Quantum Tunneling ofthe Magnetization inHigh Spin Molecules

Zaher Salman

Advisor: Dr. Amit KerenCollaboration: P. Mendels, M. Verdaguer, V. Marvaud, A.

SchuillierThis research was supported by AFIRST

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Introduction

• High Spin Molecules (HSM)consist of coupled magneticions.

• At low temperatures HSMbehave like giant spins.

• They crystallize in a latticewhere they are magneticallyseparated.

• HSM can become the small-est magnetic writing unit.

Harris, Physics World (1999)

1950 1960 1970 1980 1990 2000 2010103104105106107108109

10101011101210131014101510161017101810191020

QTM

Magnetic cores Disk file Magnetic bubble Thin fil Optical disk IBM

Ato

ms

per b

it

year

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Most Studied HSM

• Fe8 with ground spin stateS = 10.

• 6 Fe (S = 5/2) ions coupledanti-ferromagneticly to 2 Feions.

• Mn12 with ground spin stateS = 10.

• 4 Mn4+ (S = 3/2) ions cou-pled anti-ferromagneticly to8 Mn3+ (S = 2) ions.

Fe8

Mn12

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Quantum Tunneling of the Magnetization (QTM)

• QTM was first observedin Mn12 and then in Fe8,through steps at regular in-tervals of magnetic field inthe hysteresis loop.

• QTM provides a signaturequantum mechanical behav-ior in macroscopic systems.

Wernsdorfer, QMAG (1999)

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More QTM Experiments

Fominaya, PRL 79, 1126 (1997)

• The heat capacity of Mn12

single crystal shows peaks atregular intervals of magneticfield.

Tejada, Europhys. Lett. 35, 301 (1996)

• The relaxation rate of themagnetization decreasesdrastically at regular inter-vals of magnetic field.

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A Simple Model for QTM

• At low T the Hamiltonian for asingle molecule is

H = −DS2z − gµBHzSz

• Transition is due to quantum tun-neling only.

• Tunneling occurs when two stateson both sides have the same energy.

• Matching of energy levels isachieved when Hz = nD

gµB

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µSR in Fe8

• All measurements were per-formed at 40 mK.

• We apply H = 2 T for 30minutes.

• We sweep the field H →Hi → −50 Gauss.

• All measurements are per-formed at H = −50 Gauss.

• The asymmetry is differentonly if different matchingfields Hmatch = 0, 2.1, 4.2, · · ·kG are crossed.

0 1 2 3 4

-0.10.00.10.2

(a)

Time [µµµµsec]

Asy

mm

etry

2T →→→→ 50G →→→→ -50G

-0.10.00.10.2

(b)

2T →→→→ -1.5kG →→→→ -50G

-0.10.00.10.2

(c)

2T →→→→ -2.0kG →→→→ -50G

-0.10.00.10.2

(d)

2T →→→→ -3.5kG →→→→ -50G

-0.10.00.10.2

(e)

2T →→→→ -5.0kG →→→→ -50G

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Problem

• The Hamiltonian of HSM is

H = −DS2z − gµBHzSz

• H commutes with Sz

[H, Sz] = 0

therefore Sz should be conserved.

What induces tunneling ??

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Theory of QTM

• In order to have tunneling additional terms H⊥ should beadded to the spin Hamiltonian.

• Possibilities are:

1. High order spin terms and crystal field.

2. Transverse field.

3. Spin-phonon interaction.

4. Dynamic nuclear spins and dipolar interaction.

• We are interested in the dependence of tunneling on the spinvalue S.

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Proposed Mechanisms

• Crystal Field: H⊥ = − gµB2∑Nn=1 hn

(

Sn+ + Sn−)

τ−1 = DShπ

(

gµBhN (Sh)N−2

2D

)2S/N

van Hemmen, Europhys. Lett. 1, 481 (1986).

• Transverse Field: H⊥ = −gµBHxSx

τ−1 = 2Dπ[(2m−1)!]2h

(S+m)!(S−m)!

(

gµBHx2D

)2m

Chudnovsky, PRL 79, 4469 (1997).

• Spin-Phonon: H⊥ ∝ (SkSl + SlSk)

τ−1 = 12π2h4c5ρ

| < S|H⊥| − S > |2(HzS)3

Hartmann Boutron, PRL 75, 537 (1995).

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Dynamic Nuclear Spins and Dipolar Interaction

• Dipolar fields bring the spinstates of the molecules out ofresonance.

• Rapidly fluctuating hyper-fine fields bring moleculesinitially to resonance.

• Tunneling changes the dipo-lar fields and brings moremolecules to resonance.

• The tunneling rate in thiscase is

τ−1 ∝ ∆20T2|ES−E−S |

EDProkofev, PRL 80,5794 (1998).

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Strategy

• The Hamiltonian of anisotropic HSM is

H = −DS2z − gµBHzSz +H⊥

• For isotropic HSM D −→ 0

H = −gµBHzSz +H⊥

We can probe H⊥ directly!

