Tunable magnon-photon coupling in a compensating ferrimagnet - from weak to strong coupling

1.1 MB·

by  H. Maier-Flaig

Tunable magnon-photon coupling in a compensating ferrimagnet – from
weak to strong coupling
H. Maier-Flaig,1,2 M. Harder,3 S. Klingler,1,2 Z. Qiu,4,5 E. Saitoh,4,5,6,7,8 M. Weiler,1,2 S. Gepr¨ags,1,2 R.
Gross,1,2,9 S. T. B. Goennenwein,1,2,9,10 and H. Huebl1,2,9
1)Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften, Garching,
Germany
2)Physik-Department, Technische Universita¨t Mu¨nchen, Garching, Germany
3)Department of Physics and Astronomy, University of Manitoba, Winnipeg,
Canada
4)WPI Advanced Institute for Materials Research, Tohoku University, Sendai,
7 Japan
1 5)Spin Quantum Rectification Project, ERATO, Japan Science and Technology Agency, Sendai,
0 Japan
2 6)Institute for Materials Research, Tohoku University, Sendai, Japan
7)PRESTO, Japan Science and Technology Agency, Saitama, Japan
r
a 8)Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Japan
M 9)Nanosystems Initiative Munich, Mu¨nchen, Germany
10)Institut fu¨r Festko¨perphysik, Technische Universita¨t Dresden, Dresden, Germany.

0 (Dated: 13 March 2017)
1
Weexperimentallystudythemagnon-photoncouplinginasystemconsistingofthecompensatingferrimagnet

] gadoliniumirongarnet(GdIG)andathree-dimensionalmicrowavecavity. Thetemperatureisvariedinorder
i
c to tune the GdIG magnetization and to observe the transition from the weak coupling regime to the strong
s coupling regime. By measuring and modelling the complex reflection parameter of the system the effective
-
l coupling rate g and the magnetization M of the sample are extracted. Comparing g with the magnon
r eff eff eff
t and the cavity decay rate we conclude that the strong coupling regime is easily accessible using GdIG. We
m
show that the effective coupling strength follows the predicted square root dependence on the magnetization.
.
t
a
m During the last decade, the coupling of paramagnetic plingofmultipleindependentferromagneticmomentsvia
moments with microwave photons has attracted a lot of a cavity18,19, coupling a ferromagnetic material to a su-
-
d attention.1,2 Apart from the fundamental physical inter- perconducting qubit20 and probing strong coupling be-
n estinthetopic,coherenttransferofinformationbetween tween spin system and a microwave resonator by opti-
o
the two systems can be achieved when the coupling rate cal means21,22. These experiments demonstrate the po-
c
exceeds the individual relaxation rates. This is of con- tential of ferromagnetic systems for magnon-microwave
[
siderable interest for applications in quantum informa- photon and magnon-optical photon conversion as well as

3
tion processing as it opens the door to conversion be- memoryapplicationseveninthequantumlimitedregime.
v
tween traveling photonic states and long-lived states in Interestingly,however,dedicatedworkonthefoundation
9 √
spin ensembles.3 When the coupling rate between the of the coupling mechanism, the n scaling of the cou-
6
9 sub-systems exceeds their invididual loss rates, the sys- pling rate, does not exist for exchange coupled spin en-
8 tem enters the so-called strong coupling regime. While sembles. Here, we provide direct experimental proof of
√
0 the coupling between an individual spin and an electro- the n scaling in an exchange coupled ensemble of spins
. magnetic mode of a microwave resonator is typically in inaferrimagneticgadoliniumirongarnet(GdIG)sample.
