosciences Division (Grant No. DE-SC0016501). Computing
resources were provided by NYU Shanghai HPC.
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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1.3700154.pdf | A geometrical correction for the inter- and intra-molecular basis set
superposition error in Hartree-Fock and density functional theory
calculations for large systems
Holger Kruse and Stefan Grimme
Citation: J. Chem. Phys. 136, 154101 (2012); doi: 10.1063/1.3700154
View online: http://dx.doi.org/10.1063/1.3700154
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Downloaded 25 Nov 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissionsTHE JOURNAL OF CHEMICAL PHYSICS 136, 154101 (2012)
A geometrical correction for the inter- and intra-molecular basis set
superposition error in Hartree-Fock and density functional theory
calculations for large systems
Holger Kruse1and Stefan Grimme2,a)
1Theoretische Organische Chemie, Organisch-Chemisches Institut der Universität Münster, Corrensstr. 40,
D-48149 Münster, Germany
2Mulliken Center for Theoretical Chemistry, Institut für Physikalische und Theoretische Chemie
der Universität Bonn, Beringstr. 4, D-53115 Bonn, Germany
(Received 9 February 2012; accepted 15 March 2012; published online 16 April 2012)
A semi-empirical counterpoise-type correction for basis set superposition error (BSSE) in molec-
ular systems is presented. An atom pair-wise potential corrects for the inter- and intra-molecular
BSSE in supermolecular Hartree-Fock (HF) or density functional theory (DFT) calculations. This
geometrical counterpoise (gCP) denoted scheme depends only on the molecular geometry, i.e., no
input from the electronic wave-function is required and hence is applicable to molecules with tenthousands of atoms. The four necessary parameters have been determined by a fit to standard Boys
and Bernadi counterpoise corrections for Hobza’s S66 ×8 set of non-covalently bound complexes
(528 data points). The method’s target are small basis sets (e.g., minimal, split-valence, 6-31G*), butreliable results are also obtained for larger triple- ζsets. The intermolecular BSSE is calculated by
gCP within a typical error of 10%–30% that proves sufficient in many practical applications. The ap-
proach is suggested as a quantitative correction in production work and can also be routinely appliedto estimate the magnitude of the BSSE beforehand. The applicability for biomolecules as the primary
target is tested for the crambin protein, where gCP removes intramolecular BSSE effectively and
yields conformational energies comparable to def2-TZVP basis results. Good mutual agreement isalso found with Jensen’s ACP(4) scheme, estimating the intramolecular BSSE in the phenylalanine-
glycine-phenylalanine tripeptide, for which also a relaxed rotational energy profile is presented. A
variety of minimal and double- ζbasis sets combined with gCP and the dispersion corrections DFT-
D3 and DFT-NL are successfully benchmarked on the S22 and S66 sets of non-covalent interactions.
Outstanding performance with a mean absolute deviation (MAD) of 0.51 kcal/mol (0.38 kcal/mol
after D3-refit) is obtained at the gCP-corrected HF-D3/(minimal basis) level for the S66 benchmark.
The gCP-corrected B3LYP-D3/6-31G* model chemistry yields MAD =0.68 kcal/mol, which repre-
sents a huge improvement over plain B3LYP/6-31G* (MAD =2.3 kcal/mol). Application of gCP-
corrected B97-D3 and HF-D3 on a set of large protein-ligand complexes prove the robustness of
the method. Analytical gCP gradients make optimizations of large systems feasible with small basis
sets, as demonstrated for the inter-ring distances of 9-helicene and most of the complexes in Hobza’sS22 test set. The method is implemented in a freely available
FORTRAN program obtainable from the
author’s website. © 2012 American Institute of Physics .[http://dx.doi.org/10.1063/1.3700154 ]
I. INTRODUCTION
Quantum chemically computed supermolecular interac-
tion energies and molecular structures are subject to the ba-sis set superposition error (BSSE) in an incomplete, atom-
centered one-particle basis set.
1–4
We like to split the general term basis set error (BSE),
following Ref. 1, into the basis set superposition error
(BSSE) and the basis set incompleteness error (BSIE). The
BSSE arises due to an unbalanced basis set expansion ofmonomers and the ensuing multimer complex in the super-
molecular approach of calculating interaction energies. In a
dimer complex of the moieties A and B, the basis set of thecomplex is larger than the basis sets of A and B because unoc-
cupied basis functions from A can be used by B to lower the
a)Electronic mail: grimme@thch.uni-bonn.de.energy (and vice versa). In the case of complexation (interac-
tion) energies, this lowering of the energy leads to artificial-
or over-binding of complexes. The famous counterpoise (CP)
scheme by Boys and Bernadi5(BB) can be used to remedy
this unbalanced description. For a dimer it reads
/Delta1ECP=[E(A)a−E(A)ab]+[E(B)b−E(B)ab]
where a, b are the basis functions of monomer A, B (in their
frozen complex geometries). /Delta1ECP, which is always positive
in this formulation, is added to the binding energy (BE) of the
complex yielding the corrected BECP
BECP=BE+/Delta1ECP.
The BB-CP correction covers only the case of intermolecular
BSSE of non-covalently bound dimer complexes, but in fact
does any close lying molecular assembly (another monomer,
0021-9606/2012/136(15)/154101/16/$30.00 © 2012 American Institute of Physics 136 , 154101-1
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alkyl side-chains or bulky substituents) “offer” its unoccupied
(virtual) basis functions to another part of the molecule, re-
ducing the energy of the whole assembly. This extends theidea of BSSE to the intramolecular case (IBSSE), although a
uniform, clear definition is missing.
Experience shows that already at polarized triple- ζbasis
quality /Delta1E
CPis reduced to about 10% of the interaction en-
ergy in self-consistent field (SCF) type treatments. However,
ab initio studies on large biochemical systems using orbital-
based quantum chemical methods are limited by non-linearly
increasing computational costs if reasonable large triple- ζba-
sis sets (or above) are employed. Schemes to deal with this
problem either on a technical (code parallelization6,7), al-
gorithmical (linear scaling techniques,8,9divide-and-conquer
approaches10–12) or theoretical (hybrid schemes such as
QM/MM (Ref. 13)) level are still actively developed and not
yet “black box”.
Today’s best cost-accuracy ratio is provided by den-
sity functional theory14,15(DFT). For large systems
such as biomolecules,16,17nanostructured materials,18,19or
supramolecular chemistry20–22weak interactions play a domi-
nant role in their stability and reactivity. A fundamental prob-
lem of all current semi-local and conventional hybrid DFTfunctionals is their inability to provide the asymptotically
correct −C
6/R6dependence of the London dispersion inter-
action energy on the inter-atomic/molecular distance Rfor
these weak, non-covalent interaction.23–25Various approaches
to tackle this problem were proposed in the literature; for a re-
cent review see, e.g., Ref. 26. Herein, we will consider the lat-
est dispersion-correction schemes DFT-D3 (Refs. 27and28)
and DFT-NL.29,30
Very few limitations are set to the computational chemist
having the latest computer codes and the newest supercom-
puter at hand. Within typical limited resources, however, oneis forced to apply rather small double- ζbasis sets of various
sizes (6-31G*, 3-21G) often out of necessity.
31,32The hope is
that the BSSE of relative energies is tolerable if only a quali-tative understanding is sought. Validation studies are only vi-
able on smaller systems, and as no simple BSSE propagation
from smaller to larger systems can be expected, the errors forthese (very) large systems remain uncertain. Even for smaller
systems (20–80 atoms) it seems that many authors find the use
of triple- ζbasis sets prohibitive for their computational stud-
ies using hybrid density functionals, which can be seen from
the vast number of B3LYP/6-31G* applications.
The BSSE problem has many aspects that have been in-
vestigated in the past and we mention here some of which
are relevant in the present context. Different formulations
for multi-body complexes (clusters) are described in the
literature,
33–38they differ in the number of terms used to cal-
culate the counterpoise correction. A central question is, ifand how the so-called second order basis-set effects (affecting
the individual pair-interactions in the cluster) should be ac-
counted for. A critical examination of the different approachescan be found in Ref. 38. Developments of (supermolecular)
methods that exclude BSSE by constructions as the chemi-
cal Hamiltonian approach (CHA) (Ref. 39) have not found
widespread use, partially due to its involved theoretical foun-
dation and technical difficulties.
40Gill et al. propose to re-duce the BSSE in the framework of dual-basis set schemes,
using the monomer basis as primary basis and the complex
basis as secondary basis in his Hartree-Fock (HF) or den-sity functional perturbative correction (HFPC or DFPC) ap-
proach, and reports major computational savings.
41Aiming
at the biomolecular community, Merz et al. proposed a sta-
tistical, fragment-based model to estimate quickly intra- and
inter-molecular BSSE of protein structures,42but the need of
an internal classification of the fragment type makes it lessuniversal.
The problem of IBSSE,
43–47e.g., for conformational en-
ergies or isomerization reactions, can be approximated in
the framework of the BB-CP correction – if one allows
breaking of covalent bonds – by defining a suitable numberof fragments.
48,49The choice of the fragmentation scheme
is crucial and introduces an unwelcoming arbitrariness, be-
sides technical difficulties of charge and spin-state of thefragments. It can be broken down all the way towards in-
dividual atoms, leading to N
atoms additional computations.
Valdés et al. demonstrated the application of this in their
atom by atom scheme (CPaa) called approach50for esti-
mating the intramolecular BSSE in phenylalanine-glycine-
phenylalaine (FGF) tripeptide. Jensen extended this idea inhis atomic counterpoise correction (ACP(x)) (Ref. 51)b ys e -
lecting only a specific subset of the whole complex-basis set
(including basis functions from atoms x bonds apart), ar-guing that only not-directly bound atoms contribute to the
BSSE. Jensen’s ACP(1), i.e., including also directly neighbor-
ing atoms, equals Galanos CP
aamethod. It was further argued
by Jensen that the restriction to atoms reduces the computa-
tional costs of each calculation tremendously, because of ef-ficient integral screening effects and a certain locality of the
BSSE.
51One nice feature is that inter- and intra-molecular
BSSE are treated on the same conceptual level.
Many of the newly proposed BSSE correction schemes
lack discussions of nuclear gradients. These are of utmost im-
portance since a BSSE contaminated structure optimization isnot expected to yield a good interaction energy at any higher
level of theory.
Herein, we propose a correction scheme that can estimate
the inter- and intra-molecular BSSE for HF/DFT calculations
with various (small to medium sized) basis sets. The main
aims of this project are: (1) To provide a fast, conceptuallysimple energy- and gradient-correction for the BSSE in com-
putations of large molecules, where small basis sets often can-
not be avoided because of limited computer resources. (2) Tosupply a tool for a quick BSSE estimation without the explicit
need of expensive CP calculations.
The central idea is to estimate the CP correction solely
based on the Cartesian coordinates of the molecule or com-
plex, i.e., no input from its wave-function and no informa-tion about connectivity (“bonding”) are required. A strong
motivation of our scheme is certainly to keep the computa-
tional overhead as low as possible. We denote the schemegeometrical counterpoise (gCP) correction. The name im-
plies that it is fitted against the conventional BB-CP correc-
tion and that the geometry of the molecules is the only infor-mation required. It is an atomic correction in the sense that
the BSSE of a molecule or complex is evaluated by additive
Downloaded 25 Nov 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions154101-3 H. Kruse and S. Grimme J. Chem. Phys. 136 , 154101 (2012)
atomic contributions. One main assumption here is that the
BB-CP correction is “right”. Although this is still theoreti-
cally under debate,1,38,52–54clearly it works very well in par-
ticular for small basis sets, which are our primary target. Ob-
viously, in a complete basis no BSSE arises. Therefore, one
needs a measure of basis set completeness (quality) that wedefine as the energy differences between a large (nearly com-
plete) and a smaller (target) basis. We can use this measure
to estimate atomic BSSE contributions in molecules, by ac-counting for the interatomic distance decay of the BSSE with
a semi-empirical potential (four global parameters per basis
set/method combination). The potential includes the number
of unoccupied basis functions assigned to the atoms (virtual
orbitals) and an estimate for the overlap of the valence or-bitals. The parameters are fitted against the BB-CP correc-
tions for the recent S66 ×8 benchmark set
55of non-covalent
interactions. We want to stress at this point already that theproposed correction is not meant as some general substitute
for calculations that can easily be done with triple- ζquality
basis sets or higher. Large basis sets are the preferable way togo whenever this is technically possible and we are perform-
ing more and more routine calculations in our group at the
quadruple- ζlevel.
The outline of the paper is as follows: First, the theory of
the method is presented. Second, some technical details and
the fit procedure are discussed. Finally, the performance ofthe correction is demonstrated on a few illustrative examples
including inter- and intramolecular BSSE, and for geometry
optimizations.
II. THEORY
The central idea is to add in a semi-empirical fashion an
energy correction /Delta1EgCPto the energies of molecular systems
in order to remove artificial overbinding effects from BSSE.
As the focus lies on the contribution of individual atoms a nat-
ural outcome is its ability to yield also intramolecular BSSEcorrections. The parameterization is constructed such, that it
approximates the BB-CP correction /Delta1E
CPfor dimers
/Delta1ECP≈/Delta1EgCP, (1)
where, e.g., for a complexation reaction A+B→Cour cor-
rection is given by
/Delta1EgCP=EgCP(C)−EgCP(A)−EgCP(B). (2)
In practice, EgCPcan simply be added to the HF/DFT energy
Etotal=EHF/DFT +EgCP. (3)
In the following, the details of the gCP correction will
be outlined. EgCPcontains four parameters specific for a
given Hamiltonian (HF or KS-DFT) and basis set combina-
tion. The atomic contributions are globally scaled by the fit
parameter σ
EgCP=σ·atoms/summationdisplay
aEatom
a. (4)
The atomic contributions Eatom
a are obtained by multiply-
ing the energy emiss
a, which measures the incompleteness forthe chosen target basis set for atom awith a decay function
fdec(Rab) depending on the inter-atomic distance Rab
Eatom
a=atoms/summationdisplay
b/negationslash=aemiss
a·fdec(Rab). (5)
Theemiss
aterms are obtained from the atomic energy differ-
ence between a large (nearly complete) basis set and the tar-get basis set. The quantity is pre-computed and supplied for a
wide range of basis sets (see Sec. IV A ) and computed as
e
miss
a=Etarget basis −Elarge basis |F=0.6a.u., (6)
where the index F=0.6 a.u. denotes an applied electric field
of 0.6 a.u. Energy minimization of an atomic wave-functionwill generally not properly populate higher angular momen-
tum (polarization) functions in the basis set. The ground state
energies of a single hydrogen atom at the HF/SV and HF/SVPlevel of theory, e.g., are identical. To account for the pop-
ulation of (molecular) polarization functions that occur in
molecules, we apply a weak electric field Fof 0.6 a.u. in the
first Cartesian quadrant ( x=y=z=0.03464) and perform
restricted open-shell Hartree-Fock
56(ROHF) calculations to
obtain Etarget basis andElarge basis . The calculations are based on
the Turbomole code57–59without any symmetry constraints
and refer to state average solutions for 3d-transition metals.
Because of SCF convergence problems in the correspondingROKS calculations, we use the e
miss
aenergies from ROHF cal-
culations also for the DFT parameterization.
From test calculations and comparisons for H 2and CH 4
molecules, a field strength of 0.6 a.u. is found to populate thep/d-orbitals in the atomic calculations reasonably well. Note,that the correct modeling of the atomic orbital populations in
molecules is neither desired – nor necessary – here. The field
strength, if kept within reasonable limits, has only a minorinfluence on the overall performance of the model.
The second term in Eq. (5), the decay function f
dec(Rab),
is given by
fdec(Rab)=exp/parenleftbig
−α·Rabβ/parenrightbig
/radicalBig
Sab·Nvirtb
b, (7)
where the interaction factor exp( −α·Rabβ) is normalized by
the square-root of the Slater-overlap Sabtimes the number of
virtual orbitals on atom b. Many different kinds and combina-
tions of functions were tested. Starting from a more flexible
function (more fit parameters) some of them could be elimi-
nated during the fitting procedure, and it turned out that a ra-tional function of two exponentials (the overlap is considered
as a complicated exponential) yields favorable performance.
The square-root in the denominator results from one of theeliminated fit parameters. The parameters αandβare crucial
and determine the performance most strongly. The number of
virtual orbitals N
virt
bon atom bis straightforwardly obtained
by subtracting the number of electron pairs ( Nel
bbeing the to-
tal number of electrons of atom b) from the number of basis
functions Nbf
bin the target basis set for b
Nvirt
b=Nbf
b−Nel
b
2. (8)
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The overlap integral in Eq. (7)is evaluated over a single
Slater-type orbital centered on each atom. Optimized Slater
exponents ζoptare taken from an extended Hückel theory
study by Herman.60Thesandpvalence exponents are aver-
aged to get a single s-type orbital ζs. Thus, the valence overlap
Sabis calculated as
ζs=η·ζopt,s+ζopt,p
2, (9)
Sab=/angbracketleftζs|ζs/angbracketright, (10)
where ηis the last of the four fit parameters.
III. IMPLEMENTATION DETAILS
The method is implemented into a freely available
FORTRAN code.61Overlap integrals up to principle quantum
number three, i.e., up to Sab=/angbracketleft3s|3s/angbracketright, are available. For all
4th-row and higher elements 3 s-type valence orbitals are em-
ployed. The d-functions of transition metals are treated in the
same manner as the s,p-valence shells, i.e., the s,p, and
dexponents are reduced to a s-function by a simple arith-
metic mean of the three exponents. To allow for elements
that are not parameterized the program will internally sub-
stitute elements by their 4th-row homologous, e.g., Pb willbe replaced with Ga, I with Br, Os with Fe, and so on. Un-
parametrized 4th-row elements will give zero contribution in
the pair-potential. The error introduced by this fall-back al-gorithm should be small if only few atoms within a larger
complex are replaced or neglected.
The computational complexity of energy and gradient
evaluation formally scales with ( N
atoms)2. The prefactor is
very small, resulting in very fast computations (a few sec-
onds) even for large molecules with 2000 atoms. Jensen’sinvestigations
51indicate a certain locality of the BSSE by
showing that for HF with regular basis sets ACP contributions
of atoms beyond 7 Å (for augmented basis sets 10 Å) can beneglected for an accuracy of about 0.1 kcal/mol. A huge speed
up is gained by exploiting this locality with a proper cut-off
distance. We make use of a value of 60 Bohrs (conservativeupper bound), which results in practically no loss in accuracy
but leads already to a large decrease in computation time.
For each atom-pair one overlap integral has to be evalu-
ated. The analytical implementation is done via the auxiliary
integrals A and B (below including sum representation) withx
a=(ζa+ζb)/2 and xb=(ζa−ζb)/2
Ak(xa)=/integraldisplay∞
1ξke−xaξdξ, (11)
=e−xak!
(xa)k!+1k/summationdisplay
v=0(xa)v
v!, (12)
Bk(xb)=/integraldisplay+1
−1χke−xbχdχ, (13)=exbk!
(xb)k!+1k/summationdisplay
v=0(−xb)v
v!(14)
−e−xbk!
(xb)k!+1k/summationdisplay
v=0(xb)v
v!. (15)
Starting from the overlap integral for quantum number
na=nb=0
/angbracketleft0s|0s/angbracketright=1
2A0B0,
alls-type overlap integrals over A and B integrals can be gen-
erated by applying successively the following rule (Lofthus
algorithm, see Ref. 62): Ifnaincreases by 1 every AkBlrefor-
mulates into ( Ak+1Bl+AkBl+1). Similarly does the increase
ofnbby 1 lead to ( Ak+1Bl−AkBl+1).
Analytical derivatives of each overlap integral-type are
generated using the Maple algebra tool.63The resulting gCP
gradient takes only a little more (below a factor of two)time than the gCP energy correction. A test molecule with
48 827 atoms takes about 76 s for the energy and 118 s for
the gradient correction on an Intel Xeon E5430 (2.66 Ghz)workstation.
IV. COMPUTATIONAL DETAILS
Most calculations were either carried out with the
TURBOMOLE suite of programs (a locally modified ver-
sion of TURBOMOLE 5.9 and the recent version 6.3
(Refs. 57–59)) or with a development version of ORCA
(Ref. 64). The GGA functionals BLYP,65–67B97-D,68
revPBE,69the meta-GGA TPSS (Ref. 70) the hybrids
PW6B95,71B3LYP,72,73and revPBE0 (Ref. 74) are used. For
the M06-2X (Ref. 75) meta-hybrid functional GAUSSIAN 09
(Ref. 76) was used.
The DFT-D3 corrections27,28(both damping-function
variants) were applied with our group own program dftd3 .
Becke-Johnson (BJ) damping28,77,78is the default damping
function used, i.e., DFT-D3 always corresponds to DFT-
D3(BJ). In the case of M06-2X, the original D3-damping
function (denoted zero-damping) is applied. The functionals’specific parameters were determined in Refs. 27,28,74, and
79. A variety of basis sets is employed: The Ahlrichs-type ba-
s i ss e t sS V ,
80def2-SVP, def2-TZVP,81,82and def2-QZVP,82,83
Hunzingas valence-scaled version of his minimal basis MINI
(denoted MINIS) is used as provided by the EMSL ba-
sis set exchange website84,85and the Pople-style basis
set 6-31G*.86
For the def2-SVP, def2-TZVP, and def2-QZVP basis sets,
the TURBOMOLE calculations for (meta-)GGA and hybrid
functionals uses the resolution of the identity (RI-J) approxi-
mation for the Coulomb part.87ORCA employs the Split-RI-
J variant.88For def2-QZVP calculations with Turbomole and
ORCA, the RI-K approximation to the exchange integrals89
is additionally used. All auxiliary basis functions were takenfrom the Turbomole basis set library.
90,91If not denoted other-
wise the Turbomole grid m591was used.92–97The calculations
for the crambin protein employed the smaller m3grid.
Downloaded 25 Nov 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions154101-5 H. Kruse and S. Grimme J. Chem. Phys. 136 , 154101 (2012)
The development version of ORCA was used for all
calculations using the non-local (NL), density-dependent
dispersion correction DFT-NL.29,30The keywords grid4
andvdwgrid3 specify the integration accuracy of the
exchange-correlation and the NL-part, respectively. The cou-
pled electron pair (CEPA, version 1) calculations make useof the local pair natural orbital (LPNO) approximation (de-
noted LPNO-CEPA/1) (Refs. 98and99) as implemented
100
in ORCA. A complete basis set (CBS) two-point extrapola-
tion for LPNO-CEPA/1 was done using the Halkier extrapo-
lation scheme101,102(separate extrapolation of SCF and cor-
relation energy) using def2-TZVP and def2-QZVP (termed
CBS(3,4)).
The symmetry adapted perturbation theory (SAPT)
(Ref. 103) calculations are done with MOLPRO (Ref. 104)
(default settings) employing the aug-cc-pVTZ (Ref. 105)
basis set.
A. Fitting procedure
The recently introduced benchmark set S66 ×8i su s e d
to generate CP data on which the gCP parameters are fit-
ted. The set consists of 66 dimers at eight different distances.
The dimers are combinations of 14 different monomers
with one another: acetic acid, acetamide, benzene, cyclopen-tane, ethene, ethyne, neopentane, n-pentane, methylamine,
methanol, N-methylacetamide, pyridine, uracil, and water (for
a list of all 66 dimers, see Ref. 55). They have been classified
according to dominating intermolecular interactions so that 23
complexes are formed by hydrogen bonds, 23 are dominated
by dispersion interactions and 20 complexes are equally dom-inated by dispersion and electrostatic interactions. Prominent
interactions such as π-πstacking (ten systems), aliphatic-
aliphatic interactions (five systems), and π-aliphatic interac-
tions (eight systems) display a wide range of interactions typ-
ical in large organic and biological systems.
For all 66 complexes counterpoise-corrected MP2/TZVP
structures were computed. For each dimer “the closest inter-
molecular distance in the complex along an intermolecular
axis” (Ref. 55) was identified and the monomer-monomer
distances were varied (frozen monomers) along this axis
so that a dissociation curve is generated (two distances be-low and five above equilibrium distance), where the longest
distance is twice the equilibrium distance. The first five
structures of these systems were taken and by interpolat-ing MP2/CBS +/Delta1CCSD(T)/aDZ single-point energies with a
fourth-order polynomial the energetically optimal intermolec-
ular distance was determined. These “minima” constitute theS66 benchmark set used in Sec. VC for benchmarking non-
covalent interactions.
For each of the 528 dimers in the set the standard BB-CP
correction is calculated for a target method (e.g., HF/MINIS),
and the four parameters in the gCP scheme are fitted in
a least-squares sense (i.e., minimization of the root-mean-square deviation, RMSD). The weight of the errors for the
shortest distance in the set is reduced to 0.5. This focuses
the correction slightly to equilibrium geometries (important
for good structures) and longer distances (important for large
biomolecules).TABLE I. Parameters for the gCP correction and fit quality (RMS deviation
in kcal/mol) of the fit against Boys and Bernadi CP values for the S66 ×8
dimer geometries. The arithmetic mean CPof the BB-CP correction is also
given (in kcal/mol).
σηαβ CP RMSD
HF/MINIS 0.1290 1.1526 1.1549 1.1763 1.02 0.30
HF/SV 0.1724 1.2804 0.8568 1.2342 1.16 0.32HF/SVP 0.2054 1.3157 0.8136 1.2572 1.10 0.41
HF/6-31G(d) 0.2048 1.5652 0.9447 1.2100 1.02 0.40
HF/def2-TZVP 0.3127 1.9914 1.0216 1.2833 0.20 0.12B3LYP/MINIS 0.2059 0.9722 1.1961 1.1456 1.10 0.34
B3LYP/SV 0.4048 1.1626 0.8652 1.2375 1.64 0.56
B3LYP/SVP 0.2990 1.2605 0.6438 1.3694 1.64 0.65B3LYP/6-31G(d) 0.3405 1.6127 0.8589 1.2830 1.43 0.48
B3LYP/def2-TZVP 0.2905 2.2495 0.8120 1.4412 0.32 0.20
In principle, parameters have to be fitted for every com-
bination of Hamiltonian and basis set (model chemistry). The
efforts to account for the broad range of common densityfunctionals (plus Hartree-Fock) combined with a variety of
basis sets are huge and beyond the scope of this study. We de-
cided to supply a few practically useful combinations as listedin Table I. We suggest to use the B3LYP fit parameters also
for any other density functional. As will be shown later, the
accuracy gained by a re-fit for different density functionals isnegligible because different functionals provide rather similar
orbitals and the mechanism for the BSSE is the same in all
SCF methods.
As can be seen in the last column in Table I, the represen-
tation of the BB-CP correction by the gCP approach is sur-
prisingly good keeping in mind that no electronic information
has been used and only four parameters had to be determined.
With most method/basis combinations RMSDs of 0.1 to0.4 kcal/mol are obtained. This corresponds to a typical rela-
tive accuracy of 10%–30%, which is sufficient for many pur-
poses as will be demonstrated in detail below. For the largerdef2-TZVP basis, the RMSD is rather small but the relative
accuracy deteriorates although we still consider this as prac-
tically useful for a quick estimate of the magnitude of theBSSE. Not unexpectedly, the best fits (small RMSD and best
relative accuracy) are obtained with the compact minimal ba-
sis set. A further discussion of the accuracy of the gCP methodis provided in Sec. VA. Except for ηthe behavior of the four
gCP parameters is unsystematic. The global scaling factor σ
varies mostly between 0.1 and 0.5. For αvalues are found be-
tween 0.8 and 1.2 except for B3LYP/SVP that has a small α
of 0.6. The parameter βranges between 1.1 and 1.5. A sys-
tematic increase from smaller to larger basis sets is found for
η, which scales the Slater exponent (see Eq. (9)). This makes
the Slater functions more compact (less diffuse) and reducesthe overlap S
abas a result.
V. RESULTS AND DISCUSSION
A. BSSE in HF and DFT calculations for the S66 ×8 set
During the course of the project a few thousand CP cor-
rections have been calculated (data points for the fit). As a side
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0 1 02 03 04 0 50 600123456ΔE(CP) / kcal/molBLYP/MINIS
B3LYP/MINIS
HF/MINIS
0 1 02 03 04 0 50 600123456ΔE(CP) / kcal/molBLYP/6-31G*
B3LYP/6-31G*
HF/6-31G*(b)(a)
# dimer of S66x8
FIG. 1. Computed Boys and Bernadi CP correction for the S66 ×8 bench-
mark set with HF, B3LYP, and BLYP with the basis sets (a) MINIS and (b) 6-
31G*. The set consists of 66 systems (abscissa) with eight varying distances.
The CP correction for each distance (ordinate) results in 528 data points.
effect, this allows to analyze the behavior of the BSSE for a
statistically very meaningful number of data. For the two ba-
sis sets MINIS and 6-31G* CP corrections for HF, B3LYP,
and BLYP are plotted in Figures 1(a)and1(b). Shown are for
each of the 66 systems all eight CP values for the different
inter-fragment distances (eight values on the ordinate for onecomplex on the abscissa).
For the MINIS basis set the data points for HF, B3LYP,
and BLYP are hardly distinguishable, giving in all three casesa similar CP correction. Largest deviations (although still very
small in magnitude) are seen for the hydrogen-bonded sys-
tems (entries 1–23). The largest CP correction (at the shortestdistance) is obtained in most cases for BLYP. HF shows a
tendency to yield smaller CP corrections than DFT. The data
points for 6-31G* are more clearly separated. The HF CP cor-rection is the smallest, while BLYP gives again the largest
values. Comparing MINIS with 6-31G*, the former yields
only for the hydrogen-bonded systems a larger BSSE, for thedispersion and mixed-systems 6-31G* tends to larger BSSE
values. Rather small CP corrections are observed for sys-
tem 35–38 for both basis sets. These four systems (cyclopen-
tane, neopentane, and pentane in different combinations) are
less prone to BSSE due to large inter-fragment distances. As-1012Deviation ( Δ(gCP)- Δ(BB-CP)) / kcal/molB3LYP/6-31G *
0 1 02 03 04 0 50 60 70
# dimer of S66x8-1012 HF/MINIS
FIG. 2. Deviations between the Boys and Bernadi CP correction ( /Delta1BB-
CP) and the gCP correction ( /Delta1gCP) for the S66 ×8 dimers in kcal/mol for
B3LYP/6-31G* and HF/MINIS. The set consists of 66 systems (abscissa)
with eight varying distances. The deviation ( /Delta1gCP−/Delta1BB-CP) for each dis-
tance (ordinate) results in 528 data points. Positive values indicate an overes-
timation.
Figure 1demonstrates, the order of magnitude of the CP cor-
rection is HF <B3LYP <BLYP, which indicates the amount
of Fock-exchange as the major influencing factor.
These results for two different basis sets support the no-
tion that HF is slightly less affected by BSSE than DFT, and
that the DFT BSSE is reasonably invariant to the functionaltype. This is one motivation to globally adjust the gCP pa-
rameters only to HF and DFT, instead to each DFT functional
specifically. This also suggests to estimate the true CP cor-rection for hybrid-functionals with a GGA functional to save
computation time with little loss of accuracy (hybrid function-
als are a factor of 3–5 more costly with efficient density-fitting
(RI) schemes).
