# Using magnetostriction to measure the spin-spin correlation function and magnetoelastic coupling in the quantum magnet NiCl$_2$-4SC(NH$_2$)$_2$

**ABSTRACT** We report a method for determining the spatial dependence of the magnetic exchange coupling, $dJ/dr$, from magnetostriction measurements of a quantum magnet. The organic Ni $S = 1$ system NiCl$_2$-4SC(NH$_2$)$_2$ exhibits lattice distortions in response to field-induced canted antiferromagnetism between $H_{c1} = 2.1$ T and $H_{c2} = 12.6$ T. We are able to model the magnetostriction in terms of uniaxial stress on the sample created by magnetic interactions between neighboring Ni atoms along the c-axis. The uniaxial strain is equal to $(1/E)dJ_c/dx_c < S_{\bf r} \cdot S_{{\bf r}+ {\bf e}_c} >$, where $E$, $J_c$, $x_c$ and ${\bf e}_c$ are the Young's modulus, the nearest neighbor (NN) exchange coupling, the variable lattice parameter, and the relative vector between NN sites along the c-axis. We present magnetostriction data taken at 25 mK together with Quantum Monte Carlo calculations of the NN spin-spin correlation function that are in excellent agreement with each other. We have also measured Young's modulus using resonant ultrasound, and we can thus extract $dJ_c/dx_c = 2.5$ K/$\AA$, yielding a total change in $J_c$ between $H_{c1}$ and $H_{c2}$ of 5.5 mK or 0.25% in response to an 0.022% change in length of the sample.

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- Fabrizio Anfuso, Markus Garst, Achim Rosch, Oliver Heyer, Thomas Lorenz, Christian Rüegg, Karl Krämer[Show abstract] [Hide abstract]

**ABSTRACT:**Magnetostriction and thermal expansion of the spin-ladder compound piperidinium copper bromide (C5H12N)2CuBr4 are analyzed in detail. We find perfect agreement between experiments and the theory of a two-leg spin-ladder Hamiltonian for more than a decade in temperature and in a wide range of magnetic fields. Relating the magnetostriction along different crystallographic directions to two static spin-spin correlation functions, which we compute with quantum Monte Carlo, allows us to reconstruct the magnetoelastic couplings of (C5H12N)2CuBr4 . We especially focus on the quantum critical behavior near the two critical magnetic fields Hc1 and Hc2 , which is characterized by strong singularities rooted in the low dimensionality of the critical spin system. Extending our discussion in Lorenz [ Phys. Rev. Lett. 100, 067208 (2008)], we show explicitly that the thermal expansion near the upper critical field Hc2 is quantitatively described by a parameter-free theory of one-dimensional, nonrelativistic fermions. We also point out that there exists a singular quantum critical correction to the elastic moduli. This correction is proportional to the magnetic susceptibility chi , which diverges as chi˜1/T at the critical fields and thus leads to a strong softening of the crystal.Physical review. B, Condensed matter 06/2008; 77(23). · 3.66 Impact Factor

Page 1

arXiv:0705.0365v1 [cond-mat.str-el] 2 May 2007

Using magnetostriction to measure the spin-spin correlation function and magnetoelastic coupling

in the quantum magnet NiCl2-4SC(NH2)2

V. S. Zapf1, V. F. Correa,2,†P. Sengupta,1,3C. D. Batista,3M. Tsukamoto,4, N.

Kawashima,4P. Egan,5, C. Pantea,1A. Migliori,1J. B. Betts,1M. Jaime,1A. Paduan-Filho6

1National High Magnetic Field Laboratory (NHMFL),

Los Alamos National Lab (LANL), Los Alamos, NM

2NHMFL, Tallahassee, Florida

3Condensed Matter and Thermal Physics, LANL, Los Alamos, NM

4Institute for Solid State Physics,

University of Tokyo, Kashiwa, Chiba, Japan

5Oklahoma State University, Stillwater, OK

6Instituto de Fisica, Universidade de Sao Paulo, Brazil

†Now at Comisi´ on Nacional de Energ´ ıa At´ omica,

Centro At´ omico Bariloche, 8400 S. C. de Bariloche, Argentina

(Dated: February 1, 2008)

We report a method for determining the spatial dependence of the magnetic exchange coupling, dJ/dr, from

magnetostriction measurements of a quantum magnet. The organic Ni S = 1 system NiCl2-4SC(NH2)2 ex-

hibits lattice distortions in response to field-induced canted antiferromagnetism between Hc1 = 2.1 T and

Hc2 = 12.6 T. We are able to model the magnetostriction in terms of uniaxial stress on the sample cre-

ated by magnetic interactions between neighboring Ni atoms along the c-axis. The uniaxial strain is equal

to (1/E)dJc/dxc?Sr· Sr+ec?, where E, Jc, xc and ec are the Young’s modulus, the nearest neighbor (NN)

exchange coupling, the variable lattice parameter, and the relative vector between NN sites along the c-axis.

