Magnetization dynamics in the single-molecule magnet Fe8 under pulsed microwave irradiation

0.58 MB·English

by  K. Petukhov

Magnetization dynamics in the single-molecule magnet Fe under pulsed microwave
8
irradiation
K. Petukhov,1,2 S. Bahr,1 W. Wernsdorfer,1 A.-L. Barra2 and V.Mosser3
1Institut N´eel, associ´e `a l’UJF, CNRS, BP 166, 38042 Grenoble Cedex 9, France
2Grenoble High Magnetic Field Laboratory, CNRS, BP 166, 38042 Grenoble Cedex 9, France
7 3Itron France, 76 avenue Pierre Brossolette, 92240 Malakoff , France
0
0
2 WepresentmeasurementsonthesinglemoleculemagnetFe8 inthepresenceofpulsedmicrowave
radiation at 118 GHz. The spin dynamics is studied via time resolved magnetization experiments

n using a Hall probe magnetometer. We investigate the relaxation behavior of magnetization after
a the microwave pulse. The analysis of the experimental data is performed in terms of different
J
contributionstothemagnetization after-pulserelaxation. Wefindthatthephononbottleneckwith

5 a characteristic relaxation time of ∼ 10−100 ms strongly affects the magnetization dynamics. In
1 addition, the spatial effect of spin diffusion is evidenced by using samples of different sizes and
different ways of the sample’s irradiation with microwaves.
]
l
l PACSnumbers: 75.50.Xx,75.60.Jk,75.75.+a,76.30.-v
a
h
-
s INTRODUCTION microwaves.
e
Inthispaperwestudythespindynamicsofthesingle-
m
Singlemoleculemagnets(SMMs)haveattractedmuch molecule magnet Fe8O2(OH)12(tacn)6, hereafter called
.
t interest in recentyears because of their unique magnetic Fe8. This molecule contains eightFe(III) ions with spins
a
s=5/2. These spins are strongly superexchange coupled
m properties. Having a regular structure, a well defined
spin ground state and magnetic anisotropy they exhibit formingaspingroundstateS =10andthe spindynam-
-
d quantum phenomena even at macroscopic scales.[1, 2, 3] ics can be described assuming the giant spin model by
n an effective Hamiltonian [17]
Featuressuchasquantumtunnelingbetweenspinstates,
o
interference between tunneling paths or blocking of the
c H=−DS2+E(S2−S2)+O(4)−gµ S~ ·H~ (1)
[ spin orientationatvery lowtemperature showthe quan- z x y B

1 tum nature of SMMs. [4, 5, 6, 7, 8] In addition SMMs H~ is the applied magnetic field, O(4) contains fourth
are supposed to be good candidates for data storage or
v order terms of spin operators and g ≈ 2 represents the
4 quantum computing. [9] gyromagnetic factor. The anisotropy parameters D =
2 Recent works in the field of SMMs focused on spin
0.275K andE =0.046K have been determined by vari-
3 dynamics and interactions with millimeter-wave radia-
ousexperimentalmethods. [6]ClassicalEPRtechniques,
1
tion. The aima are to control the spin orientation in
0
the sample and to selectively induce transitions between
7
0 spin states. The crucial point for any application of
/ SMMs is the knowledge of the spin relaxation time and
t
a the spin decoherence time. Therefore, various experi-
m
ments have been performed in studying spin dynamics 0
- in SMMs in the presence of microwaves. Most measure-
d
ments are based on standard electron paramagnetic res- orbach
n
)
o onance (EPR)techniques [10] or onopticalspectroscopy K -10
c [11], while others are based on magnetization measure- (
v: ments of the sample. In measuring the absorption of rgy
Xi the microwavesviathe decreaseofmagnetizationwecan Ene -20
obtain information about both the magnetization of the tunnel
r
sample and EPR-like spectra. This technique also al- h
a
lows the precise control over the excitation of the sam- 1
-30
ple and makes it possible to quantify the nonresonant
heating.[12] The magnetization sensor can be either a -10 -5 0 5 10
Hall magnetometer [5, 13], a micrometer sized super-
Quantum number m
conducting quantum inference device (SQUID) [14], a S
standard SQUID [15] or an inductive pickup loop [16].
Differences in these techniques lie mainly in the rapidity FIG.1: SpinstatesandenergybarrieroftheFe8 system. For
and sensitivity of the measurement, in the possibility of an excited spin there are various processes relevant for spin
applying magnetic fields, and in the compatibility with dynamics.
2
frequency domain magnetic resonance spectroscopy and
neutron spectroscopy are complementary methods and
give similar results. [11, 17, 18, 19] The nondiagonal
terms in the Hamiltonian are responsible for the tunnel-
ing processesbetween spin states, whereas D defines the
anisotropy barrier of approximately 25 K as can be seen
in Fig. 1.