The differences between the BB-CP and our gCP cor-
rection are given in Figure 2for two representative ex-
amples (as above eight values for each complex). The
results for B3LYP/6-31G* indicate good performance forthe hydrogen-bonding dominated and the mixed-type sys-
tems. Some outliers are found for the dispersion domi-
nated systems. The three largest deviations are observed forsystems 34 (pentane ···pentane), 26 (uracil ···uracil) and 38
(cyclopentane ···cyclopentane), all showing overestimation of
the BSSE by gCP (positive values). HF/MINIS yields a muchsmaller error range with fewer outliers, though the systems 20
to 30 seem to be problematic at both levels (underestimation
of the BSSE). For the largest dimer distances (twice the equi-librium value), the HF/MINIS BB-CP correction is essentially
zero, which is well reproduced by gCP. This asymptotic decay
of the BSSE is a very important property for large systems and
should be accurately reproduced by any approximate scheme.
B. Adjustment of the London dispersion
correction DFT-D3
Although the gCP correction is proposed as an indepen-
dent, stand-alone procedure, it is clear that HF/DFT in gen-
eral requires dispersion corrections even in the basis set limit.
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We will therefore apply our DFT-D3 scheme27,28(in the de-
fault Becke-Johnson damping variant28,77,78) and also com-
ment on the use of the modified Vydrov and van V oorhis(VV10) (Ref. 30) non-local (NL), density dependent disper-
sion correction.
29
In the original publication27the dispersion correction has
been developed with AO basis sets close to the complete ba-
sis set (CBS) limit and remaining, tiny BSSE effects have
been absorbed in the DFT-D3 short-range damping functions.Hence it is in general recommended to apply the gCP cor-
rection together with the default DFT-D3 parameterization.
Nevertheless, we studied in quite detail if any re-fit of the
DFT-D3 short-range part has a positive effect. In the case
of HF-gCP-D3/MINIS, we found a large gain in perfor-mance when the DFT-D3 parameters are re-determined in
the presence of the gCP correction. The resulting level of
theory (termed HF-gCP-D3(fit)/MINIS) performs extremelywell providing a MAD for the S66 set of only 0.38 kcal/mol
which is of about the same accuracy as obtained by most
DFT-D3/large basis methods and also much better than, e.g.,MP2/CBS (see below). In a similar fashion the short-range
damping parameter b(see Ref. 29for details) in the HF-gCP-
NL/MINIS method was re-fitted, but without the same suc-cess. The MAD for the S66 drops from 0.68 to 0.62 kcal/mol.
C. Interaction energies for S66 and S22 sets
The validation of the gCP scheme is shown for the two
benchmark sets S22 and S66. The S22 set is the de-facto testset for non-covalent interactions.
106According to Hobza and
co-workers, some interaction motifs are underrepresented in
S22.55As a consequence, they recently published the S66 set
as an extension and revision of S22.55
Reference values for the interaction energies were taken
from the original work and are based on the estimatedCCSD(T)/CBS level of theory (MP2/CBS values based on
aTZ and aQZ basis set extrapolation were combined with
the difference of MP2 and CCSD(T)/aDZ correlation ener-
gies (/Delta1CCSD(T))). Very recently, revised reference values
for S66 and another extension were published; this extensionwas dubbed S66a8 and describes the same dimers at differ-
ent inter-monomeric angles.
107While the original S66 focuses
on benchmarking wave-function based methods, DFT in theframework of DFT-D3 was extensively tested on both the S66
and S66 ×8 in our group.
108The performance of the very re-
cently proposed DFT-NL method on the S66 can be found inRef. 29. Note, that the S22 set is not included in the fitting
set, while S66 is partly included through the S66 ×8 set that
has the same systems but at slightly different equilibrium dis-tances.
Results for various methods are shown in Table II.
Well performing methods have MAD values less than about0.5–0.6 kcal/mol for the S66 interaction energies. The de-
crease of the MAD after adding the gCP correction is strik-
ing, in particular for the smaller basis sets. The already high
accuracy of methods such as BLYP-D3, B3LYP-D3, and
PW6B95-D3 with def2-TZVP can be further improved toreach exceptionally small MAD values of 0.28–0.33 kcal/mol.TABLE II. Mean absolute deviations (MADs) for S66 benchmark set in
kcal/mol. The column “ +gCP” indicates that the gCP correction is added to
the DFT-D3 results. The methods are ordered according to increasing basisset quality.
Method w.o. corr.aDFT-D3 +gCP
Parametrized methods
HF/MINIS 2.95 1.86 0.51HF/MINIS
b2.95 1.54 0.38
HF/SV 2.96 2.72 1.32
HF/def2-SVP 2.83 2.06 0.83HF/6-31G* 2.72 2.08 0.89
HF/def2-TZVP 3.97 0.88 0.67
B3LYP/MINIS 3.38 2.26 1.07B3LYP/SV 3.08 2.92 1.15
B3LYP/def2-SVP 2.61 2.33 0.68
B3LYP/6-31G* 2.30 2.20 0.68B3LYP/def2-TZVP 2.96 0.57 0.33
Methods that employ the “dft” parameters for gCP
BLYP/MINIS 3.88 2.54 1.25BLYP/MINIS
c3.88 2.54 1.22
B97-D/MINIS 3.94 1.90 1.00
TPSS/MINIS 3.43 1.79 1.25M06-2X/MINIS
d1.48 1.72 0.73
PW6B95/MINIS 2.22 1.57 0.95
BLYP/SV 3.35 3.08 1.19BLYP/SV
c3.35 3.08 1.07
TPSS/6-31G* 2.22 1.76 0.66
BLYP/def2-SVP 2.91 2.53 0.72TPSS/def2-SVP 2.45 1.92 0.80
M06-2X/def2-SVP 1.60 1.41 1.15
PW6B95/def2-SVP 1.38 1.85 0.66
PW6B95/def2-SVP
c1.38 1.85 0.60
BLYP/def2-TZVP 3.71 0.46 0.32PW6B95/def2-TZVP 1.36 0.29 0.28
aResult without any dispersion or gCP correction.
bDFT-D3 parameters fitted to the respective gCP corrected level of theory.
cgCP parameters fitted for the respective level of theory.
dDFT-D3 with zero-damping.
The MAD decreases compared to the “pure” results in
most cases by adding the DFT-D3 dispersion correction.
A look at the mean deviation (MDs, see supplementary
material109) shows that the “pure” results still suffer from un-
derbinding, which is typically found for DFT/HF for non-
covalent interactions since they are incapable of describing
London dispersion interactions. However, the artificial bind-ing from BSSE becomes obvious for the smaller basis sets
after applying DFT-D3. A typical example is B3LYP/6-31G*
that gives a MD of −1.29 kcal/mol (underbinding) without
any correction and changes to 2.20 kcal/mol (overbinding) at
the DFT-D3 level. Only after correction with gCP, a very rea-
sonable MD of 0.32 kcal/mol is obtained. The relatively largebasis set def2-TZVP, which reduces the BSSE typically below
10% of the interaction energy, already leads to good results
without gCP correction.
From the first block of data in Table II, which depicts the
performance of all parameterized methods, it becomes clear
that the minimal and polarized double- ζbasis sets (MINIS,
def2-SVP, 6-31G*) work better in conjunction with gCP than
the seemingly too unbalanced SV basis set. Remarkable is the
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excellent performance of HF-gCP-D3/MINIS with a MAD of
0.51 kcal/mol, which can be further improved by adjusting the
DFT-D3 parameters for this model chemistry (see Sec. VB).
The adjusted HF-gCP-D3(fit)/MINIS method yields the
smallest MAD of 0.38 kcal/mol for basis sets below triple-
ζquality in this work and beats even MP2/CBS results55
(MAD =0.45 kcal/mol). This very good performance clearly
relies on strong, but systematic error compensation that is an-
alyzed further below.
The second data block in Table IIapplies the gCP cor-
rection with the “dft” parameters or a special fit to the re-
spective level of theory. We focus again mainly on the gCP-
D3 corrected results. The use of MINIS with a GGA or
meta-GGA cannot be recommended as the MADs are allabove 1 kcal/mol. The gCP re-fit for BLYP/MINIS shows no
significant improvement (MAD 1.25 versus 1.22 kcal/mol).
The inclusion of Fock-exchange (with contributions of 20%(B3LYP) or 28% for PW6B95) improves the results (1.07
and 0.95 kcal/mol). M06-2X-gCP-D3 (54% Fock-exchange)
gives a reasonable MAD of 0.73 kcal/mol. The gCP re-fit is also done for BLYP/SV , but again the improvement
is small. GGAs and meta-GGAs perform much better with
the more balanced def2-SVP basis and the MADs drop to0.72 (BLYP) and 0.80 (TPSS). Even better results are ob-
tained for corrected PW6B95/def-SVP and B3LYP/def2-SVP
yielding good values of 0.66 and 0.68 kcal/mol. The gCPre-fit for PW6B95/def2-SVP further improves the MAD to
0.60 kcal/mol, which is the second best MAD next to
HF/MINIS’s for the “small” basis sets (below triple- ζqual-
ity). The def2-TZVP results show that gCP works in princi-
ple also for large basis sets, but that the performance gain issmall (e.g., the MAD for HF/def2-TZVP decreases from 0.88
to only 0.67 kcal/mol). We again note here that def2-TZVP is
not the main target for gCP, but the consistent improvementof the MAD even for BLYP/def2-TZVP and PW6B95/def2-
TZVP proves the generality of the gCP scheme. One impor-
tant point is that the MAD in all tested cases is not increasedby gCP correction and a statistically positive, or at least neu-
tral, effect is obtained.
It can be argued that DFT-NL is less suited than DFT-D3
for small basis set calculations, because the dispersion con-
tribution arises through integration of the density, and this
density is also subject to basis set errors. From an empiri-cal point of view, comparing Table IIand Table III,D F T - N L
yields very similar results to DFT-D3, proving that DFT-NL
is capable to deliver good dispersion energies also with smallbasis sets. Studies on the large GMTKN30 benchmark set
have shown that DFT-NL and DFT-D3 perform statistically
at the same, high level, but individual functional/basis com-
bination will be slightly in favor of one or the other disper-
sion correction. We observe the same apparent error compen-sation for HF/MINIS as in the DFT-D3 example, such that
HF-gCP-NL/MINIS yields also a very good MAD of only
0.64 kcal/mol.
S22 results are reported in Table IV. The updated ref-
erence values from Sherill’s group are used.
110HF-gCP-
D3/MINIS and its re-fitted variant provide again the bestperformance with a MAD of 0.64 and 0.55 kcal/mol, re-
spectively. Also PW6B95-gCP-D3 shows again high accu-TABLE III. Mean absolute dev iations (MADs) for DFT-NL for the S66
benchmark set in kcal/mol. The column “ +gCP” indicates that the gCP cor-
rection is added to the DFT-NL results. The DFT-NL calculations have beenperformed non-self-consistently, except those indicated by (SC).
Method DFT-NL +gCP
HF/MINIS 1.09 0.64
HF/SV 2.06 0.96HF/def2-TZVP 0.27 0.27
B3LYP/MINIS 2.01 1.31
B3LYP/def2-SVP 2.05 0.98BLYP/MINIS 2.28 1.47
PW6B95/SV 2.48 0.97
PW6B95/def2-SVP 1.85 0.70revPBE0/SV 2.34 1.09
revPBE0/SV (SC) 2.37 1.08
revPBE/SV 2.56 1.06revPBE/def2-SVP 1.95 0.77
revPBE/def2-SVP (SC) 1.99 0.75
TPSS/MINIS 1.89 1.42TPSS/def2-SVP 1.95 0.91
racy (MAD =0.66 kcal/mol) after re-fit of gCP. Note that the
re-fit is not done on the S22, but still on the S66 ×8 dimers.
The overall picture of performance is quite similar to the S66
set. There is, however, the tendency that the S22 MADs areslightly higher than for the S66 set. The gCP-corrected model
chemistries that reach a MAD below 1 kcal/mol are, besides
the already mentioned HF/MINIS, B3LYP, and PBE0 com-bined with 6-31G*, as well as PW6B95/def2-SVP. Obviously,
the list of tested functionals and basis sets is far from exhaus-
tive and other, well-performing hybrid functionals are sure
TABLE IV. Mean absolute deviations (MADs) for S22 benchmark set in
kcal/mol. The column “ +gCP” indicates that the gCP correction is added
to the DFT-D3 or DFT-NL results. The DFT-NL calculations have been per-formed non-self-consistently.
Method w.o. corr.aDFT-D3 +gCP
HF/MINIS 3.50 2.19 0.64
HF/MINISb3.50 1.80 0.55
B3LYP/MINIS 4.33 3.22 1.70
PW6B95/MINIS 2.99 2.32 1.24PW6B95/MINIS
c2.99 2.32 1.27
B3LYP/6-31G*/CP 3.79 0.66 . . .
B3LYP/6-31G* 2.67 2.55 0.88
PBE0/6-31G* 2.13 2.17 0.77
B3LYP/def2-SVP 3.41 3.23 1.05B3LYP-NL/def2-SVP 2.94 . . . 1.37
PW6B95/def2-SVP 1.44 2.01 0.84
PW6B95/def2-SVP
c1.44 2.01 0.66
BLYP/SV 3.69 3.36 1.10
BLYP/SVc3.69 3.36 1.02
TPSS/SV 3.14 2.77 0.99TPSS/SV
d3.14 2.77 1.06
TPSS-NL/SV 2.87 . . . 1.17
aResult without dispersion or gCP correction.
bDFT-D3 parameters fitted to the respective gCP corrected level of theory.
cgCP parameter fitted for the respective level of theory.
dgCP parameters fitted for the GGA functional BLYP/SV .
Downloaded 25 Nov 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions154101-9 H. Kruse and S. Grimme J. Chem. Phys. 136 , 154101 (2012)
to be found. The re-fitting of gCP parameters for BLYP and
TPSS shows similar to the S66 only minor improvements. In
the case of TPSS/SV the standard parameters suit even betterthan the for BLYP re-optimized gCP parameters.
We also investigated in detail the performance of the very
common B3LYP/6-31G* combination for S22 and addition-ally calculated its BB-CP corrections (B3LYP/6-31G*/CP). It
can be seen that the artificial BSSE binding contributes about
1 kcal/mol to the MAD reduction. CP-corrected B3LYP/6-31G* gives 3.79 kcal/mol, while the uncorrected MAD is
2.67. This means that the widely used B3LYP/6-31G* method
benefits from BSSE in many intramolecular situations, while
its inherent accuracy for non-covalent interactions is catas-
trophic. B3LYP-D3/6-31G*/CP gives a slightly better MADthan the gCP corrected B3LYP-D3/6-31G* (0.66 versus
0.88 kcal/mol). Adding only DFT-D3 to B3LYP/6-31G* is in-
sufficient, as it gives a MAD of 2.55, such that a larger basisset or a BSSE correction is necessary for reliable results.
The amount of Fock-exchange plays a major role for the
BSSE of very small basis sets such as MINIS. The MADsfor GGA functionals with basis sets smaller def2-SVP are
usually higher than 1 kcal/mol (e.g., BLYP-gCP-D3/MINIS
gives 1.25 kcal/mol). Hybrid functional seem to perform bet-ter (PW6B95-gCP-D3/MINIS MAD of 0.95 kcal/mol), but
not uniformly (B3LYP-gCP-D3 for MINIS gives MAD of
1.07 kcal/mol). Obviously only a very small selection of theplethora of functionals is tested here, but we believe that hy-
brid functionals or HF are better suited in combination with
very small basis sets such as MINIS and SV . The Fock-
exchange is deemed necessary for proper error compensation
between induction, electrostatics and Pauli-repulsion as dis-cussed below.
The BSSE is reduced by gCP with a typical error of 10%-
30% (see Sec. IV A ), which already improves the MADs
for the S66 and S22 set in a robust manner, i.e., addi-
tion of gCP never statistically worsens the results among all
test methods. This validates small basis set calculations ofnon-covalent interaction with the combination of gCP and
DFT-D3.
D. Energy decomposition analysis of minimal
basis set HF interactions
The very good performance of HF-D3 for non-covalent
interactions was already noted in Ref. 28. That this (at least
for the here investigated systems) also holds for the gCP-
corrected HF/MINIS method is surprising and demands fur-
ther analysis, as briefly given here. Pople already observedthat the geometries of small molecules with HF/STO-3G turn
out to be excellent as well, better than HF is inherently capa-
ble of yielding.
111,112Similar observations for CP-corrected
minimal basis set HF interaction energies were reported by
Kołos,113who already enhanced HF with a correction for the
London dispersion energy.
To gain insight into the apparent error compensation an
energy decomposition analysis (EDA) as proposed by Ki-
taura and Morokuma114and implemented in GAMESS-US
(Ref. 115) is done for HF with increasing basis set sizes.
The very large def2-QZVP basis set already includes semi-TABLE V. Energy decomposition analysis of the interaction energy of the
benzene-water complex (system 54) of the S66 benchmark set in kcal/mol.The “def2” label is omitted for clarity. The individual terms are electro-static interaction ES, Pauli, or exchange repulsion EX, induction energy IND,charge-transfer energy CT, and the high-order-coupling term MIX. The gCP
correction and the Boys and Bernadi counterpoise correction BB-CP are
given. The BB-CP-correction only affects the EX,CT and MIX terms.
Term MINIS MINIS/CP SVP TZVP QZVP
ES −2.14 −2.14 −3.18 −3.37 −3.16
EX 1.86 2.21 2.70 3.38 3.45IND −0.09 −0.09 −0.39 −0.77 −1.05
CT −0.93 −0.78 −1.23 −1.05 −0.93
MIX 0.05 0.27 −0.07 0.37 0.82
INT −1.24 −0.53 −2.18 −1.45 −0.88
gCP 0.69 . . . 0.88 0.15 . . .
BB-CP 0.72 . . . 0.99 0.48 0.12
INT+CP −0.52 . . . −1.19 −0.97 −0.76
diffuse functions and gives results near the basis set limit for
HF and DFT. Investigations of BSSE effects within the EDA
(Ref. 116) demonstrated significant BSSE in the Pauli repul-
sion (less repulsion due to BSSE) and in the charge-transfer
term (more binding due to BSSE). The decomposed interac-
tion energy for the benzene-water system from S66 is pre-sented in Table V. It is classified in the original work as of
“mixed” interaction type, meaning that both, dispersion and
electrostatics, play a dominant role.
The electrostatic energy (ES) is already almost exact at
def2-SVP, but is too small in the MINIS basis. Exchange re-
pulsion (EX) is only half as strong in MINIS as in def2-QZVP.This can be understood from the high compactness of the min-
imal basis that reduces the overlap between the fragments.
The absolute induction energy (IND) slowly increases fromMINIS (below 0.1 kcal/mol) to 1 kcal/mol with def2-QZVP.
The very small value for MINIS stems from the inflexibil-
ity of the minimal basis set. The charge-transfer term (CT) isleast affected by the basis set size. The high-order-coupling
term (MIX), which takes into account the mutual influence
of each interaction term, is small (below 0.1 kcal/mol) up to
def2-TZVP where it reaches 0.37 kcal/mol. The CP-corrected
EDA raises the repulsion stemming from EX and introduces asignificant amount of term-coupling (MIX =0.27 kcal/mol).
The CT is insignificantly affected. The major effects compar-
ing MINIS/CP with def2-QZVP can be found in the loweredES and IND terms (about 2 kcal/mol), while at the same time
the EX value is reduced (1.2 kcal/mol). The behavior of the
high-order-coupling term MIX is not easily understood andit complicates the interpretation of the error compensation. It
is together with charge-transfer the smaller effect, and both
seem to partly cancel each other as well. The investigationof only a single complex limits the generality of our conclu-
sions. However, the benzene-water interaction is a representa-
tive interaction type not only in the S66 benchmark set, but ofbiomolecular systems in general. HF’s physical correctness of
Fermi-correlation and the error compensation with minimal
basis sets turns it into an intriguing method if the coulomb-
correlation is dominated by dispersion interactions, which in
turn can be accurately evaluated by DFT-D3.
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0.9 0.8 1 1.25 1.5 1.75 2.00
equilibrium distance multiplier-6-4-2024ΔE / kcal/mol
Edisp (SAPT/aug-cc-pVTZ)
Edisp (HF-gCP-D3(fit)/MINIS)
BB-CP(HF/MINIS)
gCP(HF/MINIS)
HF-gCP-D3(fit)/MINIS
LPNO-CEPA1/CBS(3,4)
FIG. 3. Potential energy curve of of a peptide-ethene complex (taken from
the S66 set, depicted in the top right corner). The dimers are displaced along
the vectors connecting their center-of-mass points by the multipliers 0.8, 0.9,
0.95, 1.00, 1.05, 1.10, 1.25, 1.50, 1.75, 2.00. The dispersion energy Edispfrom
SAPT and DFT-D3 is plotted along with the Boys and Bernadi counterpoise
(BB-CP) and gCP correction.
Figure 3presents the potential energy curve for the
peptide-ethene complex in the S66 benchmark set and com-
pares the dispersion energy obtained through SAPT treat-
ment with that from the D3 method, and the BB-CP correc-tion scheme with gCP. The monomers are displaced along the
vector that connects their center-of-mass points. Similarly to
the S66 ×8 set, multipliers (0.8, 0.9, 0.95, 1.00, 1.05, 1.10,
1.25, 1.50, 1.75, 2.00) are used to generate displaced struc-
tures from the equilibrium geometry, which is taken from
the S66 set. Reference interaction energies
117are obtained
by a two-point extrapolation to the CBS limit using def2-
TZVP and def2-QZVP (CBS(3,4)) for LPNO-CEPA/1. The
dispersion energy Edispis calculated by SAPT/aug-cc-pVTZ
and by DFT-D3 using the parameters optimized for HF-gCP-
D3(fit)/MINIS. Both curves are in good agreement, espe-
cially in the asymptotical region, where the dispersion energy
clearly dominates the interaction energy. The BB-CP correc-
tion almost vanishes already at a multiplier of 1.25 (below0.1 kcal/mol). The gCP corrections follows closely the BB-
CP curve, but is a little bit more long-ranged (gCP vanishes at
multiplier 1.5 instead of 1.25). The close agreement betweenDFT-D3 and SAPT, and the gCP and BB-CP scheme, is en-
couraging and supports our idea to built a total correction on
physically sound components.
E. Multiple fragments
The performance of the atomic (pair-wise additive) gCP
scheme for multi-fragment situations is a justified question as
the fitting procedure includes only dimeric complexes. Four
systems of the WATER27118test set of water clusters are used
as test cases. The BSSE is calculated for multiple fragments
using the simplest, the so-called site-site function counter-
poise (ssf-CP) approach,37thus neglecting second-order ba-
sis set effects. In the ssf-CP approach, the BSSE is calculated
for n-fragments by summing up the difference between theTABLE VI. Selected systems from the WATER27 set as a test case for a
cluster-type CP correction. ssf-CP and gCP values in kcal/mol are comparedfor the HF/MINIS and DFT( =B3LYP)/def2-SVP level of theory.
HF/MINIS DFT/def2-SVP
# Fragments ssf-CP gCP ssf-CP gCP
2 1.96 1.59 2.46 2.02
3 6.22 5.50 7.74 6.97
4 10.10 8.84 12.49 10.965 12.57 11.25 15.70 13.69
fragment in its own basis set and the fragment in the complex
basis set. For a trimer it reads
/Delta1ECP(ABC )=E(A)a−E(A)abc
+E(B)c−E(B)abc
+E(C)c−E(C)abc
where A, B, C denote the fragments and a,b,c the correspond-
ing basis sets.
As can be seen from Table VI, the estimates of the BSSE
by the ssf-CP and gCP approaches are in good mutual agree-
ment. The gCP correction is constantly smaller by only 10%–
20%, which demonstrates its usefulness also for clusters ofmultiple fragments for which other methods become pro-
hibitively costly for large systems.
F. Intramolecular BSSE and biomolecules
The treatment of large biomolecules is one major aspect
of the gCP scheme. Their typical system sizes make compro-
mises at the one-particle basis set inescapable. The limitationto small double- ζbasis sets is error-prone, not only regarding
energetics, but also regarding structures. An extensive study
of biomolecular systems is beyond the scope of this work,instead prove-of-principle examples are presented. One ideal
test system is the crambin protein, which finds prominent use
in many computational and experimental studies due to its
small size.
119–121We apply the GGA functional BLYP com-
bined with the basis sets MINIS, SV , and def2-TZVP to eluci-date intramolecular BSSE in a folded and extended conformer
of crambin.
BLYP-D3/def2-QZVP performs exceptionally well for
the PCONF test set
122,123of peptide conformers (MAD
=0.5 kcal/mol) and is expected to yield reliable results at
the triple- ζlevel for our model conformational problem. Ad-
ditionally to the standard B3LYP parameters, optimized gCP
parameters for BLYP have been tested, but again without any
significant impact.
After adding hydrogen-atoms to the crystal structure
(1CRN.pdb) using OPENBABEL ,124,125the generated struc-
ture is manually corrected to give a sensible startingpoint for a PM6-DH +optimization
126–128using MOPAC,129
which results in the folded conformer (see Figure 4). A
5000 steps, gas-phase molecular dynamics simulation using
Amber/GAFF130build into the ambmd program (supplied
through the MOLDEN131package) at 298 K leads to a strongly
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FIG. 4. View of the folded (top) and extended (bottom) PM6-DH +opti-
mized (gas-phase) conformers of crambin.
unfolded crambin. A sub-sequential PM6-DH +optimization
regained a somewhat tighter packaging, and yields the final,
extended conformer. The extended conformer is thus purely
artificial and serves only test purposes. The Cartesian coordi-nates can be found in the supplementary material.
In Table VII, the relative energies between both conform-
ers are computed at different levels of theory. Note that al-
ready for the def2-TZVP basis set (12 063 basis functions)
Turbomole needed to be slightly modified in order to run thecalculations (mostly raising some hard coded memory alloca-
tion limits).
The gCP uncorrected results for /Delta1Ewith SV and
def2-TZVP basis sets deviate strongly from each other (by
85 kcal/mol), which indicates a huge IBSSE (overstabilization
of the folded conformer). The MINIS and def2-TZVP valuesagree much better, which is rather unexpected and can be at-
tributed to error compensation. As mentioned in Sec. VC,
GGAs with the MINIS basis set should be used with care.The gCP correction for the SV basis is largest (about twice
the MINIS correction) while it diminishes for def2-TZVP
(−17.4 kcal/mol). Results for SV and def2-TZVP differ by
only 3 kcal/mol which can be considered as excellent in our
opinion.
The HF-gCP-D3 results are equally promising. gCP cor-
rections for MINIS amount to 60 kcal/mol, while for SV , a
seemingly too small correction of about 80 kcal/mol leadsTABLE VII. Relative energies (kcal/mol) of two crambin conformers (ex-
tended and folded) calculated with BLYP-D3 and various basis sets. Struc-tures are optimized at the PM6-DH +level of theory.
Method /Delta1Ea/Delta1gCP (≈IBSSE)
BLYP-D3/MINIS 223.1 . . .
BLYP-D3/SV 323.5 . . .BLYP-D3/def2-TZVP 238.1 . . .
BLYP-gCP-D3/MINIS 161.4 −61.7
BLYP-gCP-D3/SV 223.4 −100.1
BLYP-gCP-D3/def2-TZVP 220.7 −17.4
HF-gCP-D3/MINIS 244.5 −59.7
HF-gCP-D3(fit)/MINIS 231.4 −59.7
HF-gCP-D3/SV 290.5 −78.5
aFolded →extended.
to a too high /Delta1E. The excellent performance of HF-gCP-
D3/MINIS for the S66 set, on the other hand, is also reflected
in the very good agreement with BLYP-gCP-D3/def2-TZVP(11 kcal/mol difference corresponding to only 5% for /Delta1E).
Although this has to be verified for more systems, HF-gCP-
D3(fit)/MINIS seems to be a very promising (non-DFT) can-
didate for related biochemical problems.
One well-known example in the literature for major
IBSSE effects is the phenylalanine-glycine-phenylalanine
(FGF) tripeptide,
43,50,51and as such it became a common test
case. Table VIII shows a comparison of the ACP and the CPaa
correction for the FGF tripeptide with the gCP scheme. The
values are taken from Ref. 51where the haug-cc-pVDZ basis
set (heavy-augmented, i.e., no diffuse functions on hydrogen)has been applied. This basis is not parameterized and so in-
stead we compare against def2-SVP and def2-TZVP results.
Our comparison suffers from the differences in the basis
set, but the BSSE for def2-SVP should be slightly higher, or at
least of comparable magnitude, than haug-cc-pVDZ. For the
HF/def2-SVP parameterization gCP yields an IBSSE of 1.72while ACP(2) gives 0.76 kcal/mol and ACP(4) 1.56 kcal/mol.
ACP(1) or its equivalent CP
aaunderestimates the IBSSE for
this system (0.38 kcal/mol as discussed by Jensen51). Sim-
ilar underestimation can be ascribed to ACP(2) that gives
only slightly larger values and good mutual agreement is
seen between ACP(4) and gCP. One serious drawback of IB-
SSE calculations is the absence of a well-established refer-
ence method, thus it is difficult to estimate “correct” values.However, the performance of gCP fits well into the other ap-
proaches, having the same sign and comparable magnitudes.
TABLE VIII. Intramo lecular BSSE calculated for the FGF-tripeptide in
kcal/mol. Comparison with Jensen’s ACP approach.
Method /Delta1
gCP(HF/def2-SVP) 1.72
gCP(HF/def2-TZVP) 0.27ACP(1) or CP
aaa0.38
ACP(2)* 0.76
ACP(4)* 1.56
aEvaluated for HF/haug-cc-pVDZ (no diffuse functions on hydrogen), values taken from
Ref. 51.
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-100 -50 0 50
δ / degree012345678 ΔE / kcal/molBLYP-D3/def2-QZVP (ref)
HF-gCP-D3/MINIS
PW6B95-D3/SV
PW6B95-gCP-D3/SV
FIG. 5. Relaxed rotational energy profile of the FGF tripeptide in kcal/mol.
The dihedral angle δbetween the atoms indicated in black is varied in steps
of 20◦and held fixed in subsequent optimizations.
An additional benefit of gCP is the practically vanishing com-
putational cost.
