arxiv:1301.5788v1 [cond-mat.mtrl-sci] 24 jan 2013 · 2 investigated by time-of-ight neutron...

Phonon-Magnon coupling in CoF2 investigated by time-of-flight neutron spectroscopy

Tapan Chatterji1, Mohamed Zbiri1, Stephane Rols11Institut Laue-Langevin, B.P. 156,38042 Grenoble Cedex 9, France

(Dated: November 12, 2018)

We report the results of inelastic neutron scattering investigation on the model antiferromagnetCoF2 by time-of-flight neutron spectroscopy. We measured the details of the scattering functionS(Q, ω) as a function of temperature with two different incident neutron wavelengths. The tempera-ture and Q dependence of the measured scattering function suggests the presence of magnon-phononcoupling in almost all branches. The present results are in agreement with the strong magnetoelasticeffects observed previously.

PACS numbers: 75.25.+z

The lattice dynamics of a crystal is normally assumedto be independent of the spin dynamics of the magneticmoments associated with the atoms and ions of the crys-tal. Similarly while describing the motions of the spins ina magnetically ordered systems it is usual to assume thatthe equilibrium positions of the magnetic ions are frozen.The approximation gives a fairly well description of themagnetic system and is in the same spirit as the adiabaticor Born-Oppenheimer approximation which is normallyused for band structure calculations. In this approxima-tion the phonon and magnons are separate entities withtheir own dispersion relations and the Hamiltonian of thesystem separates out into a lattice and a magnetic part.In reality however there always exists a coupling termthat involves displacements both in the positions of theatoms in the crystal and in the orientations of the spinvectors associated with these atoms. Thus the collectiveexcitations are neither pure lattice waves nor spin wavesbut rather magnetoelastic waves. Such magnetoelasticwaves are predicted to exist theoretically long time ago1.With the advent of enormous progress in first-principlecalculations it is now possible to do phonon calculationsof magnetic solids in the magnetically ordered phase bytaking spin degrees of freedom into account. The dif-ference between the results of the calculations done bytaking and not taking magnetic degrees of freedom intoaccount can be often very significant as has been amplydemonstrated by such recent calculations on Fe-arsenidecompounds2. It is possible to study magnetoelastic ef-fects experimentally by using neutron scattering tech-niques. Our recent neutron diffraction investigations3,4

have shown strong static magnetoelastic coupling in tran-sition metal difluorides MnF2, FeF2, CoF2 and NiF2.There exist indirect evidences for dynamic magnetoe-lastic or phonon-magnon coupling in FeF2, CoF2 andNiF2

5,6. However, a systematic study of the phonon-magnon coupling in this relatively simple system is stillmissing.

CoF2 belongs to the family of transition-metal di-fluoride, which has been the subject of intensiveinvestigations18–25. CoF2, along with other transition-metal oxides MnF2, FeF2 and NiF2, crystallize with the

Co Co

Co

Co

Co

Co

Co

Co

Co

F

F

F

F

F

F

FIG. 1: The antiferromagnetic structure adopted by 3d tran-sition metal diflurides viz. CoF2 with the rutile type crystalstructure. The Co and F ions are labeled. The arrows on theCo represent moment direction of the transition metal ionsbelow the Neel temperature31.

tetragonal rutile-type structure in the P42/mnm spacegroup. However the magnetic properties of CoF2 aremore complex than those of isomorphous MnF2, becausethe Co ion has unpaired angular momentum that playsan important role in determining its magnetic proper-ties. CoF2 orders26–30 below TN ≈ 39 K with an an-tiferromagnetic structure31, shown in Fig. 1, with thepropagation vector k = 0. The magnetic moments of Coions at the corner (000) positions of the tetragonal unitcell are all parallel to the c-axis whereas those of the Co

arX

iv:1

301.

5788

v1 [

cond

-mat

.mtr

l-sc

i] 2

4 Ja

n 20

13

2

FIG. 2: (Color online) Temperature evolution of the scat-tering function S(Q,ω) measured from CoF2 using incidentneutron wavelength of λ = 1.5A.

ions at the ( 121212 ) positions are oppositely oriented. The

presence of an orbital moment in CoF2 makes it particu-larly interesting for the study of magnetoelastic couplingin this compound. We recently investigated32 the hy-perfine interaction in CoF2 by high resolution neutronspectroscopy and concluded that the presence of the un-quenched orbital moment of the Co ion in CoF2 leads toits anomalous behavior compared to that of other Co-based compounds.