In terms of the spin dynamics the giant spin model
reveals various relaxation processes that are important
for the evolution of the spin system in the time do-
main. As sketched in Fig. 1 the main parameters of the
spin system are the spin relaxation time τ1 (time scale
∼10−7 s), the excitationoverthe barrierby a thermally
FIG.2: Differentconfigurationsofirradiatingthesamplewith
activated multistep Orbach process with time constant microwaves. We used several waveguides in order to change
τorbach (timescale∼10−8 s×e∆E/kBT)where∆E isthe the electromagnetic environment of the sample, i.e. the cou-
barrier height) and the tunnel probability between de- pling of the microwaves to thesample.
generated states with time constant τ (time scale
tunnel
∼ 104 s for the ground state tunneling).[17, 20] The
use of a large crystal of the single molecule magnet Fe8 der −100 ◦C. The sample is placed on top of the 10 µm
and, in consequence, the interactions between molecules × 10 µm Hall junction with its easy axis approximately
makeitnecessarytointroducespin-phononandspin-spin parallel to the magnetic field of the solenoid. The three
interactions. Effects such as spin decoherence (typical samples used in our experiments (150 × 100 × 30 µm3,
time scaleτ2 ∼10−9 s), phononbottleneck (typicaltime 160×180×100µm3 and680×570×170µm3)areex-
scale τ ∼ 100 s) or spin diffusion (τ ) have to be posedtomicrowaveradiation. Microwavesaregenerated
ph diff
takeninto accountfor a complete descriptionof the spin by a continuous wave (cw), mechanically tunable Gunn
dynamics.[21, 22] oscillator with a nominal output power of 30 mW and a
In this paper, a series of measurements on the SMM frequencyrangeof110GHz to 119GHz. Pulsesaregen-
Fe8 is presented investigating the relaxation of magneti- erated using a SPST fast PIN diode switch (switching
zationonmillisecondandmicrosecondscales. InSection time less of than 3 ns) triggered by a commercial pulse
II we describe the experimental setup and the various generator. An oversizedcircularwaveguideof10 mm di-
experimental conditions. In Section III we present the ameter leads the microwaves into the cryostat. In some
experimental data that will be discussed in Section IV. ofourexperimentsweusetransitionpartsfromoversized
Finally in Section V, we give some concluding remarks. circular-to-rectangularWR6waveguides. Inotherexper-
iments we use only a cylindrical cone as an end piece of
the circularwaveguidethatis rightin frontofthe irradi-
EXPERIMENTAL TECHNIQUES atedsample. Thedifferentconfigurationsofthecoupling
ofthemicrowavestothesamplethatareinvestigatedand
General setup comparedareexplainedinthenextparagraph. Thecryo-
stat is filled with exchange gas that thermodynamically
couples the sample to the bath and allows a rather fast
The measurements are performed using a commercial
heatexchange. AsthesignaloftheHallmagnetometeris
16Tsuperconductingsolenoidandacryostatatlowtem-
in the range of a few microvolts a low-noise preamplifier
peraturesintherangeof1.4Kto10Kwithtemperature
isusedinordertoincreasethesignal-to-noiseratiointhe
stability better than 0.05 K. The magnetization of the
rather long coaxialcables. Finally, the signalacquisition
Fe8 sample is measured by a Hall magnetometer. The
is done by a fast digital oscilloscope having a bandwidth
Hall bars were patterned by Thales Research and Tech-
of 1 GHz and 10 G/s sample rate and is done by taking
nology(Palaiseau),usingphotolithographyanddryetch-
an average over typically 32 frames.
ing,inadelta-dopedAlGaAs/InGaAs/GaAspseudomor-
phicheterostructuregrownbyPicogigaInternationalus-
ing molecular beam epitaxy (MBE). A two-dimensional
Coupling of the microwaves to the sample
electrongasisinducedinthe13nmthickIn0.15Ga0.85As
well by the inclusion of a Si delta-doping layer in the
graded AlxGa1−xAs barrier. All layers, apart from the Conical waveguide
quantum well, are fully depleted of electrons and holes.
The two-dimensional electron gas density n is about The simplest method of coupling microwaves to the
s
8.9×1011 cm−2 in the quantum well, corresponding to sample is provided by a conical waveguide focusing mi-
a sensitivity of about 700 Ω/T, essentially constant un- crowaves from the oversized circular waveguide to the
3
sample. In our experiments a cone with an opening of Resonator Mode Frequency[GHz]
4 mm in diameter is used. This setup is sketched in TM413 108.631
Fig.2aandwillbedenotedhereafterassetupI.Theconi- TE114 110.471
calwaveguidehastheadvantagethatitconservesthepo-
TM014 111.436
larizationofthe passing light,thus allowingexperiments
TM033 114.137
depending on the angle of polarization or experiments
TE314 116.097
with circularly polarized microwaves. Due to the large
TM214 119.410
dimensionsofthewaveguidecomparedtothewavelength
TE124 120.176
the propagation of the microwaves can be considered as
quasioptical and the attenuation of microwave power is TM024 120.932
rather small.
TABLE I: Possible modes in a perfect cylindrical cavity in
the range of 108 GHz to 121 GHz and for the dimension of
the cavity specified in the text. Due to perturbations inside
Rectangular waveguide the cavity the calculated frequencies might differ from the
theoretical ones.
