Microscopic origin of the adiabatic change of the

magnetization

Hans De Raedt Applied Physics-Computational Physics, Materials Science Centre, University of

Groningen,The Netherlandsderaedt@phys.rug.nl ; http://www.compphys.org

Material for this talk taken from work done in collaboration with S. Miyashita (Tokyo), K. Michielsen (Groningen), V.V. Dobrovitski & B.N. Harmon (Ames Lab) M.I. Katsnelson (Uppsula)

Outline Introduction Microscopic mechanism(s) for energy-level

repulsions V15 (and Mn12)

Coherence of oscillations of the magnetization

Estimating the decoherence (T2) time NMR spin echo experiments on 29Si in Si powders Computer simulation

Introduction At very low temperatures, the magnetization-

versus applied field curve of single-molecule magnets such as Mn12 and V15 exhibit unusual features.

V15

K6[V15As6O42(H2O)].8H2OMn12(CH3C00)16(H2O)4O12].2CH3C00H.4H2O

Mn12

Introduction The position and size

of the steps contains information about the details of the energy-level structure of the interacting magnetic moments in the molecule.

B. Barbara et al., JMMM 200 (1999) 167

Mn12

Introduction The position of the

steps (values of the applied magnetic field) can often be described by very simple models of one (Mn12) or three (V15) spins.

I. Chiorescu et al., JMMM 221 (2000) 103

V15

Introduction At very low temperatures and for very slow

sweeps of the applied field, the change of the magnetization at a step is related to the energy splitting of the quantum states with different magnetization.

Landau-Zener-Stückelberg (LZS) transition (Miyashita 1996).E

hE

Introduction Theoretical understanding of the magnetization

dynamics of the single-molecule magnets requires detailed knowledge of the energy levels, their degeneracy and whether levels cross or repel at particular values of the applied magnetic field.

What are the microscopically relevant interactions? The calculation of energy levels/splittings of these

systems is a very challenging computational problem. Large Hilbert space, (nearly) degenerate eigenvalues,

large differences in energy scales, … Lanczos + full orthogonalization Chebyshev-polynomial-based projector method Full exact diagonalization

Modeling magnetic properties of single-molecule magnets Electronic structure: Ab initio, LSDA+U,…

Yields estimates for exchange interactions … Many-spin models

Not “many-body” but still fairly large number of degrees of freedom

Hilbert space dimension: 10 000 – 100 000 000 Single-spin models

Hilbert space dimension: 2 – 100

15xS = 1/2 frustrated antiferromagnet:

Hilbert space dimension = 32768 Full diagonalization impossible!

V15 molecule

V1

V2

V2

1413

12

11 10

15

9 7

8

1

23

4

56

J2J1

J4

J5

J6

J3

J

J’

J”V1

xy

z

V3

Dzyaloshinsky-Moriya interaction

,

,

,,

i j i j ii j i

i j i ji j

H J S S h S

D S S

V15 model parameters (1) D.W. Boukhvalov et al, JAP,

93 (2002) 7080 All J’s from LDA+U

calculation

experiment

V15 model parameters (2) N.P. Konstantinidis and D.

Coffey, PRB 66 (2001) 014408

J = -800K, J1= J’ = -225K, J2

= J” = -350K DM interaction: Dx

1,2 = Dy1,2

= Dz1,2 = 40K

Use rotational symmetry of the hexagons

5,6 1,2

5,6 1,2(240 )x x

y y

D DR

D D

3,4 1,2

3,4 1,2(120 )x x

y y

D DR

D D

V15 model parameters (3) I. Rudra et al, J.Phys.C 13

(2001) 11717 J = -800K, J1= J’ = -54.4K, J2

= J” = -160K DM interaction: Dx

1,2 = Dy1,2

= Dz1,2 = 40K

Use rotational symmetry of the hexagons

3,4 1,2

3,4 1,2(120 )x x

y y

D DR

D D

5,6 1,2

5,6 1,2(240 )x x

y y

D DR

D D

V15

Energy levels of lowest eight states may be represented by a S=1/2 model of spins on the triangle

Mz=1/23/2Mz=-1/21/2

I. Chiorescu et al., JMMM 221 (2000) 103

V15 : Adiabatic transitions

At h = 0 there is a level CROSSING.

Near h = 0 there is a level REPULSION

No LZS transition from Mz

= 1/2 to 3/2, in disagreement with experiment

I. Chiorescu et al., JMMM 221 (2000) 103

?

V15: Energy level diagram near h=0

(0,0, )h h

(0,0, )h h

( ,0,0)h h

( ,0, ) / 2h h h

Level repulsion is here, not at h=0!

No level repulsions at all

Three-spin model for V15

Dzyaloshinsky-Moriya interaction + symmetry of the triangle

-8

-6

-4

-2

0

2

4

6

8

10

-4 -3 -2 -1 0 1 2 3 4

E [

K]

h [T]

-1.5

-1

-0.5

0

0.5

1

1.5

-4 -3 -2 -1 0 1 2 3 4

<M

^z>

h

1 2 2 3 3 1 1 2 3 ,( ) ( ) i j i ji j

H J S S S S S S h S S S D S S

NO adiabatic transition at all

H-sweep

Three-spin model for V15: dependence on H-direction

(0,0, )h h

( ,0,0)h h

( ,0, ) / 2h h h

( ,0, ) / 2h h h

Level repulsion is here, not at h=0!

