Table Of ContentEqual-Interval Splitting of Quantum Tunneling in Single-Molecule Magnets with
Identical Exchange Coupling
Yan-Rong Li, Rui-Yuan Liu, Hai-Qing Liu and Yun-Ping Wang
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics,
Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
(Dated: January 23, 2014)
The equal-intervalsplitting of quantumtunneling observed in simple-Ising-model systems of Ni4
4 (3D) and Mn3 (2D) single-molecule magnets (SMMs) is reported. The splitting is due to the
1 identical exchange coupling in the SMMs, and is simply determined by the difference between the
0 two numbersof thespin-down n↓ and spin-up n↑ molecules neighboring to thetunnelingmolecule.
2 The splitting may be presented as (n↓ −n↑)JS/gµ0µB, and the number of the splittings follows
n n+1wheren=n↓+n↑ isthecoordinationnumber. Besides,sincethequantumtunnelingisheavily
dependenton local spin environment,themanipulation of quantumtunnelingmay become feasible
a
for this kind of system, which may shed new light on novel applications of SMMs.
J
2
PACSnumbers: 75.45.+j,75.50.Xx,05.50.+q75.30.Et
2
]
Single-molecule magnets (SMMs) have been used as Mn SMMsarecrystalswithIECandthemodelsystems
l 3
al model systems to study the interface between classical of simple Ising model [17]. We have observed the equal-
h and quantum behaviors, and are considered to be the interval splitting of quantum tunneling induced by IEC
- most promising systems for the applications in quantum in these two systems by ac susceptibility and hysteresis
s
e computing, high-density information storage and mag- loop measurements.
m netic refrigeration [1–5] due to the quantum tunneling Considering the low blocking temperature, we studied
. of magnetization (QTM) observed in these systems [6– quantumtunnelingeffects ofNi SMMbyacsusceptibil-
t 4
a 9]. Recentresearchesin the impact ofintermolecularex- itymeasurements,withahome-madecompensationmea-
m changecouplingsupontheQTMhavefocusedonwhether surementsetup[18]. Fig.1hasdemonstratedthetemper-
- the exchange coupling may change the quantum tunnel- ature dependence of the quantum tunneling behavior in
d ing in SMMs. SMM dimer system is reported to have Ni SMM. Apparently, the peak at zero field disappears
n different quantum behavior from that of the individual at40.75K and 0.5K, which consists with the missing step
o
SMMs,due to theintermolecularexchangecouplingsbe- at zero field in magnetization hysteresis loops at 40mK
c
[ tween the two components [10, 11]. It is also reported [13]. As a result of different orientations, the step posi-
that, in the SMM dimer with 3D network of exchange- tions are different from those mentioned in Ref[13]. We
1 couplings,theQTMisnotsuppressed[12]. Inthisletter, measured the quantum tunnelings at different orienta-
v
we demonstrate that, for the SMMs with identical ex- tions and found the resonant fields along the easy axis
4
change coupling(IEC), the quantum tunneling behavior
6
5 ismuchsimplerandtheQTMmightbeconvenientlyma-
5 nipulated by controlling of the magnetization.
.