We present magnetostriction data taken at 25 mK together with Quantum Monte Carlo calculations of the NN

spin-spin correlation function that are in excellent agreement with each other. We have also measured Young’s

modulus using resonant ultrasound, and we can thus extract dJc/dxc = 2.5 K/˚ A, yielding a total change in Jc

between Hc1and Hc2of 5.5 mK or 0.25% in response to an 0.022% change in length of the sample.

PACS numbers:

Keywords:

In many insulating magnets, the magnetic coupling is

caused by superexchange interactions created when atomic

or molecular orbitals overlap. Since the radial dependence

of the orbital wave functions can be quite steep, the over-

lap integrals and the resulting exchange coupling J depend

strongly on the interatomic bond lengths r. Past experiments

have probed dependence of J on r using hydrostatic pressure

or chemical substitution to vary the bond length, and Raman

spectroscopyto measureJ. Theseresultswerecombinedwith

high-intensity X-ray scattering measurements or elastic neu-

tron scattering to determine the bond lengths.1,2,3,4

Here we demonstrate a simple and novel approach to in-

vestigating the spatial dependence of the superexchange in-

teraction in the quantum magnet NiCl2-4SC(NH2)2(DTN).

We use applied magnetic fields to create an effective pres-

sure and measure the response of the soft organic lattice via

magnetostriction. The S = 1 Ni ions in DTN form a body-

centered tetragonal structure5shown in Fig. 1. The dominant

magnetic superexchange interaction Jc= 2.2 K is antiferro-

magnetic (AFM) and occurs along linear Ni-Cl-Cl-Ni bonds

in the tetragonal c-axis.6,7Along the a-axis, Ja = 0.18 K is

an order of magnitudesmaller and no diagonalor next-nearest

neighbor couplings have been found within the resolution of

inelastic neutron scattering measurements.6We thus treat this

compoundas a 1D system of Ni-Cl-Cl-Ni chains only weakly

coupled in the a-b plane. Because Jcis sensitive to the Ni

inter–ion bond lengths, a magnetic stress is created between

adjacent Ni spins along the c-axis. This stress depends on the

relative orientation of the two spins, e.g. on the NN spin–spin

correlation function ?Sr· Sr+ec?.

In DTN, the NN spin–spin correlation function varies

strongly with magnetic field. DTN exhibits AFM order for

applied fields along the c–axis between Hc1 = 2.1 T and

Hc2 = 12.6 T and with a maximum N´ eel temperature of

TN = 1.2 K, as shown in the phase diagram in Fig. 2. The

AFM order is confined to the a-b plane at Hc1. However, as

the field is increased from Hc1to Hc2, the spins cant along

the c-axis and finally saturate for H > Hc2. This is illustrated

in the magnetization vs field curve shown in Fig. 2.

The lack of magnetic order at zero field is due to a strong

easy–plane uniaxial anisotropy that creates a splitting D at

zero field between the Sz = 0 ground state and the Sz =

±1 excited states of the Ni ion. In applied fields parallel to

the tetragonal c-axis, the Zeeman effect then lowers the Sz=

−1 state until it becomes degenerate with the Sz = 0 state,

resulting in a magnetic ground state and AFM order below

the N´ eel temperature.7Since the Sz= −1 state is broadened

by AFM dispersion, the region of overlap between Sz= −1

and Sz= 0 extends from Hc1= 2.1 T up to Hc2= 12.6 T.

Here we show that the bare NN spin-spin correlation func-

tion can be directly proportional to the c-axis magnetostric-

tion. Since all the terms in the magnetic Hamiltonian of this

compoundhave been measured, we can calculate ?Sr·Sr+ec?

using Quantum Monte Carlo simulations to predict the mag-

netostriction response as a function of the applied magnetic

field. By combining these results with resonant ultrasound

Page 2

2

FIG. 1: Unit cell of tetragonal NiCl2-4SC(NH2)2 showing Ni (red)

and Cl (blue) atoms. The thiourea molecules have been omitted for

clarity.