Compared to the conical end piece of the waveguide
a rectangular waveguide can focus the microwaves even
better. TheopeningofatypicalD-bandWR6waveguide
is 1.7 mm × 0.8 mm, thus the cross sectionis nine times
magnetic field to increase by a few orders of magnitude
lessthanthatinthecaseoftheconicalwaveguide. How-
and thus allow better excitation of the sample even with
ever, the attenuation using the rectangular waveguide is very short pulses (< 10 µs). The Q-factor for resonant
fairly large, especially because of the transition part be- modesisnumericallycalculatedto beinthe rangeof102
tween oversized cylindrical and rectangular waveguides. to 103 depending on the mode. Therefore, the resonant
At the end of the rectangular waveguide the field distri-
modesareexpectedtohaveafullwidthathalfmaximum
butionis welldefined. This featureallowsusto irradiate in the order of 0.1 to 1 GHz. Consequently we expect
the crystal in different ways. It can be put either in the
each mode to exist in a rather broad frequency band.
central region of the waveguide (Fig. 2b, setup II), or
Themodesshouldbepresentintheresonatorevenwhen
at the edge of the waveguide, in order to partially irra-
the microwavefrequencydoes notexactlymatchthe cal-
diate the crystal (Fig. 2c, setup III). When irradiated
culated resonance frequency.
partially, the magnetization of the crystal is measured
with a Hall sensor at the nonirradiated side of the crys- Intheexperimentstwoslightlydifferentendplatesare
tal. This method allows us to point out the importance used. In the first case an unprotected Hall bar with a
of the inhomogeneous distribution of magnetization in sample on top is directly glued on top of the end plate.
the sample and spin diffusion processes that take place The position of the sample is about 2 mm off center of
in large samples. theendplate. InrespecttothesizeofthecavitytheHall
sensorandthesamplerepresentaperturbationoftheres-
onator that might be non-negligible. In consequence the
Microwave resonator
frequenciesforthe differentmodesmightslightlyandin-
homogeneouslyshiftaccordingtothecalculatedfrequen-
Insomeofourexperimentsacylindricalcavitymadeof cies. Neverthelessthedensityofmodesbetween109GHz
copperwithdetachableendplatesisused,andthesample and 120 GHz should remain rather high.
is placed on top of one of the end plates (Fig. 2d, setup
IV). The inner diameter of the resonator is 10.10 mm In the second case in order to perturb the cavity as
andtheheightis5.5mm. AstandardWR6waveguideis weakaspossible weprotected the Hallbarwitha copper
coupled to the sidewall of the cavity by a coupling hole. foil. Thepositionofthesampleinthiscaseisabout1mm
Themicrowavesfromthewaveguideenterintothecavity off center of the end plate. A small hole is drilled into
at half height and can excite various modes obeying the the end plate and the Hall sensor is placed into the hole
selection rule TEodd,∗,∗ and TMeven,∗,∗ (Table I). All and is finally coatedwith a thin copper foil (thickness of
themodeshavezerotangentialelectricalfieldontheend 10 µm). Thus only the sample placed on the copper foil
plates. The magnetic field has one or more maxima on isdirectlyexposedtothe electromagneticfieldinsidethe
the end plates according to the mode and the direction cavitywhereastheHallsensorandthecablesareoutside
of the magnetic field is always radial. The sample is the resonator. In this setup the Hall sensor is protected
mounted on one of the end plates in such a way that from the microwave radiation by the thin copper foil,
the magnetic field in the cavity and the easy axes of the however the sensitivity in measuring the sample’s mag-
sample are parallel. netizationisexpectedtobeweakerasintheunprotected
By using a resonator we expect the amplitude of ac case.
4
MEASUREMENTS
Magnetometry combined with microwaves
Whenthe sampleofSMMplacedinthe magneticfield
B is exposed to the cw microwaves, the magnetization
curves show resonant absorption dips, similar to EPR
spectroscopy spectra. The absorption of microwave ra-
diation takes place at certain field values at a given fre-
quency, when the microwave frequency matches the en-
ergy difference between two neighboring energy states,
thus the allowed transitions are ∆m = ±1. The pop-
s
ulating of the upper levels (see Fig. 1) reduces the net
sample’smagnetizationM,thechangeofwhich∆M can
be detected via Hall voltage measurements. If the ap-
plied magnetic field is ramped while the microwaves are
applied,theobtainedmagnetizationspectraclearlyshow
a series of nearly evenly spaced absorption dips, which
can be easily attributed to the appropriate transitions,
as shown in Fig. 3a by the thick solid curve. This curve
is placed on the top of ”pure” magnetization curves, i.e.
thecurvesmeasuredwithoutmicrowaves,depictedbythe
thin solid curves in Fig. 3a. These reference curves were
measuredatdifferentcryostattemperatures inthe range
from 2 K (top curve) to 20 K (bottom curve) with 1 K FIG. 3: (a) Magnetization of Fe8 as a function of magnetic
incremental step. As can be seen from Fig. 3a, the 2 K field. Thecurvesarenormalizedtothesaturationmagnetiza-
magnetization data measured without microwaves and tionvalueMS. Thethinsolidcurvesrepresentthemagnetiza-
tionmeasuredwithoutmicrowavesattemperaturesfrom 2K
the magnetization data measured in the presence of mi-
to 20 K in 1 K steps. The thick solid curve represents mag-
crowaves do not match: the latter curve lies much be-
netization curve in presence of cw microwaves of frequency
low the first curve, as the temperature during the mi- fCW = 118 GHz measured at 2 K. The thick arrow repre-
crowaveexperimentwouldbehighercomparedtothatof sents the magnetization difference ∆M between data taken
the pure magnetization experiment. The difference be- with and without microwaves at 2 K. (b) Spin temperature
tween the two curves is denoted by the magnetization TSvsappliedfieldB,calculatedusingthemappingprocedure
difference ∆M, which is a good measure of the amount describedinthetext. Thedashedcurvesdepictoff-resonance
absorption.
of microwaveradiation absorbed by the spin system.