1
0 the weak coupling regime, the strong coupling regime The theoretical description of the coupling of an or-
7 can be reached by using a spin ensemble. The coupling deredferromagnettoanelectromagneticcavityhasbeen
√
1 strength is then boosted by a factor of n with n being discussedinvariouspublications16,23–25. Weusethethe-
v: the number of polarized spins in the spin ensemble.4–7 ory developed by Cao et al.24 that approaches the prob-
i Reaching the strong coupling regime using an ensemble lem in the 1D case starting from Maxwell’s equations
X
ofnon-interacting(paramagnetic)spinsandamicrowave and describes the complex reflection parameter of a cav-
r resonatorisestablishedinavariatyofconfigurations.8–11 ity loaded with a ferromagnet as:
a √
The nscalingofthecouplingstrengthhasbeenexperi-
mentallydemonstratedbyAbeet al.9 andlaterZollitsch
A(1−κ )
et al.12. In these experiments, the number of polarized S = c . (1)
spins n was tuned by exploiting thermal depolarization 11 i(ω−ωc)−κc−ige2ff(ω−ωFMR+iγs)−1
of the spin ensemble. Recently, the approach of strongly
Here, κ and γ describe the decay rates (i.e. the half
coupling a spin ensemble to a microwave resonator has c s
widthathalfmaximumfrequencylinewidth)ofthecav-
been transferred to exchange coupled spin systems.13–17
ity and the magnon system, respectively. A is a com-
Sincethen,severalexperimentswithincreasingcomplex-
plexscalingparameterthataccountsforlossesandphase
ity have been implemented such as mediating the cou-
shifts in the setup. The effective coupling rate is de-
2
noted as g and the cavity resonance frequency is ω .
eff c Tcomp=270K
Theferromagneticresonance(FMR)frequencyωFMR de- 103
pends sensitively on the magnetic anisotropy and the Gd μH
Fe 0
applied static magnetic field H . For thin ferromag- 102
0
netic films with H0 applied in the film plane, the disper- m) 101 T(K)
sion is well described by the Kittel equation26 ω = A/ 40 70 100130160
oγfµ0th(cid:112)eHm0a(tHer0ia+l Munedffe)r. iHnverees,tiγgaitsiotnh.e gTyhreomeffaegcnteitviFecMmrRaatgio- M(k 101−001 246000000 (kA/m)
netization M = M − H is equal to the magnetiza- M
eff k 0
tion M if the shape anisotropy is the only relevant con- 10−2
0 50 100 150 200 250 300 350 400
tribution to the anisotropy while H accounts for addi-
k
T(K)
tionalanisotropiessuchasmagnetocrystallineanisotropy
or strain induced anisotropy. The effective coupling rate
geff is taken to be proportional to the square root of the FIG. 1. Net magnetization M measured at µ0H0 = 0.1T
net magnetic moment m = MV of the sample. While (appliedinthefilmplane)usingSQUIDmagnetometry(blue
in the 1D model of Ref. 24 the magnetic moment is pro- circles)normalizedtotheeffectivemagnetizationat15K. At
portional to the thickness of the sample, in the 3D case the compensation temperature Tcomp the net magnetization
considered here, it scales with the total volume V of the almost vanishes. Below Tcomp, the Gd sublattice magnetiza-
sample. In contrast to paramagnets, the magnetization tiondominatesthenetmagnetizationandisthereforealigned
with H . Above T , the net Fe magnetization dominates
of ferromagnets typically shows only a weak tempera- 0 comp
and aligns with H . The length of the arrows schematically
ture dependence for temperatures well below the Curie 0
represent the two Fe and the Gd sublattice magnetizations.
temperature. This weak temperature dependence is ad-
Inset: Within the investigated temperature range, the evo-
vantageous for applications as it makes the system more
lution of the effective magnetization M extracted from the
eff
robust against external perturbations. In order to reach FMR measurements (red data points) agrees very well with
and study different coupling regimes, however, the sam- the net magnetization M as determined by SQUID magne-
ple size typically needs to be changed15. tometry.