We have performed inelastic neutron scattering inves-tigations on CoF2 using the thermal time-of-flight neu-tron spectrometer IN4C at the Institute Laue-Langevinin Grenoble. About 5 g of CoF2 powder sample was putinside a cylindrical Al sample holder that was fixed to thecold tip of the sample stick of a standard orange cryostat.In order to cover an extended Q-range, relevant to thepresent study, and to gain in energy resolution two inci-dent wavelengths of λi = 1.1 and 1.5 A, respectively, wereused. This allowed the Stokes spectrum to be measuredat low temperature over a broader dynamical range. Thedata analysis was done using ILL software tools

FIG. 3: (Color online) Temperature evolution of the scatter-ing function S(Q,ω) from CoF2 measured with the incidentneutron wavelength of λ = 1.1A.

The present investigations have been performed withunpolarized neutrons on powder samples of CoF2. Thepresent method is therefore somewhat limited because itgives only the powder average results and does not giveinformation of detailed directional Q dependence. Alsothe unpolarised neutron scattering cannot distinguish di-rectly whether the scattering is of magnetic or structuralorigin. However and since CoF2 contains magnetic Coions, we can therefore expect inelastic magnetic scat-tering due to the interaction of the neutron spin withthe spin and orbital magnetic magnetization of the CoF2

both below and above the magnetic ordering temperatureTN ≈ 39 K. Also the neutron is sensitive to the struc-tural excitations or phonons which exist at all tempera-tures. Thus this technique will measure both excitationsand even highlights their hybrid feature described before.Magnetic and structural excitations have fortunately dif-ferent temperature and Q dependence. Strong and sharpinelastic magnetic excitations are expected only belowthe magnetic ordering temperatures. At higher temper-ature, the magnetic correlations give broad quasielastic

3

0

0.00002

0.00004

0.00006

0.00008

0.0001

-10 -5 0 5 10 15

C

Inte

nsity

(arb

. uni

ts)

Energy (meV)

y = m1+m2*exp(-(x-m3)^ 2/m4^...ErrorValue

1.8543e-73.5878e-7m1 6.9447e-79.8724e-5m2 0.00557410.047175m3 0.00830530.98782m4 5.7132e-72.9499e-5m5

0.016137.6746m6 0.024592-1.0479m7

NA5.5534e-11ChisqNA0.99908R

CoF2

T = 2 K

-0.00002

0

0.00002

0.00004

0.00006

0.00008

0.0001

-10 -5 0 5 10 15

INS-CoF2-IN4_1.5A-20K_Q0.daC

C


2.1345e-74.266e-7m1 7.777e-79.8039e-5m2

0.00628450.047694m3 0.00939130.98868m4 6.0863e-72.5407e-5m5 0.0223157.518m6

0.034411.1781m7 NA6.9557e-11ChisqNA0.99881R

CoF2

T = 20 K

Inte

nsity

(arb

. uni

ts)

Energy (meV)

0

0.00002

0.00004

0.00006

0.00008

0.0001

-10 -5 0 5 10 15


2.0737e-78.7735e-7m1 6.1952e-79.1681e-5m2 0.00533830.051356m3 0.00813520.99697m4 3.9498e-71.2613e-5m5 0.0487966.2936m6 0.0820252.0821m7


Energy (meV)

Inte

nsity

(arb

. uni

ts)

CoF2

T = 32 K

0

0.00002

0.00004

0.00006

0.00008

0.0001

-10 -5 0 5 10 15

C


4.2267e-7-4.3068e-7m1 6.8491e-77.8216e-5m2 0.00521550.031649m3

0.0104941.0162m4 2.6897e-50.00011194m5

0.34873.3792m6 0.177792.7137m7


Energy (meV)

Inte

nsity

(arb

. uni

ts)

