HARMONICALLY MODULATED STRUCTURES S. M. Dubiel * Faculty of Physics and Computer Science, AGH University of Science and Technology, PL Krakow, Poland S. M. Dubiel * Faculty of Physics and Computer Science, AGH University of Science and Technology, PL Krakow, Poland *

INTRODUCTION There exist crystalline systems with harmonic modulation of their electronic structure in a real space. The modulation occurs below a critical temperature and is known as (a) charge-density waves (CDWs), in case only the density of charge is modulated, and as (b) spin-density waves (SDWs), in case the spin- density is modulated. If both densities are modulated we speak about the co- existence of CDWs and SDWs. One of the basic parameters pertinent to such structures is periodicity, . If   n ·a, where a is the lattice constant and n is an integer, the modulation is commensurate with the lattice, if   n ·a, the modulation is incommensurate. CDWs were found to exist in quasi-1D linear chain compounds like TaS 3 and NbSe 3, 2D layered transition-metal dichalcogenides such as TaS 2, VS 2, or NbSe 2, 3D metals like  -Zr and Cr [1]. In the case of metallic Cr, which will be descussed here in more detail, SDWs originate from s- and d-like electrons and show a variety of interesting properties [2]. The most fundamental is their relationship to a density of electrons at the Fermi surface (FS). Between the Néel temperature of  313 K and the so-called spin-flip temperature, T SF of  123 K, SDWs in chromium are transversely polarized i.e. the wave vector, q, is perpendicular to the polarization vector, p. Below T SF they are longitudinally polarized.

Another peculiarity of the SDWs in chromium is their incommensurability i.e. q  2  /a. This feature can be measured by a parameter , such that q  2  (1-  )/a. The periodicity can thus be expressed as   a/(1-  ), hence  > a. In chromium  depends on temperature and  a  a/  varies between ~60 nm at 4 K and  80 nm at RT. [1] T. Butz in Nuclear Spectroscopy on Charge Density Waves Systems, 1992, Kluwer Academic Publ. [2] E. Fawcett, Rev. Mod. Phys., 60 (1988) 209 Another peculiarity of the SDWs in chromium is their incommensurability i.e. q  2  /a. This feature can be measured by a parameter , such that q  2  (1-  )/a. The periodicity can thus be expressed as   a/(1-  ), hence  > a. In chromium  depends on temperature and  a  a/  varies between ~60 nm at 4 K and  80 nm at RT. [1] T. Butz in Nuclear Spectroscopy on Charge Density Waves Systems, 1992, Kluwer Academic Publ. [2] E. Fawcett, Rev. Mod. Phys., 60 (1988) 209 Fermi Surface of chromium 3D 2D

SIMULATED SPECTRA SDWs can be described by a sinusoidal function or a series of odd harmonics, and CDWs can be described by a series of even harmonics, where   q · r and  is a phase shift. H 2i-1 and I 2i are amplitudes of SDWs and CDWs, respectively. Investigation of SDWs and CDWs with Mössbauer Spectroscopy (MS) requires that one of the elements constituting a sample shows the Mössbauer effect. If not, one has to introduce such an element into the sample matrix. In the latter case, a question of the influence of the probe atoms on SDWs and CDWs arises. Theoretical calculations show that magnetic atoms have a destructive effect i.e. they pin SDWs and/or CDWs. Consequently, such atoms are not suitable as probes. Unfortunately, 57 Fe atoms belong to this category of probe atoms. On the other hand, non-magnetic atoms hardly affect SDWs and/or CDWs, hence they can be used as good Mössbauer probe nuclei. Among the latter 119 Sn has prooved to be useful. In the following, all spectra were simulated and/or recorded on 119 Sn. and CDWs can be described by a series of even harmonics, where   q · r and  is a phase shift. H 2i-1 and I 2i are amplitudes of SDWs and CDWs, respectively. Investigation of SDWs and CDWs with Mössbauer Spectroscopy (MS) requires that one of the elements constituting a sample shows the Mössbauer effect. If not, one has to introduce such an element into the sample matrix. In the latter case, a question of the influence of the probe atoms on SDWs and CDWs arises. Theoretical calculations show that magnetic atoms have a destructive effect i.e. they pin SDWs and/or CDWs. Consequently, such atoms are not suitable as probes. Unfortunately, 57 Fe atoms belong to this category of probe atoms. On the other hand, non-magnetic atoms hardly affect SDWs and/or CDWs, hence they can be used as good Mössbauer probe nuclei. Among the latter 119 Sn has prooved to be useful. In the following, all spectra were simulated and/or recorded on 119 Sn.

