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1

A search for incipient lattice instabilities in MgB2 by anelastic spectroscopy

F.Cordero,1 R. Cantelli,2 G. Giunchi,3 and S. Ceresara3

1CNR, Istituto di Acustica “O.M. Corbino“, Via del Fosso del Cavaliere 100,

I-00133 Roma, Italy and Unita INFM-Roma1, P.le A. Moro 2,I-00185 Roma, Italy2Universita di Roma “La Sapienza“, Dipartimento di Fisica, and

Unita INFM-Roma 1, P.le A. Moro 2, I-00185 Roma, Italy3 EDISON S.p.A, Foro Buonaparte 31, I-20121 Milano, Italy

We measured the complex dynamic elastic modulus of MgB2 between 1.3 and 650 K at frequenciesincluded between 5 and 70 kHz. The main purpose of this work is to search for possible latticeinstabilities, which could contribute to enhance the electron-phonon coupling. The Young’s modulusindeed presents an anomalous softening on cooling below 400 K, and hysteresis between coolingand heating. Such anomalies, however, can be accounted for by intense relaxation processes withmaxima between 50 and 150 K, whose nature is not yet clarified, and by geometric defects generatedby the anisotropic thermal expansion between the grains during thermal cycling. Additional intenserelaxation processes are observed above room temperature, and their possible origin is discussed.The influence of the highly anharmonic in-plane vibration modes of the B atoms on the elastic andanelastic properties of MgB2 is discussed in detail.

1

The discovery of superconductivity with Tc = 39 Kin MgB2 is arousing great interest, since this intermetal-lic compound seems to be a simple BCS superconductor,without any of the other mechanisms for high-Tc super-conductivity that are hypothesized for the cuprate su-perconductors. Electronic band calculations1,2 indicatea predominant hole conduction in the two-dimensional σband from the spxpy orbitals of B, with an extremelylarge deformation potential for the B bond stretchingmode, which dominates the electron-phonon coupling.The most recent first principle calculations of the elec-tronic bands and phonons2 indicate that the zone-centeroptical mode with E2g symmetry, involving vibrationsof the B atoms within the ab plane, has very flat anhar-monic potential, and can account for the observed Tc witha calculated electron-phonon coupling constant λ ≃ 1.An experimental indication of the anharmonicity of thatmode comes from the high value of its Gruneisen parame-ter, evaluated from the shift of the corresponding Ramanband with pressure.3 Still, it is important to ascertainwhether there are other anharmonic or unstable modesthat are not easily seen by phonon DOS measurements2

and other spectroscopies, but can contribute to the largeelectron-phonon coupling. In many cases of BCS su-perconductivity with high Tc, the strongly anharmonicphonons causing the enhanced electron-phonon couplingalso produce a structural instability, which in general pro-foundly affects the acoustic properties of the material.The best known examples are the A15 superconductors,4

where a clear correlation exists between the occurrence ofa transformation from cubic to tetragonal, often assimi-lated to a martensitic transformation, and a high valueof Tc. It should be emphasized that the occurrence andthe entity of the structural instability in the A15 com-pounds is rather sensitive to the material preparation,and the easiest method for revealing it was the obser-vation of softening of the elastic moduli on cooling be-low room temperature.4 Also the superconducting per-ovskite BKBO presents structural instabilities,5,6 whileultrasonic absorption peaks in BPBO are interpreted interms of strong phonon anharmonicity.7

In the case of MgB2 a possible unstable mode is theB1g mode of the out-of plane vibrations of the B atoms,even though the first principle calculations indicate a stiffand harmonic mode.2 In fact, it has been observed thatMgB2 is close to the condition of buckled B planes in aphase diagram of the AB2 compounds versus charge den-sity and B-B distance.8 In addition, when the hole densityin the B planes is progressively reduced by partially sub-stituting Mg with Al, the c lattice parameter collapsesin correspondence with the disappearance of supercon-ductivity, suggesting that MgB2 is close to a structuralinstability at slightly higher electron density.9