CoF2

T = 40 K

0

0.00002

0.00004

0.00006

0.00008

0.0001

-10 -5 0 5 10 15


3.402e-7-9.2448e-7m1 5.3436e-77.6833e-5m2 0.00489410.027984m3 0.00829160.99545m4 2.1546e-50.00013394m5

0.259433.628m6 0.124691.7582m7


Energy (meV)

Inte

nsity

(arb

. uni

ts)

CoF2

T = 50 K

0

0.00002

0.00004

0.00006

0.00008

0.0001

-10 -5 0 5 10 15

C


2.0761e-7-1.6716e-6m1 3.1325e-77.8319e-5m2 0.0024680.043138m3 0.0044450.99521m4

1.6103e-50.00016837m5 0.197873.9965m6

0.0502110.82066m7 NA5.0062e-12ChisqNA0.99989R

Energy (meV)

Inte

nsity

(arb

. uni

ts)

CoF2

T = 100 K

FIG. 4: Results of the fit of the low-Q data (integrated overQ in the range from 0.9 to 3.56 A−1) from CoF2 in the low-energy range (λi = 1.5 A). Two Gaussian peaks have been fit-ted to the elastic peak at zero energy and the inelastic peakwhich is at about E = 8 meV at T = 2 K. The energy ofinelastic peak decreases with increasing temperature and be-comes zero at TN = 38 K giving rise to quasielastic scatteringwhich has been fitted with a Lorentzian function.

scattering instead. The intensity of the magnetic excita-tion peaks decrease with the momentum transfer Q. Thescattering due to structural excitations or phonons havedifferent temperature and Q dependences. The phononintensity in general increases with increase of the momen-tum transfer Q. Of course intensities of magnons andphonons are also governed by the magnon and phononstructure factors and also polarisations which dependson the details of the magnetic and crystalline structures.However the differences in temperature and Q depen-dence for magnons and phonons can be conveniently usedfor the sake of identification in order to separate them.

The cross section for neutron scattering by magneticsystems is discussed in details in several text books onneutron scattering33,34. The expression for neutron scat-tering cross sections are very complex when the ions haveboth spin and orbital angular momentum. It can be

0

0.000002

0.000004

0.000006

0.000008

0.00001

10 15 20 25 30 35

Inte

nsity

(arb

. uni

ts)

Energy (meV)

y = m1+m2*exp(-(x-m3)^ 2/2^ ...ErrorValue

4.9039e-87.8957e-9m1 1.3431e-79.374e-7m2

0.2455313.551m3 1.0079e-77.5973e-6m5 0.01965923.838m6

9.4975e-88.2706e-7m7 0.1651230.005m8


CoF2

T = 2 K

0

0.000002

0.000004

0.000006

0.000008

0.00001

10 15 20 25 30 35

y = m1+m2*exp(-(x-m3)^ 2/2^ ...ErrorValue

4.4391e-81.9556e-7m1 1.218e-77.217e-7m2 0.2895213.493m3

9.1772e-87.1619e-6m5 0.01908223.65m6

8.6307e-85.9664e-7m7 0.205829.604m8


Energy (meV)

Inte

nsity

(arb

. uni

ts)

CoF2

T = 20 K

0

0.000002

0.000004

0.000006

0.000008

0.00001

10 15 20 25 30 35

C


1.8646e-86.0741e-7m1 4.7271e-85.6062e-6m2

0.0165622.576m3 0.023632.1987m4

5.4319e-81.5463e-6m5 0.08074417.895m6


Energy (meV)

Inte

nsity

(arb

. uni

ts)

CoF2

T = 32 K

0

0.000002

0.000004

0.000006

0.000008

0.00001

10 15 20 25 30 35

C

Energy (meV)


1.1335e-83.6222e-7m1 0.00073097-2.4521e-6m2

73.1219.11m3 2.08633.6607m4

0.000727418.7332e-6m5 28.89819.7m6


Inte

nsity

(arb

. uni

ts)

CoF2

T = 40 K

0

0.000002

0.000004

0.000006

0.000008

0.00001

10 15 20 25 30 35

C

C


1.0017e-83.3143e-7m1 5.4811e-82.9647e-7m2

0.5773414.814m3 0.0248923.4323m4

3.2101e-86.7677e-6m5 0.02374219.884m6


Energy (meV)