A relaxed rotational energy profile is calculated for
the FGF-tripetide for fixed dihedral angles as depicted in
Figure 5. The influence of the IBSSE on peptide confor-
mations is discussed, e.g., by van Mourik et al.45,46for the
Tyr-Gly structure, where it was found that some conforma-
tions are obscured by IBSSE. For each level of theory, thecentral NCCN dihedral angle (black atoms in Figure 5)i s
varied by 20
◦. The remaining part of the tripeptide is opti-
mized, while the dihedral angle is held fixed. The rotationprofile for BLYP-D3/def2-QZVP, which serves as the refer-
ence, shows two minima: the lowest at −60
◦and a very flat
minimum at 0◦. The best agreement is reached with HF-gCP-
D3/MINIS that gives an almost quantitatively correct transi-
tion barrier between both conformers and indicates the second
minimum with a rather flat progression around 0◦. PW6B95-
D3/SV shows a too high barrier at −20◦and continues its
strong increase to 0◦, not showing the flat region of BLYP-
D3/def2-QZVP or HF-gCP-D3/MINIS. Applying PW6B95-
gCP-D3/SV yields the quasi-correct (0.2 kcal/mol deviation)
barrier and regains some of the flatness around 0◦, although
the basis set errors (intramolecular BSSE or BSIE) are still
significant in this region. HF-gCP-D3/MINIS and PW6B95-
gCP-D3/SV qualitatively reproduce the rest of the energyprofile. The high sensitivity of peptide structure towards the
level of theory is well-known
45,46,122,123and basis set ef-
fects, as well as an accurate description of dispersion inter-actions are crucial to consider for high accuracy. We suggest
gCP-corrected HF/DFT-D3 as an efficient tool in screening a
large number of peptide conformations and pre-selecting con-formations that can be subsequently subjected to high-level
calculations.
Recently, Antony et al.
132presented a set of protein-
ligand complexes, where the drug molecule together with
parts of the binding pocket of the protein within certain
radii was cut out of X-ray structures. Reference energies for
the smaller complexes could be obtained by LPNO-CEPA/1.
Good agreement was found with B97-D3/def2-TZVP calcu-14 8 1 2 16 20 24
protein-li gand complex-80-70-60-50-40-30-20-100ΔE / (kcal/mol)LPNO-CEPA/1 (ref)
HF-gCP-D3(fit)/MINIS
B97-D3/def2-TZVP (ref)
B97-D3-gCP/def2-SVP
B97-D3/def2-SVP
FIG. 6. B97-D3 and HF-D3 interaction energies for 24 protein-ligand com-
plexes with and without gCP correction. The LPNO-CEPA/1 and B97-
D3/def2-TZVP reference values are taken from Ref. 132.
lations that were proposed as reference values for the larger
complexes. The performance of B97-D3/def2-SVP calcula-tions with and without gCP correction is demonstrated in
Figure 6, where both kinds of reference values are included.
The overbinding of gCP-uncorrected B97-D3/def2-SVP isclearly visible for all interaction strengths. Relative errors
without gCP can be large, e.g., for system 2, where the error
is about 80% or about 40% for complex 18. The gCP cor-rection (B97-D3-gCP/def2-SVP) reduces the errors to 13%
and 12%, respectively. The HF-gCP-D3(fit)/MINIS results
are also plotted in Figure 6. The agreement with the reference
is better than for B97-D3-gCP/def2-SVP for this set which is
very encouraging because this benchmark is very close to typ-
ical target applications in biochemistry. However, gCP can-not correct for inherent limitations by the basis set and a few
outliers can be found; most notable system 1, where many
flourine atoms and a flourine-hydrogen bond are present.
G. S22 optimizations with small basis sets
To systematically investigate the applicability of the gCP
correction for geometry optimizations, the structures of the
well-known S22 benchmark set106were optimized with HF-
D3/MINIS, HF-D3/SV , and B3LYP-D3/6-31G*, applying astandard BB-CP optimization scheme and our gCP correction.
The resulting structures are compared with the original S22
reference geometries that are based on MP2 calculations. Wefocus on the differences in the intermolecular center-of-mass
distances as a measure of deviation from the reference struc-
ture, as shown in Figure 7. Only dispersion corrected results
are presented.
As can be seen in Figure 7(a), HF-D3/SV results in too
short distances (BSSE induced overbinding). Compared tothe reference data, CP-optimization yields too short distances
(underbinding), uncovering shortcomings of the model chem-
istry that can be in large parts addressed to a significant BSIE.The gCP correction improves the results compared to raw
HF-D3/SV in all cases and (due to favorable error compen-
sation) overall provides the best results compared to the ref-
erence data. This favorable error cancellation is not transfer-
able to other methods as the results for B3LYP-D3/6-31G*
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24 6 81 0 1 2 1 4 16 18 20 22
system number-0,3-0,2-0,100,10,20,3center-of-mass distance / ÅB3LYP-D3/6-31G*
B3LYP-D3/6-31G*/CP
B3LYP-gCP-D3/6-31G*24 6 81 0 1 2 1 4 16 18 20 22
system number-0,4-0,3-0,2-0,100,10,20,30,4center-of-mass distance / ÅHF-D3/SV
HF-D3/SV/CP
HF-gCP-D3/SV
24 6 8 1 01 21 4 16 18 20 22
system number-0,100,10,20,30,4center-of-mass distance / ÅHF-D3/MINIS
HF-D3/MINIS/CP
HF-gCP-D3/MINIS(a)
(b)too short
too long
too short(c)too short
too long
too long
FIG. 7. Comparison of BB-CP- and gCP-optimized structures with uncor-
rected ones for the S22 benchmark set. Deviation from the center-of-mass
distances of the MP2 reference structures is plotted in Å. (a) HF-D3/SV ,
(b) B3LYP-D3/6-31G*, and (c) HF-D3/MINIS.
show (Figure 7(b)), where the gCP optimization produces
slightly longer distances than the CP-optimization. However,the correction emulates the CP-optimization fairly well, hav-
ing larger deviations only for complex number eight (methane
dimer) and number 18 (benzene ···NH
3). The overall agree-
ment between CP-optimized and gCP-optimized geometries
is better than in the HF-D3/SV case. Large deviations can be
seen in both plots for system 14 (indole ···benzene complex)
that undergoes a conformational change from a parallel to a
T-shaped arrangement. HF-D3/SV does not show this peak,but this is an artifact from BSSE since the CP-optimizations
yield the other conformer. The gCP scheme is able to repro-
duce this correctly.
Figure 7(c) shows the results for HF/MINIS. Notable is
system six (2-pyridoxine 2-aminopyridine complex) that is
not affected by BSSE at all. For most systems reproducesgCP the CP-optimized structures very well. Some minor de-
viations can be seen for the mixed-type complexes. Complex
22 (phenol dimer) gives a large deviation from the reference atthis model chemistry, which is, however, specifically related
to this complex and not to the gCP procedure.
With the exception of the sensitive systems 14 and 22,
the general structural features of the S22 complexes are pre-
served with CP-optimized DFT-D3 in comparison to the MP2reference data. The computationally much faster gCP method
reproduces the CP-optimized structure well and seems a reli-
able alternative for larger systems. The good performance ofHF-gCP-D3/MINIS in most cases is also notable and suggests
this method for many problems in structural bio-chemistry as
an alternative to semiempirical methods.
H. 9-Helicene
The helical structure of helicene polycyclic aromatic hy-
drocarbons is rather sensitive to the account of dispersion in-teractions in the quantum chemical treatment and intramolec-
ular BSSE.
43Hobza et al. showed that DFT-D is able to
correct for missing dispersion interactions in DFT for thissystem.
43Geometry optimizations with gCP are tested for
HF-D3/SV on the 9-helicene structure. Artificial IBSSE sta-
bilization for this system is revealed in a compressed helicalgeometry with too short inter-ring distances. The values for
the two in Figure 8shown, representative distances are com-
pared to those of a HF-D3/def2-QZVP reference structure thatcan be considered almost IBSSE free.
The inter-ring distances for HF-D3/SV are significantly
too short by about 6 and 10 pm, respectively.
The HF-gCP-D3/SV geometry optimization, on the other
hand, yields an excellent agreement for the two non-bondedC–C distances, which are too short by only 1 and 3 pm. In
passing it is noted that it takes a few days with the def2-QZVP
FIG. 8. HF-D3/def2-QZVP optimized structure of 9-helicene. Two in-
tramolecular distances of raw and gCP-corrected HF-D3/SV structures are
compared with the shown reference structure.
Downloaded 25 Nov 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions154101-14 H. Kruse and S. Grimme J. Chem. Phys. 136 , 154101 (2012)
FIG. 9. TPSS-D3/def2-QZVP structure of the (RhCl(PH 3))2dimer.
basis to optimize 9-helicene, while the calculation with the
small SV basis finished over night (same starting structure
and a comparable number of optimization cycles on a typical
workstation).
I. A transition metal dimer
The focus of this work clearly lies on organic and
biological molecules. However, active centers in the latter
often host transition metals (TMs). Currently, the gCP pa-
rameterization for TMs is available only for the basis sets
def2-SVP and def2-TZVP and only for 3d metals. The ear-
lier mentioned fall-back algorithm substitutes the parametersfor 4d and 5d metals internally to those of the 3d series. The
proof-of-principle for this approach is shown on a model sys-
tem dimer of the square-planar Wilkinson catalyst,
133where
the PPh 3ligands are replaced by PH 3. Normally one expects
that the gross of the BSSE between two TM-complexes stems
from the bulky, stabilizing ligands and the error of our fall-back algorithm is minor. The small PH
3ligands makes the
BSSE contributions from the (two) metal centers far more
important and providing a worst case scenario with an upperbound error estimation in a system with larger ligands.
The dimer is optimized at the TPSS-D3/def2-QZVP level
of theory (depicted in Figure 9). In this example, the program
substitutes the Rh by Co atoms.
The evaluation of the BSSE for the interaction en-
ergy with gCP using the “dft/svp” parameters (fitted for
B3LYP/def2-SVP) yields a somewhat too small but reason-
able value of 5.2 kcal/mol in comparison to the “true” BB-CPresult of 8.4 kcal/mol.
VI. CONCLUSIONS
A semi-empirical approach to correct for the BSSE in HF
and DFT calculations is presented that requires only the geo-metric information of the molecule and no wave-function in-
put. The approach dubbed gCP can be applied to supermolec-
ular interaction energies in the spirit of Boys and Bernadi’s
counterpoise correction (BB-CP), but additionally is able
to correct for intramolecular BSSE. The correction is con-structed from overlap integrals over Slater functions and em-
ploys computed measures for the quality (incompleteness) of
the target basis set. A few central points of the gCP schemeare:
1. Only four adjustable parameters are necessary to pro-
duce accurate fits (RMSD mostly between 0.1 and
0.4 kcal/mol) against BB-CP corrections of the S66 ×8
dimers. Adjustments for different DFT functionals are
not necessary and we discriminate only between HF and
DFT.
2. Low-order scaling behavior with system size and a small
computational pre-factor for both, energy and gradient
evaluations results in very fast computations even for
thousands of atoms (gradient for 48 000 atoms below
2 min computation time on a typical workstation).
3. The gCP correction provides reasonable estimates for
the intermolecular BSSE in the benchmark sets S66
and S22 with typical errors of 10%-30% for the BSSE.Common dispersion-corrections (DFT-D3 and DFT-NL)
work well in combination with gCP for total interaction
energies. For the polarized double- ζbasis sets 6-31G*
and def2-SVP, HF and all considered functional types
(GGAs, meta-GGAs, and hybrid functionals) perform
quite well.
4. The analytical gradient has been applied successfully for
optimizations of the non-covalent systems in the S22
set and for the 9-helicene structure where agreement fornon-bonded C–C distances with the (almost IBSSE free)
reference is obtained within a few pm.
5. For conformers of the small protein crambin and the
tripeptide FGF, the significant intramolecular BSSE
can be efficiently removed so that, e.g., in the caseof crambin BLYP-D3/def2-TZVP, HF-gCP-D3/MINIS,
and BLYP-gCP-D3/SV are in agreement within 10%-
15% of the folding energy while deviations withoutgCP reach 50%. A relaxed rotational energy profile for
the FGF-tripeptide proves sensible towards intramolec-
ular BSSE, which could partly be removed for HF-D3/MINIS and PW6B95-D3/SV by gCP to yield results
close to BLYP-D3/def2-QZVP quality.
6. Calculated interaction energies with B97-D3/def2-SVP
and HF-D3/MINIS for a set of 24 large protein-ligand
complexes are significantly improved by removing inter-
molecular BSSE with gCP. The structures includes fluo-rine, phosphorus, and chlorine atoms, which are not in-
cluded within the gCP training set, demonstrating wide
applicability of the scheme.
7. For a few water clusters, gCP shows good agreement
with site-site function counterpoise computation of theBSSE, thus demonstrating its applicability also for
multimer-BSSE.
8. The program gcpis supplied
61as open-source to make
the correction easily accessible to the community. It pro-
vides a user friendly fall-back algorithm to compute sys-
tems with unparameterized elements ( Z>36).
9. An important finding of our detailed investigations of
various method/basis set combinations is that the mini-
mal basis set MINIS together with HF-D3 and the gCP
Downloaded 25 Nov 2012 to 128.148.252.35. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions154101-15 H. Kruse and S. Grimme J. Chem. Phys. 136 , 154101 (2012)
correction yields exceptionally high accuracy for typical
non-covalent interactions due to a relatively systematic
error compensation. Hence we recommend this so-calledHF-gCP-D3/MINIS method (or slightly enhanced vari-
ants) for studies on large bio-molecular systems as an
alternative to DFT because it does not suffer from theself-interaction error of common density functionals. A
comparison with the dispersion energy from SAPT cal-
culations and Boys and Bernadi counterpoise correctionsfor a peptide-ethene potential energy curve shows that
D3 and gCP (in HF-gCP-D3/MINIS) provide a physi-
cally sound description of the interaction energy com-
ponents.
The diverse set of examples investigated here already
provide a good initial basis for the validation of the gCP ap-
proach and future studies and applications likely will further
solidify it. One yet unattended topic, e.g., is the impact of thecorrection for force constant (vibrational frequency) calcula-
tions and the derived statistical thermodynamic corrections.
ACKNOWLEDGMENTS
This work was supported by the DFG in the framework
of the International Research Training Group 1444 “Gen-
eration of Supramolecular Functional Cavities – Container
Molecules, Macrocycles, and Related Compounds” with ascholarship to H.K.
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5.0021741.pdf | J. Appl. Phys. 128, 083903 (2020); https://doi.org/10.1063/5.0021741 128, 083903
© 2020 Author(s).Effect of interfacial oxidation layer in spin
pumping experiments on Ni80Fe20/SrIrO3
heterostructures
Cite as: J. Appl. Phys. 128, 083903 (2020); https://doi.org/10.1063/5.0021741
Submitted: 18 July 2020 . Accepted: 12 August 2020 . Published Online: 27 August 2020
T. S. Suraj
, Manuel Müller
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, Matthias Opel
, Mathias Weiler
, K.
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Journal of Applied Physics 128, 080901 (2020); https://doi.org/10.1063/5.0020752Effect of interfacial oxidation layer in spin
pumping experiments on Ni 80Fe20/SrIrO 3
heterostructures
Cite as: J. Appl. Phys. 128, 083903 (2020); doi: 10.1063/5.0021741
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Submitted: 18 July 2020 · Accepted: 12 August 2020 ·
Published Online: 27 August 2020
T. S. Suraj,1,2,a)
Manuel Müller,3,4
Sarah Gelder,3Stephan Geprägs,3
Matthias Opel,3
Mathias Weiler,3,4
K. Sethupathi,2
Hans Huebl,3,4,5
Rudolf Gross,3,4,5
M. S. Ramachandra Rao,1,b)
and Matthias Althammer3,4,c)
AFFILIATIONS
1Department of Physics, Nano Functional Materials Technology Center, Material Science Research Center,
Indian Institute of Technology Madras (IITM), Chennai 600036, India
2Low Temperature Physics Lab, Department of Physics, Indian Institute of Technology Madras (IITM), Chennai 600036, India
3Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany
4Physik-Department, Technische Universität München, 85748 Garching, Germany
5Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 München, Germany
a)Electronic mail: surajts@physics.iitm.ac.in
b)Electronic mail: msrrao@iitm.ac.in
c)Author to whom correspondence should be addressed: matthias.althammer@wmi.badw.de
ABSTRACT
SrIrO 3with its large spin –orbit coupling and low charge conductivity has emerged as a potential candidate for efficient spin –orbit torque
magnetization control in spintronic devices. Here we report on the influence of an interfacial oxide layer on spin pumping experiments in
Ni80Fe20(NiFe)/SrIrO 3bilayer heterostructures. To investigate this scenario, we have carried out broadband ferromagnetic resonance
(BBFMR) measurements, which indicate the presence of an interfacial antiferromagnetic oxide layer. We performed in-plane BBFMR exper-iments at cryogenic temperatures, which allowed us to simultaneously study dynamic spin pumping properties (Gilbert damping) and staticmagnetic properties (such as the effective magnetization and magnetic anisotropy). The results for NiFe/SrIrO
3bilayer thin films were ana-
lyzed and compared to those from a NiFe/NbN/SrIrO 3trilayer reference sample, where a spin-transparent, ultra-thin NbN layer was
inserted to prevent the oxidation of NiFe. At low temperatures, we observe substantial differences in the magnetization dynamics parametersof these samples. In particular, the Gilbert damping in the NiFe/SrIrO
3bilayer sample drastically increases below 50 K, which can be well
explained by enhanced spin fluctuations at the antiferromagnetic ordering temperature of the interfacial oxide layer. Our results emphasizethat this interfacial oxide layer plays an important role for the spin current transport across the NiFe/SrIrO
3interface.
Published under license by AIP Publishing. https://doi.org/10.1063/5.0021741
I. INTRODUCTION
Charge to spin current conversion efficiency in heavy metal
(HM)/ferromagnet (FM) bilayers has become one of the central
themes of spintronics research with the goal to manipulate themagnetization in the FM via spin –orbit torques (SOTs) induced at
the interface to the HM.
1–5Heavy metals, like Pt, W, and Ta, have
been successfully used in SOT experiments in HM/FM bilayers
owing to their large spin –orbit coupling (SOC).4,6,7Beyond these
well-established HM materials, iridium-based oxides (iridates) withtheir high spin Hall conductivity, low charge conductivity, and
large SOC ( /C250:1/C01 eV) are promising candidates for SOT
studies.8,9Among them, the 5d transition metal oxide SrIrO 3
(SIO), in particular, remains in the spotlight due to its exotic band
structure with extended 5d orbitals.8,10,11Depending on the
strength of the Coulomb interaction, SIO shows a transition from aMott insulator to a metallic ground state. Compared to metals, theSOT effects in oxide materials offer a wide tunability due to the
dependency of the electronic properties on the oxygen octahedralJournal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 128, 083903 (2020); doi: 10.1063/5.0021741 128, 083903-1
Published under license by AIP Publishing.rotation or oxygen vacancies.12A large spin Hall angle of 110%
was reported for SIO from second-harmonic Hall measurements in
NiFe/SIO.13
The study of direct SOT effects requires microlithography pro-
cesses and also demands sophisticated measurement protocols.Typically, in experiments, either magnetization dynamics are studied
upon injecting large charge current densities through HM/FM heter-
ostructures
2,14or the torque imposed on the magnetization in the
FM is detected by measuring modulations of the magnetoresistancein the heterostructure as a function of the applied chargecurrent.
1,6,15However, to achieve sizeable effects, these experiments
require high charge current densities which can only be realized in
patterned samples. In contrast, due to Onsager reciprocity,16spin
pumping experiments allow one to study the inverse SOT effects inblanket HM/FM bilayer structures. Here, magnetization dynamics isexcited in the FM via an external microwave driving field and excess
angular momentum is pumped as a pure spin current across the
interface into the HM. Absorption of this pure spin current inthe HM represents an additional contribution to the damping of themagnetization dynamics. Evaluating the change in linewidth in com-bination with the voltage generated by the inverse SOT enables one
to address and quantitatively analyze the inverse SOT effects in the
HM.
17,18Recent spin pumping experiments demonstrated a large
SOT for SIO, as compared to elemental heavy metals.13All the spin
pumping experiments carried out in SIO employed the metallic fer-
romagnet NiFe as a spin injector layer due to its low damping,
making it an ideal candidate for ferromagnetic resonance (FMR)experiments.
12However, anomalies in the Gilbert damping of NiFe
films grown on oxide substrates have been found in low temperatureFMR measurements by different groups.
19,20In particular, a pro-
nounced increase in Gilbert damping in NiFe/Al 2O3heterostruc-
tures near T¼50 K was observed and attributed to a spin
reorientation arising from thermal excitations.19In addition, it was
found that the choice of NiFe in combination with oxide substratesinvokes interfacial oxidation exhibiting antiferromagnetic ordering at
low temperatures.
20This interfacial oxidation has not been taken
into account in previous experiments with NiFe/SIO bilayers butneeds to be addressed to tune the SOT efficiency.
In this letter, we study the influence of an interfacial oxide
layer between NiFe and SIO on spin pumping experiments. To this
end, we explore the spin transport in NiFe/SIO bilayers with and
without inserting a thin NbN spacer layer between SIO and NiFe.The additional spacer layer prevents the diffusion of oxygen to theNiFe layer, while weakly suppressing spin transport. We employed
the BBFMR technique to study the magnetization dynamics and
extracted the FMR spectroscopic parameters as a function of tem-perature. A comparison of these parameters for samples with andwithout NbN spacer layer permits us to identify the potential for-mation of an interfacial oxide layer.
II. EXPERIMENTAL DETAILS
SIO thin films with a thickness of 5 nm were deposited on
single crystalline, (001)-oriented SrTiO
3(STO) substrates using
pulsed laser deposition.21Subsequently, NiFe (Ni 80Fe20) was DC
sputter-deposited ex situ on top of SIO and capped with a 3 nm
thin Al layer to prevent the top surface of NiFe from oxidation. Forcomparison, an additional trilayer sample was fabricated with a
3 nm thin NbN layer between SIO and NiFe to prevent the oxida-
tion of the NiFe layer at this interface. NbN is a well establisheddiffusion barrier for oxygen,
22and a superconductor with TC
around 18 K.23In addition, we also fabricated a NiFe thin film
grown directly on top of an STO substrate and capped with 3 nm
of Al to explore its intrinsic properties. All sputter deposition pro-
cesses were performed at room temperature in an ultrahighvacuum system (base pressure in the 10
/C09mbar range). The sput-
tering process was carried out at 5 /C210/C03mbar in an Ar (NiFe,
Al) or an Ar and N 2mixture (NbN, flow ratio of Ar to N 2:1 8 . 1 /
1.9) atmosphere. To ensure the sample quality, we performed
x-ray diffraction studies and magnetometry measurements(SQUID magnetometer). Broadband ferromagnetic resonance(BBFMR) measurements employed a vector network analyzer(VNA) in combination with a 3D-vector magnet cryostat with a
variable temperature insert.
III. RESULTS AND DISCUSSION
A. Structural characterization of SIO thin films
High-resolution x-ray diffraction measurements reveal an epi-
taxial growth of SIO on STO (see Fig. 1 ). The high crystalline
quality of the samples is confirmed by 2 θ-ωscans revealing satel-
lites around the SIO(002) reflection due to Laue oscillations, indi-cating a coherent growth [ Fig. 1(a) ]. The thin films show a low
mosaic spread, as demonstrated by the full width at half maximum
(FWHM) value of 0 :02
/C14extracted from the SIO(002) rocking curve
[the inset in Fig. 1(a) ]. In addition, the reciprocal space maps
(RSM) around the STO (002) [ Fig. 1(b) ] and STO (204) reflections
[Fig. 1(c) ] reveal a lattice matched growth of the SIO film on the
STO as both share the same reciprocal lattice units q H. Thus, the
SIO exhibits a compressive strain in the film plane.
B. Room temperature BBFMR measurements
For the investigation of the magnetization dynamics, we per-
formed BBFMR measurements using a coplanar waveguide (CPW)
with a 80 μm wide center conductor. We record the complex
microwave transmission parameter S 21at fixed microwave frequen-
cies fin the range from 5 GHz to 32 GHz as a function of the
in-plane magnetic field μ0Hextwith a VNA output power of
0 dBm. A schematic illustration of the experimental setup is shown
inFig. 2(a) . The real and imaginary part (black and red symbols)
of the recorded transmission parameter S 21are fitted (black and
red lines) to the Polder susceptibility as shown in Fig. 2(b) .25From
this fit, we extract the values of the FMR field μ0Hresand the FMR
linewidth μ0ΔHfor each frequency. Using the Kittel formula26for
the in-plane magnetization case,
μ0Hres¼/C0μ0Hani/C0μ0Meff
2þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
μ0Meff
2/C18/C192
þ2πf
γ/C18/C192s
, (1)
with γthe gyromagnetic ratio. Thus, we can extract the effective
saturation magnetization Meff¼Ms/C0Ku(with the saturation
magnetization Msand the out-of-plane anisotropy field Ku) and
the in-plane anisotropy field μ0Hanialong the CPW direction.Journal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 128, 083903 (2020); doi: 10.1063/5.0021741 128, 083903-2
Published under license by AIP Publishing.In addition, we determine the Gilbert damping parameter αas well
as the inhomogeneous line broadening μ0Hinhfrom the microwave
frequency dependence of μ0ΔHvia the relation27,28
μ0ΔH¼μ0Hinhþ22πfα
γ(2)
(see the supplementary material for BBFMR data).
To quantify the role of spin pumping in our SIO(5 nm)/NiFe
(dFM) bilayer heterostructures, we extracted the Gilbert damping
parameter αas a function of the NiFe layer thickness dFM[see
Fig. 2(c) ]. A linear dependence of αon 1 =dFMis clearly evident
and is attributed to pumping a spin current from the NiFe intoSIO.
29Moreover, the y-axis intercept allows to quantify the bulk
Gilbert damping α0, while the slope allows to determine the effec-
tive spin mixing conductance g"#
effvia30
α(dFM)¼α0þγ/C22hg"#
eff
4πMs1
dFM/C18/C19
: (3)
Here, /C22his reduced Planck ’s constant, and Ms¼740 kA =m is deter-
mined from SQUID magnetometry of our NiFe layers. We obtainα
0¼6:44/C210/C03+8/C210/C05, which corresponds well to litera-
ture studies.31–34In addition, we find g"#
eff¼5:5/C21018+2:6
/C21017m/C02for these heterostructures at room temperature.24,35
This result agrees well with values in the literature for NiFe/heavy
metal heterostructures36–38and highlights the feasibility of SIO
for SOT devices.
To investigate the role of an interfacial NiFeO xoxide layer, we
conducted BBFMR measurements on two SIO/NbN/NiFe trilayersamples. To obtain an estimate for spin pumping from these two
samples, we note that NiFe can be grown on NbN with bulk prop-
erties (see the supplementary material ) and consequently con-
strained in the linear fit the bulk damping to α0¼6:44/C210/C03.
Although one would expect that NbN acts as a spin-transparentinterlayer due to the long spin diffusion length of NbN (14 nm),
39
we observe a reduction of 40% in g"#
efffor the trilayer samples.
Assuming the formation of an interfacial oxide NiFeO xlayer when
SIO is in direct contact with NiFe, our results suggest that eitherthis oxide layer allows for more efficient spin current injection as
already reported for HMs in contact with NiFe
40or serves as a
source of spin memory loss.41However, more systematic studies
are required to unambiguously separate these two contributionsand better understand the role of the interfacial NiFeO
xlayer for
spin current transport.
T h en a t u r eo ft h ei n t e r f a c i a lo x i d el a y e ri no u rb i l a y e r sw e r e
further investigated by plotting μ0Meffvs 1 =dFMas shown in Fig. 2(d) .
This linear scaling with 1 =dFMis an indication of interfacial anisot-
ropy in the bilayer sample and may be induced via the oxidation ofNiFe.
42Interestingly, for the SIO/NbN/NiFe trilayers, we find a larger
(/difference15%) μ0Meffas compared to bilayer samples with similar dFM.T h i s
clearly indicates a change in interface anisotropy.
C. BBFMR experiments at low temperatures
To more systematically investigate the role of an interfacial
NiFeO xlayer, we investigated SIO (5 nm)/NiFe (5 nm) bilayer,
NiFe (5 nm) single layer and SIO (5 nm)/NbN (3 nm)/NiFe(5 nm)trilayer samples at 5 K /C20T/C20300 K and extracted the FMR spec-
troscopic parameters. Figure 3(a) shows the Gilbert damping αas a
function of temperature for these three samples (fitting procedures
FIG. 1. Structural properties of a SIO (30 nm) thin film grown on a (001)-oriented SrTiO 3(STO) substrate. (a) 2 θ-ωscan along the [001] direction of STO. The inset
shows the rocking curve of the SIO(002) reflection and the derived full width at half maximum (FWHM) value. (b) and (c) Reciprocal space mappings aroun d the symmetric
STO(002) and the asymmetric STO(204) reflections, respectively. The reciprocal lattice units (rlu) are related to the respective STO(001) substra te reflection.Journal of
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J. Appl. Phys. 128, 083903 (2020); doi: 10.1063/5.0021741 128, 083903-3
Published under license by AIP Publishing.are described in Fig. S2 in the supplementary material ). For both
the SIO/NiFe bilayer and the NiFe single layer samples, theGilbert damping αincreases at low temperatures, reaches a
maximum around 25 K, and then decreases with decreasing tem-perature, highlighted as a green shaded region in Fig. 3(a) .I nc o n -
trast, we only observe a weak temperature dependence for the
SIO/NbN/NiFe trilayer sample, which is in accordance withearlier reports of elemental 3d-transition FMs.
43,44We attribute
the observed maximum in Gilbert damping for the SIO/NiFebilayer and NiFe single layer samples to the antiferromagnetic
ordering
20,40of an oxide layer (with a thickness of /difference0:5n m )
formed between SIO or STO and NiFe. This interfacial antiferro-magnetic oxide layer also contributes to the damping due to mag-netic fluctuations near the Néel temperature, which enhances thespin mixing conductance across the interface and, thus, increases
the observed α. From additional temperature-dependent BBFMRmeasurements on a SIO(5 nm)/NiFe(7 nm) bilayer, we extract an
estimate for g
"#
eff(T) and find an enhancement of g"#
effaround 50 K
(see the supplementary material ). Similar results have been
reported by Frangou et al. ,20where they showed that the contribu-
tion of an interfacial antiferromagnetic oxide layer formed
between SiO 2and NiFe manifests as a peak in αnear T/difference50 K.
At these low temperatures, we also find a larger Gilbert dampingfor the SIO/NiFe bilayer sample compared to the NiFe single layersample. We attribute this observation to the fact that the SIO thinfilm has larger roughness than the STO substrate, which promotes
a higher amount of NiFe oxidation. The effect of NiFe oxidation
also manifests itself in the values extracted for M
eff,a sp l o t t e di n
Fig. 3(b) (see also Fig. S3 in the supplementary material ). For the
SIO/NiFe bilayer and NiFe single layer samples, the extracted Meff
is significantly lower as for the SIO/NbN/NiFe trilayer sample.