1 In the following, we report a unique quantum tun- 0) 1) 2) 3) 4) 1K
0 neling effect observed in the single-molecule magnets 4, 3, 2, 1, 0,
( ( ( ( (
4 of [Ni(hmp)(CH CH OH)Cl] (hereafter Ni ) [13, 14]
3 2 4 4
1
and [Mn O(Et-sao) (MeOH) (ClO )] (hereafter Mn )
v: [15, 16]. 3Ni4 SMM 3is a cryst3al with4 3D network of ex3- a.u.) 0.85K
i change coupling, in which each molecule is coupled with (
X
fourneighboringmoleculesbyCl···Clcontact(whichcon-
ar tributes to the exchange coupling) forming a diamond- 0.75K
like lattice. Ni crystal has S symmetry, which ensure
4 4
that the four exchange couplings between each molecule
andits fourneighboringmoleculesareidenticalthrough-
0.5K
out the crystal. Mn SMM is a crystal with 2D net-
3
work of exchange coupling, in which each molecule is -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
coupled with three neighboring molecules by hydrogen FIG. 1: (Color online). Field dHezp (Te)ndenceof susceptibility χ′
bonds(whichcontributestotheexchangecoupling)inab from −0.3T to 0.3T at different temperatures measured on
plain, forming a honeycomb-like structure viewed down Ni4 single crystal, with the sweeping rate of 0.001T/s. The
along the c-axis. Mn crystal has C symmetry, which quantumtunnelingpeaksmarkedbyblackdottedlinesorigin
3 3
ensure that the three exchange couplings between each from tunnelings between |−4i and |4i spin state, and the
molecule and its three neighboring molecules are identi- labelednumbersetin()besideseachdottedlineindicatesthe
calthroughout the crystal. We notice that both Ni and local spin environment(n↓,n↑) of thetunneling molecules.
4
2
of the sample are −0.21T −0.11T, 0T, 0.11T, 0.21T as ofthemarenotpopulatedatour measurementtempera-
shown in Fig.1. It is seen that the tunneling peaks ap- tures). At negative saturated field, all the molecules ini-
pearwithequalinterval. Theshiftofthetunnelingpeaks tiallyoccupy|−4iinthesameLSE(4,0)showninFig.4a
fromhigherto lowerfield withthe increasingT is due to (left). Accordingtoequation(1),|−4iand|4ispinstates
theenhancementofthe effectofthermalactivationupon in the LSE (4,0) are degenerate when the field reaches
tunneling [19, 20]. 4JS/gµ µ , therefore those molecules which occupy the
0 B
Thehigherblockingtemperatureallowsustostudythe |−4ispinstate inthe LSE(4,0)(Fig.4a)havethe same
hysteresis loops above 1.6K for Mn3 SMM. Fig.2 shows probability to undergo tunneling at 4JS/gµ0µB, leading
the typical step-like hysteresis loops of Mn3 SMM at to the resonant tunneling peaks at −0.21T as seen in
different temperatures. The blocking temperature esti- Fig.1. Following this resonant quantum tunneling, some
mated from ZFC (zero field cooling) and FC (field cool- molecules will occupy |4i spin state, and the LSE of the
ing) curves shown in the inset is around 3K. The sweep- molecules willnot be identicalany more. When the field
rate-dependent magnetization curves at 1.6K are shown reaches2JS/gµ0µB (correspondingto −0.11Tasseenin
in Fig.3, with only a dM/dH curve at the sweeping rate Fig.1), the resonant tunneling takes place from |−4i to
of 0.0005T/s presented for simplicity. A series of quan- |4i spin state in the LSE (3, 1) (Fig.4b). As a matter of
tum tunneling peaks with an equal interval of 0.36T are fact, at zero field the tunneling of the molecules in the
observed in the dM/dH curves, which is similar to those LSE (2, 2) (Fig.4c) will change neither Zeeman energy
observed in Ni SMM. nor the exchange interaction energy, which gives rise to
4
With IEC taken into account, the molecules are not themacroscopicquantumtunnelingobservedatzerofield
isolated,andthe spinHamiltonianof eachmolecule may at relatively higher temperatures shown in Fig.1. Simi-
be presented as: larly, there are quantum tunnelings taking placing from
|−4ito|4ispinstatewiththeLSE(1,3)at−2JS/gµ µ
0 B
n , and from |−4i to |4i spin state with the LSE (0, 4)
Hˆ =−DSˆz2+gµ0µBSˆzHz −XJSˆzSˆiz, (1) at −4JS/gµ0µB. The exchange interaction constant J
i=1 is calculated to be −0.019K according to the splitting
interval, which is close to the simulation value −0.02K
where D is the axial anisotropy constant, n is coordina-
obtainedfromthe experimentalAFMtransitiontemper-
tion number, J is the exchange interaction constant, Sˆ
z ature of T = 0.91K [21]. At temperatures obviously
and Sˆ are the easy-axis spin operators of the molecule N
iz belowT ,the spinsofthe molecules willbe anti-parallel
N
and its ith exchange-coupled neighboring molecule. For
to its neighbors, i.e. the molecules are in the LSE (0, 4)
Ni , S =4, D =0.86K, g =212 [13, 14]; while for Mn ,
4 · 3 and (4, 0) instead of the LSE (2, 2), which causes the
S =6, D =0.98K, g =206 [16].