Inte

nsity

(arb

. uni

ts)

CoF2

T = 50 K

0

0.000002

0.000004

0.000006

0.000008

0.00001

10 15 20 25 30 35


1.3212e-83.2915e-7m1 2.9654e-86.534e-6m2

0.0125119.759m3 0.0200443.4863m4


Energy (meV)

Inte

nsity

(arb

. uni

ts)

CoF2

T = 100 K

FIG. 5: Results of the fit of the low-Q data (integrated over Qin the range from 0.9 to 3.56 A−1) from CoF2 in the higher-energy range (λi = 1.5 A). At low temperatures below TN =38 K, three peaks could be identified and they were fittedwith three Gaussian functions. At higher temperature onlya broad peak could be identified and has feen fitted with asingle Gaussian function.

greatly simplified if the momentum transfer is less thanthe reciprocal of the radius of the magnetic shell and isgiven by

d2σ

dΩdω=kfki

1

2π

(1.91e2

2mc2

)2

(1)

×∑ij

|f(Q)|2∫ ∞−∞〈K(i, 0).K(j, t)〉

× exp [iQ.(R(i)−R(j))] exp(−iωt)dt,

where f(Q) is the form factor of the ions, K(j, t) is thepart of the magnetic moment operator, L + 2S, which isperpendicular to Q, for the ion i ar R(i) at the time t, ki

and kf are the wavevectors of the incident and scatteredneutrons.

The neutron scattering cross section of a phonon mode

4

19

20

21

22

23

24

25

0 20 40 60 80 100

E1(meV)E2(meV)

Ene

rgy

(meV

)

Temperature (K)

CoF2(b)

0

2

4

6

8

10

0 10 20 30 40 50

E(meV)HWHM

Temperature(K)

CoF2

Ene

rgy

(meV

)H

WH

M(m

eV)

(a)

FIG. 6: (Color online) (a)Temperature variation of the en-ergy and the half-width at half-maximum (HWHM) of thelow-energy inelastic magnetic peak of CoF2. (b)Temperaturevariation of the energy of the high-energy inelastic magneticpeak of CoF2 (λi = 1.5 A).

(qj) is given by

d2σ

dΩdω=kfki

1

ω(qj)(2)

|∑k

(h

2Mk

) 12

bkQ.e(k,qj)

× exp[i(Q− q).R(k)] exp(−Wk(Q))|2

×(n(qj)

n(qj) + 1)

where exp(−Wk(Q)) is the Debye-Waller factor for thekth atom, n(qj) is the population of the mode and istaken for neutron energy loss, bk is the scattering lengthof the kth atom in the unit cell, Mk its mass, and e(k,qj)is the eigen vector in the normal mode (qj). Note thateverything we stated in the previous section about thedifferent Q dependence of the magnons and phonons, fol-lows directly from equations (1) and (2)

0

5 10-5

0.0001

0.00015

0.0002

0.00025

0.0003

0 2 4 6 8 10

2 K20 K40 K70 K100 K

CoF2

Inte

nsity

(arb

. uni

ts)

Q (Å-1)

E = 7.3 meV(a)

5 10-6

1 10-5

1.5 10-5

2 10-5

2.5 10-5

3 10-5

3.5 10-5

4 10-5

4.5 10-5

0 2 4 6 8 10

Ecut-14meV_1.1A-2K

2 K20 K40 K70 K100 K

Inte

nsity

(arb

. uni

ts)

Q (Å-1)

y = m1+m2*x+m3*x^ 2+m4*cos((...ErrorValue

6.8443e-71.8934e-5m1 2.8307e-7-4.7932e-6m2 2.5954e-85.9354e-7m3 1.7192e-72.4206e-6m4 0.0239992.1185m5

0.192654.234m6 NA3.67e-10ChisqNA0.95901R


6.9026e-71.835e-5m1 2.8471e-7-4.3557e-6m2 2.6014e-85.4207e-7m3 1.7591e-72.1538e-6m4 0.0252462.0684m5