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Isotropic High Spin Molecules

CrCu6

S = 9/2JCr−Cu = 77KSCu = 1/2

CrNi6S = 15/2

JCr−Ni = 24KSNi = 1

CrMn6

S = 27/2JCr−Mn = −11KSMn = 5/2

SCr = 3/2

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Magnetic Properties

• At high H and T = 2 K the magne-tization per molecule saturates giv-ing the value of S.

• M vs. H follows the Brillouin func-tion of the corresponding S.

• At low H and low T the sus-ceptibility saturates and shows noanisotropy.

• The susceptibility follows that ex-pected from the Hamiltonian

H = −J6∑

i=1

~S0 · ~Si − gµB ~H ·6∑

i=0

~Si

0 1 2 3 4 50

3

6

9

12

15

18

21

24

27

30

CrCu6

CrNi6

CrMn6

M [

µ B ]

H [Tesla]

10 1001

10

CrCu6

CrNi6

CrMn6

χ χχχT [a

.u.]

T [K]

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Calculation of the Susceptibility

• We diagonalize the Hamilto-nian H

• We find the eigenstates|S,m > and eigenvaluesES,m.

• The susceptibility is

χ =∑

|S,m>

me−ES,mT

ZH0 1.5 3 4.5 6 7.5

−10

−8

−6

−4

−2

0

2

4

6

8

10

12

S

E S

,m/J

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Spin Lattice Relaxation Time

• It is the time that takes a probing spin to reach equilibrium

H0

H0

System and probe in Equilibrium.

P0

Probe Spin ofPolarization

t 0 ft

1 f 0T = t −t

• In µSR P0 = 1 and Pf = 0 while in NMR P0 = 0 and Pf = 1.

• Only fluctuating fields ~B(t) contribute.

• Only transverse fluctuations B⊥ contribute.

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T Dependence of the µ+ Polarization

• At H = 0 the muon polar-ization relaxation increaseswith decreasing temperaturedown to 5 K and then satu-rates.

• At H = 2 T and tempera-tures lower than ∼ 17 K therelaxation decreases with de-creasing T .

0 2 4 6

0.4

0.6

0.8

1.0

H=0-100G T[K]=2 275 4217 200

(a)P z(t

,H)

Time [µµµµsec]0 2 4 6

H=2T T[K]=1 105 17

(b)

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µSR and NMR Results

0.1 1 10 1001E-3

0.01

0.1

1

10

NMR - H[kG]= 10 20

(a)

1/T

1µ µµµ and

1/T

1H

[ µ µµµse

c-1]

Temperature[K]0.1 1 10 100

µµµµSR - H [kG] = 0-0.1 5 1 10 2 20

(b)

0.1 1 10 100

S=27/2S=15/2S=9/2

(c)

1. No T dependence

at H → 0, T → 0.

2. Scale of T varies.

3. H dependence

varies.

4. Different behavior

in different H.

• The spin lattice relaxation rate measured by µSR and NMR atthe same external field can be scaled.

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The Spin Lattice Relaxation

• In spin lattice relaxation theory one assumes a fluctuating localfield ~B(t) experienced by a local probe (muon or nucleus) ofspin ~I.

• The spin lattice relaxation time T1 in this case follows

1T1

=γ2

2

∫ ∞

−∞dt 〈B⊥(0)B⊥(t)〉 exp(iγHt).

• The correlation time τ and mean square of the transverse fielddistribution at the probe site in frequency units ∆2 are definedby

γ2 〈B⊥(t)B⊥(0)〉 = ∆2 exp (−t/τ) .

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Field-Field Correlation Time

• The spin lattice relaxationrate is

1T1

=∆τ

ω2τ2 + 1

where in our system ω =gµBH.

T1 =ω2τ2 + 1

∆τ

• The value of T1(T → 0) de-pend linearly on H2.

0 1 2 3 4 50.0

0.2

0.4

0.6

0.8

4

8

12

16

20

CrCu6

CrNi6

CrMn6

T 1µ µµµ (T

=0) [

µ µµµse

c]

H2 [kG2]

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Field-Field Correlation Time

• From the linear fit ofT1(T → 0) vs. H2 we have

τint [nsec] ∆ [MHz]

CrCu6 0.034(4) 4.9(9)

CrNi6 0.053(4) 26(2)

CrMn6 0.044(4) 38(2)

• The correlation time at alltemperatures follows

τ(T ) =(gµB)2

3kBS(S + 1)

T1(T )Tχ(T )∆2.

1 10 100

0.000

0.005

0.048

CrCu6

CrNi6

CrMn6

τ τττ [ns

ec]

Temperature [K]

1 10 10020406080

100120140160180200

4.5x5.5

<S2 >(

T)

13.5x14.5

7.5x8.5

T [K]

• The correlation time depends weekly on S at T → 0.