The anelastic spectroscopy is very sensitive to struc-tural instabilities, and therefore we measured the com-plex dynamic Young’s modulus of MgB2 between 1.3 and650 K, searching for possible signs of lattice instabili-ties not yet detected by neutron diffraction2,10 and other

spectroscopies.The sample was prepared by reacting Mg and B in a

sealed stainless steel tube, lined with Nb foil, at 950 oCfor 2 hours. Further preparative details are reportedelsewhere.11 A bar 39× 6× 1 mm3 was cut by spark ero-sion. The mass density was ρ = 2.27 g/cm3, 86% of thetheoretical one. The superconducting transition, mea-sured by resistivity, was centered at 39.5 K, with a widthof 1 K. The X-ray diffraction spectra showed the presenceof a residual minority phase of metallic Mg, presumablyat the grain boundaries. The sample was suspended withthin thermocouple wires and electrostatically excited onits flexural modes, whose resonant frequencies are givenby fn = αnh/l2

√

E′/ρ, where E (ω, T ) = E′ + iE′′ isthe complex dynamic Young’s modulus,12 l and h thesample length and thickness, and αn numerical constantsdepending on the vibration mode. The 1st, 3rd and 5thmodes could be excited, at ω/2π = 5.7, 38 and 73 kHzrespectively at 1 K. The value of the Young’s modulus atroom temperature, not corrected for the sample poros-ity and the Mg traces, is E = 167 GPa. The imaginarypart is related to the elastic energy loss coefficient Q−1 =E′′/E′, which was measured from the width of the reso-nance curves.

Figure 1 shows the anelastic spectrum between 1 and620 K, measured by exciting the first flexural mode. Afirst result is the absence of anomalous softening be-low 150 K, so excluding a scenario like that of theA15 compounds, where the strong phonon anharmonic-ity is due to a mode which becomes unstable slightlyabove Tc. Still, anomalous softening and hysteresis be-tween cooling and heating are observed between 400 and150 K. Such anomalies, however, should not be straight-forwardly attributed to a structural instability, sincemicroplastic phenomena connected with the anisotropicthermal expansion of the hexagonal crystallites are ex-pected. Neutron diffraction10 experiments have shown alarge anisotropy of the thermal expansion and compress-ibility, both being about twice larger along the c axis.These plastic phenomena have already been observedin other hexagonal polycrystals, like Zn, Co and Cd,13

whose thermal expansivity is sufficiently anisotropic andthe critical stress σc for plastic deformation sufficientlylow to yield the formation of dislocations during ther-mal cycling. The increase of the density of geometricaldefects results in a reduction of the modulus, and thecompetition between the defect generation and annihila-tion, the latter especially at high temperature, explainsthe hysteresis between cooling and heating above 130 K.Below that temperature the thermal expansion becomesso small that the stress generated at the grain boundariesis below σc and no further modification in the geometricaldefects occurs. We do not know what type of geometri-cal defects accommodates the strains generated duringthermal cycling in MgB2, but they are not exclusivelyaccommodated at the Mg impurities between the grains;in fact, after heating the sample up to 690 K in vacuum,about 8% of Mg was lost, the absorption component at

2

80 K was reduced, but the hysteresis of the modulus be-low room temperature was unchanged. Also the elasticenergy loss is partially due to the motion of the geomet-rical defects, at least above 150 K, since it has a compo-nent which increases with the cooling/heating rate, andexhibits hysteresis, as shown by the open symbols in Fig.1. The plastic phenomena can account for the irregu-larities and hysteresis of E (T ) between 150 and 400 Kbut probably not completely for the plateau of E′ (T )between these temperatures. This plateau could alsohave another origin; indeed, a small softening of somemodes appears in the phonon DOS measured by neutronscattering2 around 325 K. We shall see however, thatalso the relaxation processes observed below room tem-perature in the elastic energy loss might cause a similareffect.