0.214523.9987m6 NA3.777e-10ChisqNA0.95146R


8.8734e-73.0386e-5m1 3.6738e-7-7.5201e-6m2 3.3685e-87.6709e-7m3 2.2412e-72.5393e-6m4 0.0281732.0854m5

0.232253.877m6 NA6.1561e-10ChisqNA0.91668R


7.8053e-73.3385e-5m1 3.2219e-7-8.1731e-6m2 2.9457e-88.188e-7m3 1.9891e-72.5615e-6m4 0.0239432.0669m5

0.203033.9192m6 NA4.8245e-10ChisqNA0.93685R


9.0692e-73.4397e-5m1 3.742e-7-8.3108e-6m2

3.4214e-88.5246e-7m3 2.2974e-72.5635e-6m4 0.0287942.0884m5

0.239324.1264m6 NA6.4991e-10ChisqNA0.92745R

CoF2

E = 14 meV(b)

0

2 10-5

4 10-5

6 10-5

8 10-5

0.0001

0.00012

0 2 4 6 8 10

2 K20 K40 K70 K100 K

CoF2

Q (Å-1)

Inte

nsity

(arb

. uni

ts)

E = 23.5 meV(c)

2 10-6

4 10-6

6 10-6

8 10-6

1 10-5

1.2 10-5

1.4 10-5

1.6 10-5

0 2 4 6 8 10

Ecut-14meV_1.1A-2K

2 K20 K40 K70 K100 K

Inte

nsity

(arb

. uni

ts)

Q (Å-1)


6.3548e-71.7957e-5m1 2.6401e-7-4.5611e-6m2 2.4996e-84.3913e-7m3 1.1312e-7-6.637e-7m4 0.0622972.058m5 0.503862.1296m6



5.5286e-71.6613e-5m1 2.2859e-7-4.278e-6m2 2.1522e-84.2513e-7m3 9.8525e-8-6.3125e-7m4 0.0541191.999m5 0.470682.2742m6



4.9038e-78.1401e-6m1 2.028e-7-2.0401e-6m2

1.9095e-82.7811e-7m3 8.7823e-8-4.6008e-7m4 0.0629571.9754m5 0.552191.6955m6



5.4468e-77.1166e-6m1 2.2522e-7-1.6041e-6m2 2.1176e-82.3954e-7m3 9.8259e-8-5.072e-7m4 0.0560971.9034m5

0.5411.4239m6 NA7.3849e-11ChisqNA0.93664R


6.0786e-77.6204e-6m1 2.5193e-7-1.8333e-6m2 2.3789e-82.6548e-7m3 1.073e-7-5.5154e-7m4 0.0713652.0307m5 0.594352.2886m6


CoF2

E = 31.6 meV(d)

0

2 10-6

4 10-6

6 10-6

8 10-6

1 10-5

1.2 10-5

1.4 10-5

0 2 4 6 8 10

2 K20 K40 K70 K100 K

Inte

nsity

(arb

. uni

ts)

Q (Å-1)


1.0447e-62.4109e-6m1 4.2113e-7-3.3644e-7m2 4.0167e-81.4725e-7m3 7.6554e-8-3.3177e-7m4

0.32962.8267m5 1.377316.618m6



7.0057e-71.7952e-6m1 2.8455e-7-1.2284e-7m2

2.701e-81.2764e-7m3 8.5053e-84.1694e-7m4

0.111072.3339m5 0.689064.6224m6



6.4267e-68.3417e-6m1 2.9305e-6-2.8305e-6m2 2.9512e-73.8667e-7m3 8.3973e-7-7.9524e-7m4

1.89124.6588m5 2.669953.039m6



9.7614e-72.7939e-6m1 3.9339e-7-4.2924e-7m2

3.747e-81.4581e-7m3 7.5765e-8-4.0742e-7m4

0.25292.8602m5 1.016317.133m6



2.0412e-65.7893e-6m1 8.6258e-7-1.6943e-6m2 8.4923e-82.6974e-7m3

1.361e-73.923e-7m4 0.716063.3735m5

1.95989.5699m6 NA4.8256e-11ChisqNA0.94462R

CoF2

E = 43 meV(e)