· missing of quantum tunneling at zero field at T ≤0.75K
In Ni SMM, every Ni molecule has four AFM
4 4
exchange-coupled neighboring molecules, and hence for
each molecule there are five different kinds of local spin
environment (LSE), which may be labeled by (n ,n ),
↓ ↑
where n↓ and n↑ represent the number of the neighbor- 1.0 0) 1) 2) 3)
eainxngcditmSedozles=pcui4nle(sshtawetrheesiacfahtreeorcn|c4ouit)pcysopnSisnzids=etrae−tde4sher(erhesep,rebecaetfcitaveuerlsy|e−m(To4hsie)t 00..68 0000....000000005210T5T5TT//ss//ss (3, (2, (1, (3, 0) (2, 1)(0, (1, 2) (0, 3)
0.4 0.0002T/s u.)
a.
1.0 11..68KK M/Ms00..02 -0.54T (3, 0) 1.26T(2, 1) 1.62T(1, 2) 1.98(0, 3)T dM/dH (
0.5 22K.2K -0.2 (3, 0) (2, 1)
-0.4
Ms -0.6
M/0.0 16 -0.8 -0.18T 0.18T 0.54T 0.90T
14 FC
-0.5 -311()M 10emu46802 TB=3K ZFC -1.0-1.5 -1.0 -0.5 0.0 0Hz0 (.5T) 1.0 1.5 2.0
2 FIG. 3: (Color online). Sweep-rate-dependent magnetization
02 4 T 6(K) 8 10 (M/Ms) versus applied magnetic field µ0Hz (from −1.5T to
-1.0 2.2T) at1.6KmeasuredonMn3 singlecrystal. dM/dHcurve
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
with sweeping rate of 0.0005T/s is given. The quantumtun-
0Hz (T)
FIG.2: (Coloronline). Magnetization (M/Ms) ofMn3 single nelingpeaksmarkedbythedottedlinesofthesamecolorbe-
crystal versus applied magnetic field with the sweeping rate longtothesame tunnelingof|mii→|mfi,thelabeled num-
of 0.003T/s at different temperatures. The inset shows ZFC ber set in () besides each dotted line indicates the local spin
and FC curves. environment (n↓,n↑) of thetunneling molecules.
3
as seen in Fig.1. However, in the vicinity of the transi- Accordingtoequation(2),whenD >n|J|S and|H |<
z
tiontemperature,somemoleculesarestillinthe LSE(2, (D−n|J|S)/gµ µ , any quantum tunneling with l 6= 0
0 B
2) due to the thermal fluctuation, thus there is still an are not allowed; while according to equation(1), when
evidenceofresonantquantumtunnelingat0.85Katzero the first excitation energy of a molecule D(2S −1) ≫
field shown in Fig.1. kT,almostallmoleculeswilloccupythetwogroundspin
states of |±Si. Therefore, under the above conditions,
(a)
(4,0) equation(1) may be simplified as
n
Hˆ =gµ0µBSˆzHz −XJSˆzSˆiz, (3)
i=1
(b) (3,1)
whichis just the Hamiltonianof simple Ising model [17].
For both Ni and Mn , D > n|J|S, thus Ni and Mn
4 3 4 3
SMMs are good model systems of simple Ising model at
low temperature and low field, which are important for
the studies of quantum tunneling behavior and related
(c)
(2,2)
applications.