The intense absorption peak at 475 K is due to athermally activated relaxation process,12 namely it canbe described as a contribution δE = −∆/ (1 + iωτ) toE (ω, T ), where the relaxation time follows the Arrheniuslaw τ = τ0 exp (W/T ). The absorption peak and mod-ulus dispersion of this type of processes shift to highertemperature with increasing frequency. Due to the highdamping, the higher frequency modes could be followedonly up to 440 K, but exhibited such a shift, and couldbe interpolated by the expression

δE = −∆/ [1 + (iωτ)α] , (1)

where the parameter α = 0.65 reproduces the broaden-ing of the spectrum of the energy barriers (α = 1 in themonodispersive case), W = 0.89 eV and τ0 = 10−14 s.The dotted lines in Fig. 1 are the corresponding contri-butions to the absorption and to E′, plus the tail of an-other process at higher temperature, probably connectedwith grain-boundary motion or with the loss of Mg. Thevalue of τ0 is typical of point defect motion, as could bethe migration of Mg atoms and vacancies, but, due tothe bad quality of the higher frequency data, we cannotexclude that τ0 is actually longer and therefore connectedwith the motion of geometrical defects.

In view of the occurrence of plastic phenomena dur-ing thermal cycling above ∼ 130 K, it is likely that theabsorption contains an important contribution from therelaxation of extended lattice defects. Some contributionmay also be expected from Mg at the grain boundaries,since deformed Mg presents an extremely broad absorp-tion maximum14 reminiscent of the curves found herebelow room temperature; nevertheless, the contributionof metallic Mg is weighted with its molar fraction, i.e.few percents, and considering that the intensity of thepeak in deformed Mg14 never exceeds 2×10−3, it cannotaccount for elastic energy loss curves of Fig. 2. In addi-tion, at Tc it is evident a change of the slope of both theimaginary and real parts of E (T ), which must be con-nected with processes occurring in the superconductingbulk of MgB2. Another unusual feature of the anelasticspectrum below 200 K is that the curve simultaneously

measured at higher frequency, instead of being simplyshifted to higher temperature, as for any thermally acti-vated relaxation process, is about a factor 1.32 more in-tense than that at lower frequency. Finally, it should benoted that the spectrum below 100 K is perfectly repro-ducible during various cooling and heating cycles; con-sidering the extreme sensitivity of the dislocation peaksto the sample state, and the occurrence of plastic phe-nomena during thermal cycling, it is unlikely that thelow temperature spectrum is exclusively connected withgeometrical defects.

It is tempting to relate the low temperature anelas-tic spectrum with the highly anharmonic E2g opti-cal mode which is thought responsible for the strongelectron-phonon coupling. Indeed, a relaxation mech-anism involving the soft optical modes responsible forthe structural phase transformations in perovskites hasbeen proposed by Barrett,15 and later adopted in or-der to explain the ultrasonic absorption in the per-ovskite superconductor7 BaPb1−xBixO3 and even insome cuprate superconductors.16 The mechanism is anal-ogous to the Akhiezer-type absorption,12 but instead ofthe acoustic phonons, it is based on the existence ofoptical phonons with a large Gruneisen constant, γ =−d (lnω0) /dε, namely whose frequency ω0 strongly de-pends on a strain ε. The flexural vibration of the sam-ple causes a non homogeneous uniaxial strain ε along thesample length and modulates the energies of the phononsaccording to their Gruneisen parameters. The phononssystem goes out of thermal equilibrium if different modeshave different values of γ, and the equilibrium is es-tablished with an effective phonon relaxation time τeff.The change of the population is reflected in the dynamicelastic modulus, which acquires an out of phase imag-inary component and a temperature dependent soften-ing. It is appropriate to discuss this mechanism, sincethe Gruneisen constant of the E2g optical mode has beenfound to have a very large value at room temperature,3

γop = 13

d ln νd ln a

≃ 3.9, a being the lattice constant, whilenormal values range between 1 and 2. Barrett solved thecase in which the system is schematized with an opticalmode with frequency ω0 and large γop, and the acousticmodes with normal γac; since we are considering uniax-ial strain ε = da/a instead of volume change, we defineγ = 3 (γop − γac) ≃ 7, where γac ∼ 1.5 is assumed. Thecontribution of the phonon relaxation to the dynamicYoung’s modulus E is15