5 10-6

1 10-5

1.5 10-5

2 10-5

2.5 10-5

3 10-5

3.5 10-5

0 2 4 6 8 10

C


6.8443e-71.8934e-5m1 2.8307e-7-4.7932e-6m2 2.5954e-85.9354e-7m3 1.7192e-72.4206e-6m4 0.0239992.1185m5 0.192654.234m6


Q (Å-1)

Inte

nsity

(arb

. uni

ts)

CoF2

E = 14 meV

T = 2 K(f)

FIG. 7: Q dependence of the intensity for the inelastic peaksat E = 7.3, 14.0, 23.5, 31.6 and 43.0 meV at T = 2, 20, 40,70 and 100 K (λi = 1.1 A).

Both magnetic and structural excitations have beeninvestigated27,28 by inelastic neutron scattering on CoF2

single crystals. The magnetic properties of CoF2 aremuch more complex than those of MnF2 having the samerutile crystal structure and also the same antiferromag-netic structure. Crystal field calculations of Gladney35

shows that the L = 3, S = 3/2 atomic ground state issplit by the octahedral crystal field to give an orbital Γ4

triplet as the ground state whose properties can be de-scribed by an effective l = 1 operator. The rhombic andtetragonal crystal field distortions and the spin-orbit cou-pling causes this state to split into six Kramers doublets.The molecular field in the antiferromagnetic phase causesfurther splitting. Labeling the ground state as A andfirst three excited states as B,C and D three crystal-fieldexcitations have been labeled27 as A-B, A-C and A-Dtransitions. Alternatively these states are also labeled28

as 0, 1, 2, 3. The corresponding excitations are labeledas 0 - 1, 0 - 2 and 0 - 3. The dispersions of these exci-tations have been measured from single crystal samplesalong [100] and [001] directions. However, the anisotropy

5

in dispersion of these excitations is found to be rathersmall. The strongest excitation is A-B which is at about7.4 meV for q = 0. The other two excitations are A-Cand A-D that both lie at about 24-25 meV.

Fig. 2 shows the temperature evolution of the scat-tering function S(Q,ω) measured with incident neutronwavelength of λ = 1.5 A and Fig. 3 shows temperatureevolution of the scattering function S(Q,ω) measuredwith λ = 1.1 A. At T = 2 K we observe two inelas-tic peaks at about E = 8 meV and E = 24 meV. Theenergy of these peaks decreases with increasing temper-ature. However whereas the energy of the first peak atE = 8 meV becomes almost zero at TN = 38 K, theenergy of the second peak decreases and then attains afinite value at TN and stays constant at about 19 meVat higher temperatures. Fig. 4 shows the results of thefit of the low-Q data from CoF2 in the low-energy rangein the left two panels. Two Gaussian peaks have beenfitted to account for the elastic peak at zero energy andthe inelastic peak which is at about E = 8 meV at T = 2K. The energy of the inelastic peak decreases with in-creasing temperature and becomes zero at TN = 38 Kgiving rise to quasielastic scattering, which has been fit-ted with a Lorentzian function. The results of the fitwith Gaussian functions to the higher-energy peaks areshown in Fig. 5. At low temperatures below TN = 38 K,three peaks could be identified and they were fitted withthree Gaussian functions. At higher temperatures only abroad peak could be identified and has feen fitted with asingle Gaussian function. Fig. 6 (a) shows the temper-ature variation of the energy and the half-width at half-maximum (HWHM) of the low-energy inelastic magneticpeak of CoF2. In Fig. 6 (b) the temperature variation ofthe energy of the higher-energy inelastic magnetic peakof CoF2 has been shown. The temperature variation ofthe low energy inelastic peak which is at about 8 meVat 2 K shows clearly that this peak is due to magneticexcitations. The energy of this peak decreases continu-ously and becomes almost zero at TN = 38 K giving riseto quasielastic scattering of Lorentzian form. The higherenergy peak at about 24 meV on the other hand seemsto be a hybrid peak due to the magnon-phonon coupling.The peak persists at temperature higher then TN and itsenergy remains constant at about 19 meV.

