Low- and high-energy excitations in the incommensurate antiferromagnet, chromium

Journa1 of Magnetism and Magnetic Materials !77-181 (1998) 1343-i346

ELSEVIER

~,~ Journalof madgnetlsm magnetic materials

Invited paper

Low- and high-energy excitations in the incommensurate antiferromagnet, chromium Yasuo Endoh*, Tatsuo Fukuda Department of P/o'sics, Tohoku UniversiO,, Aramaki Aza Aoba, Aoba-ku, Sendai, 980-7Z Japan

Abstract

Dynamical susceptibility of metallic chromium (Cr) has been elucidated by using neutron magnetic scattering technique. Systematic studies in a wide energy range at various temperatures revealed that magnetic excitations in the antiferromagnetic state of Cr well below N~el temperature are not interpreted by a simple spin-wave mode like in the conventionally ordered magnet. The most prominent feature of spin dynamics in the incommensurate spin density wave (SDW) state is that magnetic excitations consist of two components having incommensurate and commensurate wave vector in the reciprocal space. Energy and temperature dependence of such magnetic excitations is also unusual, which may reflect the characteristic band structure of Cr. ~C,. 1998 Elsevier Science B.V. All rights reserved. Keywords: Excitation - spin wave; Incommensurate phase; Itinerant electrons - antiferromagnetism; Neutron scattering - inelastic; Neutron scattering - polarized; Spin density waves; Two-band model

1. Introduction

Although the antiferromagnetism of metallic chromium (Cr) is put in a category of the older subject in modern magnetism, the dynamical susceptibility or magnetic excitation cannot be understood on the basis of the itinerant antiferomagnetism [-1, 21. Recent inelastic neutron scattering (INS) experiments, in particular, near the N+eI temperature, brought a renewal of interest, since the low-energy magnetic excitations have a really complicated dynamical structure [3 6]. Furthermore, the fluctuation mode of low-energy magnetic excitations is not so simple that the mode switching from the mode of mainly longitudinal to isotropic seems to occur at a threshold energy around 8-9 meV [5]. High-energy INS using pulsed neutron source revealed that magnetic excitations persist even at an excitation energy of 0.5 eV, which can be detected as a relatively sharp peak appeared at the antiferromagnetic superlattice points in Q space [7]. These are just typical examples of many mysterious experimental facts, which must be elucidated by more

* Corresponding author. Tel.: + 81 22 217 6485; fax: + 81 22 217 6489.

systematic INS measurements from a single-domain high-quality, single crystals of Cr. We present here mainly our recent results of the INS experiments together with important other experiments [8-1 i] and then we discuss the dynamical feature of SDW in Cr in the end.

2. Magnetic excitations in the L - S D W state 2.1. Low-energy excitations

A spin-flip transition occurs at TSF = 121 K, below which spin polarization (S) of SDW is parallel to the SDW wave vector, Q~ defined as the longitudinal (L-) SDW. Therefore, for a single-domain single crystal, the elastic magnetic scattering at (1 + c50 0) is forbidden due to the scattering law of $111¢, 7¢ is also defined as the scattering vector. On the other hand, the reflection at ( _+ ~51 0) is allowed because of SLTc. However, the detailed analysis of inelastic magnetic scattering signal at both the reflections immediately tells us neither a simple spin-wave mode nor a simple amplitude fluctuation mode alone. Furthermore, an additional magnetic signal develops at the commensurate or antiferromagnetic superlattice reflection points, (1 0 0) and (0 1 0), which

0304-8853/98/$19.00 1998 Elsevier Science B.V. All rights reserved PII S 0 3 0 4 - 8 8 5 3 ( 9 7 ) 0 0 8 6 0 - 3

1344

Y. Endoh, T. Fukuda /Journal of Magnetism and Magnetic Materials 177-181 (7998) 1343-t346 Z"

q,o~)

15c

~CrLSDW

× 30

state

I

~. 7 ~

k/"