This change in Meffmay be attributed to an additional surface
FIG. 2. Room temperature BBFMR measurements. (a) Schematic of the experimental setup and illustration of the sample stack m ounting on a CPW. (b)
Experimental data (symbols) of the real (black) and imaginary part (red) of S 21from the SIO/NiFe(5 nm) sample, excited with f¼5 GHz. The lines are fits according
to Ref. 24. (c) Gilbert damping parameter, α, as a function of 1 =dFMin the in-plane geometry with a linear fit (black line) indicating the intrinsic damping of NiFe with
spin pumping contribution for SIO/NiFe heterostructures. For comparison, the values obtained from two SIO/NbN/NiFe trilayers have been added. For the tril ayers,
we assumed the same bulk damping α0as for the bilayers. (d) Effective m agnetization, M eff, as a function of 1 =dFMwith linear fit (line) indicating the presence of
interface anisotropy in the SIO/NiFe heterostructures. For comparison, data points for two SIO/NbN/NiFe trilayers are also added (for the tril ayers we constrained to
the same intercept as for the bilayers).Journal of
Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 128, 083903 (2020); doi: 10.1063/5.0021741 128, 083903-4
Published under license by AIP Publishing.anisotropy, which originates from the formation of the NiFeO xlayer
or the direct contact of NiFe to the SIO (see the supplementary
material for further discussion). Most interestingly, we also find a
dramatic change in the temperature dependence for the SIO/NiFebilayer, and NiFe single layer samples in μ
0Hinh[Fig. 3(c) ]a n d μ0Hani
[Fig. 3(d) ] as compared to the SIO/NbN/NiFe trilayer sample. In the
case of in-plane BBFMR, a non-linear frequency dependence of theFMR linewidth can arise due to an additional dissipation channelfrom two-magnon scattering.
45We can rule out the enhancement
of two-magnon scattering for the observed increase in μ0Hinhsince
we extract similar values in out-of-plane BBFMR measurements
(see the supplementary material ).
D. Magnetization measurements
To further investigate the role of the interface oxide layer, we
conducted magnetization measurements on SIO (5 nm)/NiFe(3:5 nm) bilayer and SIO (5 nm)/NbN (3 nm)/NiFe (3 :5 nm) trilayer
samples. We reduced the thickness of the NiFe as compared to the
samples above used for the temperature-dependent BBFMR studies to
enhance the contributions from the thin oxidized NiFe interfaciallayer. The magnetization measurements for the SIO/NiFe bilayer
and the SIO/NbN/NiFe trilayer sample are shown in Figs. 4(a)
and 4(b), respectively. As evident from these magnetization vs
field measurements, we find a reduction by 15% in the saturationmagnetization (extracted by using the nominal thickness of the
NiFe layer) for the SIO/NiFe bilayer as compared to the SIO/NbN/
NiFe trilayer. If we assume this reduction originates only from oxi-dization, the NiFeO
xthickness is /difference0:5 nm. Indeed, we extract for
thicker NiFe films identical values for the saturation magnetizationfor the bilayers and trilayers (see the supplementary material ),
which indicates an interfacial origin for the reduction in saturation
magnetization. However, we find a lower saturation magnetizationthan for bulk NiFe in all our samples, which we attribute to a com-bination of interdiffusion
46,47and experimental uncertainties in
the determination of the volume of the NiFe layer. Moreover, we
observe a strong increase in coercive field μ0Hcfor the SIO/NiFe
bilayer sample as compared to the SIO/NbN/NiFe trilayer samplefor temperatures below 100 K. We attribute these larger μ
0Hc
values to an exchange bias effect, where the ferromagnetic
domains of NiFe are pinned by the antiferromagnetic NiFeO x
phase.48,49We note that the hysteresis curve is not shifted when an
FIG. 3. BBFMR spectroscopic parameters as a function of temperature. (a) Gilbert damping α, (b) effective magnetization ( Meff), (c) inhomogeneous line broadening
(μ0Hinh), and (d) anisotropy field μ0Haniplotted as a function of temperature for SIO(5 nm)/NiFe(5 nm) (blue symbols), NiFe(5 nm) (red symbols), and SIO(5 nm)/NbN
(3 nm)/NiFe(5 nm) (green symbols) samples. For comparison, μ0Hcderived from SQUID magnetization measurements is also shown in (d) for a SIO(5 nm)/NiFe(3 :5 nm)
(orange symbols).Journal of
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J. Appl. Phys. 128, 083903 (2020); doi: 10.1063/5.0021741 128, 083903-5
Published under license by AIP Publishing.external magnetic field is applied while the sample is cooled down.
In addition, we extract a similar temperature dependence of thecoercive field as compared to the determined BBFMR parametersfor the SIO/NiFe bilayer sample. To illustrate this, we plotted thecoercive field μ
0HcinFig. 3(d) as orange symbols. A similar tem-
perature dependence is found for the BBFMR extracted μ0Hani
parameter and μ0Hc, indicating the same physical origin of both
phenomena, i.e., an oxide layer formed at the SIO/NiFe interface.
IV. CONCLUSIONS
In summary, our work provides an additional perspective on
spin current transport across metallic ferromagnet/SOT-active oxideinterfaces. In particular, our results show that a NiFeO
xlayer at the
interface of NiFe/SIO heterostructures leads to an enhanced spinpumping at room temperature. This enhancement can be either
attributed to an enlarged spin mixing conductance or an increase in
spin memory loss mediated by the NiFeO
xlayer. In our low temper-
ature BBFMR measurements, we find for the bilayer a significantenhancement of the Gilbert damping parameter around 50 K. Weattribute this observation to the antiferromagnetic ordering of the
NiFeO
xlayer at 50 K, which leads to an enhancement of the spin
mixing conductance via magnetic fluctuations.50Moreover, μ0Hinh
andμ0Haniexhibit an increase at low temperatures. We compared
these results to a SIO/NbN/NiFe trilayer sample and found that theformation of the oxidation layer can be avoided by inserting a thin
NbN spacer. Additional magnetization data showed an exchange
bias effect between NiFe and the antiferromagnetic oxide layer anda reduction in the saturation magnetization for the bilayer. Furtherstudies are required to analyze whether this interfacial oxide layer is
detrimental or beneficial and can be exploited to tune the spin
current transport across the SIO/NiFe interface.SUPPLEMENTARY MATERIAL
See the supplementary material for details on the growth
parameters and supporting BBFMR data.
ACKNOWLEDGMENTS
We acknowledge financial support by the German Academic
Exchange Service (DAAD) via Project No. 57452943 and by theDFG via Project No. AL 2110/2-1. T.S.S. and M.S.R.R. acknowledgefunding from DST (No. PH1920541DSTX002720) and Grant No.
SR/NM/NAT/02-2005. M.S.R.R. and K.S. acknowledge funding
from SERB (No. EMR/2017/002328).
DATA AVAILABILITY
The data that support the findings of this study are available
from the corresponding author upon reasonable request.
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Applied PhysicsARTICLE scitation.org/journal/jap
J. Appl. Phys. 128, 083903 (2020); doi: 10.1063/5.0021741 128, 083903-7
Published under license by AIP Publishing. |
1.5130003.pdf | AIP Advances 10, 015124 (2020); https://doi.org/10.1063/1.5130003 10, 015124
© 2020 Author(s).Theoretical study on lowering loss of skin
effect suppressed multi-layer transmission
line with positive/negative (Cu/NiFe)
permeability materials for high data-rate
and low delay-time I/O interface board
Cite as: AIP Advances 10, 015124 (2020); https://doi.org/10.1063/1.5130003
Submitted: 03 October 2019 . Accepted: 23 December 2019 . Published Online: 10 January 2020
Y. Aizawa , H. Nakayama , K. Kubomura , R. Nakamura , and H. Tanaka
COLLECTIONS
Paper published as part of the special topic on 64th Annual Conference on Magnetism and Magnetic Materials
Note: This paper was presented at the 64th Annual Conference on Magnetism and Magnetic Materials.
AIP Advances ARTICLE scitation.org/journal/adv
Theoretical study on lowering loss of skin effect
suppressed multi-layer transmission line with
positive/negative (Cu/NiFe) permeability materials
for high data-rate and low delay-time I/O
interface board
Cite as: AIP Advances 10, 015124 (2020); doi: 10.1063/1.5130003
Presented: 7 November 2019 •Submitted: 3 October 2019 •
Accepted: 23 December 2019 •Published Online: 10 January 2020
Y. Aizawa,1H. Nakayama,2,a)K. Kubomura,1R. Nakamura,2and H. Tanaka2
AFFILIATIONS
1Advanced Course of Production and Environment System, National Institute of Technology, Nagano College, 716 Tokuma,
Nagano, Nagano 381-8550, Japan
2Department of Electronics and Control Engineering, National Institute of Technology, Nagano College, 716 Tokuma, Nagano,
Nagano 381-8550, Japan
Note: This paper was presented at the 64th Annual Conference on Magnetism and Magnetic Materials.
a)Electronic mail: nakayama@nagano-nct.ac.jp.
ABSTRACT
This paper proposes a new application of skin effect suppression technology for long wiring on high-speed & low-delay I/O board. This
proposal will overcome the difficulty of further reducing the transmission losses on the I/O board with vert >vert 50 Gb/s data rate. In
previous research, it was demonstrated that suppression of the skin effect by electroplated conductor/magnetic multi-layer, and estimated
that the degree of transmission loss decrease at 16 GHz would be 5 %. A major challenge in this paper is to propose an electromagnetic
field calculation theory for rectangular multi-layer transmission line, verify it under the same conditions, clarify a lower loss structure by
changing thickness of each layer. Also it is expanded to low loss design technology. Cu and NiFe were selected as metal conductor material
and negative permeability magnetic material, respectively. The Cu and NiFe films are alternately stacked to form the multi-layer. The top and
bottom surface layers are Cu layers. The loss suppression was compared under the following conditions. 1) Total number of layers was 33
and total thickness was 12.67 μm by a constant ratio, Cu: tN= 0.51μm and NiFe: tF= 0.25μm. 2) Optimal stacking determined by changing
the thickness of each layer. Compared to conventional thickness by a constant ratio 1), in our proposal 2), we estimated that the loss would
dropped to 92% in optimal thickness. By offsetting the phase change of current density, a lower loss structure could be determined. Compared
with Cu conductor, the top and bottom surface current densities become low, and depth center current density becomes slightly high for the
multi-layer, showing the skin effect is suppressed.
©2020 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license
(http://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/1.5130003 .,s
I. INTRODUCTION
A loss reduction in a high-frequency transmission line is
needed for high-speed and low-delay I/O board. The skin effect loss
is a problem because the loss increases at high frequency. The skin
effect, which is known as the current crowding to the surface ofa conductor, raises up ac resistance by reducing the effective area
flowing the current. Therefore, it degrades the device performance.
As a technique for suppressing skin effect loss, a multilayer transmis-
sion line using a negative permeability material, which has negative
permeability because the response of the magnetic moment inside
the material is delayed with respect to the magnetic field cange at
AIP Advances 10, 015124 (2020); doi: 10.1063/1.5130003 10, 015124-1
© Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv
high-frequency, is proposed. Rejaei and Vroubel1proposed a design
to set the volume average of permeability to be 0 in an alternately
multi-layered structure of metal/magnetic thin film, taking advan-
tage of negative permeability beyond the FMR frequency. Also,
Yamaguchi2proposed a electroplated Cu/NiFe multi-layer, instead
of sputter-deposited thin film in literature,3–5in order to meet
high throughput mass productivity requirements. it was estimated
that the degree of transmission loss decrease at 16 GHz would
be 5 %.
The purpose of this study is to propose a lower loss design
method as compared with previous research2by using a calculation
method for transmission line. The method takes into account mag-
netic material loss and copper loss including skin effect. A major
challenge in this paper is to propose an electromagnetic field calcula-
tion theory for rectangular multi-layer transmission line, and verify
it under the same conditions, clarify a lower loss structure by calcu-
lating models in which thickness of each layer is changed. Also, it is
expanded to low loss design technology.
II. RF COMPLEX PERMEABILITY
Frequency dispersion of permeability gives the basis of mag-
netic thin-film applications in the RF range. The physical description
of the permeability consists of real and imaginary parts, μr′andμr′′,
respectively
μr=μ′/μ′
0=μ′
r−jμ′′
r (1)
The Landau-Lifschitz-Gilbert (L.L.G.) equation6gives the formulas
of frequency-dependent permeability as follows:
μ′
r=4πMs
Hkω2
r(ω2
r−ω2)
ω2r(ω2r−ω2)2+(4πλω)2+ 1 (2)
μ′′
r=4πMs
Hkω2
r(4πλω)2
ω2r(ω2r−ω2)2+(4πλω)2(3)
λ=αγMs (4)
ωr=⌟roo⟪⟪op
⌟roo⟪mo⟨⌟roo⟪mo⟨⌟roo⟪⟨o⟪γ2
μ0MsHk (5)
whereωris the angular FMR frequency, Msis the saturation mag-
netization, Hkis the anisotropic magnetic field, γis the gyro-
magnetism constant, αis the Gilbert damping constant. A magnetic
material with high MsandHkhigh is essential to realize higher fFMR.
Regarding the dynamics, the L.L.G. equation reminds us that μ′
r
=Ms/Hk, in a static and low-frequency magnetic field as well as
fFMR=(γ/2π)×(MsHk/μ0)1/2andμ′
r×f2
FMR=constant for a mag-
netic film with uniaxial anisotropy. Broadband permeameter7can
be used for this measurement, but accurate measurement beyond
10 GHz is still a challenge.8–10The complex permeability was calcu-
lated from the above. The parameters necessary to estimate higher
frequency complex permeability were extracted as 4 πMs= 10000 G,
Hk= 20 Oe, FMR frequency=1.25 GHz, and damping constant α=
0.1. The damping constant is fairly large but still acceptable for the
electrodeposited film bi-layered with the conductive metal. Theseparameters were applied to (2)-(5) to calculate the frequency pro-
file. This calculation leads μ′
r=−2.05 at 16 GHz where | Q| becomes
the maximum 3.78. The imaginary part of the complex permeability
is the magnetic loss.
III. SKIN EFFECT SUPPRESSION BY NEGATIVE
PERMEABILITY
The calculation theory of electromagnetic field distribution in
consideration of the skin effect in the rectangular multi-layer trans-
mission line stacked with multiple materials was derived by extend-
ing the theory of a single rectangular structure. In the following, the
theoretical formulas are described together with the derivation pro-
cess of the theory. Fig. 1 shows a model of a rectangular transmission
line which represents the theoretical formula in the structure stacked
multiple materials. The AC current Jzwith an angular frequency ωin
thez−direction is defined as flowing inside a rectangular conductor
with an infinite width in the x−direction . This conductor is defined
to be isotropic and homogeneous, and its permittivity, permeability,
and conductivity to be ε,μ, andσ, respectively. Maxwell’s equations
in a metal conductor are following equation (6) and (7). The current
density is defined as flowing only in the z−direction . it is expressed
by equation (8).
∇×⃗H=σ⃗E (6)
∇×⃗E=−jωε⃗H (7)
Jx(x,t)=Jy(x,t)=0 ,Jz=Jz(y,t) (8)
Therefore, Maxwell’s equation (6) and (7) has only the xdirection
component. From the above results, equations (6) and (7) are as
follows:
∂Hx(y)
∂y=σEz(y) (9)
FIG. 1 . Model of rectangular transmission line.
AIP Advances 10, 015124 (2020); doi: 10.1063/1.5130003 10, 015124-2
© Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv
∂Ez(y)
∂y=−jωμHx(y) (10)
Eliminating Hxfrom both equations above, the differential equa-
tion (11) for Ezis derived. Here, kis a constant and is expressed by
equation (12). Solving this differential equation, the solution is equa-
tion (13). In addition, the relational expression (14) is established
from (10).
∂2Ez(y)
∂y2−k2Ez=0 (11)
k2=jωσμ (12)
Ez(y)=Ae+ky+Be−ky(13)
∂
∂y(Ae+ky+Be−ky)=−jωμHx(y) (14)
Solving this for Hx, the solution is equation (15). AandBare
both constants. The origin of the ycoordinate is defined to be the
center of the conductor, and the magnetic field at ( y= 0) is zero.
Hx(y)=−k
jωμ(Ae+ky−Be−ky) (15)
Hx(0)=0 (16)
When constants AandBare calculated, A=B=C, and equations
(13) and (15) are as follows:
Ez(y)=C(e+ky+e−ky) (17)
Hx(y)=−Ck
jωμ(e+ky−e−ky) (18)
The spatial distribution of current density in a single-layer transmis-
sion line is derived from these equations and the ⃗J=σ⃗Erelationship.
By extending the theory of a single-layer, the electric field Ezn, cur-
rent density Jzn, and magnetic field Hxnof the n-th layer from the
center in the multilayer transmission line model can be obtained. kn
is a coefficient based on each material property and can be expressed
by (22).
Ezn(y)=Ane+kny+Bne−kny(19)
Jzn(y)=σnEzn(y) (20)
Hxn(y)=−kn
jωμn(Ane+kny−Bne−kny) (21)
kn2=jωσnμn (22)
From the above, the loss of each layer of the rectangular multi-layer
transmission line can be obtained by the equation (23). The first term
is the copper loss obtained from the current density and conductivity
of each layer. The second term is the magnetic loss due to the imag-
inary part of the complex permeability of the negative permeability
material. The magnetic loss can be obtained from the local magnetic
field H(r) and the imaginary part μ” of the complex magnetic per-
meability. The total loss can be evaluated by adding the copper lossand the magnetic loss. Thus, the loss of the entire transmission line
can be obtained from the equation (24).
Loss n=∫yn
yn−1∣Jzn(y)∣2
σndy+∫yn
yn−11
2ωμ”
rn∣Hn(y)∣2dy (23)
Loss=n
∑
i=1Loss i (24)
IV. VERIFICATION PROCEDURE
We investigated a rectangular multi-layer transmission line
using the electromagnetic field theory. In order to evaluate under
the same conditions as the previous research, Cu, which is a general
metal conductor material, was selected as the positive permeability
material, and NiFe was selected as the negative permeability mate-
rial. Table I shows the material values, total current, total thickness,
and frequency conditions.
The loss of the following models was compared by the electro-
magnetic field calculation theory. 1) Copper layer model: A single-
layer rectangular transmission line made only of copper was tar-
geted, and the loss with the skin effect was obtained. We normalize
based on this loss value and compare the loss of each model. 2) Con-
ventional model: The Cu and NiFe films were alternately stacked
to form the Cu/NiFe multi-layer, as shown in Fig. 2. The top and
bottom surface layers are Cu layers. Total number of layers was
TABLE I . Conditions for Consideration.
Total current I 1 A
Frequency f 16 GHz
Total thickness t 12.67μm
CuConductivity σp 5.81×107S/m
Relative permeability μrp +1
NiFeConductivity σn 6.00×106S/m
Relative permeability μrn -2.05-j0.542
FIG. 2 . Structure of conductor/magnetic film multi-layer.
AIP Advances 10, 015124 (2020); doi: 10.1063/1.5130003 10, 015124-3
© Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv
33 and total thickness was 12.67 μm by a constant ratio, Cu thick-
ness tN= 0.51μm, NiFe thickness tF= 0.25μm. 3) Optimal stacking
model: In order to find a lower loss structure, it was determined by
changing the thickness of each layer from the conventional model.
4) 17-layer model: Consider whether the same loss reduction effect
can be obtained even if the number of layers is reduced.
V. RESULTS AND DISCUSSION
Fig. 3(a) shows the current density distribution ∣Jz(y)∣, and
Fig. 3(b) shows the current density phase ∠Jz(y) from the surface.
Table II shows a comparison of loss under each model. It was esti-
mated that the loss of 59.9% in the conventional model 2), 55.1%
in the optimal stacked model 3), and 56.3% in the 17-layer model
4), compared to the copper layer model 1). Compared to conven-
tional thickness by a constant ratio 1), it was estimated that the loss
is 92% in optimal thickness we proposed 2), and the loss is 94% in
17-layer model. Loss suppression effect by stacking negative per-
meability material was expected. In addition, we were able to find
a lower loss structure than the conventional model stacked con-
stant ratio. It was estimated that the same loss suppression could be
obtained even under condition 4) where the number of layers was
reduced. By offsetting the phase change of current density, a lower
loss structure could be determined. Compared with Cu conductor,
Fig. 3(a) shows that the top and bottom surface current densities
become low, and depth center current density becomes slightly high
for the multi-layer, showing the skin effect is suppressed.
FIG. 3 . Calculation result; (a) Current density (b) Current density phase.TABLE II . Comparison of loss under each model.
Models Normalization loss
1)Cu single layer 1
2)Constant ratio 0.599
3)Optimal stacking 0.551
4)Optimal stacking in 17layer 0.563
VI. CONCLUSION
We described the electromagnetic field calculation theory for
the transmission line with the rectangular conductor structure, and
based on the theory, it was investigated skin effect loss suppression
by stacking negative permeability material. Compared to conven-
tional thickness by a constant ratio, in our proposal, we estimated
that the loss would dropped to 92% in optimal thickness. By offset-
ting the phase change of current density, a lower loss structure and
structure with a reduced number of layers could be determined. In
order to expanding low loss design technology, we will calculate the
surface roughness which is conductor loss as well as skin effect next
step.
ACKNOWLEDGMENTS
This work was supported in part by the JSPS joint bilateral
research project, “Verification of the suppression theory of skin
effect loss in transmission lines and establishment of optimum
design method,” JSPS KAKENHI Grant Number 17K14674, and
JSPS KAKENHI Grant Number 19K04521.
REFERENCES
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superlattice conductors,” J. Appl. Phys. 96, 6863 (2004).
2M. Yamaguchi, T. Yanai, H. Nakayama et al. , “Skin effect suppressed ni-fe/cu
electroplated multilayer wiring for high data-rate and low delay-time i/o interface
board,” IEEE Trans. Magn. 54, 1–5 (2018).
3Y. Zhuang, B. Rejaei, H. Schellevis, M. Vroubel, and J. N. Burghart, “Magnetic-
multilayered interconnects featuring skin effect suppression,” IEEE Electron
Device Lett. 29, 319–321 (2008).
4M. Yamaguchi, N. Sato, and Y. Endo, “Skin effect suppression in multi-layer thin-
film spiral inductor taking advantage of negative permeability of magnetic film
beyond fmr frequency,” in Proc. 40th Eur. Microw. Conf. (2010), 1182–1185.
5N. Sato, Y. Endo, and M. Yamaguchi, “Skin effect suppression for cu/cozrnb
multilayered inductor,” J. Appl. Phys. 111, 07A501-1–07A501-3 (2012).
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material),” Koudansha Sci. (1999).
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thin-film permeameter using a side-open tem cell and a planar shielded-loop coil,”
Trans. Magn. Soc. Jpn. 3, 137–140 (2003).
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AIP Advances 10, 015124 (2020); doi: 10.1063/1.5130003 10, 015124-4
© Author(s) 2020 |
1.1812360.pdf | Suppression of skin effect in metal∕ferromagnet superlattice conductors
Behzad Rejaei and Marina Vroubel
Citation: Journal of Applied Physics 96, 6863 (2004); doi: 10.1063/1.1812360
View online: http://dx.doi.org/10.1063/1.1812360
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129.24.51.181 On: Fri, 28 Nov 2014 19:12:05Suppression of skin effect in metal/ferromagnet superlattice conductors
Behzad Rejaeia)and Marina Vroubel
High Frequency Technology and Components Group, Delft Institute of Microelectronics and Submicron
Technology, Faculty of Electrical Engineering, Mathematics, and Computer Science, Delft University ofTechnology, Feldmannweg 17, 2628 CT Delft, The Netherlands
(Received 22 July 2004; accepted 14 September 2004 )
We propose and theoretically investigate a superlattice consisting of alternating normal-metal and
ferromagnetic layers as a low-loss conductor for realization of planar-integrated radio-frequencydevices. At sufficiently high frequencies, the negative permeability of the ferromagnetic filmseffectively compensates the positive permeability of the nonmagnetic metal layers, leading to anoverall suppression of the skin effect. Simulations based on realistic material parametersdemonstrate the effectiveness of this approach regardless of intrinsic relaxation losses in themagnetic films. © 2004 American Institute of Physics .[DOI: 10.1063/1.1812360 ]
I. INTRODUCTION
A major drawback of monolithic-integrated radio-
frequency (rf)passive components such as inductors and
transmission lines is their proneness to significant conductorand substrate losses.
1Whereas substrate loss can be largely
suppressed by the use of highly insulating materials2or even
local removal of the substrate beneath the device,3efforts to
reduce the conductor loss at high frequencies are impeded bythe skin effect. The use of thick metal layers reduces theconductor loss at low frequencies, but has virtually no effectat high frequencies where current effectively flows near thesurface of the conductor within the skin-depth
d
=˛2/vm0mNs. Here v=2pfwithfthe frequency of the rf
field, m0is the vacuum permeability, mNis the relative per-
meability of the nonmagnetic metal s.1d, and sis the metal
conductivity.
From the above equation, it is apparent that skin effect
can be (partially )suppressed by reducing the permeability
experienced by the electromagnetic field inside the conduc-tor. In this paper, we theoretically demonstrate that this canbe effectively achieved in a composite conductor built from asuperlattice consisting of thin normal-metal and ferromag-netic films. The permeability of a ferromagnetic film, withthe rf magnetic field applied perpendicular to the direction ofmagnetization in the film, becomes negative at frequenciesbetween the ferromagnetic resonance (FMR )and antireso-
nance frequencies.
4This property can be used to compensate
the positive permeability mNof the nonmagnetic metal layers
in the superlattice.The field, generated by the flow of currentalong the direction of magnetization, experiences (on aver-
age)a much smaller permeability, leading to a suppression of
the skin effect.
Calculations for aluminum (Al)/permalloy (NiFe )super-
lattices presented in this paper show the effectiveness ofskin-effect suppression, which yields up to a threefold reduc-tion of the conductor sheet resistance at high frequenciescompared to Al lines of the same total thickness. The thick-ness of each layer in the superlattice, however, must be muchsmaller than the skin depth within that layer so that perme-
ability averaging can take place. We will demonstrate that
the inclusion of intrinsic magnetic relaxation losses in NiFedoes not significantly change this picture. As a practical ex-ample, we will investigate how skin-effect suppression influ-ences the characteristics of a microstrip transmission linebuilt from an Al/NiFe superlattice.
It is important to mention that rf properties and applica-
tions of magnetic multilayers and superlattices have beensubject to studies by various authors. Potential rf applicationsof metal/ferromagnetic multilayers include GHz inductors,
5,6
microstrip filters,7–9and devices based on the giant magne-
toimpedance effect.10–12Of particular importance to our
study is the work by Camley and Mills,13who investigated
the microwave propagation in a dielectric/ferromagnetic su-perlattice. They found a sharp attenuation dip close to theantiresonance frequency of the ferromagnetic layers, as a re-sult of opening up of the skin depth. The dielectric/ferromagnetic system can potentially be used as the core of aband-pass filter. In our work, however, the normal-metal/ferromagnetic superlattice is proposed as a low-loss conduc-tor for integrated rf devices, operating below antiresonancefrequency where the permeability of ferromagnetic films isnegative.
II. ANALYSIS
A. The configuration
Figure 1 illustrates a stripline made from a superlattice
consisting of alternating normal-metal and ferromagnetic
a)Electronic mail: b.rejaei@ewi.tudelft.nl
FIG. 1. Astripline built from a superlattice consisting of alternating normal-
metal and ferromagnetic (shaded )layers. The ferromagnetic films are mag-
netized along the strip (in thexdirection ).JOURNAL OF APPLIED PHYSICS VOLUME 96, NUMBER 11 1 DECEMBER 2004
0021-8979/2004/96 (11)/6863/6/$22.00 © 2004 American Institute of Physics 6863
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129.24.51.181 On: Fri, 28 Nov 2014 19:12:05layers magnetized along the strip (in thexdirection ). The
relative permeability tensor inside the magnetic films isgiven by
mˇ=310 0
0m−ima
0ima m4,
m=1+vHvM
vH2−v2,
ma=vMv
vH2−v2, s1d
vM=gMs,vH=gH0,
where gis the gyromagnetic ratio, Msis the saturation mag-
netization, and H0is the anisotropy field. To study the prop-
erties of the system, uniformity is assumed in the xdirection,
and the electric field Eand current density Jare assumed to
have anxcomponent only. In such a configuration, with E
parallel to the magnetization everywhere, the ferromagneticlayers are characterized by the effective permeability
4
m’=m−ma2
m=v2
AR−v2
v2
FMR−v2,
vFMR=fvHsvH+vMdg1/2, s2d
vAR=vH+vM,
where vFMRis the (thin-film )FMR frequency and vARis the
antiresonance frequency at which m’=0. The permeability
m’becomes negative in the frequency range of
vFMR,v,vAR. In on-chip devices, where application of
large external bias magnetic fields is not feasible, the aniso-tropy field H
0only consists of internal magnetocrystalline
anisotropy and demagnetization fields (shape anisotropy ).14
In magnetic materials such as NiFe where those fields are
small compared to Ms, one has vH!vMandvFMR!vAR,s o
that m’,0 over a broad frequency range. For permalloy
with a saturation magnetization of ,1T, the antiresonance
frequency is around 27 GHz, while the FMR frequencyhardly exceeds several GHz’s for typical thin films.
15
The negative value of m’can be used to “average out”
the permeability inside the superlattice as the followingqualitative argument demonstrates. At low frequencies, faraway from the edges of the strip, the average relative perme-ability for in-plane magnetic fields along ydirection in the
superlattice is
mav=mNtN+m’tF
tN+tF, s3d
wheretNandtFdesignate the thickness of the normal and
ferromagnetic layers, respectively. This relationship remainsvalid at high frequencies, if the magnetic field is not stronglyshielded by individual films, i.e., if t
NandtFare much
smaller than the skin depth in their respective layers. Return-ing to the argument presented in the introduction, one canthen expect the skin effect to become effectively suppressed
inside the superlattice, if the positive permeability of thenormal-metal layers is averaged out against the negative per-meability of the ferromagnetic layers, i.e., if
mav,0. As a
result, current density in the superlattice will be evenly dis-tributed in the vertical direction, reducing the overall ohmicloss in the system.
A complete study of this effect in the structure of Fig. 1
requires, in principle, the (numerical )solution of the electro-
magnetic field on the cross section of the strip. In conven-tional on-chip applications, however, the lateral size of thestrip is much larger than its thickness so that the zero-thickness (or planar )approximation can be used.
16Here the
finite-thickness strip is modeled as an infinitely thin conduc-tor characterized by a constant surface impedance
zs. The
complex quantity zs, whose real part represents the ohmic
losses in the conductor, replaces the bulk conductor resistiv-ity in further computations of the electromagnetic propertiesof the strip or any other device built from the superlattice.Therefore, for our purpose, it suffices to investigate thesurface impedance of the normal-metal/ferromagneticsuperlattice.
17
B. Surface impedance
The surface impedance zsof the superlattice can be cal-
culated by looking for uniform (plane-wave )solutions of the
Maxwell equations for the fields
E=xˆExszd,H=yˆHyszd+zˆHzszd, s4d
in a system infinitely extended in the x-yplanes (Fig. 2 ).
Herexˆ,yˆ, andzˆ, denote the unit vectors in the x,y, andz
directions, respectively. The total complex power per unitlength delivered to a lateral segment of the superlattice withthe width wis
18
P=−1
2R
CE3H*·nˆdl
=−w
2suExHy*uz=t/2−uExHy*uz=−t/2d, s5d
whereCis the boundary of the segment, nˆis the unit vector
normal to C, andtis the overall thickness of the conductor.