(a) (b)
FIG.4: (Coloronline). Sketchmapsofthreepairsofspincon-
figurationswith differentLSE(n↓,n↑) inNi4 SMM,in corre-
spondencetothetunnelingsoccurringat−0.21T,−0.11Tand
0TinFig.1,respectively. Otherequivalentspinconfigurations (3, 0) (2, 1)
are not listed here for simplicity. The tunneling molecule is
markedinblackwithblackarrowindicatingitsspinstate,its (c) (d)
fourneighborsaremarkedingray,withgreenandredarrows
indicating spin-upand spin-down state respectively, the blue
lines between molecules represents theexchange couplings.
Mn SMM displays a finer quantum tunneling behav- (1, 2) (0, 3)
3
ior than Ni SMM. Every Mn molecule has three AFM
4 3
FIG. 5: (Color online). Sketch maps of four spin configura-
exchange-coupled neighboring molecules, and hence for
each molecule there are four different kinds of local spin tionswith differentLSE(n↓,n↑)in Mn3 SMM,otherequiva-
lentspinconfigurationsarenotlistedhereforsimplicity. The
environment as shown in Fig.5, labeled as (3, 0), (2, 1),
tunneling molecule is marked in black, its three neighboring
(1,2),(0,3)respectively,thus,therearequantumtunnel-
molecules are marked in grey, with green and red arrows in-
ings occurring at 3JS/gµ0µB, JS/gµ0µB, −JS/gµ0µB, dicating spin-upandspin-down state respectively. Theblack
−3JS/gµ0µB from|−6ito|6ispinstate,whichiscorre- lines between molecules represent the exchangecouplings.
sponding to the four tunneling peaks marked by the red
dotted lines shown in Fig.3.
Since the intermolecular exchange couplings are iden-
InbothMn andNi SMMs,withthepresenceofIEC,
3 4 tical in the system, the magnitude T of a tunneling may
the tunneling betweentwo groundspinstates of|±Siis be simply factorized into intermolecular contribution
splitted by equal-interval field of 2|J|S/gµ µ . Gener-
0 B N and intramolecular contribution P ,
ally, according to equation(1), the tunneling from |−Si (n↓,n↑) |mii→|mfi
to |S−liis splitted by the same equal-intervalfield, and T =αN P , (4)
(n↓,n↑) |mii→|mfi
the splitted tunneling field may be simply expressed as
where N is the number of molecules with the LSE
(n↓,n↑)
H =lD/gµ µ +(n −n )JS/gµ µ . (2) (n ,n ), P is the tunneling probability of the
z 0 B ↓ ↑ 0 B ↓ ↑ |mii→|mfi
moleculefromthespinstate|m ito|m i,andαisacon-
i f
The first term comes from the internal spin states in stant. N stronglydependsonthemagnetizationM
(n↓,n↑)
each molecule, and the second term is of the tunnel- andmay be easily modulated, while P is deter-
|mii→|mfi
ing splitting induced by IEC. The splitting is simply de- mined by the tunneling barrier between |m i and |m i
i f
termined by the difference between the two numbers of inside molecules and is hardly to be controlled. There-
thespin-down(n↓)andspin-up(n↑)moleculesneighbor- fore, for SMMs-with-IEC, with the dependence of T on
ingto the tunnelingmolecule. Accordingto equation(2), N , the manipulation of quantum tunneling should
(n↓,n↑)
the number of splittings equals the number of different be rather simple.
kinds of (n↓, n↑) LSEs, and hence may be expressed as The quantum tunnelings from the same initial states
n+1 =n+1 by combinatorics, where n=n +n . |m i to the same final states |m i but with different
(cid:0) 1 (cid:1) ↓ ↑ i f
4
LSEs are referred to as a tunneling set. The five tun- the contribution of the tunneling from |−6i to |5i spin
neling peaks of Ni SMM in Fig.1, belong to the same state with the LSE (3, 0), eventually.