δE = −γ2

v0

TCop

1 + iωτeff

1 + (ωτeff)2

(2)

τeff = τop +Cop

Cac

τac (3)

where v0 is the cell volume, Cac and Cop are the specificheat of the acoustic and optic modes per molecule andτac and τop are their relaxation times. The elastic energyloss coefficient is, for ωτeff ≪ 1,

3

Q−1 =E′′

E′=

γ2

v0ETCopωτeff = ∆(T ) ωτeff ; (4)

where Cop has the usual expression

kB [(hω0/2T )/ sinh (hω0/2T )]2

with3 hω0/kB = 830 K,E = 167 GPa and v0 = 29 × 10−24 cm3; the result-ing relaxation strength ∆ is 6.6 × 10−5 at 100 K and6.5 × 10−3 at 300 K, but has to be multiplied by thefactor ωτeff, which is very small. In fact, the phonon re-laxation rates and therefore τ−1

eff should exceed 1012 s−1;τac can be deduced from the lattice thermal conductivityκ through the collision formula κ = 1

3Cacv

2τac. Assum-

ing κ ≃ 0.1 W/(cm K) found17 above 100 K, an averagesound velocity somewhat lower than that deduced from√

E/ρ = 8× 105 cm/s, and a Debye lattice specific heat,it results τac < 10−12 s; τop can be assumed to be of thesame order of magnitude or even smaller. Since our mea-suring frequency is only ω ∼ 104 − 105, the ωτeff factormakes the imaginary part negligibly small, and thereforethe absorption peaks around 100 K cannot be attributedto an Akhiezer type mechanism involving the anharmonicoptic phonon. Still, the real part of δE contributes to themodulus softening with a substantial δE/E = −∆(T );the continuous curve in Fig. 1 is calculated with theabove parameters, assuming a temperature independentγ = 7.

Although the mechanisms producing the acoustic ab-sorption below 150 K remain unclear, it is worth checkingwhether they can also be the origin of the anomalous soft-ening observed below 400 K. The imaginary part of thedynamic modulus should be describable in terms of a su-perposition of relaxation processes like Eq. (1), wherethe phenomenological parameter α ≤ 1 describes thebroadening of the spectrum of relaxation times aroundan average value τ , and the relaxation strength is gen-erally ∆ ∝ 1/T or nearly constant. In order to repro-duce the experimental curves, we need a ∆ (T ) stronglyincreasing with temperature, and therefore we adopt∆ (T ) ∝ 1/

[

T cosh2 (E/2T )]

, appropriate18 for relax-ation occurring between states which differ in energy byE. The solid lines roughly interpolating the absorptiondata in Fig. 2 are obtained with the following parame-ters, τ = 5×10−14 exp (800/T ) s, α = 0.2, E/kB = 230 Kfor the peak at 70-80 K, and 5 × 10−15 exp (2900/T ) s,α = 0.4, E/kB = 130 K for the peak at 150 K. Wedo not attribute a particular meaning to these values,since it is possible that the two peaks are superimposedto a broader maximum, which has not been considered.The fit is therefore only indicative, but it shows that theanomalous softening below 400 K may be accounted forby the same relaxation processes which cause absorptionat low temperature; in fact, the continuous line in theupper panel of Fig. 2 is the real part corresponding tothe curves in the lower panel and even exceeds the ex-perimental softening. The fact that the magnitude of thenegative step of E′ is much larger than the amplitude ofthe E′′ peaks is a consequence of the non additivity ofthe amplitudes of the elementary peaks, whose maxima

occur at different temperatures.The presence of these intense relaxation processes also

masks the negative jump and positive change of theslope of the elastic moduli due to the superconductingtransition;4 in fact, the latter are of the order of 10−4

or less, while the low temperature relaxation processescause a modulus defect of the order of 10−3 at 40 K, andthey are affected by the transition to the superconductingstate, as observed in both the real and imaginary parts ofE. In order to analyze the intrinsic changes of the mod-uli at Tc one should be able to subtract the relaxationalcontribution. For this reason, although we can see theexpected jump and change of slope of E′ (T ) below Tc,we omit the discussion of this issue.