Here, we take a simple, robust and continuously tun-
able approach that allows for an in-situ manipulation
of g by simply adjusting an external control param- tion shows a paramagnetic-like behavior, i.e. its mag-
eff
eter. Wevarythenetmagneticmomentofacompensat- netization increases towards low temperatures following
ing ferrimagnet by an order of magnitude by changing a brillouin like function. At room temperature, the net
temperature. Compensating ferrimagnets are a partic- magnetization of GdIG is dominated by the magnetiza-
ular class of ferrimagnets containing two or more mag- tionoftheironsublatticesandthereforepointsalongthe
netic sublattices, where at least one of the sublattices largerFesublatticemagnetization. However,asthemag-
consists of internally weakly interacting moments, which netization of the Gd sublattice strongly increases with
can be thought of as acting like a paramagnetic spin en- decreasingtemperature,thenetremanentmagnetization
semble. Therefore, thermal polarization and hence the decreases and vanishes at T , where the sublattice
comp
netmagnetizationofthissublatticechangessignificantly magnetizationsjustcompensateeachother. BelowT
comp
with temperature. When this sublattice is antiferromag- thenetmagnetizationincreasesagainduetotheincreas-
netically coupled to the other sublattices, magnetization ingthermalpolarizationoftheGdsublatticeandreaches
compensation can occur at the so-called compensation a value of approximately 595kAm−1 at 5K.27 We inves-
temperature T .27 At T , the individual magneti- tigate the latter temperature range in the following.
comp comp
zations of all involved sublattices cancel each other and Theinvestigatedsampleisat=2.6µmthickGadolin-
the net remanent magnetization of the ferrimagnet van- ium iron garnet film grown by liquid phase epitaxy on
ishes. The macroscopic net magnetization that couples gadolinium gallium garnet (GGG) with lateral dimen-
tothecavityphotonscanthusbetunedbytemperature. sionsl=5mmandb=2mm. ThenetmagnetizationM
Gadolinium iron garnet (GdIG) is a compensating of the sample was measured using SQUID magnetome-
ferrimagnetic insulator with three magnetic sublattices. try at an external magnetic field of 0.1T (see Fig. 1).29
Just like in the ubiquitous yttrium iron garnet, GdIG We find a compensation temperature of Tcomp = 270K,
contains two iron sublattices that are strongly antiferro- which is lower than the bulk value of 285K27 in agree-
magnetically coupled and effectively form a single (net) ment with literature suggesting that Tcomp is slightly re-
iron sublattice with reduced magnetization. The mag- duced in thin films.30,31
netization of this iron sublattice shows only a weak For the magnon-photon coupling experiments, we
temperature dependence below room temperature. A placethesampleinthemagneticfieldanti-node(electric
third sublattice is formed by the magnetic moments field node) of the TE mode of a 3D microwave cavity
011
of the gadolinium ions that couple weakly to the iron (BrukerFlexlineMD5dielectricringcavityinanOxford
moments and are aligned antiparallelly to the net iron Instruments CF935 gas flow cryostat). The identical ex-
magnetization.28 The gadolinium sublattice magnetiza- perimental setup was used in Ref. 32 which contains a
3
more detailed description of the setup. showssomedeviationoffitanddataforlowtemperatures
We measure the complex (phase sensitive) reflection [Fig.2(25K)]. Weattributethisslightdiscrepancytothe
scatteringparameterS aroundtheresonancefrequency presenceofasecondresonancelineforourGdIGsample,
11
of the cavity mode while applying a variable external which is apparent at low temperatures upon close ex-
magnetic field H in the film plane. The applied probe amination. The second resonance might originate from
0
power is chosen to be small (0 dBm) so that non-linear spatial inhomogeneities in the sample. The data, the
processes do not play a role. At this power, the num- analysis scripts and results are publicly accessible under
ber of photons excited in the cavity is approximately Ref. 35 for further analysis and evaluation.
NPh = P/((cid:126)ωcκc) = 2×1013 and is several orders of We first discuss the effective magnetization Meff ex-
magnitude lower than the minimum effective number of tracted from the fitting procedure and displayed in the
spins in the sample (1×1017 for the highest analyzed insetofFig.1(reddatapoints). Thetemperatureevolu-
temperature). In the microwave cabling connecting the tionoftheeffectivemagnetizationM determinedusing
eff
cavity and the vector network analyzer, standing waves FMRandthenetmagnetizationM measuredbySQUID
can form that cause a field independent background sig- magnetometry agree well, indicating that the dominant
nal. In order to efficiently remove this background, we anisotropy contribution in our GdIG thin film is indeed
analyzethefieldderivative∂BS11 = ∂(∂µS01H10) asafunction given by shape anisotropy. We therefore take M = Meff
of H . Typical data for the absolute value of ∂ S is inthefollowing. TheobservedslightdifferenceofM and
0 B 11
showninFig.2fortwodistincttemperatures. Whenthe M can be explained by a small increase of the g-factor
eff
external magnetic field is adjusted to tune the FMR fre- with decreasing temperature as indicated in Ref. 34.