4
setof|−4i→|4iandhasthesameP , thusthe Due to the strong dependency of a tunneling on the
|mii→|mfi
intensities of the five peaks is proportional to N , N based on equation(4), the subsequent quantum
(n↓,n↑) (n↓,n↑)
which means that N may be monitored by macro- tunnelingheavilydependsonthetheprecedingquantum
(n↓,n↑)
scopic measurements of the tunneling peaks. For Mn tunnelings in SMMs-with-IEC. As shown in Fig.3, tun-
3
SMM, the AFM exchange coupling constant J is calcu- neling at −0.54T (from |−6i to |6i spin state with the
lated to be J = −0.041K according to the field interval LSE (3, 0)) is inherited by tunneling at −0.18T (from
of the |− 6i → |6i tunneling set(Fig.3). However the |−6i to |6i spin state with the LSE (2, 1)), the tunnel-
axial anisotropy constant D = 0.98K [16] of Mn SMM ingsat−0.54T,−0.18Tarefurthercarriedonbythenext
3
happens to be close to 4|J|S, which results in the over- tunneling,andtheprocesscontinuesastheLSEchanges.
lap of two adjacent tunneling sets demonstrated by the In fact, the history dependence is not prominent for Ni
4
overlapped dotted lines shown in Fig.3. The tunneling SMM, due to that the measurements were performed at
steps at 0.18T and 0.54T are the combinations of the temperaturesmuchhigherthanitsblockingtemperature,
tunnelings from |−6i to |6i spin state with the LSEs whilethermalactivatedeffectruinthememoryofhistory.
(1, 2) and (0, 3) (marked by red dotted lines) and quan- Apparently, the subsequent quantum tunneling is more
tum tunnelings from | − 6i to |5i spin state with the heavily dependent on the preceding quantum tunnelings
LSEs (3, 0) and (2, 1) (marked by blue dotted lines) re- in SMMs-with-IEC when the thermal activated effect is
spectively. Similarly, all subsequent tunneling steps are severelysuppressedasthetemperaturedropsadequately.
combinations of quantum tunnelings in different tunnel- Thisindicatesanewwayformanipulatingquantumtun-
ing sets with different local spin environments. It may neling.
be worth a mention that the tunnelings are expected to In summary, we performed detailed ac susceptibility
occur at 1.62T and 1.98T (marked by green and orange and hysteresis loop measurements on Ni and Mn sin-
4 3
dottedlines)atlowertemperaturesaswell,althoughnot gle crystals, respectively, and have observed the equal-
observed in these curves. interval splitting of quantum tunneling in both sys-
Of the overlapped tunnelings mentioned above, due tems, the splitting of quantum tunneling is presented
to the dependence of tunneling on the local spin en- by (n −n )JS/gµ µ ; and the number of splitting fol-
↓ ↑ 0 B
vironment, the contribution of the individual tunneling lows n+1, where n =n +n is the coordination num-
↓ ↑
changes as the field sweeping rate varies. For example, ber. Since the splitting is induced by the IEC between
thetunnelingstepat0.18Tisthecombinationoftunnel- the molecules, the rules should be universally applica-
ing from |−6i to |6i spin state with the LSE (1, 2) and ble to all single-molecule magnets with IEC. Besides, it
tunneling from |−6i to |5i spin state with the LSE (3, is demonstratedthat, the manipulation ofquantum tun-
0), therefore the tunneling magnitude is determined by nelingmaybecomefeasibleforthiskindofsystem,which
N P +N P ,whereN andN may shed new light on novel applications of SMMs.
(3,0) |−6i→|5i (1,2) |−6i→|6i (3,0) (1,2)
strongly depends on the magnetization M. As shown in We thank Prof. Dianlin Zhang, Lu Yu, and Li Lu
Fig.3,forthetunnelingat0.18T,M/M isincreasingwith for helpful discussions. We also thank Shaokui Su for
s
thedecreasingoffieldsweepingrate,whichsuggeststhat experimental assistance. This work was supported by
N is increasing while N is decreasing, and hence the National Key Basic Research Program of China
(1,2) (3,0)
the contribution of the tunneling from |−6i to |6i spin (No.2011CB921702)andtheNaturalScienceFoundation
state with the LSE (1, 2) is taking the dominance from of China (No.11104331).
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