0

0.3

0.6

0.9

1.2

0

0.2

0.4

0.6

0.8

1

051015

MCE

Specific Heat

QMC

data

QMC

TN (K)

M/Msat

H (T)

AFM/BEC

DTN

H || c

Hc1

Hc2

FIG. 2: Temperature T - Magnetic field H phase diagram for H||c

determined from specific heat and magnetocaloric effect (MCE)

data, together with the result of Quantum Monte Carlo (QMC)

simulations.6,7The magnetization vs field measured at 16 mK and

calculated from QMC is overlayed onto the phase diagram.8

spectroscopy to determine the elastic moduli, we are also able

to extract the leading linear term in the spatial dependence of

the exchange interaction Jc(r) along the tetragonal c-axis.

We first present magnetostriction measurements that were

performed on single crystals of DTN down to 25 mK in a 20

T magnet at the National High Magnetic Field Laboratory in

Tallahassee, FL, as described in Ref. 9. The magnetostric-

tion as a function of H for H||c is shown in Fig. 3 for both

the a and c-axes of the crystal. The c-axis magnetostriction

∆Lc/Lcshows sharp shoulders at the boundaries of the or-

deredstate at Hc1andHc2, andnonmonotonicbehaviorin be-

tween. Thenet differencebetweenthe c-axis lattice parameter

at Hc1and Hc2is 0.022%. The nonmonotonicbehaviorof the

magnetostrictioncontrastswiththe roughlylineardependence

of the magnetization M(H) in the region of AFM order be-

tween Hc1and Hc2, as shown in Fig. 2. It also contrasts with

the magnetostriction observed in the Cu dimer spin gap sys-

tem KCuCl3, in which the magnetostriction closely tracks the

magnetization.10

The a-axis lattice parameter decreases monotonicallyby an

amount that is an order of magnitude smaller than the change

in the c-axis parameter, reflecting the fact that Ja<< Jc. The

a-axis behavior is more difficult to explain since the exchange

interaction is mediated by an unknown and likely convoluted

path, and because the a-axis is subject to significant Poisson

-0.010

0.000

0.010

0.020

02

Hc1

4681012

Hc2

14

∆L/L (%)

c-axis

a-axis

H || c

1 10-4

H (T)

-2 10-4

-1 10-4

0 100

1234

Hc1

a-axis

FIG. 3: Normalized percentage length change %∆L/L as a function

of magnetic field measured along the crystallographic c-axis (solid

blue lines) and a-axis (dashed red lines). The data is taken at T = 25

mK with the magnetic field applied along the c-axis. The inset shows

the feature at Hc1in %∆La/La in greater detail, and a straight line

has been subtracted from the inset data for clarity.

forces from the larger c-axis distortion.

We thus focus on the c-axis magnetostriction and we sug-

gest a straightforward explanation for its nonmonotonicfield-

dependence between Hc1and Hc2. The canted AFM order

results in two competing forces on the c-axis of the lattice.

Near Hc1, the Ni spins order antiferromagnetically, thus cre-

ating an attractive force. By reducing the c-axis lattice pa-

rameter, the system can increase Jc, and thereby lower the

energy of the antiferromagnetically aligned spins. However,

with increasing field the spins cant, resulting in a ferromag-

netic component to the order. The ferromagnetic component

stretches the lattice, thereby reducing Jc. Once the magnetic

field exceeds ∼ 5.5 T, the ferromagnetic component wins and

the lattice expands.

We now test this model by calculating the expected c-axis

magnetostriction. The energy density of the system can be

written as the sum of magnetic and lattice components, ǫ =

ǫe+ ǫm, with

ǫe =

1

2E

?xc− co

c

?2

ǫm =

2

Na2c?Hm?

(1)

Here a and c are the lattice parameters at zero field. E is

Young’s modulus along the c-axis and N is the total number

of Ni sites. The factor of 2 is required because there are two

Ni atoms per unit cell of volume a2c. The variable cois the

value of the lattice parameter along the c-axis in the absence

of magnetic interactions and external pressure. The variable

xcis the new valueof the lattice parameterwhen the magnetic

interactions are included. We have neglected the effect of a-

axis strain inducingchanges in the c-axis via the Poisson ratio

since it is a 1 % effect.