Fig. 1. Constant o scans across the (1 0 0) reciprocal lattice point for 0-40 meV energy transfers• The intensity scale for the co = 0 scans is different from the inelastic scans [8]. makes the total magnetic scattering profile in the Q space rather complicated. We present here typical INS spectra in the L-SDW state as a function of magnetic excitation energy, co in Fig. 1. Essential feature of these magnetic scattering is represented as the sharp response in the Q space for the excitations centered at the S D W wave vector, Qo (incommensurate scattering) and the broad one in Q for the antiferromagnetic superlattice point, QK (commensurate scattering). Combined with the detailed polarization analysis by using polarized-neutron-scattering technique, we found that the fluctuation mode of sharp incommensurate scattering is more or less isotropic, except low-energy part less than the threshold energy. On the other hand, the broad commensurate scattering is dominated by the lonNtudinal fluctuation component• We also found that the Q~ value gradually shifts toward the commensurate position with increase of the excitation energy, which is clearly seen in Fig. 2. These observations are not entirely new but were reported in previous neutron measurements by several authors, particularly, in the vicinity of the N6el temperature (TN = 311 K). It is emphasized here that these systematic as well as unexpected dynamical aspects are for the first time observed in the welt-ordered state far below TSF and T > We will discuss more about this later in the final section.

Tsukuba, Japan. Though the K E N S n e u t r o n source is about 50 times weaker than ISIS, we could observe essentially a lower-energy part (<0.1 eV) of the previous results at ISIS. Combined with the data of a lower-energy part below 40 meV, which were measured on the TOP A N three-axis spectrometer installed at JRR3 reactor at JAERI, we found that the scattering intensity continuously grows to the higher-energy region. The quantitative energy dependence of the scattering intensity was derived converting it to ~dQz'(Q,v)), which shows a broad maximum at around 40-60meV. The other energy dependent quantities such as the structure in Q can hardly be derived with the current performance of the pulsed neutron source. In summary, our view of the magnetic excitations or the dynamical susceptibility in the L-SDW state of Cr is as follows. The incommensurate excitations are clearly visible only in low energies, and in other words, the sharp incommensurate excitations are taken over by the broad commensurate magnetic excitations in higher energies. The crossover occurs approximately at 30 meV, judging from the peak intensity plot with respect to the excitation energy shown in Fig. 3, since the incommensurate peaks Incommensurate Peak Positions

q=~o~m %

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.

~,

~';;

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• O

5

O O

0.94

0.96

0.98

i

1.02

1.04

1,06

(r.l.u.)

Fig. 2. Incommensurate scattering peak positions. The inset is the constant w scans at large-energy transfer and T = 130 K [12], 4

T=54 K

2.2. High-energy excitations

The present systematic studies were motivated by the joint experiments with Mitchel et al. searching high-energy magnetic excitations from antiferromagnetic Cr on the HET spectrometer at ISIS in 1991 [7, 131. We were very surprised and also excited to see a clear evidence of magnetic excitations centered at the commensurate point, whose energy is greater than 0.3 eV. Since a quantitative analysis is time consuming mainly due to the limit of neutron intensity, I switched to elucidate lower-energy INS, which was just summarized in the preceding section. We carried out similar scans at the INC spectrometer installed at KENS, the K E K pulsed neutron source at

y. ,

L 10

I 20

Energy transfer ~ (m3e~/)

'

40

Fig. 3. Energy dependence of the intensity at the incommensurate position (open circles) and the commensurate position (cLosed circles) at 54 K in the LSDW state. The solid lines are guide to the eye [9].

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Y. Endoh, T. Fukuda / Journal of Magnetism and Magnetic Materials 177-181 (I998) 1343-1346

become weaker and broaden beyond the eminent intensity peak at around 20 meV.

3. Temperature dependence of magnetic excitations

Majority of our experiments were performed at low temperatures well below Tsv and TN, since the dynamical susceptibility at the ground state of the SDW in Cr was focused in our studies at least in the initial stage. Then, we naturally have made a search of thermal evolution by selecting designated excitation energies across Tsv, above which spin polarization turns around re/2 with respect to the propagation vector, becoming the transverse (T-) SDW state. As the general trend of the temperature dependence is shown in Fig. 4, the principal feature was readily understood by energy dependence at the lowest temperature. For instance, the commensurate scattering predominates as the temperature increases. The wave vector, Q~, tends to approach to the commensurate wave vector, QK. The appearance of the so-called Fincher-Burke (FB) mode, which is an unusual excitation branch in the lowenergy range less than 6 meV and within two incommensurate peak positions in Q, is still a subtle problem. Though overall dynamical feature shows a little temperature dependence up to 220 K, about 100 ° below TN, it dramatically changes above this temperature, in particular, at lower energies and near TN. We certainly confirmed the appearance of the FB mode, besides further dominant contribution of the commensurate scattering. In this respect, we saw a clear contribution of the incommensurate scattering in the large intensity of the commensurate peak below 233 K for the scanned energies larger than 6 meV, When the temperature approaches further toward TN, incommensurate peak intensity decreases further, but commensurate scattering governs so much that it is impossible to definitely determine the contribution of the incommensurate scattering.

