Ultrasonic studies of the magnetic phase transition in MnSi
aa r X i v : . [ c ond - m a t . s t r- e l ] D ec epl draft Ultrasonic studies of the magnetic phase transition in MnSi
A.E. Petrova and S. M. Stishov (a)
Institute for High Pressure Physics - Troitsk, Moscow Region, Russia
PACS – Phase transitions and Curie point
PACS – Elastic moduli
PACS – Static properties
Abstract. - Measurements of the sound velocities in a single crystal of MnSi were performed inthe temperature range 4-150 K. Elastic constants, controlling propagation of longitudinal wavesreveal significant softening at a temperature of about 29.6 K and small discontinuities at ∼ The nature of the helical magnetic phase transition inMnSi, occurring at about 29 K at ambient pressure, hasbeen a subject of significant controversy. For a long timethe phase transition in MnSi at ambient pressure was gen-erally believed to be second order or continuous, thoughthere have been no proper proofs for that conclusion. Inthis context, the evolution of magnetic susceptibility ofMnSi at high pressure [1] also seemingly indicated a tri-critical point on the phase transition line, where the sec-ond order phase transition became first order. As a result,the idea was borne that tricritical behavior was a genericfeature of itinerant ferromagnets at low temperatures [2].Recent precise measurements of the lattice constants ofMnSi apparently confirmed a first order character of thephase transition in MnSi at high pressure and low temper-ature [3]. However, studies of thermodynamic and trans-port properties of a high quality single crystal of MnSi [4,5]strongly suggested a first order nature of the magneticphase transition in MnSi even at ambient pressure, a re-sult in direct contrast with the previous assumption.But arguments presented in [4, 5] were not quite com-plete because in real solids inevitable imperfections willalmost always smear first order phase transitions over cer-tain temperature and pressure intervals. Consequently,variations of heat capacity and thermal expansion in thebest case would assume the form of a broadened δ –function (a) E-mail: [email protected] that might be interpreted as evidence for divergence typ-ical of second order phase transitions.In Refs. [4, 5], sharp peaks in heat capacity, thermalexpansion coefficient and temperature derivative of resis-tivity in a temperature interval less than 0.1 K signifiedthe magnetic phase transition in MnSi. Sharpness of thepeaks and obvious lack of mean-field-like anomalies favor afirst order nature of the phase transition. To confirm thispoint, it is desirable to perform an experiment that mini-mizes effects of smearing the phase transition. Ultrasonicstudies of the phase transition in MnSi seems most appro-priate in this sense, because sound velocities, which arerelevant to this case, are defined by local values of elasticconstants and their evaluation does not involve a proce-dure that requires probing certain ranges of temperatureor pressure. The high sensitivity and potential accuracyof ultrasonic measurements make them a valuable tool forstudying subtleties of phase transitions.There are three possible scenarios for the behaviors ofsound velocities and sound attenuation at a phase transi-tion [6–8]:a) Finite jumps in sound velocity and attenuation coef-ficient are expected at a second order, mean-field phasetransition.b) A power-law divergence in sound velocity and attenua-tion coefficient are characteristic of a second order phasetransition with strong fluctuations. The correspondingcritical exponent for sound velocity is expected to be equalp-1. E. Petrova et al. that of heat capacity.c) At a first order phase transition, there will be a finitejump of sound velocity and δ –function like behavior of theattenuation coefficient.
20 40 60 80 1003.223.243.263.283.30
Temperature (K) E l a s t i c m odu li ( d y n / c m ) C L =(c +c +2c )/2
20 40 60 80 1003.123.143.163.183.20 C
28 29 30 313.1303.1353.140
28 29 30 313.2243.2283.232
Fig. 1: Elastic moduli of MnSi, controlling propagation of lon-gitude sound waves.
20 40 60 80 1001.2601.2631.266 E l a s t i c m odu li ( d y n / c m ) Temperature (K)C
20 40 60 80 1001.1801.1821.184
C’=(C -C )/2
28 29 30 311.18001.18051.18101.1815
Fig. 2: Shear elastic moduli of MnSi.