Spin temperature the magnetization curve (magnetization spectrum) ob-
tained under the use of microwaves onto underlying ref-
Although the difference of magnetization ∆M can be erence magnetization curves,measuredat different cryo-
used to qualify the amount of absorbed microwave pho- stat temperatures without microwaves. For eachmagne-
tons, it is rather inconvenient to speak in terms of rela- tization point of the absorption spectra one finds, at the
tiveunitsof∆M. Anothermoresignificantcomplication correspondingfieldB, the temperature TS thatgivesthe
in the use of ∆M for quantitative characterizationscon- same magnetization measured without microwave radia-
cerns the loss of sensitivity of ∆M close to zero field. tion[Fig.3a]. Thetemperaturesinbetweenthereference
As the magnetic field B goes to zero, the magnetization magnetizationcurves are obtained with a linear interpo-
also goes to zero, and hence the sensitivity of detection lation. A typical result of such a mapping is depicted in
of absorption peaks goes to zero as well. Therefore, we Fig. 3b. TS can be called the spin temperature because
needtoperformatransformationofthemagnetizationto the irradiation time is much longer than the lifetimes of
a physical quantity which does not depend on the mag- the energy levels of the spin system which were found to
netic field B. bearound10−7seconds[23]. Thephononrelaxationtime
Such a quantity called spin temperature was explicitly τph from the crystal to the heat bath (cryostat) is much
introduced in our earlier paper [12] as a perfect mea- longer, typically between milliseconds and seconds [22].
sure of the amount of microwave radiation absorbed by Thespinandphononsystemsofthecrystalaretherefore
an SMM spin system. The concept of spin temperature in equilibrium.
can be easily understood from Fig. 3a. We can map Figure 3b shows spin temperature data calculated for
5
the magnetization measurements at cryostat tempera-
tures of 2 K and 10 K, performed at frequency of cw
microwavesoff =118GHz. FromFig.3bwecancon-
CW
clude that the obtained spin temperatures TS are much t
larger than the cryostat temperature T. This is asso-
ciated with a strong heating of the spin system. This
effect is more prominent at lower T: the baseline of 2 K
TS spectrum is around 7 K, while 10 K TS spectrum’s
background is very close to the nominal cryostat tem- M
0
perature T = 10 K, as depicted by the dashed curves in [4]
Fig. 3b. We also see that both backgrounds are not flat, [1]
but are magnetic field dependent. The presence of the
M
spectra’s nonflat background is due the presence of off-
resonance absorption of microwaves, which takes place
[3]
between the resonant absorption peaks. The nonreso-
M
nant(orbackground)absorptionismodulatedinthe fol- MW
[2]
lowingway: ithaslargercontributionwheretheresonant
absorptionhaslargerspectralweight,i.e. higherpeaksof 0 10 20 30 40
TS. Interestingly, the off-resonance absorption was also
evidenced in the EPR spectra of SMMs, but its origin Time (ms)
remains undiscussed[19, 24]. On the topof the nonreso-
nant backgroundone can see a perfect EPR-like absorp- FIG.4: Typicaloscillogramofapulsedexperiment. Themag-
tion spectra, and the TS peak positions exactly match netizationwasmeasuredasafunctionoftimeforamicrowave
the magnetic field values, corresponding to |∆m | = 1 pulselength of t=10 ms at temperature T = 10 K.
S
transitions (see Fig. 3).
tiondifference ∆M defined fromthe PWmeasurements,
Pulsed microwave measurements as shown in Fig. 4, one can perform spin temperature
TS calculations. SuchcalculationsforFe8 havebeenper-
Anotherwaytoperform∆M measurementsinorderto formed in our previous work for the PW configuration
calculate the spin temperature TS is to utilize a pulsed and it has been shown that obtained spin temperatures
microwave (PW) radiation [12]. This method gives di- TS are much closer to the cryostat temperature T than
rect information about ∆M at a given magnetic field that for cw experiments [12]. The linewidths and shapes
value B, at a given temperature T, and at a given mi- of PW TS spectra depend on the pulse length, but in
crowave power, a measure of which is a pulse length t. general the peak positions of cw and PW configurations
This advanced method also drastically reduces the heat- areidentical. Incontrasttothecwexperiments,thePW
ing of the sample with microwaves, since the repetition method can successfully resolve absorption peaks near
rate of microwave pulses (typically 200 ms in our ex- zero field [12].
periments) is much larger than the pulse length values, Unlike cw measurements, PW magnetization profiles
typically t ≤ 10 ms. Restoration of the after-pulse mag- contain information not only about ∆M but also about
netization to the equilibrium value M0 normally takes the magnetizationdynamics. Letus assumethat the ap-
less than 100 ms, as can be seen in Fig. 4. pliedmagneticfieldB andthemicrowavefrequencyf
PW
The scheme of possible pulsed experiment is depicted do match the resonance condition, i.e., the in-resonance
inFig.4. ThetoppartofFig.4 schematicallyshowsmi- microwave pulse is applied. Since the sample’s magneti-
crowavepulseofdurationt=10ms,andthebottompart zation is connected to the spin state level occupancy of
of the figure shows time-resolveddevelopment ofmagne- SMMs,themagnetizationdynamicsshouldbeconnected
tization data collected during such pulsed experiment. to the level’s lifetime. The spin-spin relaxation time τ2
The magnetization before and at the end of the pulse is usually much shorter than the spin-phonon relaxation
hasvaluesM0andMMW,respectively. Thedifferencebe- time τ1; if τ1 obtained through magnetization dynamics
tweentheunperturbedmagnetizationvalueM0andmag- is short enough, it can determine the upper limit of τ2.