quency ω close to the unperturbed cavity frequency
FMR Ascentralresult,weconfirmthescalingoftheeffective
ω ,theexcitationsofthemagneticsystem(magnons)and
c coupling rate g with the magnetic moment (or magne-
eff √
the microwave cavity (photons) start hybridizing with a
tization). As mentioned above, we expect g ∝ M
eff eff
signatureinS11describedbyEq.1. Forthecaseofstrong in analogy to the paramagnetic case. Fig. 3 shows g2
coupling (g >κ ,γ ) [Fig. 2 (25K)], Eq. 1 describes a eff
eff c s as a function of M where a straight line indicates the
eff
characteristic anti-crossing of the H -independent cavity
0 expected scaling with g = 0 at M = 0. The data
eff eff
mode and the H -dependent spin resonance. The split-
0 follows this behavior accurately over an order of mag-
ting can be used to determine the coupling strength. In
nitude of the magnetization. As noted above, for low
the weak coupling regime [Fig. 2 (110K)], the cavity is
temperatures(largemagnetization)asecondresonanceis
only marginally disturbed. Here, the coupling strength
indicated that we do not fit separately. The fit therefore
can be determined by analyzing the change in the line
slightly overestimates the coupling in this regime which
width of the microwave cavity.33 This case is equivalent
can also be seen in Fig. 3. From the slope, we can cal-
to conventional FMR and therefore plotting the S pa-
culate the single spin – single photon coupling rate to
rameter at the cavity frequency as a function of the ex- (cid:0) √ (cid:1)1/2
g /2π = g /2π µ 3/V = 0.072Hz with the Bohr
ternalfieldshowsthetypicalFMRabsorptionlineshape. s 0 B
magneton µ and the sample volume V and assuming a
Note that by taking the field derivative of S , the un- B
11 spin1/2particlewithag-factorof2. Thisvalueisinrea-
perturbed cavity absorption peak vanishes (Fig. 2) as ω
c sonable agreement with the values of g /2π = 0.043Hz
is field independent. s
determined for paramagnetic ensembles9.
Becauseweaimtoobservethetransitionofthesystem
In order to coherently exchange excitations between a
from the weak coupling regime (γ > g ) to the strong
s eff
magnonic system such as GdIG and a photonic system,
couplingregime(κ ,γ ≤g ),theapproximatesolutions
c s eff
strong coupling (i.e. κ ,γ ≤g ) is required. Fig. 4 dis-
for the extreme cases are not sufficient. The system has c s eff
playstherelaxationrateofthecavityκ andthespinsys-
to be modeled by the complete reflection characteristic c
tem γ for comparison. The cavity decay rate κ should
given by Eq. 1. To remove the field independent back- s c
ideally be temperature independent but depends on the
groundofS ,wefirstnumericallycalculatethemagnetic
11 coupling of the feed line to the cavity.32 As the temper-
field derivative of the complex S parameter. We then
11
ature of the cavity is decreased, it thermally contracts
performafull2Dfit(i.e. allfitsofallcutsatconstantH
slightly leading to a shift in resonance frequency and
have shared parameter values) using the magnetic field
requires a change of the mechanical adjustment of the
derivative of Eq. 1 for every temperature. From this fit,
coupling mechanism. Hence, the decay rate of the cavity
we extract M , g , κ and γ . The g-factor is fixed to
eff eff c s
variesslightlywithtemperature. Themeanvalueofκ is
g =2overthewholetemperaturerangebasedonRef.34 c
shownasgreenlineinFig.4,whileitsstandarddeviation
thereby reducing the number of free parameters further.