We report results of ultrasonic studies of a single crystalof MnSi in the temperature range 4-150K. In the courseof these studies, several runs were performed using digi-tal pulse-echo techniques (see [9] and references therein).Three samples of MnSi of 4.7, 2.15 and 5.05 mm thick-nesses and with orientations along [110] and [100] were cutfrom a high quality big single crystal, characterized pre-viously [4, 5]. The corresponding surfaces of the sampleswere made optically flat and parallel. The 36 ◦ Y ( P-wave)and 41 ◦ X (S-wave) cut
LiN bO transducers were bondedto the samples with silicon grease. Temperature was mea-sured by a calibrated Cernox sensor with an accuracy of 0.02 K.A sinusoidal pulse of ∼
50 MHz was sent to the trans-ducer that excites a sound wave, experienced multiple re-flections inside the sample. The distance between two ar-bitrary reflections corresponds to a multiple of the roundtrip travel time, which is determined by performing across-correlation between two selected reflections. Thespeed of sound and elastic constants are calculated us-ing the known thickness and density of the samples andthe relationship c ij = ρV . The precision of the soundvelocity determinations is no worse than one part in 10 ;whereas, the absolute accuracy is about 0.1% due to un-certainty connected with a phase shift at the transducer–sample bond. The precision and accuracy of the calculatedelastic constants are basically comparable but may be notas good when calculated using data from different runswith the samples of different orientations.The elastic constants c and c and the combinations C L = ( c + c + 2 c ) / C ′ = ( c − c ) / /L log ( A /A
2) (fig. 4). The precision of the attenua-tion data is about 0.5 %, though the absolute accuracy isuncertain. These data are qualitatively comparable withlow resolution results on sound velocity [10] and soundattenuation measurements [11]. It is of interest to com-pare the present data on MnSi with elastic properties ofits structural analog FeSi [12]; numerical values of the cor-responding elastic constants appear to be rather close inboth materials.As is shown in fig. 1, the elastic constants c and C L = ( c + c + 2 c ) /
2, controlling propagation of lon-gitudinal waves in the [100] and [110] directions, revealpronounced rounded dips at about 29.6 K, with disconti-nuities at the low temperature side of the dips at ∼ c and C ′ = ( c − c ) / C ′ = ( c − c ) / K = ( c + 2 c ) /
2, the average shear mod-ulus G = ( c − c + 3 c ) / V /V o . It is clearly seen in fig. 3 that the small drop inG is associated with the volume increase resulting frommagnetic ordering and does not carry any traces of soft-p-2itle
20 40 60 80 1001.181.201.221.24
20 40 60 80 1001.2281.232
20 40 60 80 100-7.2-7.0-6.8-6.6 ∆ V / V - Temperature (K)
20 40 60 80 1001.561.581.601.62 K ( d y n / c m ) K=(c +2c )/3
28 29 30 311.5701.5751.580 G ( d y n / c m ) G=(c -c +3c )/5 Fig. 3: Bulk modulus (K), average shear modulus (G) and vol-ume change (∆
V /V o ) of MnSi [5]. Averaging is done accordingto the Voigt approximation [13]. The dash–dot line and arrowindicate position of the rounded minimum of K. ening as the bulk modulus certainly does. Moreover, thetotal variation of the shear modulus G is an order of mag-nitude less than the variation of the bulk modulus K inthe phase transition region. Bearing in mind the itinerantnature of magnetism in MnSi, this drastic difference inbehavior of the longitudinal and transverse elastic prop-erties may imply the essential decoupling of electron andionic subsystems in this material in respect to the mag-netic transformation.Now we turn to the sound attenuation. Two curves il-lustrating attenuation of longitude waves in [100] and [110]directions are shown in fig. 4. The double peak struc-ture makes these curves look like almost exact copies ofthose characterizing behavior of the heat capacity, ther-mal expansion and resistivity in the vicinity of the phasetransition in MnSi [4, 5]. It needs to be pointed out thatthe sharp peaks in sound attenuation are quite symmetricand do not look like they should to be associated with asecond order phase transition [6–8]. This behavior of theattenuation most probably is connected with violation ofadiabatic conditions arising from a finite entropy changeat the first order phase transition in MnSi. In this case,the sound attenuation should behave similar to a heat ca- Our data on attenuation of transverse waves indicate the exis-tence of small but distinct maxima at a temperature of about 29 K.