netization at the final edge of the pulse MMW (i.e. the Finally,sincethereisanincreaseofthesample’stemper-
heightofthemagnetizationresponse)∆M =MMW−M0 ature due to the heating with microwaves, the phonon
is identical to the magnetization difference ∆M defined relaxation time τ from the crystal to the heat bath
ph
for the cw microwaves case, as graphically explained by (cryostat) can be systematically studied from the mag-
Fig. 3a. Thus, having a set of reference magnetization netization ”cooling” after long enough pulses in PW ex-
curves, shown by thin curves in Fig. 3a, and magnetiza- periments. Thus, the detailed consideration of the time-
6
resolved magnetization profile, depicted in Fig. 4, might
provide information about the τ1, τ2, and τph relaxation
times.
Let us consider the magnetization behavior during a
PW experiment in detail in Fig. 4. At the beginning
of the pulse, the magnetizationrapidly decreases(region
[1]) and starts to saturate (region [2]) until the end of
the pulse. We need to note that a complete saturation
isobservedonlyforratherlongpulsesofseveralseconds.
After the microwave pulse is switched off, the magneti-
zationrestoresback to the equilibriumvalueM0. At the
beginning of its restoration the magnetization increases
rapidly (region [3]), several millisecond later magnetiza-
tion increase changes to the slowerbehavior (region[4]),
untilitlevelsoutatM0. Thisslowrestorationlastslong,
up to a hundred of milliseconds, but we were able to
follow it completely, since the typical repetition time of
pulses was 200 ms. This brings us to the conclusion,
FIG. 5: Fit of magnetization restoration data (open circles)
thatregion[4]mightcomprisetheinformationaboutthe
with a single exponential relaxation (solid line). The inset
cooling of the sample after the microwave pulse, i.e. the
shows the fit with two exponents: fast relaxation with relax-
phonon relaxation time τph from the crystal to the heat ationtimeτshortandcoolingwithrelaxationtimeτlong. Note
bath (cryostat). Exactly this relaxationtime is typically thatthedepictedmagnetization datacurvecontainsonly2%
of the order of magnitude of several tens of milliseconds oftheexperimentalpointstakenduringthesingleexperiment,
up to seconds [22]. The fast-running beginning of region i.e. only every50th point is shown.
[3]couldcontainthelongitudinalrelaxationtimeτ1(typ-
ically ∼ 10−7 seconds [23]). There was another interest-
ingobservationinregion[3]ofthemagnetizationcurvein time, which can be found right after the pulse is much
Fig.4: duringsomeofourexperimentswehaveobserved shorterthanthecoolingτ relaxationtime,whichcanbe
ph
that right after the pulse was switched off the magneti- amajorcontributioninthetailofmagnetizationrestora-
zationcontinuedtodecreaseforsometimeandonlythen tion. Therefore, we have separately considered two dif-
started to increase to the equilibrium value. Similar be- ferent regions of the magnetization restoration curve, as
haviorofmagnetizationaftermicrowavepulsesinFe8was depicted in the inset of Fig. 5. Firstly, we have assumed
observed in the recent work of Bal et al. [13]. Below we that the magnetization data, obtained right after the
willexplicitlyinvestigatesuchanovershooting ofmagne- pulse was switched off (typically, within the time frame
tization after the microwave pulse. In principle, regions of 10-20% of the pulse length), could contain the infor-
[1]and [2]are verysimilar to the regions[3] and[4], cor- mation about the spin-lattice τ1 relaxation time. This
respondingly. The problem with the use of this part of data can be described by a fast exponential relaxation,
magnetization evolution curve is that regions [1] and [2] and we will denote the corresponding relaxationtime by
are limited by the pulse duration, and therefore it seems τ , as shown in the inset of Fig 5. Another valu-
short
to be problematic to estimate relaxationtimes from this able contribution to the overall magnetization restora-
part of the data, especially the long-lasting τph. In this tion comes from the cooling of the specimen after the
paper, we pay attention to the time-resolvedbehavior of microwavepulse, such a process can be described by the
the magnetization after the pulse in a PW experiment, long-lastingrelaxationprocesswithrelaxationtimeτ ,
long
i.e. to regions [3] and [4] as determined in Fig. 4. taken later after the pulse was switched off (typically, 3
to 4 pulse length values later after the pulse edge until
the end of the magnetization data), as depicted in the
The model inset of Fig 5. If the slow relaxation τ is responsible
long
for the sample’s cooling, it can only be sensitive to the
First, in order to analyze the behavior of the mag- sample size and its thermal coupling to the bath, with
netization after the microwave pulse, we have tried to both parameters unchanged during an experimental set.