Since we observed little change of the scattering nature through Tsv, we consider that the spin-flip transition has no major contribution in the subject we have discussed. Note that the threshold energy of about 9 meV is very close to TSF equivalent, below which the longitudinal spin fluctuations dominate.

4. Magnetic excitations near and above TN

We measured paramagnetic scattering just above TN (up tO 330 K). We confirmed isotropic magnetic scattering and the major magnetic signal is confined at QK. As shown in Fig. 5, the scattering intensity shows a large energy width with a peak at co = 0, which essentially duplicates the previous studies by Grier et al. [i4"1. The Q width as well as the peak position in Q are quite insensitive to the excitation energy, which significantly differs from paramagnetic scattering in conventional ferromagnets. As for the latter (ferromagnetic) case, appreciable peaks appear at finite Q in the Q scan at fixed energy, though the intensity always shows maximum at co = 0 [-15"1. The X"(Q, co) in this case is well explained by the double Lorenzian in both Q and co. The energy width, F, is approximated by Aq e5 power law. A is correlated with the spin-wave stiffness constant. In general statement, the commensurate scattering itself shows a unique feature that it spreads over large energies, but less temperature-dependent across TsF. On the other hand, the incommensurate scattering reflects more sensitively the SDW ordering.

5. Discussion - challenge to the two-band model

The nesting of the flat surfaces at the Fermi energy in the antiferromagnetic ordered state, or the SDW state in Cr 1-161, creates a pseudo gap. Then the major contribution

Cr30'-30'-30'-60'PG after E~14.7meV T-330K q=(100) 10000 t . . . . . . . . . . . . . . . . . . . . . . . . .

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8000

6000

6000

4000

4000

2000

2000

50

150 r,..O LC0 50 0

090 095 IIN) 105 110 h (r.l.u.)

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Fig. 4. The scans across (1 0 0) by fixing ~) = 15 meV at three designated temperatures of: (a) 54; (b) 76; (c) 133 and; (d) 235 K. The solid lines are the fitted curves and the dashed and dotted curves are the components [9].

0

0 0

5

10

I5 20 Energy (meV)

25

30

Fig. 5. Intensity as a function of energy transfer v) at ~/= (1 0 0) and T ~ 330 K. The solid line is a resolution-broadened Lorentzian fit to the data.

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Y. EJTdoh, Z Fukuda / Journal of MctgJ~e~ismand Magnetic Mawrials 177-181 (1998) 1343-1346

to the total Z"(~f,~o) in the lower-energy range at least below TN, is electron-hole pair excitations connecting the branches between the energy gap. According to the mean-field theories (RPA) in terms of the two-band model which succeeded in the interpretation of the SDW instability, the spin-wave mode is only allowed below the energy gap [17]. As shown in our INS experiments, however, the incommensurate scattering seriously violates the theoretical predictions, though the recent theory suggests similar complicated dynamical structure in the incommensurate S D W phase [18]. Nonetheless, unlike any conventional antiferromagnet, the SDW in Cr should be considered to be a special ordered state of the amplitude modulation along Q~ (T < TsF) or transversally to Q~ (Tsv < T < TN). The incommensurate SDW state associates with excitations of quasi-particles of electron-hole pairs and then the fact that observed incommensurate scattering is no longer spin waves, is not strange. Since the commensurate scattering is dominated over a wide-energy range, the dynamical susceptibility in the S D W state in Cr is characterized by excitations centered at the commensurate position. Also due to the fact that the commensurate scattering has a rather less sensitive temperature feature, spin excitations might arise in the Fermi sea, namely, Stoner continuum. Though we do not know an?, calculated result of the dynamical susceptibility beyond the RPA, we speculate a distinct peak corresponding to the antiferromagnetic wave vector, if the electron correlations are not weak. Therefore, it is very important to see to what extent this commensurate peak occurs in energy. In any event a reliable calculation which must be based on the realistic energy band of the Cr S D W state and also the electron correlation should be taken into account beyond the RPA. We are indebted to R.A. Cowley, E. Fawcett, R. Fishman, S. Liu, S. Werner and G. Shirane for their stimulating conversations. The present paper is essentially based on our recent neutron experiments at T o h o k u University. We thank K. Yamada, M. Takeda, K. Hirota, K.

Kakurai, and K. Nakajima for their cooperation throughout the experiment. The work has been supported by a Grant-in-Aid of Scientific Research from the Ministry of Education, Science, Sports and Culture (Monbusho).

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