The surface impedance zsis then evaluated by equating Pto
FIG. 2. Cross section of a normal-metal/ferromagnet superlattice with the
total thickness t.The thickness of the normal-metal and ferromagnetic layers
istNandtF, respectively.The configuration is assumed to be symmetric with
respect to z=0. The boundary of a segment with the width wand height tis
denoted by C.zkdesignates the interface between the k-th and sk−1d-th
layers.6864 J. Appl. Phys., Vol. 96, No. 11, 1 December 2004 B. Rejaei and M. Vroubel
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129.24.51.181 On: Fri, 28 Nov 2014 19:12:05P=1
2I*IZs,Zs=s1/wdzs, s6d
whereZsis the impedance per unit length of the segment,
andIis the total current flowing through the segment, given
by the Ampere’s law
I=R
CH·dl=wfHys−t/2d−Hyst/2dg. s7d
Since the Maxwell equations do not allow any discontinuity
in the tangential component of the electric field as onecrosses a zero-thickness conductor, one has to impose thecondition E
xst/2d=Exs−t/2dto ensure a consistent derivation
of the surface impedance.19To further simplify the analysis,
we restrict ourselves to vertically symmetric structures (i.e.,
with respect to z=0)and consider field distributions satisfy-
ingExs−zd=ExszdandHys−zd=−Hyszd. From Eqs. (5)–(7)the
surface impedance zscan then be written as
zs=−1
2UEx
HyU
z=t/2. s8d
With uniformity assumed in both xandydirections, the
electric and magnetic fields satisfy the one-dimensionalequations
]z2Ex−jN2Ex=0,
Hy=−sivm0mNd−1]zEx,
Hz=0, s9d
jN2=−v2m0seN−isN/vd,
inside the normal-metal layers, while inside the ferromag-
netic films, one has
]z2Ex−jF2Ex=0,
Hy=−sivm0m’d−1]zEx,
Hz=−sima/mdHy, s10d
jF2=−v2m0m’seF−isF/vd.
In the above equations, sN,sFandeN,eFdenote the conduc-
tivity and dielectric constant of the normal-metal and ferro-magnetic layers, respectively. The solution for E
xandHy
inside each layer is
Exszd=Akexpsjkzd+Bkexps−jkzd,
Hyszd=−jk
ivm0mkfAkexpsjkzd−Bkexps−jkzdg, s11d
wherekis the layer index, AkandBkare constants, mk=mN,
jk=jNif the layer is a normal metal, and mk=m’,jk=jFif
the layer is ferromagnetic.
The surface impedance zscan now be calculated as fol-
lows. Due to the symmetry in the zdirection, we restrictourselves to the upper half of the superlattice sz.0d. Num-
bering the layers according to Fig. 2, we define the imped-
ances
zk=−Exszkd
Hyszkd, s12d
wherezkdenotes the interface between the k-th and the
sk+1d-th layers. Note that the fields ExandHyare continu-
ous across each interface. From the solution (11), one can
then show that
zk=zk−1+hktanh sjktkd
1+hk−1zk−1tanh sjktkd,
hk=ivm0mk
jk, s13d
wheretkdenotes the thickness of the kth layer. In order to
use Eq. (13), we assume that the layer at the center of the
superlattice is (fictitiously )divided at z0=0 into two layers of
the same thickness. Since Hys0d=0 due to the symmetry of
the problem, we have z0=‘from Eq. (12). The impedances
zk,k=1,2,... can then be recursively calculated from Eq.
(13). The surface impedance of the superlattice is given by
zs=zM/2, where Mis the index of the topmost layer with
zM=t/2. In the next section, we use the calculation outlined
above to study the suppression of skin effect in the normal-metal/ferromagnetic superlattice.
III. RESULTS AND DISCUSSION
A. Skin-effect suppression
Surface impedance of superlattices consisting of Al and
NiFe films was calculated assuming a saturation magnetiza-tion ofM
s=1 T and an anisotropy field of H0=1 0O ei nt h e
NiFe films. This corresponds to FMR frequency of,0.85 GHz and antiresonance frequency of 27 GHz. The
conductivities of NiFe and Al layers were taken as
sF=6
3106S/mand sN=3.7 3107S/m, respectively.
In order to demonstrate the behavior of ohmic losses in
the superlattice, in Fig. 3 we have plotted the sheet resistance
rs=Reszsdof several superlattices as a function of frequency.
FIG. 3. Sheet resistance srsdof 6- mm-thick Al/NiFe superlattices as a func-
tion of frequency. The simulation was carried out for a total number of N
=9, 13, 21, and 33 layers, with the ratio tN/tFkept constant at 2.5. The
parameters used where Ms=1T,H0=10 Oe, and sF=63106S/m for the
NiFe films while sN=3.7 3107S/m inAl layers. The dashed line shows the
result for a 6- mm-thick Al conductor. The dc value of sheet resistance srdcd
for the superlattice is shown by the dotted line.J. Appl. Phys., Vol. 96, No. 11, 1 December 2004 B. Rejaei and M. Vroubel 6865
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129.24.51.181 On: Fri, 28 Nov 2014 19:12:05The structures have the same overall thickness of t=6mm
but different number of layers. The ratio r=tN/tFwas kept
constant at 2.5. For comparison, the result for a 6- mm-thick
Al conductor is also shown. In all cases a reduction of rsis
seen above ,12 GHz with a minimum around 14.5 GHz.At
this frequency the effective permeability of the NiFe layers is
m’=−2.46. Note that, since tN/tF=2.5 in the simulation, the
minimum in loss roughly corresponds to the frequency atwhich
mav[Eq.(3)]vanishes (note that mN.1).
Figure 4 shows the vertical distribution of current den-
sity in the Al/NiFe superlattice with 21 layers at 14.5 GHz.Current in the superlattice is mainly concentrated in the non-magnetic (Al)layers due to their much higher electric con-
ductivity compared to NiFe. Furthermore, the fluctuation incurrent density increases as one approaches the surface of thestrip. Nevertheless, the overall distribution of current densityis far more uniform than that in a pure Al conductor, whichexplains the strong reduction of the sheet resistance
rs.
As argued in the previous section, permeability averag-
ing only makes sense if electromagnetic shielding by indi-vidual layers of the superlattice is negligible, i.e., if thethickness of each layer is much smaller than the skin depthinside that layer. This fact is reflected in the increased effec-tiveness of skin-effect suppression as the thickness of indi-vidualAl and NiFe layers is decreased with increasing num-ber of layers (Fig. 3 ). At 14.5 GHz, the sheet resistance of
the superlattice with 33 layers is almost 3 times smaller thanthat of the Al conductor and is close to the dc sheet resis-tance
rdc. This implies an almost uniform distribution of cur-
rent density inside the layers, as in the dc case. Note, how-ever, that below 12 GHz the loss in the superlattices exceedsthat of theAl conductor due to the large negative value of
m’
and, therefore, very small values of skin depth in the ferro-magnetic layers.The electromagnetic field is then completelyshielded by few ferromagnetic layers at the bottom and topof the stack, accompanied by high ohmic losses in thoselayers.
Figure 5 shows
rsas a function of frequency for the
samples with the same total thickness s6mmdand number of
layers sN=33dbut with different ratios r=tN/tF.A srin-
creases, larger negative values of m’are required for ensur-
ing a zero average permeability [Eq. (3)]. Therefore, the
minimum in loss is shifted to lower frequencies, as can beseen from Eq. (2). Thus, in actual applications, the superlat-
tice can be “optimized” at the desired frequency by adjustingthe ratiot
N/tF.
B. Effect of magnetic loss and shape anisotropy
In the results presented so far, we neglected the effect of
intrinsic magnetic relaxation loss on the overall loss of thesuperlattice. This effect can be studied by replacing
vHin
Eq. (2)byvH+iav, where ais the Gilbert damping
parameter.4This results in a complex permeability m’=m’8
−im’9, where m’9s.0daccounts for the magnetic losses in
the ferromagnetic layers. Figure 6 shows rsversus frequency
for the superlattice of Fig. 2 with N=33, for different values
ofa. As can be seen from Fig. 6, magnetic losses become
remarkable when a.0.02. The reported experimental values
ofain NiFe films are in the range between 0.008 and
0.01220,21so that magnetic losses do not significantly con-
tribute to the overall loss in this case. Note, however, thatmagnetic losses increase with increasing volume of the mag-netic material. Figure 7 shows
rsfor a superlattice composed
of 0.2- mm-thick Al and 0.08- mm-thick NiFe layers at
14.5 GHz as a function of the total thickness t. Comparison
between the curves with a=0 and 0.01 demonstrates the in-
creasing share of magnetic loss as the total thickness of thestack increases.
Another issue is the influence of the anisotropy field H
0
on the rf properties of the superlattice. In the simulations, so
far, we assumed a total anisotropy field of H0=10 Oe inside
the magnetic films. In practice, as no strong external dc bias
FIG. 4. Vertical distribution of the current density uJxuin Al (continuous
line)and an Al/NiFe superlattice with 33 layers (broken line )at 14.5 GHz.
The thickness of both conductors was 6 mm. The discontinuities in Jxin the
superlattice are due to different conductivities of Al and NiFe.
FIG. 5. Sheet resistance of a 6- mm-thick Al/NiFe superlattice as a function
of frequency for different ratios r=tN/tF. The stack consists of 33 layers.
The dashed line shows the result for a 6- mm-thick Al layer.
FIG. 6. Sheet resistance of an Al/NiFe superlattice (t=6mm,N=33, and
tN/tF=2.5 ), as a function of frequency for a=0.0, 0.01, 0.02, 0.05, and 0.1.
The dashed line shows the result for a 6- mm-thick Al conductor.6866 J. Appl. Phys., Vol. 96, No. 11, 1 December 2004 B. Rejaei and M. Vroubel
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129.24.51.181 On: Fri, 28 Nov 2014 19:12:05field is applied, H0depends on the sample shape, as well as
the internal magnetocrystalline anisotropy, which may be af-fected by film deposition conditions. Nevertheless, at fre-quencies much higher than
vFMR,m’8is practically indepen-
dent from vH(and therefore H0)and is given by
m’8.1−SvM
vD2
. s14d
Therefore, the properties of the system become independent
of the anisotropy field, provided that the sample remainsmagnetized along the direction of the flow of current. Inparticular, the frequency at which
mav[Eq.(3)]vanishes only
depends on the saturation magnetization Ms(through vM)
and on the ratio between tNandtF. This point is of crucial
importance for reliable design of passive components basedon the superlattice, since M
sis a material parameter, and film
thicknesses tNandtFcan be easily controlled in the inte-
grated circuit technology.
C. Application to a microstrip line
In order to demonstrate the potential impact of skin-
effect suppression on the performance of integrated rf pas-sive components, we used the calculated values of
zsand
employed the zero-thickness approximation to study thecharacteristics of a microstrip transmission line. The simula-tions were performed using an in-house computer program.Figure 8 shows the resistance and inductance per unit lengthof a microstrip line consisting of a 30-
mm-wide superlattice
strip built on a dielectric substrate. The substrate is made ofa 30-
mm-thick dielectric on a ground plane. The latter is
assumed to be a perfect conductor to simplify the calcula-tion. The Al/NiFe superlattice used in the simulation is6-
mmthick and consists of 21 layers with tN=0.4 mmand
tF=0.16 mm. Results for a conventional 6- mm-thick Al strip
are also shown for comparison. The inductance per unitlength of the superlattice microstrip is only slightly lowerthan that of the conventional device below 18 GHz. Thissmall deviation is caused by the imaginary part of
zsbut
hardly affects the operation of the device. On the other hand,a significant reduction of ohmic loss is observed in the su-perlattice microstrip near 14.5 GHz due to skin-effect sup-pression.To verify the results obtained, we also carried out a com-
putationally intensive three-dimensional (3D)electromag-
netic analysis of the microstrip using
ANSOFT’S HFSS (Fig. 8 ).
At both low and high frequencies, 3D simulations yield up toa 25% lower resistance compared to that of the zero-thickness model. Considering the fact that the height of thesuperlattice s6
mmdis not much smaller than its width
s30mmd, this discrepancy may be caused by the zero-
thickness approximation. Nevertheless good agreement be-
tween the two calculations is observed close to 14.5 GHzwhere skin-effect suppression occurs.
Finally, it should be mentioned that application of the
metal/ferromagnet superlattice is not necessarily restricted tomicrostrip lines but can encompass other devices such ascoplanar waveguides and spiral inductors.
IV. CONCLUSIONS
Superlattices consisting of normal-metal and ferromag-
netic films offer a low-loss alternative to conventional con-ductors for the implementation of planar rf devices. Thenegative in-plane permeability of ferromagnetic films at highfrequencies can be used to compensate the positive perme-ability of the nonmagnetic layers, leading to the (partial )sup-
pression of skin effect. Calculations based on realistic mate-rial parameters show a two to three fold reduction of rf sheetresistance, even in the presence of intrinsic magnetic relax-ation losses. The frequency at which total compensation ofthe skin effect takes place, only depends on the thickness of
FIG. 7. Sheet resistance of an Al/NiFe superlattice at 14.5 GHz as a func-
tion of total thickness tof the stack for a=0.0 and 0.01. The superlattice is
built from 0.2- mm-thickAl, and 0.08- mm-thick NiFe layers.The dashed line
shows the results for an Al conductor.
FIG. 8. Inductance and resistance per unit length for a microstrip line. Thewidth and thickness of the stripline are 30 and 6
mm, respectively. The
substrate consists of a 30- mmdielectric layer on a perfectly conducting
ground plane. Both results for a superlattice and an Al strip are shown. TheAl/NiFe superlattice is made of 21 layers with t
N=0.4 mmandtF
=0.16 mm. In the NiFe layers a damping constant of a=0.01 was assumed.
Markers show the results obtained from 3D HFSSsimulations for the super-
lattice (triangles )and the Al (squares )microstrip lines.J. Appl. Phys., Vol. 96, No. 11, 1 December 2004 B. Rejaei and M. Vroubel 6867
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129.24.51.181 On: Fri, 28 Nov 2014 19:12:05the layers and on the saturation magnetization of the mag-
netic films. This facilitates reproducible design and realiza-tion of rf devices based on the superlattice conductor.
ACKNOWLEDGMENTS
The authors would like to thank Y. Zhuang and J. N.
Burghartz from Delft Institute of Microelectronics and Sub-micron Technology for their many useful discussions.
1See. e.g., N.M. Nguyen and R.G. Meyer, IEEE J. Solid-State Circuits 25,
1028 (1990 ); J.N. Burghartz, M. Soyuer, and K.A. Jenkins, IEEE Trans.
Electron Devices 43, 1559 (1996 ).
2J.N. Burghartz, D.C. Edelstein, K.A. Jenkins, and Y.H. Kwark, IEEE
Trans. Microwave Theory Tech. 45, 1961 (1997 ).
3K.J. Herrick, J.G. Yook, and L.P.B. Katehi, IEEE Trans. Microwave
Theory Tech. 46, 1832 (1998 ).
4A.G. Gurevich and G.A. Melkov, Magnetization Oscillations and Waves
(CRC Press, New York, 1996 ).
5V. Korenivski and R.B. van Dover, J. Appl. Phys. 82,5 2 4 7 (1997 ).
6A. Gromov, V. Korenivski, K.V. Kao, R.B. van Dover, and P.M. Mank-
iewich, IEEE Trans. Magn. 34, 1246 (1998 ).
7H. How, T.-M. Fang, and C. Vittoria, IEEE Trans. Microwave Theory
Tech.43, 1620 (1995 ).
8N. Cramer, D. Lucic, D.K. Walker, R.E. Camley, and Z. Celinski, IEEE
Trans. Magn. 37,2 3 9 2 (2001 ).
9Y. Zhuang, B. Rejaei, E. Boellaard, M. Vroubel, and J.N. Burghartz, IEEEMicrow. Wirel. Compon. Lett. 12, 473 (2002 ).
10D.P. Makhnovskii, A.N. Lagarkov, L.V. Panina, and K. Mohri, J. Appl.
Phys.84, 5698 (1998 ).
11L.V. Panina, D.P. Makhnovskii, and K. Mohri, J. Magn. Soc. Jpn. 23,9 2 5
(1999 ).
12D.P. Makhnovskii, A.N. Lagarkov, L.V. Panina, and K. Mohri, Sens. Ac-
tuators, A 81, 106 (2000 ).
13R.E. Camley and D.L. Mills, J. Appl. Phys. 82, 3058 (1997 ).
14Despite the absence of an external bias field, the easy axis of the ferro-
magnetic films be aligned along the strip by subjecting the films to a largedc field during deposition (magnetocrystalline anisotropy ). The alignment
is further maintained by the shape anisotropy of the sample, although 90°domains may appear near the ends of the strip.
15M. Vroubel, Y. Zhuang, B. Rejaei, and J.N. Burghartz, J. Magn. Magn.Mater.258–259, 167 (2003 ).
16see, e.g., J.C. Rautio and V. Demir, IEEE Trans. Microwave Theory Tech.
51,9 1 5 (2003 ).
17For a calculation of the surface impedance of a ferromagnet/normal-metal/
ferromagnet multilayer see A.S. Antonov and I.T. Yakubov, J. Phys. D 32,
1204 (1999 ).
18R.E. Collin, Field Theory of Guided Waves (IEEE Press, NewYork, 1991 ).
19By contrast, the normal component of the electric field and the tangential
component of the magnetic field are usually not continuous across thesurface of a zero-thickness conductor. This is because of the presence ofsurface charges, respectively, flow of surface current in the conductor.
20Y. Tserkovnyak, A. Brataas, and G.E.W. Bauer, Phys. Rev. Lett. 88,
117601 (2002 ).
21O. Acher, S. Queste, M. Ledieu, K.-U. Barholz, and R. Mattheis, Phys.
Rev. B68, 184414 (2003 ).6868 J. Appl. Phys., Vol. 96, No. 11, 1 December 2004 B. Rejaei and M. Vroubel
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1.4862080.pdf | Compositional dependence of magnetic and high frequency properties of nanogranular
FeCo-TiO2 films
Yicheng Wang, Huaiwu Zhang, Luo Wang, and Feiming Bai
Citation: Journal of Applied Physics 115, 17A306 (2014); doi: 10.1063/1.4862080
View online: http://dx.doi.org/10.1063/1.4862080
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/115/17?ver=pdfcov
Published by the AIP Publishing
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130.225.243.124 On: Sun, 27 Apr 2014 05:26:36Compositional dependence of magnetic and high frequency properties
of nanogranular FeCo-TiO 2films
Yicheng Wang, Huaiwu Zhang,a)Luo Wang, and Feiming Baia)
State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and
Technology of China, Chengdu 610054, People’s Republic of China
(Presented 5 November 2013; received 22 September 2013; accepted 17 October 2013; published
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A series of soft magnetic nanogranular FeCo-TiO 2films with different compositions of FeCo were
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make the films an excellent candidate for microwave applications.
VC2014 AIP Publishing LLC .
[http://dx.doi.org/10.1063/1.4862080 ]
INTRODUCTION
Soft magnetic thin films have been wildly used for
microwave devices.1With the development of electronic
industry and the increase of operational frequency and degree
of integration, the requirements for materials become increas-ingly critical. The basic requirements for soft magnetic
films include high permeability, high saturation magnet-
ization, high ferromagnetic resonance (FMR) frequency,and high electric resistivity. Previously, a lot of research
efforts have been taken to satisfy these requirements.
2–5
Among all candidates, it is shown that metal-insulator
granular films consisting of magnetic metal nanoparticles
embedded in an insulator matrix is one of the best choo-
ses because of the high electric resistivity and the excel-
lence high frequency soft-magnetic properties.6The usual
composition of metal insulator granular films can bedescribed by F-M-X, where F are magnetic metals (Fe,
Co, Ni, and their alloys), M are nonmagnetic elements
(Al, Si, Hf, Zr, Mg, etc.), X denotes N, O, or F element,and M-X forms the insulating matrix.
7–11It is well known
that the saturation magnetization and resistivity are lim-
ited by the volume fraction of the insulating phase, i.e.,lower fraction of insulating phase (thin layer thickness)
causes high saturation magnetization and permeability but
poor resistivity and vice versa. To increase the resistivityfor given layer thickness, the selection of insulating mate-
rial is very important. In this work, we choose TiO
2as
nonmagnetic material to improve the soft magnetic prop-erties and high frequency performance. TiO
2, like
well-studied HfO 2,12is a high-k material, which may
reduce high frequency leakage current when serving asthin insulating layer. In such case, we may increase the
volume fraction of magnetic nanograins, thus improve the
saturation magnetization and permeability of the nanogra-nular film.EXPERIMENTS
A series of FeCo-TiO 2thin films were deposited on Si
(100) substrates by RF magnetron sputtering using a Fe 65Co35
target with TiO 2chips symmetrically placed on the erosion
race-track. The number of chips was changing from 6 to 16 toadjust the composition of nanogranular films. The thickness
was fixed at 350 nm for all films. The base pressure was better
than 2.0 /C210
/C04Pa. The pressure of Ar gas and the sputtering
power were optimized to 0.25 Pa and 250 W, respectively.13A
magnetic field of about 500 Oe was applied in the plane of the
films during deposition to induce a uniaxial anisotropy.
The film structures were characterized by X-ray
diffraction (XRD, Bede TM 2000, UK) and transmission elec-
tron microscopy (TEM), respectively. The local microcomposi-
tions of the films were analyzed by energy dispersive x-ray
spectroscopy (EDS, GENESIS-2000). The thickness of filmswas measured by a step profiler. The magnetic hysteresis loops
were measured at room temperature by a vibrating sample
magnetometer (VSM, BHV-525, Japan). Permeability spectrawere obtained with an Agilent network analyzer (N5230A),
using a shorted microstrip transmission-line perturbation
method without any external magnetic field.
14The ferromag-
netic resonance linewidth was measured by FMR with modula-
tion of magnetic bias field in the plane of films at a fixed
frequency of 9 GHz.15
RESULTS AND DISCUSSIONS
Figure 1shows the XRD patterns and TEM results of
FeCo-TiO 2thin films with different volume fraction xof
FeCo alloy. The XRD patterns of the thin films exhibit only
one (110) peak which belongs to the FeCo nanocrystalline
phase. No peaks related to oxides are found. The EDS analy-sis shows that the composition ratio of Ti and O is close to
1:2, which means the film may contain bcc FeCo nanograins
embedded in a TiO
2amorphous matrix. The average size of
FeCo nanoparticles is calculated by the Scherrer’s formula,
which increases with the volume fraction x, as shown in
Table I. This indicates that the addition of TiO 2can reducea)Authors to whom correspondence should be addressed. Electronic mail
addresses: hwzhang@uestc.edu.cn and fmbai@uestc.edu.cn.
0021-8979/2014/115(17)/17A306/3/$30.00 VC2014 AIP Publishing LLC 115, 17A306-1JOURNAL OF APPLIED PHYSICS 115, 17A306 (2014)
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130.225.243.124 On: Sun, 27 Apr 2014 05:26:36the FeCo grain size effectively. From the TEM results, it is
also found that the films form the metal-insulator granularstructure, and the grain size decreases with the addition of
TiO
2, consistent with the XRD results. Meanwhile, with the
decrease of x, the (110) diffraction peak slightly shifts to the
lower angle side, indicating that Ti atoms with a larger
atomic size may enter FeCo lattices and result in an increase
in the lattice constant.
Figure 2shows the magnetic hysteresis loops along the
easy and hard axis of films with different volume fraction x
of FeCo alloy. The magnetic and high frequency propertiesof the films are also summarized in Table I. It is seen that the
easy axis coercivity ( H
ce) of films decreases with decreasing
xin the range of 0.90 <x<0.68, reaching the minimum
Hce¼5.3 Oe when x¼0.68, then increases when xis lower
than 0.68. On the other hand, the anisotropy field ( Hk) has an
opposite trend reaching the maximum Hk¼95 Oe at
x¼0.68. This excellent property can be understood by
exchange coupling between FeCo particles. Based on the
random anisotropy model,16,17the magnetic property is
determined by the grain size. When the grain size is below
the exchange length, magnetic exchange coupling interaction
between grains will force the magnetic moments of neigh-boring grains to align parallel, leading to a cancellation of
local anisotropies and demagnetization effect of individual
grains, and the smaller the grains, the stronger the exchangecoupling. From the XRD analysis, the added TiO
2in the
films can limit the growth of FeCo nanograins, so that the
increasing exchange coupling causes the film having a goodmagnetic property. But as the volume faction xcontinues to
decrease, the property become worse. The reason is that the
increasing TiO
2content will increase the distance of FeCo
particles, reduce their exchange coupling, and disturb thearrangement of the magnetic moments, thus deteriorate soft
magnetic property.
Figure 3shows the change of the saturation magnetiza-
tion (4 pMS) and resistivity ( q) with different volume fraction
xof FeCo alloy. As an intrinsic property of material, the sat-
uration magnetization is only determined by the volume fac-
tion of magnetic element. So it can be found that the
saturation magnetization almost linearly decreases from 19.1kGs to 11.3 kGs with decreasing x. With the decrease of x,
the resistivity increases slowly from 356 lXcm to 408 lX
cm when x¼0.82, then sharply increases to 1732 lXcm
when x¼0.58. This can be explained by the percolation
theory.
18In the metal insulator composite material, with the
decrease of conductive phase, in a critical content, the con-ductive path connected by the FeCo nanograins will be
blocked by the insulator phase, so that the resistivity of the
composite films will increase rapidly.
Figure 4shows the permeability spectra of the films as a
function of xfrom the experimentally measured results and the
calculated values using the Landau–Lifshitz–Gilbert (LLG)equation.
19,20In the calculation, the measured values of satura-
tion magnetization and the anisotropy field were used, and the
gyromagnetic ratio cwas set as 17.6 MHz/Oe for FeCo mag-
netic metal. The FMR frequency frand initial permeability li
are also shown in Table I. The behavior of frandlican be
described with the Kittle equation fr¼c=2p(4pMSHk)1=2and
FIG. 1. XRD patterns of films with different volume fractions of FeCo (a),
TEM images of FeCo-TiO 2films at x¼0.90 (b), and x¼0.68 (c).
TABLE I. The magnetic and high frequency parameters of films with differ-
ent FeCo volume fraction x.
Volume faction x 0.90 0.82 0.76 0.68 0.61 0.58
Grain size (nm) 10.5 9.6 8.2 6.2 2.9 2.1
q(lXcm) 365 408 540 1068 1526 1732
Hce(Oe) 17.9 11.6 6.8 5.3 6.2 6.4
Hk(Oe) 13 39 65 95 94 82
fr(GHz) 1.4 2.3 2.9 3.2 3.0 2.7
li … 438 252 148 134 139
FIG. 2. The M-H loops of FeCo-TiO 2thin films with FeCo volume fraction
x¼0.76 (a), 0.68 (b), 0.61 (c), 0.58 (d).
FIG. 3. CoFe alloy volume faction xdependence of MSandq.17A306-2 Wang et al. J. Appl. Phys. 115, 17A306 (2014)
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130.225.243.124 On: Sun, 27 Apr 2014 05:26:36the Snoek’s limit ( li/C01)fr2¼4c2MS2.21From Figure 4and
Table I, it can be seen that the FMR frequency increases with
the increase of xin the range of 0.90 >x>0.68, reaching the
maximum fr¼3.2 GHz, and then decreases when xis below
0.68. The initial permeability lihas the opposite trend related
to FMR frequency and a relatively high initial permeabilityabout 148 is obtained at x¼0.68.
It is seen from Figure 4that the full width at half maxi-
mum (FWHM, Df) of the imaginary part decreases with
decreasing volume fraction x.SinceDfis determined by the
effective damping factor a
effbyDf¼c=2pMSaeff,22aeffcan be
calculated from permeability spectra shown in Figure 4.A s
shown in Figure 5,aeffdecreases first and then become almost
invariant with the decrease of x.A relatively low effective
damping factor about 0.026 is obtained at x¼0.68. Figure 5
also gives the results of FMR damping factor ( aFMR) calcu-
lated by the FMR linewidth ( DH) which represents the loss of
films. It is found that as xdecreases, aFMRalso decreases rap-
idly first, and then becomes unchanging. This may be attrib-
uted to the change of grain size and more homogeneous
distribution of FeCo nanograins, both of which cause thebetter alignment of FeCo nanograins’ magnetic moment dur-
ing deposition and the improved Larmor procession.
CONCLUSION
In summary, we have studied the magnetic and high fre-
quency properties of a series of soft magnetic nanogranular
FeCo-TiO 2films with different composition of TiO 2. The
added TiO 2in the films can limit the growth of FeCo grains
and improve the properties of films. A high saturation mag-
netization of 14 kGs, a low coercivity of 5.3 Oe, and a high
anisotropy field of 95 Oe were obtained when the FeCo vol-ume factor x¼0.68. As a result, the FMR frequency of films
was up to 3.2 GHz and the initial permeability was as high as
148. In addition, the film resistivity is larger than 1000 lX
cm. Such an excellent electromagnetic property makes the
films having a great potential in microwave applications.
ACKNOWLEDGMENTS
The authors would like to acknowledge financial support
from the National Basic Research Program of China under
Grant No. 2012CB933104, the Foundation for InnovativeResearch Groups of the National Natural Science Fund of
China under Grant No. 61021061, and the National Science
Fund of China under Grant No. 61271031.
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FIG. 4. Permeability spectra of FeCo-TiO 2thin films with FeCo volume
fraction x¼0.76 (a), 0.68 (b), 0.61 (c), 0.58 (d).
FIG. 5. Compositional dependences of the effective damping factor ( aFMR),
ferromagnetic resonance damping factor ( aFMR) and FMR linewidth ( DH).17A306-3 Wang et al. J. Appl. Phys. 115, 17A306 (2014)
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1.3441405.pdf | Island shape controls magic-size effect for heteroepitaxial diffusion
Henry H. Wu, A. W. Signor, and Dallas R. Trinkle
Citation: J. Appl. Phys. 108, 023521 (2010); doi: 10.1063/1.3455848
View online: http://dx.doi.org/10.1063/1.3455848
View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v108/i2
Published by the American Institute of Physics.
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Downloaded 01 May 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsIsland shape controls magic-size effect for heteroepitaxial diffusion
Henry H. Wu, A. W. Signor, and Dallas R. Trinkle
Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, Illinois
61801, USA
/H20849Received 31 March 2010; accepted 15 May 2010; published online 30 July 2010 /H20850
Lattice mismatch of Cu on Ag /H20849111/H20850produces fast diffusion for “magic sizes” of islands. A size- and
shape-dependent reptation mechanism is responsible for low diffusion barriers. Initiating thereptation mechanism requires a suitable island shape, not just magic sizes. Shape determines thedominant diffusion mechanism and leads to multiple clearly identifiable magic-size trends fordiffusion depending on the number of atoms whose bonds are shortened during diffusion, whichultimately affects the self-assembly of islands. © 2010 American Institute of Physics .