fit the magnetization data in regions [3] and [4] by a Therefore,weexpectthiscontributiontobetemperature
single exponential relaxation. We have found that in andpulse lengthindependent. Nevertheless,the slow re-
many cases a single exponential description was unsat- laxation τ contribution (i.e. sample thermalization)
long
isfactory, as shown in Fig. 5. This is not surprising in canbecomedominatingoverthefastrelaxationτ on
short
the framework of consideration concerning different re- increase of the temperature and/or for very long pulses,
laxation times given above. Indeed, the τ1 relaxation sincetheτshortdrasticallyshortensundersuchconditions
7
and can be unresolved. In this case, a single exponential
relaxation could be suitable for the magnetization data

description and it could give solely the relaxation time
4.0
τ . In our experiments we avoid such a situation and
long
we carefully adjust the experimental condition to have setup I
3.5
two clearly distinguishable regions [3] and [4], where the 0.05 ms
uncontroversialanalysisbymeansofτ andτ can 0.1 ms 0.1 ms
short long ) 3.0 0.5 ms
be performed. This analysis wasapplied in the following s
m 1 ms
magnetization relaxation measurements. (rt2.5 0.5 ms 0.05 ms 51 0m mss
o
h 1 ms
s2.0 10 ms
Relaxation of magnetization
5 ms
1.5
We have studied the relaxation of magnetization em-
ploying different sample irradiation configurations, de- 1.0
scribed in the ”Experimental setup” part. In all the
cases, the applied magnetic field was set to 0.2 T and 1 2 3 4 5 6 7
the frequency of microwavesduring pulses was 118GHz.
T (K)
Thus,belowwedescribethestudiesofthemagnetization
dynamics of the first transition from the ground state
m = −10 to the first excited state m = −9, since the FIG. 6: Temperature dependence of relaxation time τshort
S S
measured for a small sample with a conical waveguide. The
givenmagnetic field andfrequency values matchthe res-
microwavepulsesofrepetitiontimeof200msandofdifferent
onanceconditionforFe8 placedintomagneticfieldalong lengths were applied: 0.05 ms (dotted curve), 0.1 ms (solid
its easy axis [12]. While the temperature and the pulse curve), 0.5 ms (dashed curve), 1 ms (dashed-dotted curve),
length were changedduring the PW experiments, shown 5ms(boldsolidcurve),and10ms(bolddashedcurve). Color
below, the repetition time of microwave pulses was al- online.
ways set to 200 ms.
Measurements with conical waveguide
Measurementswithaconicalwaveguideareperformed
onatinysampleplacedinsetupI configuration,asshown
in Fig. 2a. The volume of the sample is 150 × 100 × 30
µm3. We have investigated the τ and τ relax-
short long
ation times as a function of temperature T for different
values of pulse length. The results obtained from such
PW experiments for the short relaxation time τ are
short
depicted in Fig. 6.
The fast relaxation shows the relaxation time τ
short
of the order of magnitude of 1-1.5 ms. The tempera-
ture dependence of τ is not strong, but is clearly
short
pronounced: onwarmingfrom2Ktoapproximately4K
therelaxationtimeτ decreases,reachesitsminimum
short
near 4 K, and then smoothly increases on further warm-
ing(seeFig.6). Thedecreaseofrelaxationtimeτ on
short
warmingfrom2 K toapproximately4K is muchsteeper
for shorter pulses, while the τ behavior above4 K is FIG. 7: Generic plot depicting temperature dependence of
short
similar for all the pulse length values. fast relaxation time τshort and slow relaxation time τlong for
differentsamplesizesanddifferentsample’sirradiationconfig-
For the same sample and experimental configuration
urations: setupI (solidcurves),setupII forabigandasmall
setup I, the slow relaxation time τ is approximately
long sample (big and small solid circles, respectively), setup III
one order of magnitude greater, that is, its value lies
(open triangles), and setup IV (solid rhombus with dotted
around 20 ms. This long-lasting relaxation time is tem- curve). NotethattheLatinnumbersstandforcorresponding
perature independent within 5% in the measured tem- setup notation.
perature range, as shown in Fig. 7, where both τ
short
and τ obtained from the 10 ms data using setup I
long
8
configuration are shown by solid curves for comparison.
This behavior of τ perfectly fits into the above given
long
explanation of τ -process as a cooling of the sample
long
after the microwave pulse. Such a cooling rate is only
defined by the sample thermal coupling to the bath, and
therefore,itispermanentforagivenexperimentalsetup.
Figure 7 shows the generic plot of relaxation times
τ and τ , obtained from allthe experimentalcon-
short long
figurations at 10 ms pulse length over 200 ms pulse rep-
etition time. Such a rather long pulse length was chosen
mainly due to the following reasons. At first, at rather
long pulses the contribution of the τ -process,i.e. the
long
sample thermalizationafter the pulse, becomes valuable,
making better separation of τ and τ data inter-
short long
vals possible, and thus both fast and long relaxations
can be estimated and comparedfor the same systemun-
der the same experimental condition. Secondly, at long
pulses, a high signal-to-noiseratio enables better fitting,
evenforthelimitedregionsofexperimentaldata. Finally,
as will be shown below, at such long pulses there are no
overshooting phenomenaobservedforbothsmallandbig FIG. 8: Magnetization loops measured at several cw mi-
samples at all the temperatures applied; with the pres- crowave frequencies using a cavity. Note, that the curves are
equally scaled.
ence of overshooting, the analysis of magnetization data
in terms of relaxation exponents becomes controversial.