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Calculation of T1

• Assuming a coupling A~I · ~S between the probe ~I and themolecular spin ~S, the spin lattice relaxation rate is

1T1

=A2

2

∫ ∞

−∞〈S−(t)S+(0)〉 eiωtdt

• For the spin states of H we obtain

1T1

=A2

2Z∑

|S,m>

(S(S + 1)−m(m+ 1))

(

τS,me−ES,mT

1 + ω′2τ2S,m

)

where τS,m is the lifetime of the level |S,m > andω′ = (γ − gµB/h)H ' −gµBH/h.

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1/T1 Due to Spin-Phonon Interaction

• Assuming that the finite life-time of the levels is due to spin-phonon interaction

τ−1sp =

C(ES,m − ES′,m′ )3

exp[

(ES,m − ES′,m′ )/T]

− 1

• The behavior is similar to theexperimental data at high T butdiffers at T → 0.

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Additional Constant Fluctuations

• To reproduce 1/T1 at low temperatures and low fields weassume 1

τS,m= 1

τsp+ 1

τintwhere τint is constant.

• With this we improve the agreement.

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Discussion

• The correlation time τint depends weekly on S, ruling outcrystal field and dipolar interactions.

• We have seen that spin-phonon interaction cannot account forthe finite 1/T1 at low T .

• We are left with fluctuating hyperfine interaction~Bmol = a~i(t) · ~S.

Consider the Bloch equation

d~S

dt= gµB

[

a~i(t)× ~S]

which is independent of S.

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Conclusion

• The spin lattice relaxation in HSM is temperature independentat low temperatures.

• The spin-spin correlation time at low temperatures dependsweekly on the molecular spin S.

• The molecular spin dynamics at high temperature is driven byspin phonon interaction, while dynamically fluctuatinghyperfine fields induce the molecular spin dynamics at very lowtemperatures.

H⊥ ∝~i · ~S

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Landau-Zenner Tunneling

• Landau Zenner model isused for experiments withswept external magneticfield.• The tunneling probability is

P = 1−exp

[

−π∆

4hgµBS(dH/dT )

]

where ∆ is the gap at thelevel crossing (tunnel split-ting).

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The µSR Technique

• The muons are 100%polarized.

• The positrons areemitted parallel tothe muons polariza-tion.

• Histograms ofpositron countsvs. time are collectedin the F/B counters.

B

B FC

ounts

Counts

Tim

e

Ase

mm

etry

Time

Random

T12π/ω

Ordered

Tim

e HL

Beam

zµ

e

+

+

• The Asymmetry = F−BF+B is proportional to the polarization.

• Zero field, Longitudinal or transverse field can be applied.Back

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µSR in Fe8

• We fit the asymmetry to

A(t) = A10 sin(ω10t)e−λ10t +A±10 sin(ω±10t)e−λ±10t

• A10, ω10 and λ10 representsthe fraction of µ+ near m =+10 state.

• A±10, ω±10 and λ±10 repre-sent the fraction of µ+ nearm = ±10 state.

-5 -4 -3 -2 -1 07.07.58.08.59.09.5

10.0(b)

ω ωωω ± ±±± 1

0 [M

Hz]

H i [kG]

0.06

0.08

0.10

0.12

0.14 (a)

A 10

A ±±±± 10

Am

plitu

de

• Steps in the value of ω±10, A10 and A±10 coincide with thematching field values.

Back

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Muon Spin Relaxation in a Static Field

Time scale Recovery in Decoupling in high

of relaxation zero field longitudinal field

t−1 ' γ√

< B2loc >

Bloc

Pµ

Bloc

PµBloc

Pµ

H

Btot

P

Bloc

µ

limt→∞ Pz(t) = 13 limH→∞ Pz(t) = 1

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H Dependence of the µ+ Polarization

• The µSR results in CrNi6 atT = 50 mK show that

1. There is no recovery, un-like the static local fieldcase.

2. The relaxation time scaleis 1 µsec, and therefore√

B2loc should be ∼ 10 G.

3. Decoupling should occurat H ∼ 100 G, but evenat 5 kG there is no decou-pling.

0 1 2 3 4 5 60.00

0.20

0.40

0.60

0.80

1.00

T=0.05 HL=0G 2kG 5kG

P z(t,H

)

Time (µµµµsec)

We conclude that even at 50 mK

the CrNi6 spins are dynamically

fluctuating.

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Principle of NMR

I

H0

I I

H1

Ballerina watch-ing spin rotating.

Ballerina andspin are rotating.From her pointof view the spinis fixed so thereis no field.

A rotating fieldH1 is applied.Ballerina sees afixed field so thespin is rotatingaround it.

Back

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Principles of NMR

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1H

0H

OscilloscopeTra

nsm

itter

Coil

Sample

Back

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The Polarization in Isotropic HSM

• In these HSM the muon could occupy many different sites inthe sample as a result one must average over ∆.

• The polarization of a local probe, in the fast fluctuationτ∆� 1, is given by

P (H, t) = (P0 − P∞) e[−t/T1]β + P∞

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The Spin Dependence of the Tunneling Rate

Interaction Spin dependence of the

tunneling rate τ−1

High order spin terms S2 or higher

Spin-phonon S3

Static transverse field higher than S2

Dynamic hyperfine and dipolar 1/S

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