The magnetic Hamiltonian Hmis:

Hm=

?

r,ν

JνSr· Sr+eν+

?

r

[D(Sz

r)2− gµBHSz

r],

(2)

Page 3

3

where eν= {aˆ x,bˆ y,cˆ z} are the relative vectors between NN

Ni ions along the a, b and c–axis respectively. In this Hamil-

tonian, the magnitude of the various parameters D, Ja, Jc,

and g have been measured experimentally via ESR and neu-

tron diffraction in combination with Quantum Monte Carlo

simulations.6,7

We cannowobtainthe valueofxcas a functionofmagnetic

field by minimizing the total energy with respect to xc:

∂xcǫ =E

c2(xc− xo) + ∂xcǫm= 0.

(3)

We assume that the only term in Hmthat depends on xcis the

AFM Heisenberg coupling along the c–axis. The single–ion

anisotropy D typically has a much smaller dependence on xc.

In addition, since the temperature at which the magnetostric-

tion measurements were performed (25 mK) is much lower

that any characteristic energy of the system, we will assume

that T = 0 K. Under these conditions we obtain:

∂xcǫm=

2

a2c∂xcJ|xc=c?Sr· Sr+ec?.

(4)

In Eq. 4 we have applied the Hellman–Feynmanand assumed

that ∂xcJ ≃ ∂xcJ|xc=cbecause the relative distortion is very

small. Substituting into Eq. (3) we find that:

E

c2(xc− xo) +

1

a2c∂xcJ|xc=c?Sr· Sr+ec? = 0.

(5)

We know that xc= c for H = 0 and thus,

E

c2(c − xo) +

1

a2c∂xcJ|xc=c?Sr· Sr+ec?H=0= 0,

(6)

where ?Sr· Sr+ec?H=0indicates that the mean value is com-

puted for a field H = 0. By taking the difference between

Eqs. 5 and 6 we obtain:

∆L

L

= −∂xcJ|xc=c

a2E

[?Sr·Sr+ec?H=0−?Sr·Sr+ec?H], (7)

where ∆L/L = (xc− c)/c. Thus our measured c-axis mag-

netostriction is proportional to the NN spin-spin correlation

function with a proportionality constant of

κ =

1

a2E∂xcJ|xc=c.

(8)

We can therefore model the experimental magnetostriction

data with the parameter κ as the only fitting parameter. We

have determined the NN spin-spin correlation function us-

ing Quantum Monte Carlo simulations on a 8 × 8 × 24 lat-

tice and the parameters: Jc = 2.2 K, Ja = 0.18 K, and

D = 8.6 K.7The results of our model are shown in com-

parison with the measured magnetostriction in Fig. 4, with

κ = 1.00 × 10−5. The agreement between theory and ex-

periment is very good and confirms our hypothesis that the

spatial dependence of D is much smaller than the spatial de-

pendence of J and can thus be neglected. This is to be ex-

pected since J results from the overlap integral between ad-

jacent molecular wave functions, which can have large radial

-0.010

0.000

0.010

0.020

theory

experiment

024681012

Hc2

14

∆Lc/Lc (%)

H || c

H (T)

Hc1

FIG. 4: Comparison of experimental c-axis magnetostriction data as

a function of H for H||c with the model described in the text.

FIG. 5: Mechanical resonances of DTN at room temperature. Inset:

Temperature-dependence of the major peak near 500 kHz between 5

K and 300 K. The line is a fit to the usual Einstein oscillator model

equation11and used to extrapolate the resonance to 0 K.

dependencies with high power-laws, whereas D depends on

crystalline electric fields that change more weakly with lat-

tice distortions. For instance, previous experimental and the-

oretical works on other compounds have modelled the spa-

tial dependence of superexchangeinteractions as a power-law

J(r) = br−nwhere r is the relevant spacing between mag-

netic ions. Values for the exponent n of 10-14 have been

reported for metal halides,2,3and 2-7 for cuprates.1,4In this

work,we aredeterminingthe leadinglinearterm in theexpan-

sion of J(r), e.g. ∂rJ ≈ −nJ/r. We have neglected higher

order terms because the relative change of the lattice param-

eter c that results from the magnetic stress is always smaller

than 0.03% as shown in Fig.3.

We take our analysis one step further and quantitatively de-

termine the spatial dependence of the AFM exchange interac-

tion dJc/dxcfrom Eq. 8. The lattice parameters a = 9.558˚ A

and c = 8.981˚ A are known from published X-ray diffraction

measurements at 110 K.5That leaves Young’s modulus E as

the remainingquantity to be determinedbeforewe can extract

dJc/dxc.