Measurements with a rectangular waveguide
The irradiationof the sample with a piece of the rect-
angular waveguide WR6 is advantageous in the sense
that the distribution of electromagnetic field is known with setup I. Indeed, this relaxation time decreases dur-
at the waveguide cut edge. Another advantage of the ing the temperature increase from 2 K to 4 K, where
use ofthe WR6 waveguideis thatthe areaofits opening it reaches its minimal value of around 1.5 ms, as shown
(1.36 mm2) is approximately 9 times smaller than the by the small solid circles in Fig. 7. On the consequent
area of the 4 mm opening of the conical waveguide, so increase of the temperature above 4 K, we see the in-
we could expect a higher density of microwaves exposed crease of τ with temperature T. The T-dependence
short
to the sample. On measurements employing rectangu- ofthe fastrelaxationparameterτ forthe big sample
short
lar waveguide, setup II configuration of irradiating the is qualitatively similar to that of the small sample, the
sample with microwave radiation is used. In this con- corresponding data is depicted with big solid circles in
figuration, the sample is placed in the geometrical cen- Fig.7. Here,the relaxationtime alsorevealsa minimum
ter of the waveguide opening, as schematically shown in ataround4K,buttheabsolutevaluesofτ forthebig
short
Fig.2b. SetupI configurationprovidesthepointofmax- sampleareapproximatelyfourtimeshigher,andbothde-
imal magnetic field for the propagating TE10 mode in a pendencescanbeperfectly scaledoneontoanother. The
rectangular waveguide [25]. τ valuesforboththesmall andbig samplesareshown
long
Inourmagnetizationstudy employingsetup II config- in Fig. 7 by the small and big solid circles, respectively;
uration, we have used two samples of different volumes: boththeserelaxationtimesliearound25-30msandthey
a big sample with a volume 680×570×170 µm3 and a are temperature independent, similar for the earlier de-
smallersamplewithavolumeof160×180×100µm3. We scribedconfigurationsetupI.Thisisnotsurprising,since
will refer these samples hereafter as big and small, cor- in both configurations of coupling of the samples to the
respondingly. The measurements were performed upon electromagnetic (EM) radiation, the surrounding of the
irradiationwithpulsedmicrowavesof118GHzandatap- sample was the same: we have used the same sample
pliedmagneticfieldof0.2T,theseconditionscorrespond holder in both cases, and the same amount of exchange
to the first transition from the ground state -10 →-9 for gaswas containedin the sample volume chamber for the
Fe8 system along the easy axis. We have found that the sample’sthermalization. This wasnotthe casewhenthe
relaxation time τ for the small sample behaves very sample was placed onto the interior wall of the massive
short
similar to the relaxation time τ of the sample used copper cavity resonator.
short
9
the magnetization relaxation measurements we choose
the frequency of 118 GHz, since the mode working at
this frequency shows the largest amplitudes of absorp-
tion peaks in the positive magnetic-field direction; so,
all the measurements presented below are taken at the
frequency of 118 GHz.
In general, we find that there is no increase of the
EM-field amplitude exposed on the sample in compar-
ison with the case, when the sample is irradiated with
microwaves without a cavity. The best way to quanti-
tatively characterize the amount of EM radiation (pho-
tons) absorbed by the sample is to consider the spin-
temperature TS growth due to the exposure of a mi-
crowave pulse. Using the mapping procedure described
above, we convert the magnetization data obtained at a
pulse lengthof0.1ms intoTS forthe differentconfigura-
tions of coupling of the sample to the microwaves. The
plot, representing spin temperature TS calculated when
FIG.9: Spintemperaturecalculated duringapulseoflength no cavity is used (setup I and setup II) and in cavity-
of 0.1 ms at nominal cryostat temperature T = 2 K for the employedconfiguration(setup IV), is depicted in Fig. 9.
different ways of irradiating sample with microwaves. Color
The plot presented in Fig. 9 shows no evidence of en-
online.
hancement of absorption of microwaves, when the reso-
nant cavity is used, although the best performing mode
is chosenfor this spin temperature comparison. Instead,
Measurements with a cavity
both waveguide-employed configurations clearly show a
betterperformance. ThesamesampleisusedforsetupII
In order to increase the amplitude of the EM radi-
and setup IV configurations,while the smaller sample is
ation exposed to the sample, we have used the cylindri-
usedinconfigurationsetup I;details ofthe sample’s size
calcoppercavityresonator,constructiondetailsofwhich
are given above.
are given above. The corresponding configuration is de-
noted as setup IV and it is shown in Fig. 2c. The use
ofamicrowavecavityassumesthe useofdifferentmodes Theuseofacavityalsosignificantlyextendstherelax-
compatible with the cavity geometry and the microwave ation of magnetization after the microwave pulse. This
frequency. The general problem of the cavity usage is slowing of magnetization restoration is also clearly visi-
that the modes of the cavity are coupled to the sample blefromthespin-temperaturedynamics,showninFig.9,
in a different way, i.e. the electromagnetic environment as comparedto the experiment without a cavitysetup II
of the sample inside the cavity is strongly dependent on on the same sample. We have summarized the relax-
the mode and the frequency. This leads not only to dif- ationparameterswhenthecavitywasusedonthegeneric
ferent amplitudes of exposed (and therefore, absorbed) plotshownin Fig.7 andcomparedthese relaxationtime
microwaves, but also to the irregularities of the absorp- values to the parameters, obtained from the magnetiza-
tion spectra. Figure 8 shows the microwave absorption tion restoration during pulsed microwave measurements
spectraofmagnetizationobtainedatseveralfrequencies, on the same sample without cavity. The relaxation time
separated just 1.5 GHz from each other; the spectra are dataobtainedduringcavity-employedexperimentarede-
obtained by the use of cw microwaves with setup IV. noted as ”τ, IV” on Fig. 7. The obtained relaxation
The sample used for the studies with the cavity has di- time τ is around 100-200ms, which is one order of mag-
mensions of 680×570×170 µm3. Unfortunately, when a nitude larger than the slow relaxation time τ2, obtained
smaller sample is used in the cavity-employedconfigura- in no-cavity experiments. We cannot unambiguously at-
tion setup IV, the sensitivity is reduceddrastically. This tribute the obtained relaxation time τ to the fast relax-
is due tothe factthatthe sample’schangeofmagnetiza- ation τ , although the data for its calculation were
short
tion is sensedby the Hall bar separatedby a copper foil, taken right after the pulse, as what was done for the
as described above. τ definition in no-cavity setups. We think that the
short
Itcanbeseenthatthemodesdiffernotonlyintheam- calculated relaxation time τ rather corresponds to the
plitude of absorption peaks, but some of them are also mixture of τ and cooling of the sample thermally
short
highly distorted (peaks instead of dips, amplitude-phase coupled to the massive copper cavity, i.e. the relaxation
mixing, asymmetry for opposite field directions). For time τ .