/H20851doi:10.1063/1.3455848 /H20852
I. INTRODUCTION
Control of thin-film morphology relies on understanding
multiple ongoing processes during deposition and growth. Inparticular, diffusion of small atom clusters on surfaces play acritical role in thin-film growth, especially in early stages.The diffusion kinetics of small islands in heteroepitaxial sys-tems is less well understood than that of homoepitaxial dif-fusion, for which much experimental
1–4and theoretical5–8
work has been done. Strain is known to govern the mesos-
cale morphology in self-assembling systems.9While predic-
tions about the role of size and misfit for small islands goback over a decade,
10only recent experiments have captured
and quantified the rapid diffusion at “magic sizes” in theheteroepitaxial Cu/Ag /H20849111/H20850system.
11However, the experi-
mental observations of rapid diffusion from several distinctsizes of islands does not easily comport with the simplemodel of Hamilton for magic sizes. Large-scale, automatedcomputational studies are now exploring some of the unusualdiffusion mechanisms in heteroepitaxial systems.
12A missing
element in explaining the atomistic diffusion mechanism isthe role of island shape in controlling diffusion. Understand-
ing the trends of diffusion barriers for small islands withisland size and shape for the common cubic /H20849111/H20850surface
provides insight for experimental measurements and thin-film growth, and provides the fundamental understanding tocontrol morphology.
Hamilton predicted a magic-size effect for heteroepi-
taxial islands with a one-dimensional /H208491D/H20850Frenkel-
Kontorova model
13and a corresponding two-dimensional
/H208492D/H20850equivalent.10The 1D model describes a chain of atoms
harmonically coupled to their neighbors and interacting witha rigid periodic substrate potential. The lattice misfit is thefractional difference between the equilibrium spring lengthand the substrate periodicity, and the island misfit straingrows linearly with the number of atoms in the island chain.The diffusion barrier has nonmonotonic behavior with size,showing a minimum at the “magic size” equal to inverse ofthe lattice misfit. At this size, the island ground-state con-figuration contains one dislocation, where island atoms sit ata peak of the substrate potential instead of a valley. Islandsbelow the magic size are dislocation-free with a large energybarrier to nucleate and propagate a dislocation for island dif-
fusion, while larger islands contain a dislocation and requirean increasing energy barrier to move this dislocation for is-land diffusion. Using the embedded atom method /H20851EAM
/H20849Ref. 14/H20850/H20852for Ag/Ru /H208490001 /H20850, Hamilton considered a 2D
equivalent with closed-shell islands, and the magic size cor-responds with the ground-state configuration containing onedislocation.
10The ground state has atoms displaced from fcc
to hcp sites on the hexagonal Ru /H208490001 /H20850surface. The disloca-
tion line marks the separation between fcc and hcp sectionsof the island. Diffusion for these islands proceeds as all at-
oms in the island collectively glide to continuously propa-gate the dislocation.
Reptation—first proposed for small island diffusion in
homoepitaxial systems—relies on dislocation movement.
15
Unlike Hamilton’s collective glide mechanism, reptation pro-ceeds as the motion of an island section from fcc to hcp siteson the /H20849111/H20850surface, forming a dislocation where the island
atomic bonds are stretched. Diffusion is completed after theremaining island section subsequently follows in the samedirection to complete the transition. Therefore, the reptationdislocation propagates through sequential motion of islandsections while the collective glide dislocation propagates bythe continuous motion of the entire island.
We find that the reptation diffusion mechanism exhibits a
magic-size effect in Cu/Ag /H20849111/H20850controlled by island shape
that explains experimental observations of anomalous diffu-sion. Using an optimized EAM potential with molecular dy-namics and the dimer method, we calculate island diffusionbarriers up to 14-atom islands. The diffusion barriers of dif-ferent island sizes and shapes is a nonmonotonic function ofthe island misfit strain, and separates into simple groups de-pending on the geometry of the island. The shape effect isexplained by the continuity of bonds during diffusion. Wefind that the reptation mechanism is competitive over theglide mechanism for all islands except those with closed-shell shapes. After considering the modulating effect of is-land geometry and the competition between the diffusionmechanisms, we find multiple magic sizes each diffusingwith a single dislocation. By considering island shape, wepredict a series of rapidly diffusing islands, each as their ownmagic size.JOURNAL OF APPLIED PHYSICS 108, 023521 /H208492010 /H20850
0021-8979/2010/108 /H208492/H20850/023521/4/$30.00 © 2010 American Institute of Physics 108 , 023521-1
Downloaded 01 May 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsII. METHODOLOGY
Our study of island diffusion relies on an EAM potential
optimized for the prediction of island geometries, energetics,and kinetics for Cu/Ag /H20849111/H20850.
16The potential was optimized
using monomer and dimer density-functional theory /H20849DFT /H20850
energies and geometries. The optimized EAM predicts theDFT monomer diffusion barrier and the DFT energy differ-ence between all-fcc trimers and all-hcp trimers. The poten-tial /H20849and DFT /H20850overestimates the monomer and dimer diffu-
sion barriers at 93 meV and 88 meV compared toexperimental
17values of 65 /H110069 meV and 73 meV , respec-
tively. This is a carryover from DFT which has been shownto overestimate surface adsorption energies.
18Due to this
discrepancy, we present our calculated island diffusion bar-riers relative to the monomer diffusion barrier E
diffmonomerof 93
meV .
We anneal to determine island ground-state structures,
and use molecular dynamics,19dimer-search,20and nudged
elastic band21method to find island diffusion transitions and
determine diffusion barriers. Multiple annealing runs foreach island size reveal compact islands with all fcc-siteground-states for small sizes while mixed fcc/hcp ground-states exist only for the 13- and 14-atom islands, the largestin our study. We run direct molecular dynamics at high tem-perature /H20849600 K /H20850over several nanoseconds to explore pos-
sible transitions. We also use the dimer method which ran-domly searches through phase-space for possible transitionsfrom a starting state. Nudged elastic band finds the minimumenergy pathway between the starting and ending states ofdiscovered transitions and extracts the diffusion barrier.
III. RESULTS
Different island shapes are possible for each island size
in 2D, while two different sites on the /H20849111/H20850surface allows
different energies for the same shape. The islands form com-pact ground-state configurations that maximize atomic coor-dination and minimize island strain. The triangular /H20849111/H20850sur-
face lattice contains two hexagonal sublattices—fcc andhcp—each with lattice spacing equal to that of the Agnearest-neighbor distance nn
Ag=2.89 Å. A fcc-site is sur-
rounded by three hcp-sites in the /H20855112¯/H20856directions, while the
next nearest fcc-sites are in the close-packed /H208551¯10/H20856direc-
tions. Cu islands, with bulk nearest-neighbor distance nn Cu
=2.56 Å, experience large misfit strains if all the island at-
oms sat exclusively on one sublattice. However, since fcc-sites are closer to neighboring hcp-sites, the total islandstrain can be reduced if some atoms in the island sit in amixed fcc-hcp /H20849FH/H20850bond with a lattice site distance of only
/H208491/
/H208813/H20850nnAg=1.67 Å rather than the longer fcc-fcc or hcp-
hcp bonds. For a system with an opposite lattice mismatch
we would expect the next nearest-neighbor FH-bonds with a
lattice site distance of /H208492//H208813/H20850nnAg=3.34 Å; this is also the
case in the Pt/Pt /H20849111/H20850system.15
We measure the substrate strain in each island in terms
of the island misfits in Fig. 1. We relax ground-state configu-
rations of different islands and calculate the surface strain asa function of distance away from the island’s center-of-mass.The atomic strain tensor for each surface Ag atom is an av-
erage over the change in all nearest-neighbor vectors.
22In
Fig. 1/H20849a/H20850, the tensile normal strain—sum of /H9280xxand
/H9280yy—drops off as the inverse squared radial distance from the
island’s center-of-mass and scales with island size; we definethe scaling coefficient as the island misfit, which has units ofarea. This island misfit with the substrate varies with the sizeand shape, and in Fig. 1/H20849b/H20850, we find a linear correlation with
the island size. We use the island misfit as a measure of thestrain in the island because it contains information aboutboth the island size and shape. Other island growth studiesused the a related measure of the substrate stress instead ofstrain,
23which is linearly related. Substrate strain accounts
for the effect of island shape, and relates the 2D system backto the simpler 1D model.
Islands diffuse by the passage of a dislocation, with two
different forms in 2D: collective glide /H20851Fig.2/H20849b/H20850/H20852or reptation
/H20851Fig. 2/H20849a/H20850/H20852. Collective glide is the mechanism described by
Hamilton, where the dislocation propagates with the continu-
FIG. 1. /H20849Color online /H20850Cu islands and strain in the Ag /H20849111/H20850surface. /H20849a/H20850
Magnitude of the atomic strain plotted against the radial distance from the10-atom island center-of-mass. The tensile strain /H9255
xx+/H9255yyfollows the inverse
squared radial distance times the island misfit /H9255i. The inset shows the hy-
drostatic strain induced by the ground-state 10-atom Cu island in the Agsurface. Compressive strain is in red /H20849strains exceed /H110021.0% under the island
and are not shown /H20850, and the compensating tensile strain is in blue. /H20849b/H20850The
island misfit /H9255
iof ground-state island configurations grows linearly with
island size N as 0.028 Å2/H11003N, while modulations show the influence of
island shape.023521-2 Wu, Signor, and Trinkle J. Appl. Phys. 108 , 023521 /H208492010 /H20850
Downloaded 01 May 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsous motion of the entire island moving from fcc to hcp sites,
or vice versa. Closed-shell configurations /H208493-, 7-, and 12-
atom islands /H20850favor the collective glide mechanism, which
maintains neighboring bonds during diffusion. Reptation in-volves sequential motion of island sections to hcp sites, via ametastable dislocated structure. The fcc portion of the meta-stable state is separated from the hcp portion by a dislocation
with a /H208551
¯10/H20856-type line direction. We identify this dislocation
line direction to be characteristic of the reptation mechanism.
Island shape is the critical criteria to determine whether a 2Disland prefers the collective glide mechanism or the reptationmechanism.
Figure 2shows the geometric requirement to undergo
the reptation mechanism at a low energy barrier: no atomshould be left without a bond across the dislocation line. Thereptation mechanism slips part of the island onto hcp sites
where the number of /H208551
¯10/H20856island rows sheared by the dis-
location line form the same number of heterogeneous FH-
bonds /H20851Fig. 2/H20849a/H20850,2/H20852. The energy cost to form FH-bonds and
move atoms to hcp sites is compensated by the reduction inisland strain from the smaller island area of the dislocatedstate. For the 10-atom island shown, the island misfit is re-duced from 0.27 Å
2/H20851Fig. 2/H20849a/H20850,1/H20852to 0.22 Å2/H20851Fig. 2/H20849a/H20850,2/H20852.
An island with unequal number of atoms across the disloca-tion line—two facing three in the 7-atom island—has toovercome the additional barrier to break a bond during island
shear /H20851Fig. 2/H20849c/H20850,2/H20852. The dimer dissociation energy for Cu on
Ag/H20849111/H20850is 370 meV /H110154E
diffmonomer, which increases the repta-
tion mechanism energy barrier for closed-shell islands abovethe collective glide mechanism barrier. Ultimately, the ab-sence of bond-breaking during slip is the deciding factor inallowing a low energy reptation mechanism.
Figure 3groups barriers for islands with equal numbers
of sheared /H208551
¯10/H20856rows, demonstrating the shape-modulated
magic-size effect. For reptation-allowed islands, this groups
islands with the same number of FH-bonds in the dislocatedstate, and for reptation-disallowed islands this is the number
of/H208551¯10/H20856rows in the diffusion direction. The 2- and 3-row
groups appear very similar to the characteristic magic-size
effect plot for 1D island chains diffusion.10The 7- and
8-atom island configurations in the 2-row group are notground-states and are constructed to test the continuation ofthe 2-row magic-size effect. The shapes of these two islandsinduce larger strains than in their ground-states and promotethe dramatic reduction in diffusion barrier. The shape effect
from sheared /H208551
¯10/H20856rows gives different magic-size regions
even though only one dislocation is present in all cases.
FIG. 2. /H20849Color online /H20850Geometric requirements for low barrier island repta-
tion diffusion. A /H17005of Ag surround the fcc surface site and /H17006for the hcp
surface site. /H20849a/H20850An allowed reptation transition involves transforming ho-
mogeneous bonds into heterogeneous bonds without leaving an unbonded
Cu atom. This is a necessary condition for the reptation dislocation-mediated island diffusion. /H20849b/H20850The collective glide dislocation-mediated dif-
fusion mechanism observed for the 7-atom island. Atoms shuffling in thedirection of diffusion propagate the passing dislocation /H208492/H20850./H20849c/H20850A disallowed
reptation transition for the 7-atom island leaves one Cu atom without a bondacross the dislocation line /H208492/H20850and is high in energy. The energy barrier
pathway for all three transitions is shown below normalized with respect tothe EAM monomer diffusion barrier of 93 meV .FIG. 3. /H20849Color online /H20850The island diffusion barriers versus island misfit /H9255i
for different islands showing a shape-modulated magic-size effect. The dif-
fusion barrier is normalized with respect to the EAM monomer diffusionbarrier of 93 meV . The open symbols represent islands that allow reptationaccording to their shape, while the filled symbols are islands that do not /H20849the
configurations shown are oriented with respect to the dislocation line direc-
tion in the inset /H20850. Islands with the same number of sheared /H208551
¯10/H20856rows along
the dislocation line follow a “magic-size” trend in barriers as a dislocation
aids in diffusion—with blue for 1-, green for 2-, black for 3-, and red for4-rows. The distinct groupings of magic-size regions highlight the key roleof shape in the diffusion of small islands with a large lattice misfit. The 7-and 8-atom 2-row configurations in green are not energetically favorable,but show the continuation of the 2-row magic-size effect. Note: the glyphsshown for the 6-, 9-, and 11-atom islands are not the ground-state configu-rations for that size, but instead represent the shape before a diffusiontransition.023521-3 Wu, Signor, and Trinkle J. Appl. Phys. 108 , 023521 /H208492010 /H20850
Downloaded 01 May 2012 to 132.236.27.111. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissionsIV. DISCUSSION
The transition pathways for 2D islands of different row
groups in Fig. 3also follow the trends seen for the 1D island
chains. In the 2-row group, the dislocated state is not meta-stable for the 4-atom island and becomes more and morestable with increasing island size. The dislocated state of the7-atom 2-row is almost equal in energy with the undislocatedstate, while the dislocated state of the 8-atom 2-row is theground state and possesses a higher diffusion barrier. In the3-row group, the dislocated state is not metastable for theboth the 8- and 9-atom islands and becomes metastable at10-atoms. Continuing the 3-row group, the dislocated stateof the 11-atom island is still metastable, but the diffusionbarrier is higher due to asymmetric island structure. Finally,the dislocated state for the 13-atom island is the ground-stateand the diffusion barrier becomes even higher. The minimafor ground-state diffusion barriers—6- and 10-atomislands—corresponds well with experimental observations,as well as the immobility of 7-atom islands.
11
Island shape controls the 2D magic-size effect for Cu/
Ag/H20849111/H20850where a combination of island geometry and misfit
produces multiple magic sizes even for single dislocation-mediated diffusion. We find that the reptation diffusionmechanism allows for greatly reduced diffusion barriers forheteroepitaxial systems compared with the collective glidemechanism. The criteria for the reptation mechanism requirethe island shape to be such that no atom is left unbondedacross the dislocation line. This mechanism predicts multiplemagic sizes even for small /H20849/H1102120/H20850island sizes, which quan-
titatively agrees with experimental observations of Cu/
Ag/H20849111/H20850. We expect similar effects in other heteroepitaxial
systems with large lattice misfits, and for the magic-size is-lands to affect the growth morphologies for low coverages.ACKNOWLEDGMENTS
The authors thank John Weaver for helpful discussions.
This research is supported by NSF/DMR Grant No. 0703995and 3M’s Untenured Faculty Research Award.
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1.2695060.pdf | The reversible susceptibility tensor of synthetic antiferromagnets
Dorin Cimpoesu, Alexandru Stancu, and Leonard Spinu
Citation: J. Appl. Phys. 101, 09D112 (2007); doi: 10.1063/1.2695060
View online: http://dx.doi.org/10.1063/1.2695060
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Downloaded 25 Apr 2013 to 141.161.91.14. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsThe reversible susceptibility tensor of synthetic antiferromagnets
Dorin Cimpoesu
Advanced Materials Research Institute (AMRI), University of New Orleans, New Orleans, Louisiana 70148
Alexandru Stancu
Faculty of Physics, Alexandru Ioan Cuza University, Iasi 700506, Romania
Leonard Spinua/H20850
Advanced Materials Research Institute (AMRI), University of New Orleans, New Orleans, Louisiana 70148
and Department of Physics, University of New Orleans, New Orleans, Louisiana 70148
/H20849Presented on 8 January 2007; received 31 October 2006; accepted 16 November 2006;
published online 2 May 2007 /H20850
In this work we propose to study the reversible susceptibility tensor /H20849including the transverse
susceptibility components /H20850of a synthetic antiferromagnetic /H20849SAF /H20850structure. This study is motivated
by the fact that knowing the magnetic anisotropy of SAF structure is essential for the devicefunctioning and the transverse susceptibility is known as one of the best methods to determinemagnetic anisotropy and switching properties of magnetic systems. The starting point of this workis the magnetization’s equation of motion Landau-Lifshitz-Gilbert for the coupled two layers. Bysolving the inhomogeneous system of linear equations on the magnetization’s spherical coordinatedeviations, the susceptibility tensor of the SAF structure is obtained. Simplified equations for thevery low frequency case are given. By comparing the plots of the susceptibility versus the appliedfield with the regular hysteresis loops one concludes that susceptibility experiments are moreappropriate for switching investigations of the SAF systems. © 2007 American Institute of Physics .
/H20851DOI: 10.1063/1.2695060 /H20852
Synthetic antiferromagnet /H20849SAF /H20850structures are key ele-
ments in modern devices based on tunnel magnetoresistance/H20849TMR /H20850or giant magnetoresistance /H20849GMR /H20850effects as reading
heads and magnetic random access memories /H20849MRAMs /H20850.
SAF consists of two ferromagnetic layers that are coupledthrough a nonmagnetic spacer whose thickness is tuned toprovide an antiferromagnetic coupling. View their configura-tion SAF has a reduced magnetic moment, reduced shapeanisotropy, and reduced switching field. In this work we pro-pose to study the reversible susceptibility tensor /H20849including
the transverse susceptibility components /H20850of SAF structure.
This study is motivated by the fact that knowing the mag-
netic anisotropy of SAF structure is essential for the devicefunctioning and the transverse susceptibility is known as oneof the best methods to determine magnetic anisotropy andswitching properties of magnetic systems.
The theoretical study of transverse susceptibility /H20849TS /H20850
was not subject to substantial changes after the classical ar-ticle of Aharoni et al.
1in which the comprehensive theoret-
ical description of the reversible susceptibility tensor for theStoner-Wohlfarth model was given. Subsequent theoreticalapproaches of TS overcame some limitations of the approachof Aharoni et al. by taking into account magnetic systems
with different types of anisotropy
2or studying the effect of
interactions and thermal relaxation. What is common to allpapers published previously and dealing with transverse sus-ceptibility modeling is the theoretical approach used. Thetransverse susceptibility for a specific free energy expressionis evaluated employing a free energy minimization method.
The disadvantage of this approach is the lack of generalityand the difficulty of describing reversible transverse suscep-tibility /H20849RTS /H20850for magnetic systems with more complex free
energy expressions. Recently we proposed a generalapproach
3to theoretically determine the susceptibility tensor
for virtually any magnetic systems if an expression of its freeenergy is known. In this paper we apply this approach toSAF structure. Expression for reversible susceptibility tensorfor SAF is given and plots of the field dependence are alsopresented.
In a susceptibility experiment it is required to apply two
magnetic fields: a dc field H
dcwhich can be varied in a quite
large range, and a small perturbing ac field Hac=Hac,0ei/H9275t,
and the magnetization variation is measured. In the absenceof thermal agitations the dynamics of the magnetization Mof
a single-domain particle is governed by the Gilbert equationthat can be written in the Landau-Lifshitz form as
dM
dt=−/H9253M/H11003Heff−/H9251/H9253
MsM/H11003 /H20849M/H11003Heff/H20850, /H208491/H20850
where /H9253=/H20841/H92530/H20841//H208491+/H92512/H20850,/H20841/H92530/H20841=2.211 /H11003105m/A s is the gyro-
magnetic ratio, /H9251is the phenomenological Gilbert damping
constant, and Heff=−/H11509F//H20849/H92620V/H11509M/H20850+Hacis the effective field
and it incorporates the effects of different contributions in the
free energy F/H20849that contains the dc field /H20850and the small time-
dependent ac field, with Vthe volume of the particle. In
order to preserve the magnetization’s magnitude the Landau-Lifshitz-Gilbert /H20849LLG /H20850equation can be expressed better in
spherical coordinates /H20849M
S,/H9258,/H9272/H20850. The singularities of the
spherical coordinates along the polar axis can be avoided bya/H20850Author to whom correspondence should be addressed; electronic mail:
lspinu@uno.eduJOURNAL OF APPLIED PHYSICS 101, 09D112 /H208492007 /H20850
0021-8979/2007/101 /H208499/H20850/09D112/3/$23.00 © 2007 American Institute of Physics 101 , 09D112-1
Downloaded 25 Apr 2013 to 141.161.91.14. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionsan adequate choice of the Ozaxis. Thus the time variation of
the magnetization direction is given by
d/H9258/dt=/H9253/H20849H/H9272+/H9251H/H9258/H20850,
d/H9272/dt=/H9253/H20849/H9251H/H9272−H/H9258/H20850/sin/H9258, /H208492/H20850
where H/H9258=−F/H9258//H20849/H92620VM S/H20850+Hac,/H9258 and H/H9272=
−F/H9272//H20849/H92620VM Ssin/H9258/H20850+Hac,/H9272are the polar and azimuthal com-
ponents of the effective field, respectively, with F/H9258=/H11509F//H11509/H9258
and F/H9272=/H11509F//H11509/H9272. The time varying ac field will produce a
dynamic component of the magnetization /H9254M
=/H208490,MS/H9254/H9258,MSsin/H9258/H9254/H9272/H20850from the static equilibrium position.
The generalized SAF consists of two ferromagnetic lay-
ers 1 and 2 that have the thickness t1andt2, and magnetiza-
tions M1andM2, respectively, coupled through a nonmag-
netic spacer. Assuming that the two magnetic layers behavelike two single domain particles, the time evolution of thesystem is described by the coupled equations of motion foreach magnetization. Following a similar derivation as in theferromagnetic resonance /H20849FMR /H20850,
4by solving the inhomoge-
neous system of linear equations on the deviations /H9254/H9258i,/H9254/H9272i
from the static equilibrium position, the susceptibility tensor
/H9273in the laboratory reference system /H20849x,y,z/H20850, defined by
/H9254M=/H9273·Hac, with /H9254Mthe variation of the total magnetic
moment, can be obtained as
/H9273=/H92620M12V12/H20851T1T2/H20852DA−1/H9011D/H20875T1t
T2t/H20876, /H208493/H20850
where
A=/H9011/H20849/H9014/H20850ij+i/H9275
/H9253/H92620M1V1Diag /H208491,− 1, mt,−mt /H20850,
/H9011= Diag /H20849/H90110,/H90110/H20850,/H90110=/H20875/H92511
−1/H9251/H20876,
D= Diag /H208491,1,mt,mt /H20850,
/H9014ij=/H20875F/H9258i/H9258jF/H9258i/H9272j/sin/H9258j0
F/H9258j/H9272i/sin/H9258i0F/H9272i/H9272j/sin/H9258i0sin/H9258j0/H20876,
Ti=/H20900cos/H9258i0cos/H9272i0− sin/H9272i0
cos/H9258i0sin/H9272i0cos/H9272i0
− sin/H9258i0 0/H20901,
andm=M2/M1,t=t2/t1,V1=St1is the volume of first layer,
with Sthe area, F/H9258i/H9258j,F/H9258i/H9272j,F/H9272i/H9272jare the second derivatives of
free energy density at the static equilibrium position /H20849/H9258i0,/H9272i0/H20850,
and Diag is the diagonal matrix. The element /H9273lpof the ten-
sor /H208493/H20850is thus describing the response of the total magneti-
zation in the lth direction from an incremental change in the
pth direction of the applied field. If the Ozaxis is parallel to
the direction of the biasing dc field then the diagonal ele-ments are the parallel susceptibility /H20849PS /H20850
/H9273zzmeasured in the
field direction, and the two transverse susceptibilities /H20849TS /H20850
/H9273xxand/H9273yy, measured perpendicular to the bias field direc-
tion. As it results from /H208493/H20850the susceptibility is complex anddepends on the ac field frequency /H9275and the damping con-
stant/H9251.
For low values of the frequency of the ac field /H20849/H9275→0/H20850it
may be assumed that the magnetization lies in a minimum of
the free energy at any moment, and the change of the appliedfield provides only reversible changes of magnetization /H20849i.e.,
processes which involve no loss of energy /H20850, as is the case in
the model of Aharoni et al. In this case the reversible sus-
ceptibility tensor is given by
/H9273=/H92620M12V12/H20851T1T2/H20852D/H20849/H9014/H20850ij−1D/H20875T1t
T2t/H20876, /H208494/H20850
where no imaginary part and no dependence of damping con-
stant/H9251exist.
Equations /H208493/H20850and /H208494/H20850, due to their generality, are able to
describe the susceptibility of a generalized SAF coupledmagnetic systems with virtually any type of anisotropy, anyorientation of the anisotropy’s axes, and any type of interac-tions between the two layers. Moreover, using this generalapproach it is possible to calculate not only TS and PS but allthe susceptibility tensor’s components and/or the susceptibil-ity along any direction. This is very useful if the susceptibil-ity along different directions, not only transverse to dc field,is measured.
5,6
If we assume that the film sample lies in the xOz plane,
both layers have a uniaxial anisotropy with the easy axes inthe sample’s plane, and the dc field is also applied in the
FIG. 1. Major hysteresis loop /H20849MHL /H20850and/H9273xxfor symmetric SAF structure
for two orientations of the dc field /H9258H=15° and /H9258H=60° and for two cou-
pling strengths: /H20849a/H20850hJ=0.2, and /H20849b/H20850hJ=1.09D112-2 Cimpoesu, Stancu, and Spinu J. Appl. Phys. 101 , 09D112 /H208492007 /H20850
Downloaded 25 Apr 2013 to 141.161.91.14. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jap.aip.org/about/rights_and_permissionssample’s plane, then due to symmetry reasons the magneti-
zations Miwill lie in the xOz plane and the energy density
per unit area can be expressed by7
W=2Ku1t1/H20877−/H20849hxcos/H92581+hysin/H92581/H20850−mt /H20849hxcos/H92582
+hysin/H92582/H20850−1
2/H20851cos2/H20849/H92581−/H9258K1/H20850+ktcos2/H20849/H92582−/H9258K2/H20850/H20852
+hJcos /H20849/H92581−/H92582/H20850/H20878,where /H9258K1and/H9258K2denote the easy axes angle, Ku1andKu2
the anisotropy constants, respectively, hx=Hx/Hk1,hy
=Hy/Hk1the applied field, Hk1=2Ku1//H92620M1,k=Ku2/Ku1,
andhJthe exchange coupling strength between the two lay-
ers /H20849with hJ/H110210o rhJ/H110220 for ferromagnetic/antiferromagnetic
coupling /H20850. The diagonal elements of the reversible suscepti-
bility tensor can be written as
/H9273xx=/H92620M12V12F/H92582/H92582cos2/H92581−2mtF/H92581/H92582cos/H92581cos/H92582+m2t2F/H92581/H92581cos2/H92582
F/H92581/H92581F/H92582/H92582−F/H92581/H925822,
/H9273yy=/H92620M12V12F/H92722/H92722sin2/H92581−2mtF/H92721/H92722sin/H92581sin/H92582+m2t2F/H92721/H92721sin2/H92582
F/H92721/H92721F/H92722/H92722−F/H92721/H927222,
/H9273zz=/H92620M12V12F/H92582/H92582sin2/H92581−2mtF/H92581/H92582sin/H92581sin/H92582+m2t2F/H92581/H92581sin2/H92582
F/H92581/H92581F/H92582/H92582−F/H92581/H925822. /H208495/H20850
From Eqs. /H208495/H20850we observe that for SAF systems the diagonal
reversible susceptibility varies inverse proportionally withthe discriminant of the free energy for the coupled magneticthin films, and their singular points are zeros of the freeenergy discriminant, namely, that the susceptibility is a mea-sure of the “curvature” of free energy.
In Fig. 1are shown the field variation of the normalized
total magnetic moment and the variation of the reversiblesusceptibility
/H9273xxfor a symmetric SAF structure with the
easy axes of the two layers parallel to each other and alongOxaxis for two orientations of the dc field and for two cou-
pling strengths. Depending on coupling strength and appliedfield orientation the magnetization switching of the two lay-ers from the SAF structure can be more or less visible in thehysteresis loops. For example, as it is shown in Fig. 1/H20849a/H20850, for
a field applied along a direction which makes an angle of
/H9258H=60° with the hard axis the switching is very well defined.
However, for /H9258H=15° the switching events are not very well
visible, and experimentally this can be very difficult to beobserved from regular hysteresis loops. On the other hand,one observes that the susceptibility versus field curvespresent sharp peaks at the field values at which magnetiza-tion switches for the entire angular domain of the appliedfield. Thus, plotting on a polar chart the switching field ver-sus the field orientation one can obtain the SAF criticalcurve,
7i.e., the corresponding of the astroid from the Stoner-
Wohlfarth model. For larger coupling strengths switchingevents could become difficult to observe even in the suscep-tibility curves, as can be observed in Fig. 1/H20849b/H20850. In this case
the switching will determine only a shoulder /H20849still more vis-ible than in the hysteresis loops /H20850and not a very well defined
peak in the susceptibility versus field curve. From additionalstudy of these particular situations we concluded that thesusceptibility does have its denominator /H20849and discriminant of
the free energy /H20850equaled to zero at the switching point, and
the numerator has a very small value. Consequently, forthese particular situations the zeros of the free energy dis-criminant will not determine a very sharp peak seen in thesusceptibility signal. To make these switching points visiblea strategy to follow is to use field scans that do not passthrough origin.
Work at AMRI was supported by DARPA under Grant
No. HR0011-05-1-0031. One of the authors /H20849A.S. /H20850acknowl-
edges the support from CNCSIS-RO, Grant Ac-NANOCONS.
1A. Aharoni, E. H. Frei, S. Shtrikman, and D. Treves, Bull. Res. Counc.
Isr., Sect. A: Math., Phys. Chem. 6, 215 /H208491957 /H20850.
2L. Spinu, A. Stancu, C. J. O’Connor, and H. Srikanth, Appl. Phys. Lett.
80, 276 /H208492002 /H20850.
3L. Spinu, I. Dumitru, A. Stancu, and D. Cimpoesu, J. Magn. Magn. Mater.
296,1 /H208492006 /H20850.
4S. V . V onsovskii, Ferromagnetic Resonance: The Phenomenon of Reso-
nant Absorption of a High-frequency Magnetic Field in FerromagneticSubstances , 1st ed. /H20849Pergamon, Oxford, 1966 /H20850.