Fig. 7 shows the Q dependence of the intensity forthe inelastic peaks at E = 7.3, 14.0, 23.5, 31.6 and 43.0meV at T = 2, 20, 40, 70 and 100 K. Fig. 7 (a) showsthe Q dependence of the peak at E = 7.3 meV for sev-eral temperatures. The intensity of the peak at E = 7.3meV at T = 2 K decreases continuously with increasingQ and then becomes more or less constant at higher Q.This indicates that the inelastic peak is of mostly mag-netic origin. This is also supported by its intensity andQ dependence at T = 100 K. However even at T = 100 Kthe intensity is not zero. This suggests that there existssome phonon contribution to this peak as well. The Qdependence at T = 100 K shows that the intensity in-creases slightly at higher Q and this also confirms that

there exist some phonon contribution to this peak. Fig.7 (b) shows the Q dependence of the inelastic peak at E= 14 meV at several temperatures. The intensity of thispeak first decreases with increasing Q and then increasesvery much at higher Q. This suggests that the peak hasa hybrid phonon and magnon character. The Q depen-dence of such a hybrid peak is also expected to be moreexotic. Fig. 7 (c) shows the Q dependence of the inelasticpeak at E = 23.5 meV. Its Q dependence is very similarto the peak at E = 7.3 meV shown in Fig. 7 (a). Evenits temperature dependence is very similar. The inten-sity of the peak at all Q is higher at lower temperature.The decrease of its intensity with increasing Q suggeststhat it is magnon like. However like the peak at E = 7.3meV the peak at E = 23.5 also shows slight tendency ofincreasing in intensity at higher Q suggesting that thepeak has some phonon like character. Fig. 7 (d) showsthe Q dependence of the inelastic peak at E = 31.6 meV.It looks very similar to that at E = 14 meV and con-sists clearly of both phonon and magnon contributions.Lastly the Q dependence of the inelastic peak shown inFig. 7 (e) shows that this peak is almost pure phononlike because its intensity increases with Q and tempera-ture. We have fitted the Q dependence of the intensityfor the peaks at 14 and 31.6 meV assuming them to con-sist of both magnon and phonon contributions and alsoassuming that the Q dependence has oscillatory cosinewave form due to the periodicity of the lattice. We fittedthe intensity by the equation

I(Q) = a1 + a2Q+ a3Q2 + a4 cos

(2π

a5Q+ a6

)(3)

which consists of a polynomial plus an oscillatory cosinefunction. Here a1, a2, and a3 are coefficients of a sec-ond order polynomial and a4 is the amplitude and a5 isthe periodicity of the oscillatory part superimposed onthe polynomial function and a6 is the phase. This is amuch simplified function but since it contains term pro-portional to Q describing Q dependence of the magneticintensity and a term proportional to Q2 describing theQ dependence of the phonon intensity and has also a su-perimposed oscillatory term describing the effect of lat-tice periodicity, it describes the intensity variation ratherwell. The fitted periodicity is in agreement with the aver-age lattice periodicity. Since the fits for different temper-ature in Fig. 7 (b) is not clearly visible we show a typicalfit at T = 2 K separately in Fig. 7 (f). The fit looksrather convincing indeed. In fact the Q dependences ofthe intensities for all five energies shown in Fig. 7 (a-f)show oscillations but the statistics and also the resolu-tion of the data shown in Fig. 7 (a), 7 (c) and 7 (e) arenot good enough for reasonable fits.

In conclusion we investigated the phonon and spin dy-namics of CoF2 by inelastic neutron scattering on powdersamples by time-of-flight neutron spectroscopy with un-polarized neutrons. The temperature and the Q depen-dence of the intensity of the inelastic signals enabled us toidentify their phonon, magnon or hybrid origin at least to

6

a first approximation. We note that there exist consider-able phonon-magnon interaction in this simple antiferro-magnetic system also evidenced by the strong magnetoe-lastic coupling observed by neutron diffraction3. How-

ever, in order to be certain about the hybrid nature ofthese inelastic peaks and evaluate quantitatively phonon-magnon interaction strength, polarized neutron scatter-ing investigation is desirable.

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arxiv:1301.5788v1 [cond-mat.mtrl-sci] 24 jan 2013 · 2 investigated by time-of-ight neutron...

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