Microscopic origin of the energy gaps ? Dzyaloshinsky-Moriya interaction breaks rotational

symmetry and seems therefore to be a good candidate to explain the origin of energy gaps in the single-molecule magnets.

Our calculations for 3- and 15-spin V15 models demonstrate that the energy gaps are anisotropic with respect to the direction of the applied field.

Only for special directions of the applied field, adiabatic changes of the magnetization are possible.

Adopting the extended (Kaplan-Shekhtman-Entin-Wohlman-Aharony) form of the interaction does not change these conclusions.

Alternative mechanisms …

hE ?

Summary The Dzyaloshinsky-Moriya interaction

yields level splittings that depend on the direction of the applied magnetic field

Does not seem compatible with present experimental findings

For Mn12 we find that DM + local anisotropy (in x-y-z direction) does not yield observable energy gaps.

Microscopic origin of energy gaps remains an open question

Quantum dynamics of the magnetization: Decoherence Energy level repulsions: oscillations

of the spins Presently very relevant for experimental

demonstration of qubit operation Questions:

What does the observation of oscillations mean in terms of coupling of the spins to their environment

Decoherence, T2

Concrete example Spin-1/2 objects interacting with a “bath” of

spin-1/2 objects . “Simple” model:

In general, the interactions between the spins are often anisotropic

Quantum dynamics of the subsystem: Magnetization: Reduced density matrix: Variance on the subsystem entropy:

Characterizes “mixing” of the state: decoherence

L

nnn SSIJSSJH

121

2210 )()(

iHtzk

iHtk eSetM )(

]))(Tr()(Tr[)( 22 tttS

)()()()(Tr)( ttttt

Simulation of two S=1/2 spins coupled to a S=1/2 bath Model: Random Jn Initial state of subsystem: Initial state of bath:

random We observe:

fast decoherence followed by a resurrection

of magnetization oscillation

a very slow decay of the amplitude of these oscillations

L

nnn SSIJSSJH

121

2210 )()(

MagnetizationVariance ofthe entropy

Magnetizationenvelope

S1S2

correlations

V.V. Dobrovitski et al,Phys. Rev. Lett. 90, 210401 (2003)

L=14

Simulation of two S=1/2 spins coupled to a S=1/2 bath Model:

Initial state of subsystem: Fast decoherence,

followed by oscillations with a very slow decay of the amplitude

Spin-spin correlation vanishes fast

Presence of oscillations in an observable of the subsystem does not say much about decoherence

It is necessary to measure CORRELATION

L

nnn SSIJSSJH

121

2210 )()(

L=20

What is the decoherence time of 29Si in silicon* ? 29Si in silicon: dilute dipolar-

coupled spins in a solid Model Hamiltonian (rotating frame)

i : random magnetic shift aij=-2bij : dipolar interaction between

spins on random lattice positions

1

( )L L

z z z x x y yi i ij i j ij i j ij i j

i j i

H I a I I b I I b I I

*A.E. Dementyev et al, Phys. Rev B 68, 153302 (2003)

NMR on 29Si in silicon* Standard NMR measurement of T2:

Hahn echo sequence 90X-(TE/2)-180Y-(TE/2)-ECHO

TE = variable delay time Carr-Purcell-Meiboom-Gill (CMPG)

sequence 90X-{(TE/2)-180Y-(TE/2)-ECHO}repeat n times

Common wisdom: CPMG sequence should not excite echos beyond T2

(Hahn

sequence)

*A.E. Dementyev et al, Phys. Rev B 68, 153302 (2003)

NMR on 29Si in silicon* Experiments* at T=4.2K:

Hahn echosignal

TE=1.12ms

TE=2.65ms

*A.E. Dementyev et al, Phys. Rev B 68, 153302 (2003)

CMPGechosignal

Computer simulation of NMR on 29Si in silicon (1)

1. Make Si crystallites containing L (29Si) spins (= 29Si atoms)

2. Determine dipolar interactions,…3. Solve the time-dependent

Schrödinger equation for the CPMG or Hahn pulse sequence

4. Average over many (10-100) crystallites

Computer simulation of NMR on 29Si in silicon (2) Si crystallites with

L (29Si) spins (29Si atoms), determine dipolar interactions etc.

Yellow: 29Si (4.67% n.a)

Red: Selected as one of the L spins

Computer simulation of NMR on 29Si in silicon (3) Simulation results (L = 20; 10

samples)

-5

0

5

10

15

20

0 10 20 30 40 50 60 70 80 90t

Re<M_y(t)M_z(0)>

-5

0

5

10

15

20

25

0 10 20 30 40 50 60 70 80 90t

Re<M_y(t)M_z(0)>

We set Hdip-dip=0 during the

pulses: “perfect pulses”

Only possible in computer simulation

Physical picture: Spin k feels a random field k and interacts with a “bath” of other spins via a dipole-dipole interaction. This interaction leads to de-phasing:T2

T2 as obtained from different pulse sequences (e.g. Hahn or CMPG) may be substantially different.

Simulation results give a clue to understand these results: During the 180Y pulses, the interaction modifies the time evolution and this has to be taken into account.

Conclusion