5L. Spinu et al. , Appl. Phys. Lett. 86, 012506 /H208492005 /H20850.
6C. Radu, D. Cimpoesu, E. Girt, G. Ju, A. Stancu, and L. Spinu, “Revers-
ible susceptibility studies of magnetization switching in FeCoB synthetic
antiferromagnets,” Paper No. AT-20, J. Appl. Phys. /H20849these proceedings /H20850.
7H. Fujiwara, S. Y . Wang, and M. Sun, Trans. Magn. Soc. Jpn. 4, 121
/H208492004 /H20850.09D112-3 Cimpoesu, Stancu, and Spinu J. Appl. Phys. 101 , 09D112 /H208492007 /H20850
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1.3236572.pdf | Spin torque-driven switching of exchange bias in a spin valve
Xiao-Li Tang, Huai-Wu Zhang, Hua Su, Yu-Lan Jing, and Zhi-Yong Zhong
Citation: Journal of Applied Physics 106, 073906 (2009); doi: 10.1063/1.3236572
View online: http://dx.doi.org/10.1063/1.3236572
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/106/7?ver=pdfcov
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132.174.254.155 On: Tue, 23 Dec 2014 11:36:31Spin torque-driven switching of exchange bias in a spin valve
Xiao-Li T ang,a/H20850Huai-Wu Zhang, Hua Su, Yu-Lan Jing, and Zhi-Yong Zhong
State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science
and Technology of China, Chengdu, Sichuan 610054, China
/H20849Received 30 May 2009; accepted 29 August 2009; published online 7 October 2009 /H20850
We show that the strength and direction of the exchange bias in a spin valve can be changed and
recovered by applying a spin-polarized current pulse. In other words, once the exchange bias hasbeen changed by a spin-polarized current pulse with the external magnetic field antiparallel to theexchange bias direction, it can be returned to the initial state by a current pulse of the samemagnitude with reversal of the external field. Furthermore, the exchange bias field reverts to thesame magnitude, irrespective of whether one or multiple current pulses are applied. Based on amodel for spin-momentum transfer, the experimental observations can be rationalized in terms ofchanging micromagnetic distributions at the ferromagnet/antiferromagnet interface. © 2009
American Institute of Physics ./H20851doi:10.1063/1.3236572 /H20852
I. INTRODUCTION
Electrons flowing in a magnetic multilayer composed of
a ferromagnetic /H20849FM/H20850and a nonmagnetic metal are strongly
affected by the relative orientation of the magnetic momentsin the FM layer. This is associated with a change in theresistance of such a multilayer, which is known as giant mag-netoresistance /H20849GMR /H20850.
1,2Conversely, electron scattering
within the FM layer can also affect the moments in the layer,which is known as spin-transfer torque /H20849STT /H20850.
3–5GMR and
STT have been extensively studied as interesting comple-mentary phenomena with important industrial applications.They exemplify such interconnections in FM materials. Itwas recently predicted that corresponding effects ought tooccur in antiferromagnetic /H20849AFM /H20850materials.
6–10Since AFM
materials are commonly used in GMR and STT research toprovide exchange bias /H20849EB/H20850,
11the occurrence of these corre-
sponding effects in AFM material will have important con-sequences for spintronic devices.
At present, several groups are researching the effects of
AFM GMR and STT on AFM materials. Their results haveproved that EB in a spin valve /H20849SV/H20850can be changed by a
polarized current, but understanding of the effects of polar-ized current on EB is still not complete.
9,10,12In this work,
we report on the effects of a polarized current on an AFMlayer incorporated into a SV. The polarized current not onlychanges the EB but can also recover the EB to its initialstate. In addition, we investigated the effect of applying dif-ferent numbers of pulses on the AFM to better understandthe variation in micromagnetic distributions at the FM/AFMinterface.
II. EXPERIMENTAL PROCEDURE
The basic structure used in this study was Ta /H2084910 nm /H20850/
NiFe /H2084910 nm /H20850/Cu /H208494n m /H20850/NiFe /H20849tnm/H20850/FeMn /H2084915 nm /H20850on a
5/H110035m m2Si substrate, where t=6 and 10 to achieve differ-
ent EB fields Hex. The multilayers were deposited by dc
magnetron sputtering at room temperature /H20849RT/H20850in purifiedAr at 7 /H1100310−4mbar. A constant magnetic field of /H11011300 Oe
along the substrates was applied during film growth to de-velop the EB. The GMR responses were measured using acomputer-controlled four-point probe system. During thetests, the magnetic field and electronic current were appliedalong the film plane, i.e., current-in-plane geometry. Magne-tization hysteresis loops were measured by means of a BHV-525 vibrating sample magnetometer /H20849VSM /H20850. All tests were
performed at RT.
To study the effect of current on the AFM layer, the
GMR effects were tested at a small current I
0/H20849100/H9262A/H20850after
a pulse of current Ipwith external magnetic field Hphad been
applied. The pulse treatment time was only 100 ms. Hpwas
chosen as 1.5 kOe. This large value was used to suppresscurrent-induced reversal of the pinned NiFe layer and tomaintain its moments along the H
pdirection during pulse
application.
Figure 1shows GMR curves for the sample with t
=10 nm as a function of the sweeping magnetic field. Mea-surements were carried out as follows. First, the GMR curvewas measured before the pulse was applied /H20849designated as
initial /H20850. Second, a pulse /H20849e.g., 200 mA /H20850was applied to the
sample with H
pantiparallel to the EB direction of /H20849desig-
nated as state I /H20850and then the GMR curve was measured.
Finally, a pulse of the same magnitude was applied with theH
pdirection reversed /H20849designated as state II /H20850and then the
GMR curve was measured. The process was repeated forpulses of 250, 300, and 350 mA. Figure 1reveals that H
ex
decreased with a pulse of current, as already discussed in
previous reports.7,13Furthermore, if a pulse of the same mag-
nitude with reverse Hpdirection was applied, the magnitude
and direction of the EB could be recovered. In other words,once H
exhas been changed by a current pulse, it can be
returned to the initial state by applying a pulse of the samemagnitude and the reverse H
pdirection.
In further research, we used a sample with a pinned NiFe
layer of t=6 nm, which had the relative large Hex. First, we
measured the GMR curve before a pulse was applied. Sec-ond, the GMR curve was measured after a 400 mA pulsewith H
pantiparallel to the EB direction applied to the
sample. Finally, a series of reverse pulses Ip/H20849200–500 mA /H20850a/H20850Electronic mail: tang7tang@yahoo.com.JOURNAL OF APPLIED PHYSICS 106, 073906 /H208492009 /H20850
0021-8979/2009/106 /H208497/H20850/073906/5/$25.00 © 2009 American Institute of Physics 106 , 073906-1
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132.174.254.155 On: Tue, 23 Dec 2014 11:36:31was applied with the opposite Hpdirection. The GMR curves
shown in Fig. 2were measured following application of each
pulse. It is obvious that pulses of lower magnitude than theinitial pulse cannot return H
exto its initial state, but they canlead to partial recovery of Hex. When the pulse magnitude is
equal to or greater than that of the initial pulse /H20849400 mA /H20850,Hex
can be returned to its initial state.
Reviewing our own experiments on the effects of polar-
ized current on the EB, as well as those of other groups,7,13
we noted that GMR curves have always been measured im-
mediately after application of a single pulse. Hence, thequestion arose as to what would happen if we switched froma single pulse to multiple pulses. To investigate this issue,experiments were performed. For clearer observation, we se-lected the sample with small EB /H20849t=10 nm /H20850for the H
ex
change process and the sample with large EB /H20849t=6 nm /H20850for
theHexreturn process. During the change process, 1, 5, and
10 continuous pulses of the same magnitude /H20849300 mA /H20850were
applied for 100 ms with Hpantiparallel to the EB direction at
a pulse interval of 1 s. The GMR curves obtained are shownin Fig. 3. For the return process, H
exwas first set using a
pulse of 300 mA with Hpantiparallel to the EB direction.
Then, 1, 5, and 10 continuous pulses of 200 mA, which islower than the initial pulse, or of 300 mA were applied to the-200 -100 0 100 2007.307.327.347.367.387.407.427.44R(/CID1/CID1)
H( O e )INITIAL
STATE I
STATE II
-200 -100 0 100 2007.307.327.347.367.387.407.427.44R(/CID1/CID1)INITIAL
STATE I
STATE II
H( O e )
(a) (b)
-200 -100 0 100 2007.307.327.347.367.387.407.427.44R(/CID1/CID1)
H (Oe)INITIAL
STATE I
STATE II
-200 -100 0 100 2007.307.327.347.367.387.407.427.44
H( O e )R(/CID1/CID1)INITIAL
STATE I
STATE II
(c)( d)
FIG. 1. /H20849Color online /H20850Typical variations in GMR curves measured at I0for initial, state I, and state II after pulses of /H20849a/H20850200, /H20849b/H20850250, /H20849c/H20850300, and /H20849d/H20850350
mA.
-300 -200 -100 0 100 200 3006.526.546.566.586.606.626.646.66
H(Oe)R(/CID1/CID1)
FIG. 2. /H20849Color online /H20850Typical variations in GMR curves measured at I0
after different return pulses.073906-2 T ang et al. J. Appl. Phys. 106 , 073906 /H208492009 /H20850
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132.174.254.155 On: Tue, 23 Dec 2014 11:36:31sample. The time interval between pulses was again 1 s. The
GMR curves obtained are shown in Figs. 4/H20849a/H20850and4/H20849b/H20850.I ti s
evident that multiple pulses of lower magnitude than the ini-tial pulse cannot return H
exto its original value. The obser-
vations are identical to those applicable to a single pulse.
It is evident from Figs. 3and4that the change in Hexis
only correlated with the pulse magnitude, whether in thechange or return process. The number of pulses applied has
no effect on Hex. In other words, a certain pulse magnitude
corresponds to a certain EB state.
III. RESULTS AND DISCUSSION
Since the Hexmagnitude in a coupled FM/AFM system
is highly correlated with the moment arrangements near theFM/AFM interface, a change in H
exreflects the variety of
spin orientations at the interface.14,15Although the spin ori-
entation near the FM/AFM interface is complex and has stillnot been fully characterized, uncompensated magnetic mo-ments, which are responsible for the net spin moments, pro-vide a schematic illustration of EB and the effect of a polar-ized current on EB in a SV.
8,13,16,17
According to the uncompensated spin model, at the
NiFe/FeMn interface /H20849the transitional layer /H20850, the frustration
and canting of FeMn spins in the first few atomic layers ofthe FeMn layer induced by the saturated NiFe layer are re-sponsible for the EB. This is shown schematically in Fig. 5.
To analyze our observations, we treat the transitional layer asa special magnetic layer. According to the Landau–Lifshiz–Gilbert equation,
18the threshold current Icfor flipping of a
magnetic moment due to spin transfer can be expressed as4
Ic=2e/H9251mV/H20849Hk+H/H20850//H9257/H6036sin/H9272, /H208491/H20850
where eis the electron charge, mandVare the magnetic
moment and the volume of the magnet, respectively, /H9251is the
Gilbert damping parameter, Hkand Hare the anisotropic
field and applied field, respectively, /H9257is the spin-polarization
factor, and /H9272is the angle between the polarized electron and
the magnetic moment m.
In our experiments, electrons flowing in the film plane at
the FeMn layer are polarized by the nearby NiFe layer andtheir spin orientations are not parallel to the moments at thetransitional layer. Therefore, the polarized electrons can in-duce torques on the transitional layer and change the EB. Inaddition, the direction of the torques is dictated by the spinorientation of the polarized electrons. According to Eq. /H208491/H20850,
when a pulse magnitude greater than the threshold current I
c
is applied and the spin orientations of the polarized electrons-200 -100 0 100 2007.307.327.347.367.387.407.427.447.46R(/CID1/CID1)
H (Oe)INITIAL
1 time
5 times
10 times
FIG. 3. /H20849Color online /H20850GMR curves measured at I0after application of
multiple pulses of 300 mA in the Hexchange process.
-200 -100 0 100 2006.526.546.566.586.606.626.64 INITIAL
300 mA pulse to set Hex
1 time of 200mA pulse to set back Hex
5 times of 200mA pulse to set back Hex
10 times of 200mA pulse to set back Hex
H (Oe)R(/CID1/CID1)
(a)
-200 -100 0 100 2006.526.546.566.586.606.626.646.66R(/CID1/CID1)
H (Oe)INITIAL
300 mA pulse to set Hex
1 time of 300mA pulse to set back Hex
5 times of 300mA pulse to set back Hex
10 times of 300mA pulse to set back Hex
(b)
FIG. 4. /H20849Color online /H20850GMR curves measured at I0after application of
multiple pulses of /H20849a/H20850200 and /H20849b/H208503 0 0m Ai nt h e Hexreturn process.(a)(a)(a)The moments for FMThe moments for FMThe moments for FM
The moments for AFMThe moments for AFMThe moments for AFM
The moments for initial stateThe moments for initial stateThe moments for initial state
The moments for setting stateThe moments for setting stateThe moments for setting state
The polarized electronThe polarized electronThe polarized electronFMFMFMAFMAFMAFM
TransitionalTransitionalTransitional
layerlayerlayer
HHHpppTorqueTorqueTorque
(b)(b)(b)FMFMFMAFMAFMAFM
TransitionalTransitionalTransitional
layerlayerlayer
HHHpppTorqueTorqueTorqueT h em o m e n t sf o rF MT h em o m e n t sf o rF MT h em o m e n t sf o rF M
The moments for AFMThe moments for AFMThe moments for AFM
The moments for initial stateThe moments for initial stateThe moments for initial state
The moments for setting back stateThe moments for setting back stateThe moments for setting back state
The polarized electronThe polarized electronThe polarized electron
FIG. 5. /H20849Color online /H20850Schematic illustrations of the FM/AFM interface and
the influence of polarized electrons on EB for /H20849a/H20850the change process and /H20849b/H20850
the return process.073906-3 T ang et al. J. Appl. Phys. 106 , 073906 /H208492009 /H20850
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132.174.254.155 On: Tue, 23 Dec 2014 11:36:31are antiparallel to the transitional layer, the torque tends to
reverse some moments at the transition layer with angle /H9272,
while other moments cannot be changed. Since reversal de-creases the parallel net spin in the AFM layer along its initialdirection, as shown in Fig. 5/H20849a/H20850, the EB is decreased. If the
spin orientations of the polarized electrons are parallel to themoments in the transitional layer on H
preversal, the torque
will reverse the moments, as shown in Fig. 5/H20849b/H20850. Therefore,
the EB reverts to its initial state. This gives a rationale forFig.1, which revealed that EB can be changed and returned
by applying pulses of the same magnitude with only achange in the direction of H
p.
In addition, according to Eq. /H208491/H20850, it is clear that the
threshold current Icis inversely proportional to the angle
between the polarized electron and the magnetic moment m.
This means that application of a small pulse in our experi-ments can only reverse the net spin moments with large
/H9272;
for moments with a small angle /H9272, a large pulse is needed for
reversal. In other words, a pulse of a certain current magni-tude can only reverse moments with a corresponding angle
/H9272. Since the net spin moments in the transitional layer have
different angles /H9272, as shown in Fig. 5, a return pulse of lower
magnitude than the initial pulse can only reverse some of thenet spin moments to their initial state. Therefore, as observedin Fig. 2, a pulse of lower magnitude than the initial pulse
cannot return H
exto its initial state. Furthermore, one angle /H9272
corresponds to one threshold current Ic. Once a certain pulse
magnitude has reversed the moment for a correspondingangle, it will have no effect on the other moments, whichneed a larger I
cfor reversal. Therefore, application of mul-
tiple pulses has no effect on Hex. This provides a rationale for
the observations in Figs. 3and4.
In addition, the features observed in Figs. 3and4also
provide evidence that the motion of net spin moments is indirect reverse and that they do not undergo gradual rotationto change the EB. If the change in EB were due to gradualrotation of net spin moments in the transitional layer, thenmultiple pulses of the same magnitude would affect H
ex
since the angle /H9272changes during rotation. To provide more
compelling evidence of the motion of net spin moments, wemeasured the angular dependence of EB for the sample witht=10 nm before and after application of pulses of 200 mA
with H
pantiparallel to the EB direction by VSM. During
measurements the sample was rotated about an axis perpen-dicular to the sample plane and the applied field was at anangle
/H9258to the deposition field. When the applied field was
parallel to the EB direction, the maximum Hexvalue should
have been achieved. Hence, if the pulse were to decrease theEB by rotation of the net spin moments, then different trendsin the angular dependences of EB and the maximum H
ex
value would be achieved at different angles.19,20However,
according to the results in Fig. 6/H20849a/H20850and6/H20849b/H20850, variations in the
hysteresis curve were the same and the maximum Hexvalue
in each case was achieved at /H9258=0. This indicates that the
direction of net spin moments is along /H9258=0 and a pulse with
an external field only changes the EB magnitude. This pro-vides indirect evidence that the motion of net spin momentsis along the initial EB directionIV. CONCLUSIONS
In summary, it was observed that the EB can be changed
and returned by application of a pulse with an external field.The change in EB is only correlated with the pulse magni-tude and shows no correlation with the number of pulsesapplied. The observations can be rationalized in terms ofcurrent-induced excitation of AFM interface moments byspin transfer. The observations also provide a clearer sche-matic picture of the motion of the net spin moments at FM/AFM interfaces. The results should be useful for the researchand design of new devices based on STT in AFM materials.
ACKNOWLEDGMENTS
This work was supported by the National Natural Sci-
ence Foundation of China /H20849Grant Nos. 60721001, 60771047,
and 60801027 /H20850and the National High Technology Research
and Development Program of China /H20849863 Program, Grant
No. 2009AA01Z111 /H20850.
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-200 -100 0 100 200-1.0-0.50.00.51.0M/Ms
H (Oe)-200 -100 0 100 20 0-1.0-0.50.00.51.0M/Ms
H( O e )
FIG. 6. /H20849Color online /H20850Hysteresis curves for a Ta /H2084910 nm /H20850/NiFe /H2084910 nm /H20850/Cu
/H208494n m /H20850/NiFe /H2084910 nm /H20850/FeMn /H2084915 nm /H20850multilayer at various angles /H9258/H20849a/H20850be-
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EB direction.073906-4 T ang et al. J. Appl. Phys. 106 , 073906 /H208492009 /H20850
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132.174.254.155 On: Tue, 23 Dec 2014 11:36:31 |
1.1759431.pdf | The strange eigenmode in Lagrangian coordinates
Jean-Luc Thiffeault
Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 14, 531 (2004); doi: 10.1063/1.1759431
View online: http://dx.doi.org/10.1063/1.1759431
View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/14/3?ver=pdfcov
Published by the AIP Publishing
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155.33.16.124 On: Sun, 30 Nov 2014 11:39:12The strange eigenmode in Lagrangian coordinates
Jean-Luc Thiffeaulta)
Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
~Received 16 March 2004; accepted 20 April 2004; published online 18 June 2004 !
For a distribution advected by a simple chaotic map with diffusion, the ‘‘strange eigenmode’’ is
investigated from the Lagrangian ~material !viewpoint and compared to its Eulerian ~spatial !
counterpart. The eigenmode embodies the balance between diffusion and exponential stretching bya chaotic flow. It is not strictly an eigenmode in Lagrangian coordinates, because its spectrum isrescaled exponentially rapidly. © 2004 American Institute of Physics. @DOI: 10.1063/1.1759431 #
There are two main types of coordinates used to repre-
sent fluid flow and dynamical systems. Eulerian or spa-
tialcoordinates are fixed in space, while Lagrangian or
materialcoordinates follow parcels of fluid. Strange
eigenmodes are persistent patterns in mixing—they candecay slowly, and hence remain visible in the concentra-tion field for a long time. So far, these have been studiedfrom the Eulerian viewpoint. Here we describe the natureof the strange eigenmode in Lagrangian coordinates for asimple map. It is not a true eigenmode because its wave-length is continuously rescaled in time.
I. INTRODUCTION
The enhanced mixing of a passive scalar is one of the
most direct consequences of chaos: a flow whose trajectoriesexhibit sensitivity to initial conditions will lead to rapid mix-ing. There are powerful theories based on the distribution ofLyapunov exponents
1–3that link the mixing rate of the
passive scalar with the chaotic properties of the flow. It hasbeen recently suggested, following earlier work ofPierrehumbert,
4–10that the mixing properties of the flow can
often be elucidated only by solving a full eigenvalue problemfor the advection-diffusion operator, in an analogous mannerto what is done for the kinematic dynamo.
11The resulting
eigenfunctions have been dubbed strange eigenmodes by
Pierrehumbert, and are closely related to Pollicott–Ruelleresonances in ergodic theory,
12–14which describe the long-
time decay of correlations in mixing hyperbolic dynamicalsystems. Strange eigenmodes have also been observedexperimentally.
15,16They are often called persistent patterns
orlarge-scale eigenfunctions .
The strange eigenmodes reflect a balance between ad-
vection and diffusion. On its own, advection is incapable ofachieving mixing: it shuffles the concentration field of thepassive scalar but does not decrease its fluctuations. The roleof advection is to stir the concentration field, thereby creat-ing sharp gradients in concentration. Physically, these gradi-ents are reflected in the filamentation experienced by a blobof dye when it is stirred. The sharp gradients enhance therole of diffusion tremendously, and this allows mixing toproceed. As the diffusivity of the scalar is made smaller, the
scale at which this mixing occurs decreases, so the concen-tration field appears very rough. In the limit of arbitrarilysmall diffusivity, the concentration field is not smooth: itconsists of a superposition of strange eigenmodes.The domi-nant one among these eigenfunctions is called thestrange
eigenmode, although there may be several of comparableimportance.
In the present work we tie the two types of theories
together ~i.e., Lyapunov exponent-based and strange eigen-
mode !for a specific system, which was already studied in
Ref. 8 from the strange eigenmode viewpoint. The strangeeigenmode represents a fundamentally Eulerian ~spatial !
view of mixing, whereas the Lyapunov exponent view isLagrangian ~material !, following as it does the stretching his-
tory of fluid elements. In the strongly chaotic systems wedeal with in the present context, Lagrangian and Euleriancoordinates are related by a convoluted transformation. Thistransformation is so complex that its specific form is inac-cessible ~even numerically !after some time, a reflection of
sensitivity to initial conditions. For long times, the twoframes must be regarded as essentially independent: we can-not simply solve a problem in Eulerian coordinates andtransform to Lagrangian coordinates ~or vice versa !.Thus we
believe it is worthwhile to take a chaotic system whose so-lution has already been obtained in Eulerian coordinates andsolve it in Lagrangian coordinates. As indicated above, thislinks two views of mixing together, and in particular illus-trates the nature of strange eigenmodes in Lagrangian coor-dinates. This also indicates the source of the breakdown oflocal theories. We will show that there exists a kind of La-
grangian strange eigenmode , which is not quite an eigen-
mode but which exhibits similar features: specifically, it is aneigenmode if an appropriate time-dependent rescaling of co-ordinates is performed ~exponential in time !. This rescaling
is closely related to the ‘‘cone’’ involved in diffusive prob-lems in the presence of an exponentially stretching flow.
17
We call it the cone of safety , because modes inside it are
sheltered from diffusion at a given time.
We introduce the system to be studied, the perturbed cat
map, in Sec. II. We find its finite-time Lyapunov exponentsand eigenvectors using first-order perturbation theory. In Sec.III we partially solve the advection-diffusion equation for ourmap, again using perturbation theory. Some numerical work
a!Electronic mail: jeanluc@imperial.ac.ukCHAOS VOLUME 14, NUMBER 3 SEPTEMBER 2004
531 1054-1500/2004/14(3)/531/8/$22.00 © 2004 American Institute of Physics
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155.33.16.124 On: Sun, 30 Nov 2014 11:39:12is needed to complete the solution, and this is described in
Sec. IV. Finally, a few concluding remarks are offered inSec. V.
II. THE PERTURBED CAT MAP
The strange eigenmode has for the most part been stud-
ied in maps, because these present great advantages for ana-lytical work. This is also reasonable since experimental workhas so far focused on time-periodic flows.
15,16As in Ref. 8,
we consider the map
M~x!5Mx1f~x!, ~2.1!
defined on the unit two-torus, T25@0,1#2. Here Mis a matrix
of integers with unit determinant, and f~x!is a doubly
periodic function, so that Mis a diffeomorphism; specifi-
cally, we take
M5S21
11D;f~x!5K
2pSsin2px1
sin2px1D, ~2.2!
where f~x!is chosen such that Mis area-preserving. For
K50,~2.1!is the usual cat map of Arnold.18The map ~2.1!
inherits much of the simplicity of the cat map, but the per-turbation allows for more complex—and less singular—behavior. The action of the map is depicted in Fig. 1. ForsmallK, the map is very close to the cat map, but the impli-
cations of the perturbation for mixing are profound, as wewill now discuss.
We are interested in the mixing properties of the map
M. In Ref. 8 mixing in this map was investigated from an
Eulerian perspective: advection alternated with diffusion andthe central object was the distribution of the concentrationfield in Eulerian coordinates. For K50, mixing occurs super-
exponentially in the map, due to the lack of dispersion inFourier space. The concentration in a given Fourier mode ismapped entirely to one mode of higher wave number, and soon to ever higher wave numbers. This sequence of wavenumbers have exponentially growing magnitudes. Becausediffusion is exponential in the wave number, the net result is
superexponential decay ~i.e., the exponential of minus an ex-
ponential in time !.
ForKÞ0, the situation is radically different. The map
now disperses concentration among many Fourier modes ateach iteration. In particular, some concentration is alwaysmapped back to the lowest allowable wave number ~the
grave mode !, on which the weak diffusion is ineffective. The
decay of the scalar is then limited by how much concentra-tion is mapped to this grave mode at each iteration. Thegrave mode thus forms the seed of a strange eigenmode,since an eigenmode is by definition a recurring feature. The
concentration field thus settles into the slowest-decayingeigenfunction of the advection-diffusion operator—thestrange eigenmode—analogously to earlier work.
4–7,9,10
Here we wish to solve the same problem as in Ref. 8, but
in Lagrangian coordinates. This means that we focus on fol-lowing fluid elements and describing how they deform underthe action of the map. To solve the advection-diffusion prob-lem in Lagrangian coordinates, it is necessary to have ex-pressions for the finite-time Lyapunov exponents of themap
19~or equivalently the coefficients of expansion !and
their associated characteristic directions, as a function of La-grangian coordinates, and not just their distribution ~we will
see why this is so in Sec. III !. Because the finite-time
Lyapunov exponents are easily derived for the cat map ( K
50), we shall proceed perturbatively, assuming Kis small.
First let us give the Lyapunov exponents and associated
characteristic directions for the unperturbed cat map. TheLyapunov exponents are the logarithms of the eigenvalues ofM, and the characteristic directions are the corresponding
eigenvectors. It is convenient to introduce an angle
uin
terms of which the eigenvectors of Mare
~uˆsˆ!5Scosu2sinu
sinucosuD ~2.3!
with cos2u51
2(111/A5) and sin2u51
2(121/A5). Then the
corresponding eigenvalues of Mare
Lu5L51
2~31A5!511cotu5cot2u, ~2.4a!
Ls5L2151
2~32A5!512tanu5tan2u. ~2.4b!
These equalities are specific to this particular angle, as is the
relation tan u5cotu21. Theuˆdirection is associated with
stretching, and sˆwith contraction.
The coefficients of expansion ~given by LnandL2n
afterniterations of the map !and characteristic directions for
the linear cat map are uniform in space. Now we derive theirvalue for Knonzero but small, using perturbation theory.The
problem is to find the eigenvalues and eigenvectors of thematrixg
(n), with components
gpq(n)“(
m]xm(n)
]Xp]xm(n)
]Xq, ~2.5!
often called the metric tensor ~or Cauchy–Green strain ten-
sor in fluid mechanics !. HereXis the Lagrangian label ~co-
ordinate !andx5x(n)(X) is thenth iterate of the point X
under the action of the map ~2.1!~so thatx(0)(X)5X). The
FIG. 1. Action of the perturbed cat map for K50.4.~a!Initial pattern; ~b!
first iterate; ~c!second iterate; ~d!third iterate. The gray shading shows the
action of the unperturbed cat map.532 Chaos, Vol. 14, No. 3, 2004 Jean-Luc Thiffeault
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155.33.16.124 On: Sun, 30 Nov 2014 11:39:12metric tensor describes the stretching experienced at the nth
iteration by a fluid element initially at X. Its eigenvalues and
eigenvectors give the shape and orientation at the nth itera-
tion of an ellipsoid representing an initially spherical infini-tesimal element of fluid.
Before we can apply perturbation theory to the metric
tensor, we must find the form of the perturbation itself. Tofirst order in K, thenth iterate of the map ~2.1!is
x
(n)5Mn~X!5MnX1(
m50n21
Mmf~Mn2m21X!.~2.6!
The Jacobian matrix of this transformation is
]x(n)
]X5Mn1(
m50n21
Mm]f
]x~Mn2m21X!Mn2m21,~2.7!
where the numerator corresponds to rows and the denomina-
tor to columns of a matrix.We must now construct the metrictensor ~2.5!, which to leading order in Kis
g
K(n)5M˜nMn1$h(n)1h(n)g} ~2.8!
with
h(n)“M˜n(
m50n21
Mm]f
]x~Mn2m21X!Mn2m21~2.9!
and the tilde denotes the transpose of a matrix. The unper-
turbed metric is g0(n)5M˜nMn, with eigenvalues L2nand
L22nand eigenvectors given by ~2.3!. The perturbation is
the bracketed term in ~2.8!. Finding the eigenvalues and
eigenvectors of the symmetric matrix g(n)to leading order in
Kis a straightforward application of perturbation theory for
symmetric matrices, familiar from quantum mechanics. Formore details we refer the reader to standard texts on thesubject.
20,21
To leading order in K, the coefficient of stretching is
written as
LK(n)~X!5Ln~11Kh(n)~X!!, ~2.10!
where Lnis the coefficient of stretching of the unperturbed
cat map, and the correction LnKh(n)(X) is obtained from the
perturbation by contraction with the unperturbedeigenvectors,
20,21
LnKh(n)~X!51
2uˆ$h(n)1h(n)g}uˆ5uˆh(n)uˆ.~2.11!
Observe that because the coefficient of stretching is the
square root of the largest eigenvalue of gK(n), there is an extra
factor of 1/2 to leading order in K. Using the fact that, for M
symmetric, Muˆ5uˆM5Luˆ, we find Eqs. ~2.9!and~2.11!
give
h(n)5sinucosu(
m50n21
cos~2p~MmX!1!, ~2.12!
where we have substituted the specific form of the map,
given by ~2.2!. The subscript ‘‘1’’ in ~2.12!indicates the x1
component of a vector.
The perturbed eigenvectors can be written asuˆK(n)~X!5uˆ1Kz(n)~X!sˆ,sˆK(n)~X!5sˆ2Kz(n)~X!uˆ,
~2.13!
whereKz(n)may be regarded as a small angle of rotation.