isdepictedasgreenshadedarea. TheFMRlinewidth,γ
The resulting fit is virtually indistinguishable from the s
significantly increases towards T , in good agreement
data in Fig. 2. We therefore present the residual of the comp
withreportsinliterature.34 Finally,g isplottedonthe
fittothedataintheinsetsonthesamescaleasthedata. eff
same scale and shows the increase towards low tempera-
A vertical cut (dashed grey lines) of the 2D data and
turesalsoshowninFig.3. Comparingthethreerates,we
the fit is shown exemplarily for the static magnetic field
findthatthesystemisintheso-calledhighcooperativity
corresponding to ω = ω . The 2D-fit is very good
for high temperaturFeMsR[Fig. 2c(110K)], but the residuum regime (ge2ff >1) for temperatures below 110K (shaded
κcγs
4
∂ S (1) ∂ S (1)
| B 11| T | B 11| T
3.719 1.980 1.586 3.719 1.222 0.1920.199 1.222
− − − −
25K 110K
9.85
)
z
H
G 9.80
(
π
2 9.75
/
ω
9.70
135 140 145 150 155 160 165 170 0 2 210 220 230 240 250 260 270 0 2
µ0H0 (mT) |∂BS11|(T1) µ0H0 (mT) |∂BS11|(T1)
FIG. 2. Magnitude of the magnetic field derivative of the reflection parameter S at two distinct temperatures. The coupling
11
visibly increases at low temperatures. The horizontal orange dotted line marks the resonance frequency ω /2π of the unper-
c
turbedcavity,thereddottedlinemarkstheresonancefrequencyoftheunperturbedspinsystemω /2π. Inset: Residualsof
FMR
the fit to Eq. 1 on the same scale. Line cuts: Data (blue) and fit (green line) at the field where the unperturbed cavity mode
and magnon mode are degenerate (dashed vertical grey line in the adjacent colorplot).
102 109
III II I
2Hz) T = 170 K T = 5 K Hz) 108
M e (
2g (eff data Rat 107
geff= Meff(T)g0 ∙c=2¼ geff=2¼ °s=2¼
101 p 106
102 103 0 20 40 60 80 100 120 140 160 180
Meff (kA/m) Temperature (K)
FIG. 3. Effective coupling rate ge2ff as a function of the ef- FIG. 4. Magnon relaxation rate γs, cavity relaxation rate
fective magnetization. The data√agrees very well with the κc and effective coupling rate geff as a function of tempera-
expected scaling behavior g ∝ M (red line). ture. The shaded green area denotes the standard error of
eff eff
thecavityrelaxationrate. Thecouplingrateincreasestolow
temperatures while the magnon relaxation rate drops. Thus,
the system transitions from weak coupling (shaded region I)
regionIIinFig.3)andentersthestrongcouplingregime to a high cooperativity regime (region II) and enters strong
(region III) for liquid helium temperatures. By choosing coupling at low temperatures (region III).
a slightly larger sample size or specially shaped cavity
modes17,strongcouplingandthusthecoherentexchange
of information between a 3D cavity and GdIG is feasible pling strength for a system with vanishing net rema-
even at higher temperatures. nent magnetization, as found in compensating ferrimag-
Inconclusion,weinvestigatedthemagnon-photoncou- nets at the compensation point or for the exchange res-
pling in a system consisting of a compensating ferrimag- onances in anti-ferromagnets, poses an interesting non-
net(GdIG)anda3Dmicrowavecavitybymeasuringand linear problem36 that has yet to be addressed. In order
analyzing the full complex reflection spectra quantita- torealizethetransitionfromthestrongtotheweakcou-
tively. We control the magnetization of GdIG with tem- pling regime without modifying the sample or setup, we
peratureandextractthescalingofthecouplingstrength useasamplesizethatisjustsufficienttoreachthestrong
with the net magnetization of the sample. We thereby coupling regime. We emphasize, however, that strong
√
confirm the expected scaling behavior g = g M of coupling is easily accessible with GdIG / 3D microwave
eff 0
the ferrimagnet–cavity system. This result proves that cavity system.
the description used for paramagnets is equally appro- M. Harder acknowledges support from the NSERC
priate for exchange coupled spin systems. The cou- MSFSS program. We gratefully acknowledge funding
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