long
10
performed using configuration setup I at several tem-
peratures [Fig. 10a] and at several pulse length values

[Fig. 10b].
Both figures show that the long relaxation time τ
long
100 PL 10 ms (a)
remains pulse length and field independent within the
long
s) noise bandwidth and equals to approximately 25 ms;
m
there is no resonance structure evidenced in τ field-
( long
e 10 8 K 10 K dependence. This is consistent with our considerationof
m
the slowrelaxationasa coolingofthe system. Note that
i
T above approximately 0.5 T the magnetization deviation
amplitudesarereducedforshortpulsevaluesandthecor-
1
responding τ curves become very noisy in Fig. 10b.
short 2 K 4 K 6 K long
τ follows the resonance behavior and clearly pro-
short
nouncedresonantdipscanbeseeninbothfigures. Wesee
100 T = 2 K (b)
) thatoff-resonanceandin-resonancerelaxationtimeτshort
s
m long 0.5 ms valuesliewithinthe sameorderofmagnitude(around1-
( 1 ms 3 ms), while τ is one order of magnitude larger. For
long
me 10 2 ms the pulse lengthvalue of0.5 ms [Fig.10b], the difference

i 5 ms betweenthein-resonancevalueatH =0.2Tandtheoff-
T 10 ms
short resonance value of H =0.38 T is a factor of 2, while for
1 thepulseswithadurationof10msthisfactorisreduced
to 1.2.
0 0.5 1.0 1.5
Magnetization overshooting
(T)
Aswehavementionedabove,insomeofourPWexper-
FIG.10: (a)Relaxationtimeτ andτ asafunctionof imentsweobservedanovershooting ofthemagnetization
short long
appliedmagneticfieldinPWmeasurementsofpulselengthof after the microwave pulse, when the magnetization con-
10 ms and frequency f =118 GHz. The measurements were tinued to decrease even after the pulse is switched off.
performed using setup I at several temperatures. (b) Relax- This phenomenon, for example, can be clearly seen in
ation time τ and τ as a function of applied magnetic
short long the magnetization data recalculated into spin tempera-
field at temperature T = 2 K and frequency f = 118 GHz.
ture for setup II and setup IV, as shown in Fig. 9. A
The measurements were performed using setup I at several
similar effect is also evidenced in the work of Bal et al.
pulselengthvalues: 0.5ms(solidcurve),1ms(dotted-dashed
curve),2ms(dottedcurve),5ms(boldsolidcurve),and10ms [13]. Such an overshooting, however, is not observed in
(dashedcurve). Therepetitiontimeofthepulseswas200ms. magnetization measurements employing setup I, where
Color online. we have performed pulsed microwave experiments with
pulsesofthelengthof10msdownto1µs. Notethattwo
different samples are used with setup II, and the volume
Background absorption
of the small sample used in this study is approximately
6timeslargerthanthe volumeofthecrystal,usedinthe
All the measurements, described above are performed measurement with setup I.
at a resonance condition corresponding to the transition In measurements employing rectangular waveguide,
from the ground state, i.e. -10 → -9. At 118 GHz, a two configurations setup II and setup III are possible.
frequency which is used for current studies, the appro- In the former configuration, the sample is placed in the
priate applied magnetic field is always set to 0.2 T, and geometricalcenterofthewaveguideopening;inthelatter
thusthe resonantconditionisfulfilledforthe Fe8 system configuration,thesampleisplacedatthemidpointofthe
along the easy axis. Thus, only the resonant absorption shortestwallofthewaveguide,asschematicallyshownin
is detected. Nevertheless, as we have mentioned above, Fig. 2. Both configurations provide the points of maxi-
thereisalsoasignificantoff-resonant,orbackground,ab- mal magnetic field for the propagating TE10 mode in a
sorption. rectangularwaveguide [25]. In setup II configurations,a
In Fig. 10 we present the relaxation time τ and sample of any size can be used, while only a sample of
short
τ as a function of the applied magnetic field. The large enough volume can be used in setup III, because a
long
magnetic field is set in discrete steps from zero field tiny sample cannotbe properlyplacedatthe edge ofthe
to 1.5 T with an increment of 0.05 T. These relax- waveguide for partial irradiation with microwaves.
ation times are calculated from the PW measurements Inorderto findthe natureofthe overshootingofmag-