27 30 333456 A tt enua t i on ( d B / c m ) Temperature (K) <100><110>
Fig. 4: Attenuation of the longitudinal sound waves in MnSi.No background is subtracted. pacity curve at a slightly rounded first order phase tran-sition that can be described as a smeared δ –function [4].To this end, all features observed match to the case c),corresponding to a first order phase transition.At the same time, the current ultrasonic study exposesanother important fact, making obvious that the magneticphase transition in MnSi, occurring at 28.8 K, is just a mi-nor feature of the global transformation that is marked bythe rounded maxima or minima of heat capacity, thermalexpansion coefficient, sound velocities and absorption, andthe temperature derivative of resistivity [4,5]. Behavior ofthe bulk modulus of MnSi (fig. 3) indicates a tendencytoward a volume instability that may characterize thistransformation as an incomplete second order transition.But, the nature of the global transformation still remainsa puzzle, though a number of plausible spin structures(blue quantum fog [14], spin crystal [15] and skyrmiontextures [16]), suggested for the paramagnetic phase ofMnSi may be pertinent. On the other hand, relevant ex-perimental data are too scarce to solve the problem un-ambiguously. What we actually know from neutron scat-tering experiments [17–19] is that, on cooling MnSi andits closest analog FeGe, a quasi-long-range helical order(QLRO) forms at temperatures slightly above the phasetransition temperature. However, for yet unknown reasonsthe state with QLRO can not reach a true long range or-der in the continuous way. This may suggest identificationof the QLRO state as a frustrated magnetic state, possi-bly emerging as a result of competition between differentdirections of the spin helix [5, 19]. Consequently, a firstorder phase transition is needed to unlock the situation. ∗ ∗ ∗ In conclusion, we would like to acknowledge valuableadvice of Cristian Pantea, Steven Jacobsen, Baosheng Lip-3. E. Petrova et al. and Yuri Pisarevskii concerning the ultrasonic equipment.Elena Gromnitskaya, Cristian Pantea and Izabela Stroemade some preliminary measurements of the elastic prop-erties of MnSi. Technical help of Vladimir Krasnorusski isgreatly appreciated. Thomas Lograsso provided the singlecrystal of MnSi. Authors are thankful to J.D. Thompsonfor valuable remarks. We appreciate support of the Rus-sian Foundation for Basic Research (grant 09-02-00336),Program of the Physics Department of RAS on StronglyCorrelated Systems and Program of the Presidium of RASon Physics of Strongly Compressed Matter.
REFERENCES[1]
C. Pfleiderer, G.J. McMullan and
G.G. Lonzarich , Physica B , (1995) 847.[2] D. Belitz, T.R. Kirkpatrick and
Thomas Vojta , Phys.Rev.Lett. , (1999) 4707.[3] C. Pfleiderer, P. B¨oni, T. Keller, U.K. R¨oler and
A. Rosch , Science , (2007) 1871.[4] S.M. Stishov, A.E. Petrova, S. Khasanov, G.Kh.Panova, A.A. Shikov, J.C. Lashley, D. Wu and
T.A. Lograsso , Phys.Rev. B , (2007) 052405.[5] S.M. Stishov, A.E. Petrova, S. Khasanov, G.Kh.Panova, A.A. Shikov, J.C. Lashley, D. Wu and
T. A.Lograsso , J. Phys.Condens. Matter , (2008) 235222.[6] L.D. Landau and
I.M. Khalatnikov , Dokl. Acad. NaukSSSR , (1954) 469.[7] C.W. Garland , Physical Acoustics , edited by
W.P.Morse and R.N. Thurston , Vol.
VII (AcademicPress)1970[8]
B. L¨uthi, T.J. Moran and
R.J. Pollina , J. Phys.Chem. Solids , (1970) 1741.[9] Cristian Pantea, Dwight G. Rickel, AlbertMigliori, Robert G. Leisure, Jianzhong Zhang,Yusheng Zhao, Sami El-Khatib and
Baosheng Li , Rev.Sci.Inst. , (2005) 114902.[10] G.P. Zinov’eva, L.P. Andreeva, Z.A. Istomina and
P.V. Geld , Solid State Physics , (1977) 1164.(in Rus-sian)[11] S. Kusaka, K. Yamamoto, T. Komatsubara and
Y. Ishikawa , Solid State Communications , (0) , 925(1976).[12] J.L. Sarrao, D. Mandrus, A. Migliori, Z. Fisk and
E. Bucher , Physica B , (1994) 478.[13] W Voigt , Lehrbuch der Kristallphysik (Teubner, Leipzig)1928.[14]
Sumanta Tewari, D. Belitz and
T.R. Kirkpatrick , Phys.Rev.Lett. , (2006) 047207.[15] B. Binz, A. Vishwanath and V.Aji , Phys.Rev.Lett. , (2006) 207202.[16] U.K. R¨oler, A.N. Bogdanov and
C. Pfleiderer , Na-ture , (2006) 797.[17] Y. Ishikawa and
M. Arai , J.Phys.Soc.Jpn , (1984)2784.[18] S. V. Grigoriev, S. V. Maleyev, A. I. Okorokov,Yu. O. Chetverikov, R. Georgii, P. B¨oni, D. Lam-ago, H. Eckerlebe and
K. Pranzas , Phys. Rev. B , (2005) 134420. [19] B. Lebech, J. Bernhard and
T. Freltoft , J. Phys.Condens. Matter , (1989) 6105.(1989) 6105.