In conclusion, we measured the complex dynamic elas-tic modulus of MgB2 with the principal aim of searchingfor possible lattice instabilities, which might contributeto enhancing the electron-phonon coupling without beingyet revealed by other spectroscopies. The Young’s mod-ulus indeed presents an anomalous plateau on coolingbetween 400 and 150 K, and hysteresis between coolingand heating. The softening, however, can be accountedfor by intense relaxation processes with maxima between50 and 150 K, whose nature is not yet clarified. Thehysteresis is explainable in terms of generation and anni-hilation of geometric defects due to the stresses generatedby the anisotropic thermal expansion between the grainsduring thermal cycling. The influence of the highly an-harmonic vibration modes of the B atoms on the elasticand anelastic properties of MgB2 is discussed in detail;such anharmonic phonons produce negligible acoustic ab-sorption at the low frequencies of the present study, butthey may substantially contribute to the total softeningof the elastic modulus with increasing temperature.

Edison acknowledges the Lecco Laboratory of theCNR-TeMPE, Milano, Italy, for making available its fa-cilities for materials preparation.

1 J.M. An and W.E. Pickett, Phys. Rev. Lett. 86, 4366(2001).

2 T. Yildirim et al., cond-mat/0103469.3 A.F. Goncharov et al., cond-mat/0104042.4 L.R. Testardi, Physical Acoustics. ed. by W.P. Mason andR.N. Thurston, p. 193 (Academic, New York, 1973).

5 S. Zherlitsyn et al., Eur. Phys. J. B 16, 59 (2000).6 M. Braden et al., Phys. Rev. B 62, 6708 (2000).7 S. Mase et al., Solid State Commun. 57, 227 (1987).8 A. Bianconi et al., cond-mat/0102410.9 J.S. Slusky et al., cond-mat/0102262.

10 J.D. Jorgensen et al., cond-mat/0103069.11 Edison S.p.A patent pending.12 A.S. Nowick and B.S. Berry, Anelastic Relaxation in Crys-

talline Solids. (Academic Press, New York, 1972).

4

13 M.H. Youssef and P.G. Bordoni, Phil. Mag. A 67, 883(1993).

14 R.T.C. Tsui and H.S. Sack, Acta metall. 15, 1715 (1967).15 H.H. Barrett, Physical Acoustics. ed. by W.P. Mason and

T.N. Thurston, p. 65 (Academic Press, New York, 1970).16 Y. Horie et al., J. Phys. Soc. Jap. 58, 279 (1989).17 E. Bauer et al., cond-mat/0104203.18 F. Cordero, Phys. Rev. B 47, 7674 (1993).

FIG. 1. Real and imaginary parts of the anelastic spectrumof MgB2, measured exciting flexural vibrations at 5 kHz. Thearrows and numbers indicate the order of the cooling andheating runs. The dotted lines in the lower and upper panelare a fit to the high temperature relaxation peaks and the cor-responding modulus defect, respectively. The dashed line isthe estimated softening from the anharmonic E2g optic modeof the in-plane B vibrations, with the value of the Gruneisenparameter measured at room temperature. The gray curveis the experimental Young’s modulus after subtraction of theabove contributions.

FIG. 2. Real and imaginary parts of the anelastic spectrumof MgB2, measured exciting flexural vibrations at 5.7 kHz and73 kHz during the same cooling run. The continuous lines inthe lower panel are a fit assuming broad thermally activatedrelaxation processes with the parameters given in the text;the line in the upper panel is the corresponding contributionto the real part of the elastic modulus. The arrows indicatethe superconducting Tc, at which the curves change the slope.

5

0 100 200 300 400 500 600

10-3

10-2

-0.2

-0.1

0.0

f(0 K) = 5.7 kHz

MgB2

E"/

E'

T (K)

(E

'-E0)

/E0

32

1

optic phonon high-T relaxations optic+relaxation

10 1000

1

2

3

-0.06

-0.04

-0.02

0.00

Tc

5.7 kHz 73 kHzx10-3

E"/

E'

T (K)

MgB2

(E'-E

0)/E

0