Again we follow standard matrix perturbation theory,20,21so
that the angle of rotation is given by
Kz(n)(X)5uˆ$h(n)1h(n)g}sˆ
L2n2L22n, ~2.14!
which after some reduction and the use of ~2.2!yields
z(n)51
L2n2L22n~z1(n)1z2(n)!, ~2.15!
with
z6(n)“1
2~cos2u71!(
m50n21
L62(n2m)cos~2p~MmX!1!.
~2.16!
Note that the asymptotic direction ( n@1) is dominated by
z1(n), so that
z(n).cos2u(
m50n21
L22mcos~2p~MmX!1!,n@1.
~2.17!
In this form it is easy to check that uˆ„z(n)5sˆ„h(n)for
n@1, as required by the differential constraint „sˆK(n)1sˆK(n)
„logLK(n)50.22–24~The derivatives are taken with respect
to the Lagrangian coordinates X.!
The first-order perturbative solution for the coefficient of
stretching h(n)is compared to numerical results in Fig. 2,
and similarly for the eigenvector sˆK(n)in Fig. 3. Unlike the
perturbed coefficient of stretching, which eventually divergesfrom the numerical solution because of the sensitivity to ini-tial conditions, the perturbed eigenvectors converge very rap-idly and are always close to the numerical result.
FIG. 2. Numerical solution ~solid!and first-order solution h(n)~dashed !
from Eq. ~2.12!forK51025. The solutions diverge after several iterations
because we are perturbing off a chaotic trajectory.533 Chaos, Vol. 14, No. 3, 2004 Strange eigenmode in Lagrangian coordinates
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155.33.16.124 On: Sun, 30 Nov 2014 11:39:12It will be more convenient to express the metric tensor in
terms of its eigenvalues and eigenvectors. The metric tensor~2.5!can be written in terms of the coefficients of expansion
and characteristic directions as
g
(n)5@L(n)#2uˆ(n)uˆ(n)1@L(n)#22sˆ(n)sˆ(n). ~2.18!
To leading order in K, we have
gK(n)5L2nuˆuˆ1L22nsˆsˆ12Kh(n)~L2nuˆuˆ2L22nsˆsˆ!
1Kz(n)~L2n2L22n!~uˆsˆ1sˆuˆ!. ~2.19!
The only dependence on Xin~2.19!is contained in h(n)and
z(n).
III. ADVECTION AND DIFFUSION
Having derived the coefficients of expansion and char-
acteristic directions of stretching ~to leading order in K),we
can now solve the advection-diffusion equation in Lagrang-ian coordinates. We will first discuss the case for an incom-pressible flow, and then make the transition to a volume-preserving map.
The advection-diffusion equation for the concentration
of a scalar, Q(x,t), advected by an incompressible velocity
field
vis
]tQ1v]xQ5k]x2Q, ~3.1!
where kis the diffusion coefficient. We define the transfor-
mationx(X,t) from Lagrangian coordinates Xto Eulerian
coordinates xby
x˙5v~x,t!,x~X,0!5X, ~3.2!
where the overdot denotes a time derivative at fixed X.W e
can then transform ~3.1!to Lagrangian coordinates X,19,22
Q˙5]X~D]XQ!, ~3.3!
where we reused the same symbol for Q(X,t). The aniso-
tropic, nonhomogeneous, time-dependent diffusion tensor D
isD“kg21,gpq“(
m]xm
]Xp]xm
]Xq, ~3.4!
wheregis the metric tensor encountered in Sec. II. By con-
struction, the advection term has disappeared from ~3.3!. The
flow enters ~3.3!indirectly through the metric tensor in ~3.4!,
reflecting the enhancement to diffusion due to the deforma-tion of fluid elements.
19,22
We now make the leap from a flow to a map: because the
velocity field does not enter ~3.3!directly, we may regard the
time dependence in Das given by a map rather than a flow,
and use the metric ~2.5!in the diffusion tensor D.27We also
write Q(n)(X) for Q(X,t), wherendenotes the nth iterate of
the map.
Since our map is defined on the torus, we can expand
Q(n)(X) in Fourier components Qˆk(n); the resulting map, ob-
tained by first Fourier transforming and then solving ~3.3!,i s
Qˆk(n)5(
,exp~G(n)!k,Qˆ,(n21), ~3.5!
where
Gk,(n)524p2TE
T2~kD(n),!e22pi(k2,)xd2x, ~3.6!
withTthe period of the map. This is an exact result, but the
great difficulty lies in calculating the exponential of G(n).
Again, we shall accomplish this perturbatively.
For the torus map introduced in Sec. II, from Eq. ~2.19!
we obtain
@gK(n)#215L2nsˆsˆ1L22nuˆuˆ12Kh(n)~L2nsˆsˆ2L22nuˆuˆ!
2Kz(n)~L2n2L22n!~uˆsˆ1sˆuˆ!, ~3.7!
to leading order in K, where the only functions of Xareh(n)
andz(n). Inserting ~3.7!into~3.6!,w efi n d
Gk,(n)5Ak,(n)1KBk,(n), ~3.8!
where
Ak,(n)“2e~L2nks21L22nku2!dk, ~3.9!
and
Bk,(n)“2e~2~L2nks,s2L22nku,u!hk,(n)2~ku,s1ks,u!
3~z1(n)
k,1z2(n)
k,!!. ~3.10!
Here we have defined
e“4p2kT ~3.11!
to agree with the notation in Ref. 8, as well as
ku“k"uˆ,ks“ksˆ, ~3.12!
and similarly for ,uand,s.
Upon making use of the Fourier-transformed ~2.12!,
~2.15!, and ~2.16!in~3.10!,w efi n d
Bk,(n)521
2e(
m50n21
Bk,nm~dk,,1eˆ1Mm1dk,,2eˆ1Mm!~3.13!
whereeˆ1is a unit vector in the x1direction, and
FIG. 3. Numerical solution ~solid!and first-order solution ~dashed !for the
first component of the asymptotic eigenvector sˆK(‘)withK50.1, using the
asymptotic result ~2.17!in~2.13!. The error is of order K2.534 Chaos, Vol. 14, No. 3, 2004 Jean-Luc Thiffeault
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155.33.16.124 On: Sun, 30 Nov 2014 11:39:12Bk,nm“sin2u~L2nks,s2L22nku,u!1~ku,s1ks,u!
3~L2(n2m)sin2u2L22(n2m)cos2u!. ~3.14!
To obtain the full solution, we must now exponentiate
~3.8!to give the transfer matrix in ~3.5!. Fortunately, for A
diagonal there is a simple expansion,
@exp~A(n)1KB(n)!#k,5eAkk(n)dk,1KEk,(n);
Ek,(n)“Bk,(n)eAkk(n)2eA,,(n)
Akk(n)2A,,(n), ~3.15!
valid to first order in K. We say fortunately because without
such a formula it is very difficult to compute this matrixexponential—even numerically—due to the large size of thematrices ~i.e., infinite !and their magnitude ~i.e., growing ex-
ponentially in time !.
TheL
2nterm inAkk(n)seems to imply that Qˆk(n)decays
superexponentially fast as exp( 2eL2nks2). From Eulerian
considerations,8we know that for KÞ0 the decay is actually
exponential after a short superexponential transient. This isbecause the E
(n)term must be taken into account: it breaks
the diagonality of G(n), so that given some initial set of wave
vectors, the concentration contained in those modes can betransferred elsewhere. In particular, it can transfer concentra-tion to modes aligned with the unstable direction. We willsee how this avoids superexponential decay in Sec. IV.
IV. NUMERICAL RESULTS
A. The numerical method
At this point, solving ~3.5!and~3.15!numerically seems
like the only way forward. Clearly, attempting the solve thison a grid in Fourier space is hopeless: very high wave num-ber modes are quickly populated so the resolution is ex-hausted very rapidly. Instead, the procedure we use involveskeeping track of a list of excited Fourier modes ~i.e., those
that are nonzero to machine precision !. We now describe this
scheme.
First, an initial wave number is seeded with some initial
concentration. This mode will be damped by the diagonalpart of the matrix in ~3.15!, and will also excite two new
modes as seen in ~3.13!. Repeating this, starting now from
three modes, we see that the number of excited modes growsexponentially. Thus it would seem that this procedure is notvery advantageous; however, after a few iteration the diffu-sivity @the diagonal part in ~3.15!#will damp most modes
becauseA
kk(n)is growing exponentially. Thus the modes that
have been damped beyond redemption can be removed fromthe list. In this manner the number of excited modes eventu-ally reaches a constant, though they consist of ever higherwave numbers. Thus one can think of a ‘‘packet’’ of modescascading through Fourier space towards larger wave num-bers. It is this packet that is the Lagrangian analogue of thestrange eigenmode in Eulerian space, as we will discuss inSec. IVB.
Let us first present some results. Figure 4 shows the
decay of the scalar variance for K50.001 and four values of
the diffusivity. The agreement with the Eulerian results isexcellent for early times, but inevitably breaks down later.
~In fact the variance eventually begins to increase, which is
forbidden. !The agreement is also worse for smaller diffusiv-
ity. Both of these disagreements are a manifestation of thewave number dependence of the perturbation in ~3.15!: fork
too large the perturbation becomes large, invalidating the ap-proach. Nevertheless, Fig. 4 clearly validates the calculationfor times that are not too long.
Another validation is shown in Fig. 5, where we plot the
difference between the Eulerian and Lagrangian results as afunction of K. The difference clearly scales as K
22, showing
that the two agree at leading order, as required for a first-order asymptotic result.
We now interpret our results in greater detail, and look
for a manifestation of the strange eigenmode in Lagrangiancoordinates.
FIG. 4. Decay of the variance for K50.001 and different values of e, com-
pared to the result from Eulerian coordinates ~dotted lines !. The dashed line
shows the exact result for superexponential decay ( K50) for e50.1.
FIG. 5. Relative difference in the variance between the Eulerian and pertur-
bative Lagrangian result as a function of K, at the fourth iteration. The
dashed line indicates an K2dependence, showing that the two agree to first
order inK.535 Chaos, Vol. 14, No. 3, 2004 Strange eigenmode in Lagrangian coordinates
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155.33.16.124 On: Sun, 30 Nov 2014 11:39:12B. The Lagrangian strange eigenmode
The mechanism described in Sec. IVA is similar to that
originally introduced ~in the context of the kinematic dy-
namo problem !by Zeldovich et al.:17they basically solved
the advection-diffusion equation in Lagrangian coordinatesfor a linear velocity field, and found that in order to avoidrapid superexponential decay one had to restrict attention toa ‘‘cone’’ of wave numbers that are closely aligned with theunstable manifold of the flow ~a similar approach was used
later in Refs. 1–3 !. The exponential shrinking in time of this
‘‘cone of safety’’ leads to an exponential decay of scalarvariance at a rate given by the Lyapunov exponents.
The problem with that approach is that a linear velocity
field offers no possibility of dispersion in Fourier space. The
wave numbers in the cone of safety must have some concen-tration associated with them initially. What our numericalresults show is that if one considers dispersion in Fourierspace @of the type allowed by the E
(n)term in ~3.15!#then it
is possible for concentration to be moved inside the conefrom elsewhere. The Lagrangian equivalent of the strangeeigenmode must live within the cone of safety, otherwise itwould decay away superexponentially. But unlike Ref. 17 thedecay rate in the present case is not determined by theshrinking of the cone: it is set by how much variance getstransferred into the cone at each iteration.
Figure 6 shows a plot of the power spectrum of concen-
tration. The magnitude of the concentration (log
10) is plotted
vs the magnitude of the wave number normalized by ikimax
~its maximum value !, which is proportional to Ln. The con-
centration is normalized at each iteration such that the modewith largest concentration has unit magnitude. The iterationsplotted are n56,7,8,9,10,11 ~circles !andn512~black dots !.
Most of the circles appear as large black dots, because allthese points lie on top of each other. Hence, the concentra-tion is in an eigenstate, given that the wave number has beenrescaled by a factor proportional to L
n~i.e., such that the
dominant peak is at unit rescaled wave number !.This is what
we interpret as the Lagrangian equivalent of the strangeeigenmode ~a good name might be ‘‘stretched eigenmode’’in
Fourier space !.
Some points in Fig. 6 exhibit a decay with iteration num-
ber~appearing as columns of circles, with higher iteration
numbers lower on the graph !: they belong to a more rapidly
decaying eigenfunction. Note that the peaks do not sharpenwith iteration number, but more points are added to some ofthe tails.The eigenfunction appears extremely rough and dis-continuous, though the peaks are indicative of some under-lying continuum behavior. The seemingly isolated points ac-tually tend to line up with a peak far below. Finally, note thatthe relative height ~but not position or shape !of the peaks
depends on K: the whole shape is stretched vertically as Kis
made smaller. This is because the term proportional to K
controls the transfer of concentration ‘‘vertically’’ ~with re-
spect to Fig. 6 !in the eigenmode at each iteration.
V. DISCUSSION
The Lagrangian strange eigenmode has some intriguing
features: ~i!It is rescaled exponentially in time, in order to
remain within the cone of safety ~so it is not a true eigen-
mode !;~ii!its power spectrum is very discontinuous, in
sharp contrast to its Eulerian counterpart;8~iii!its decay rate
is set by how much concentration is moved into the ‘‘new’’cone of safety at each iteration ~since the cone is shrinking !.
In theAppendix we present an analytic result for a two-modesystem which gives a simplified representation of the cone ofsafety.
In the map analyzed here the exponential time-rescaling
gives a proper eigenmode, since only the constant stretchingis important at leading order. In a generic map the stretchingis a strong function of space, so the necessary time-rescalingwould be position-dependent.
It is hard to see how the Lagrangian approach presented
here could be used in more realistic problems: perturbationtheory was used extensively ~which would not be applicable
in most real situations !, both for computing the finite-time
Lyapunov exponents and the matrix exponential ~3.15!.W e
believe the approach is instructive nonetheless, giving as itdoes a picture of the strange eigenmode in Lagrangian coor-dinates.
Our approach does not yield much information about the
long-time behavior of the decay. There is currently a debateas to whether the mechanism presented in Refs. 1–3 gives alower bound on the decay rate.
25,26Our perturbation expan-
sion breaks down before this question can be answered.
ACKNOWLEDGMENT
The author wishes to thank Steve Childress for stimulat-
ing discussions.
APPENDIX: THE TWO-MODE SOLUTION
Though we have not found a general method of solution
of~3.5!with the exponential given by ~3.15!, there is at least
an approximate solution available that illustrates the broad
FIG. 6. The power spectrum of the Lagrangian strange eigenmode for n
56,...,11 ~circles !andn512~black dots !. The large black dots are points
that are the same for all iterations ~after rescaling !: this is the dominant
strange eigenmode in Lagrangian coordinates. Both axes have been rescaled
such that the dominant peak has unit amplitude and wave number @ku(n22)is
defined in Eq. ~A1!#. The hollow circles are due to an admixture of another,
faster-decaying eigenfunction.536 Chaos, Vol. 14, No. 3, 2004 Jean-Luc Thiffeault
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155.33.16.124 On: Sun, 30 Nov 2014 11:39:12features of a full solution. It also shows how the decay rate
of the variance can become independent of the diffusivity inthe Lagrangian viewpoint, as in Ref. 17.
The method is based on defining a class of ‘‘aligned’’
wave numbers ~i.e., that live inside the cone of safety !, and
retaining only two of these modes. These wave numbers k
(p)
are defined by
ku(p)5Lpsinu,ks(p)5L2pcosu, ~A1!
that is,k(p)5Mpk0, wherek0is any initial wave number for
large enough p. Thenksatisfies
k(p)2k(p21)5eˆ1Mp21, ~A2!
so that with the choice k5k(p),,5k(p21), the first Kro-
necker delta in ~3.13!is unity for m5p21.
At thenth iteration, assume that only two wave numbers
are important: k(n2d)andk(n2d21). The number dis a ‘‘lag’’
from the current iteration and will be adjusted later.
Define
Ad“expAk(n2d)k(n2d)(n)
5exp~2e~L2dcos2u1L22dsin2u!! ~A3!
which is independent of n, since we have defined prelative
to the current iteration of the map. It can be shown that thenthe coupling from k
(n2d21)tok(n2d)takes the simple form
Ed“KEk(n2d)k(n2d21)(n)521
2K~Ad2Ad21!. ~A4!
To recapitulate: at the nth iteration, the mode k(n2d)is
mapped to itself with coupling amplitude Ad, andk(n2d21)
is mapped to k(n2d)with amplitude Ed. It is easy to show
that in this two-mode situation the decay rate is determinedby the magnitude of E
d. All that remains is to find d.
The ‘‘lag’’ dis obtained by maximizing Edoverd;dhas
to be large enough that L2dovercomes the tiny diffusivity in
~A3!—so the two Aterms do not cancel in ~A4!—but not so
large that Adis damped. We are thus justified in approximat-
ingAd.exp(2eL2dcos2u)@the other term in ~A3!is
smaller by a factor L24d, which is small even for d51].We
then have
Ed51
2K~Y2YL2!,Y“Ad21, ~A5!
sinceAd5(Ad21)L2. This is easily extremized over Y: the
maximum uEduis achieved for Y5L22/(L221).0.7198, for
which uEdu.0.3074K. The lag is then given by solving for d
in terms of the extremizing Y,
d5111
2 log LlogSlogY21
ecos2uD.0.5902 10.5195log e21,
~A6!
which scales logarithmically with the diffusivity. Note that
the decay rate is now completely independent of the actualvalue of the diffusivity: the lag adjusts itself to compensate,introducing a separation of scale between the dominant wavenumberk
(n2d)and the largest wave number in the system,
k(n).
The actual decay rate as e!0~from the Eulerian solu-
tion in Ref. 8 !,i s0 . 5Kfor small K, compared to the two-
mode Lagrangian solution 0.3074 K. Thus, most of the im-portant behavior is captured by the two-mode solution. The
two modes can be seen in the spectrum of the strange eigen-
mode in Fig. 6: the dominant peak at iki/ku(n22)51i sk(n2d)
(d52 in this case !, and the peak at iki/ku(n23)5L21
.0.3820 is k(n2d21). The other peaks are modes that could
be included to get a more accurate expression for the decayrate.
The two-mode solution also nicely illustrates the idea of
the cone of safety: both modes are always inside it, andbecause the cone is shrinking by a factor L
21at each itera-
tion then the modes have to follow suit. The key differencewith Ref. 17 is that here the concentration in the modes ismappedfrom one cone to another at each iteration, and is not
part of the initial condition.
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This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
155.33.16.124 On: Sun, 30 Nov 2014 11:39:1224J.-L. Thiffeault, ‘‘Derivatives and constraints in chaotic flows:Asymptotic
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1.2018290.pdf | Use of ultrasonic spectroscopy to characterize calcified arterial lesions
J. A. Rooney , P. M. Gammel , and J. Hestenes
Citation: The Journal of the Acoustical Society of America 67, S56 (1980); doi: 10.1121/1.2018290
View online: https://doi.org/10.1121/1.2018290
View Table of Contents: https://asa.scitation.org/toc/jas/67/S1
Published by the Acoustical Society of AmericaGeneralized Griineisen parameters which measure the strain
derivatives of the lattice vibrational frequencies are often used to
test atomic theories of crystalline solids and to model their
anharmonic properties. The relationship between these parameters
and the solid nonlinearity parameters measured directly in ultrasonic
harmonic generation experiments is derived using a generalized
approach valid for normal mode acoustic wave propagation in any
crystalline direction. The resulting Griineisen parameters are purely
isentropic in contrast to the Brugger-Grfineisen parameters which
are of a mixed thermodynamic state. The equations are specialized
to the case of longitudinal wave propagation in the three pure mode
directions of cubic crystals in order to compare available
experimental data.
9:30
X3. Acoustic streaming in superfluid helium. R. F. Carey, J. A.
Rooney, and C. W. Smith (Department of Physics, University of
Maine, Orono, ME 04469)
Evidence for the existence of acoustic streaming in superfluid
helium has been obtained using a quartz wind transducer configura-
tion operating at 3 MHz. The detection of the streaming and
determination of its velocity were made by measuring the deflection
of a collimated ion current in liquid helium. The PZT-4 transducer
was mounted parallel to the axis of the ion beam and displaced
from it. Measurement of the distribution of current along a linear
array of small collectors as a function of sound amplitude
permitted determination of the streaming velocity. At low amplitudes
the streaming velocity increased with the square of the amplitude
as predicted by the classical theory for the quartz wind. Above the
threshold amplitude for turbulence the streaming velocity increased
more slowly but still depended on the square of the amplitude.
Evaluation of the rates of increase of the streaming velocity showed
that the effective viscosity of the helium was larger by a factor of
five above the threshold for turbulence. [This work was supported
in part by the Air Force Office of Scientific Research.]
9:45
X4. Acoustic streaming in a nematic liquid crystal. S. Candau and
G. Waton (Laboratoire d'Acoustique Moltculaire, Universit6 Louis
Pasteur, 4 Rue Blaise Pascal, Strasbourg, France) and S. Letcher
(Department of Physics, University of Rhode Island, Kingston,
RI 02881)
The acoustically induced birefringence has been studied in homeo-
tropic cells of nematic liquid crystal. The observed features of the
optical pattern reveal the coexistence of several streaming
mechanisms. From the spatial distribution of the optical signal, both
the velocity and the flow pattern have been experimentally deter-
mined for different test cell configurations. At oblique incidence of
the acoustic beam on the cell, the dominant acousto-optic effect
has been shown to be due to a two-dimensional streaming in the
plane of the liquid crystal film. At normal incidence of the sound
beam, the streaming is in the direction of the sound propagation.
The dependence of the optical pattern on the acoustical intensity and
on the thickness of the liquid crystal layer was found in good agree-
ment with the previously developed models based on acoustic
streaming IS. Nagai, A. Peters, and S. Candau, Rev. Phys. Appl.
12, 21-30 (1977); S. Candau et al. Mol. Cryst. and Liquid Cryst.
(to be published)]. [Work partially supported by DRME and NATO.]
10:00
X5. Pulse response of a nematic liquid crystal acousto-optic conver-
sion cell. W. Hamidzada and S. Letchef (Department of Physics,
University of Rhode Island, Kingston, RI 02881) and S. Candau
(Laboratoire d'Acoustique Mol(culaire, Universit( Louis Pasteur,
67070 Strasbourg, France) We have studied the response of a nematic liquid crystal acousto-
optic conversion cell to a pulsed ultrasonic signal. It has been estab-
lished (see previous abstract) that a streaming mechanism is respon-
sible for the acousto-optic effect when using a cw signal, but a pulsed
signal has been little studied in spite of its practical importance. We
report on the effect of pulse width, pulse amplitude, and repetition
rate of a 4-MHz wave on the integrated optical transmission and on
the spatial representation of the sound beam. [Work partially
supported by NSF and NATO.]
10:15
X6. Amplification of sound by gas-phase reactions. R. M. Detsch
and H. E. Bass (Department of Physics and Astronomy, University
of Mississippi, University, MS 38677)
The amplification of sound in a reacting gas mixture has been
observed. Energy for the amplification was provided by a photo-
induced, exothermic H:/CI: reaction. A sound wave traveling through
the gas can be regarded as a modulation of temperature about some
steady-state. value. As most chemical reaction rates increase with
temperature, the regions of higher temperature experience an
increase in the rate of the H:/CI: reaction. Since the H:/CI: reaction
is exothermic, this increase in reaction rate causes an additional
increase in temperature. Conversely, in regions of low temperature,
the H:/CI: reaction is slowed causing a relative decrease in tempera-
ture. Thus the initial acoustic signal is amplified. The measured gain
was 1.8 at 2 kHz compared to a gain of 4 predicted by Ellis and
Gilbert [J. Acoust. Soc. Am. 62, 245-249 (!977)]. The source of this
difference has not been resolved. [Work supported by NSF.].
10:30
X7. Ultrasonic analysis of cumulative internal damage in filled
polymers. G. C. Knollman, R. H. Martinson, and J. L. Bellin
(Lockheed Research Laboratory, Palo Alto, CA 94304)
An ultrasonic technique is presented for studying dewetting and
cumulative internal damage in filled polymers. A quantitative
theoretical model is developed relating measured sound speed and
attenuation in a polymer material to specific internal damage
parameters such as effective vacuole size and number density. Ultra-
sonic assessment of cumulative damage has been conducted for
applied uniaxial tension, compression, and shear strains. Estimates
of the acoustic measures of damage obtained by applying the
analytical model to experimental data on filled polymer materials
are in excellent agreement with independent microscopic observa-
tions. The ultrasonic assessment approach which is presented for
determining cumulative internal damage in filled polymers can
provide a basis for age determination and estimation of residual
lifetime. [Work supported by the Lockheed Independent Research
Program.]
10:45
X8. Use of ultrasonic spectroscopy to characterize calcified arterial
lesions. J. A. Rooney (Department of Physics, University of Maine,
Orono, ME 04469) and P.M. Gaminell and J. Hestenes (Jet
Propulsion Laboratory, California Institute of Technology, Pasa-
dena, CA 91103)
Analysis of the minima in the spectra received from an unknown
reflector as a function of angle can be used to determine the size
and orientation of the reflector. The theory for received spectral
minima resulting from the directivity patterns of reflectors was
generalized to include arbitrary orientations of the target with respect
to the receiver. Tests of the theory were performed using test
objects and a reflective-mode time-delay spectrometer operating in
the frequency range from 1 to 9 MHz. Results obtained for spectra
from stainless steel rod targets with diameters from 0.16 to 0.64 cm
S$6 J. Acoust. Soc. Am. Suppl. 1, Vol. 67, Spring 1980 99th Meeting: Acoustical Society of America S$6 were in good agreement with the theory. Measurements were made
as a function of angle of the frequencies for which minima occurred
in the spectra received from calcified lesions in in vitro arterial
specimens. Computer analysis of the data yielded accurate values
for the effective diameter and orientation of the calcified lesions.
[This work was supported in part by the National Institutes of
Health via contract NO1-HV-799301 and the National Research
Council through its Research Associateship Program.]
11:00
X9. Vibrational dynamics of gaseous bodies trapped in microscopic
pores at megahertz frequencies. Douglas L. Miller and Wesley L.
Nyborg (Department of Physics, University of Vermont, Burlington,
VT 05405)
We have reported previously on the significance of microscopic
gas-filled pores to the action of ultrasound on biological cells at low
intensity levels. In this paper we describe progress in characterizing
the oscillations set up in such pores when their dimensions are of the
order to a few microns or less and frequencies are in the megahertz
range. In one promising model for the gaseous oscillator the air-
liquid interface vibrates as a membrane, fixed at its periphery, with
tension equal to the air-liquid surface tension. Expressions for the
resonance frequency, damping coefficients, and vibrational ampli-
tude for cylindrical pores which are filled or partially filled with gas
have been obtained. In addition, the expected scattering characteris-
tics were determined for this model. Comparisons were made with
experimental results on backward and forward scattering from com-
mercial pore-containing membranes. [Work supported by NIH Grant
GM-08209.]
11:15
X10. Ultrasonic off-normal imaging techniques for under sodium
viewing. T. F. Michaels and J. E. Horn (Hanford Engineering
Development Laboratory, P.O. Box 1970, Richland, WA 99352) Methods have been developed for constructing images of objects
from ultrasonic data. Feasibility of imaging surfaces which are off-
normal to the sound beam has been established. Laboratory results
are presented which show a complete image of a typical nuclear
reactor core component. Using these techniques, surfaces up to
75 ø off-normal have been imaged. Details of equipment and
procedures used for this image construction are described. The
eventual application is imaging of objects under molten sodium in a
liquid metal fast breeder reactor.
11:30
X ll. Velocity and attenuation measurements in metal waveguides.
H. M. Frost and J. H. Prout (Applied Research Laboratory, The
Pennsylvania State University, Box 30, State College, PA 16801)
Noncontacting electromagnetic-ultrasound transducers (EMT's)
have been used on metal rods and tubes lH. M. Frost, Physical
Acoustics, Vol. XIV, edited by W. P. Mason and R. N. Thurston
(Academic, New York, 1979), Chap. 3]. Work is scarce, though, on
wire or on inhomogeneous or roughly surfaced waveguides. Too,
scientific use of EMT's is slowed by measurement errors from
EMT-waveguide gaps varying during scanning. We report on a two-
transducer (sender-receiver) setup for (nonferromagnetic) wire with
«- to 1-mm diameters. Nondispersive torsional pulses with ---10-/xs
durations are transduced by Lorentz forces on stationary and moving
wire, without access to wire ends. Velocities for alloy wire and
graphite/aluminum metal matrix composite wire agree well with prior
shear wave values at megahertz frequencies for corresponding
"bulk" material. Oscilloscope and x-y plotter data for moving wire
illustrate measurement sensitivity to artificial flaws yet insensitivity
to transducer-wire gap variations. Early work on three- and four-
transducer setups indicates even greater capability for measurement
precision and accuracy and application to nondestructive testing
and to surface acoustic waves. lThis work was supported in
part by NAVSEA 62R4.]
THURSDAY MORNING, 24 APRIL 1980 ANSLEY ROOM, 9:00 A.M. TO 12:00 NOON
Session Y. Psychological Acoustics IV--General (Poster Session)
Lawrence L. Feth, Chairman
Department of Audiology and Speech Sciences, Purdue University, West Lafayette, Indiana 47907
Contributed Papers ß
Y1. A contribution towards a model for an artificial voice. George F.
Kuhn (Vibrasound Research Corporation, 4673 S. Zenobia Street,
Denver, CO 80236) and Chris Saner (1820 Carson Street, Rock
Springs, WY 82901)
The acoustic pressure amplitudes radiated from the mouth of a
storeroom manikin were measured in a 6 cm x 6 cm vertical plane
immediately forward of the lips. The sound was produced by an
acoustic driver in the manikin's head. The lips were immoveable,
forming an approximately trapezoidal aperture with an area of 6.5
cm 2. Pressure measurements were made at the l-Octave band center
frequencies from 1.0 to 6.3 kHz. The measured results at 1.25 kHz
agree well with the speech measurements at 1.3 kHz reported by
Leman [4th Int. Conf. Acoust., abstr G36 (1962)]. As pointed out by
Leman, the spreading of the acoustic wave, on axis, can be described by an ideal point source displaced 1 cm horizontally from the tip of
the lips. Below the lips, near the chin surface, the acoustic pressure
does not spread spherically. Also, the farfield azimuthal directivity of
speech measured by Moreno and Pfretzschner [Acoust. Lett.
1, 78-84 (1979)] is shown to be predicted reasonably well by a point
source on a sphere, except in an approximate _+20 ø sector centered at
the shadowed pole. However, the median plane vertical directivity
differs from that of a point source on a sphere due to body diffraction.
[This work was done while the authors were with the Electrical
Engineering Department, University of Wyoming, Laramie, WY
82071.]
Y2. A hybrid adaptive procedure for estimation of psychometric
functions. J. L. Hall (Acoustics Research Department, Bell Labora-
tories, Murray Hill, NJ 07974)
S$7 J. Acoust. Soc. Am. Suppl. 1, Vol. 67, Spring 1980 99th Meeting: Acoustical Society of America S$7 |