Page 4

4

Thus, we have also measured Young’s modulus using Res-

onant Ultrasound Spectroscopy (RUS) between 300 K and 5

K.12,13Mechanical resonances of a roughly cube-shaped sin-

gle crystal of DTN were determined at zero field in a He-

cooled Oxford Instruments flow-cryostat. The six indepen-

dent elastic moduli were determined between 300 K and 200

K and their values at room temperature are shown in Table I.

For lower temperatures, determination of all of the reso-

nancesusedinthefitting procedurebecameambiguous. How-

ever,twogoodresonancescouldbeidentifieddownto5 Kand

based on the temperature dependencies of these resonances,

weextrapolatedthevalueofYoung’sModulusto0K.Young’s

modulus E in a tetragonal crystal is given by:

E33= C33−

2C2

13

C11+ C12,

(9)

yielding E = 7.5 ± 0.7 GPa at 0 K.

Now we can use equation (9) to calculate ∂xJ|x=xo=

dJc/dxc = 2.5 K/˚ A, yielding a total change in Jcbetween

Hc1and Hc2of 5.5 mK or 0.25%. This in turn results in a

0.1% shift in Hc2relative to its value in the absence of mag-

netostriction effects. The dominant uncertainty in these cal-

culations comes fromthe 10% errorbar in estimating Young’s

modulus E due to the softness of the crystal.

Previous papers about DTN have assumed that Jcis con-

stant when calculating the critical fields, the magnetization,

and other field-dependent measurable quantities.6,7Since Jc

only varies by 0.25%, these assumptions are quite reason-

able and well within experimental error. An open question

remains, however, whether the symmetry of the crystal is

affected by the magnetostriction. DTN has previously at-

tracted interest because the field-induced phase transition at

Hc1likely belongs to the universality class of Bose-Einstein

Condensation (BEC).6The tetragonal symmetry of the lattice

creates a necessary condition for conservation of the equilib-

rium number of bosons, and therefore structural deviations

away from tetragonal crystal symmetry could disallow the

Bose-Einstein Condensation picture. Since the magnetostric-

tion effects occur gradually at fields above Hc1, the BEC pic-

ture would hold right at Hc1as reported,6but become less

valid at high fields as the structure becomes increasingly dis-

torted. This possibility is currently being further investigated

via elastic neutron diffraction and ESR measurements.

It has been suggested that sound attenuation studies, which

probe magnon-phonon coupling, are another means of prob-

ing the magnitude of J(r).14However, as demonstrated in

measurementsof a similar antiferromagneticquantummagnet

TlCuCl3,15the wave vector k of the probing phonons is van-

ishingly small compared to the wave vector of the magnons,

and thus the contribution of J(r) to the magnon-phononcou-

pling is negligible compared to the contribution of D(r).

To our knowledge, the superexchange interaction in Ni-Cl-

Cl-Ni chains has not been previously investigated experimen-

tally or theoretically. For DTN, we speculate that the Cl-Cl

bond determines the magnitude of J along the Ni-Cl-Cl-Ni

chains, since it is the weakest link, being nearly 2x longer

than the Ni-Cl bond (4.1˚ A vs 2.4 or 2.5˚ A). Early X-ray scat-

tering studies have also implied5that the lowest-energylattice

TABLE I: Tetragonal elastic moduli of DTN at room temperature.

elastic moduli (GPa)

c11 = 26.1 c12 = 15.3

c33 = 14.2 c44 = 11.2

c23 = 12.4 c66 = 4.3

vibrations consist of the NiCl2-4SC(NH2)2molecule moving

as a unit, thus supporting the idea that the Cl-Cl bonds that

link adjacent molecules are more susceptible to pressure than

the Ni-Cl bonds within a molecule.

In summary, we have measured magnetostriction of the or-

ganic quantum magnet NiCl2-4SC(NH2)2and we have mod-

elled the magnetostriction data by treating the compound as

a 1-D magnetic system in which the strong dependence of

the superexchange interaction on the bond lengths along the

c-axis results in a magnetic stress. To our knowledge, this

is the first work in which the NN spin-spin correlation func-

tion is shown to be directly proportional to an experimentally

measurable quantity. It also presents a new and straightfor-

ward method for determining the spatial dependence of the

exchange coupling over small distances.

Acknowledgments

This work was supportedby the DOE, the NSF, and Florida

State University through the National High Magnetic Field

Laboratory. A.P.F. acknowledges support from CNPq (Con-

selho Nacional de Desenvolvimento Cientfico e Tecnolgico,

Brazil). We would like to thank S. Haas and N. Harrison for

stimulating discussions.

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