INCOMMENSURTATE CDWs and SDWs CDW = I 0 · sin  – effect of I 0 SDW = H 1 · sin  – effect of H 1 INCOMMENSURTATE CDWs and SDWs CDW = I 0 · sin  – effect of I 0 SDW = H 1 · sin  – effect of H 1 J. Cieslak and S. M. Dubiel, Nucl. Instr. Meth. Phys. Res. B, 101 (1995) 295; Acta Phys. Pol. A, 88 (1995) 1143

INCOMMENSURTATE CDWs CDW = I 0 · sin  + I 2 · sin (2  +  ) - Effect of I 2 >0 and  INCOMMENSURTATE CDWs CDW = I 0 · sin  + I 2 · sin (2  +  ) - Effect of I 2 >0 and  J. Cieslak and S. M. Dubiel, Nucl. Instr. Metyh. Phys. Res. B, 101 (1995) Sn simulated spectra and underlying distributions of the charge-density for I 2 > 0 and  = 0 o, (a) and (b), respectively, and for  = 90 o (c) and (d). I 0 = 0.5.

INCOMMENSURTATE SDWs SDW = H 1 · sin  + H 3 · sin 3  - Effect of H 3 and its sign G. LeCaer and S. M. Dubiel, J. Magn. Magn. Mater., 92 (1990) 251; J. Cieslak and S. M. Dubiel, Acta Phys. Pol. A, 88 (1995) 1143 Simulated spectra for (a) H 3 > 0 and (b) H 3 < 0 with various amplitudes of H 3 shown, and underlying distributions of the spin-density. H 1 = 60.

SINGLE-CRYSTAL CHROMIUM S. M. Dubiel and G. LeCaer, Europhys. Lett., 4 (1987) 487; S. M. Dubiel et al., Phys. Rev. B, 53 (1996) 268 First ME determination of H 3 and its sign (left) RT and LHT spectra and underlying shapes of SDW and CDW, and (right) corresponding distributions of the spin- and charge densities.

POLYCRYSTALLINE CHROMIUM S. M. Dubiel and J. Cieslak, Phys. Rev. B, 51 (1995) Sn spectra recorded at 295 K on: (a) single- and (b) – (d) polycrystalline chromium with various size of grains in a decreasing sequence (left) and underlying distributions of spin- and charge densities (right). Note an increase of the maximum hf. field and appearance of zero-field peak. Both effects can be largely explained in terms of H 3 < 0.

INFLUENCE OF VANADIUM S. M. Dubiel, J. Cieslak and F. E. Wagner, Phys. Rev. B, 53 (1996) 268 Spectra recorded at 4.2 K (left) and 295 K (right) on single-crystal samples of CrV x with (a) x = 0, (b) x =0.5, (c) x =2.5 and (d) x =5. The quenching effect of V is clearly seen. cm 4.2 K 295 K

IMPLANTED CHROMIUM S. M. Dubiel et al., Phys. Rev., 63 (2001) (R), J. Cieslak et al., J. Alloys Comp., 442 (2007) 235 Right: (a) CEMS spectrum recorded at RT on a single-crystal chromium implanted with 119 Sn ions of 55 keV energy together with the underlying distribution of the spin-density, and (b) a spectrum recorded in a transmission geometry on a similar sample doped with 119 Sn ions by diffusion. Left: Hyperfine field vs. average implantation depth, ; triangles stand for the maximum and circules for the average hf. field values in the implanted samples, while the solid straight lines indicate the same quantities for the bulk sample. Right: (a) CEMS spectrum recorded at RT on a single-crystal chromium implanted with 119 Sn ions of 55 keV energy together with the underlying distribution of the spin-density, and (b) a spectrum recorded in a transmission geometry on a similar sample doped with 119 Sn ions by diffusion. Left: Hyperfine field vs. average implantation depth, ; triangles stand for the maximum and circules for the average hf. field values in the implanted samples, while the solid straight lines indicate the same quantities for the bulk sample.

CONCLUSIONSCONCLUSIONS Harmonically modulated structures (SDWs and CDWs) can be studied in detail with 119 Sn-site Mössbauer spectroscopy, because spectral parameters, hence a shape of spectra, are very sensitive to various parameters pertinent to SDWs and CDWs, and in particular to: periodicity,  < 17a (for commensurate SDWs) amplitude and sign of higher-order harmonics phase shift Several real applications were demonstrated for metallic chromium, and, in particular, the following issues were addressed: third-order harmonic in a single-crystal interaction of SDWs with grain boundaries (polycrystalline Cr) quenching effect of vanadium enhancement of spin-density (implanted single-crystal) Harmonically modulated structures (SDWs and CDWs) can be studied in detail with 119 Sn-site Mössbauer spectroscopy, because spectral parameters, hence a shape of spectra, are very sensitive to various parameters pertinent to SDWs and CDWs, and in particular to: periodicity,  < 17a (for commensurate SDWs) amplitude and sign of higher-order harmonics phase shift Several real applications were demonstrated for metallic chromium, and, in particular, the following issues were addressed: third-order harmonic in a single-crystal interaction of SDWs with grain boundaries (polycrystalline Cr) quenching effect of vanadium enhancement of spin-density (implanted single-crystal)

MORE TO READ E. Fawcett et al., Rev. Mod. Phys., 66 (1994) 25 S. M. Dubiel, Phys. Rev. B, 29 (1984) 2816 R. Street et al., J. Appl. Phys., 39 (1968) 1050 S. M. Dubiel, J. Magn. Magn. Mater., 124 (1993) 31 S. M. Dubiel in Recent Res. Devel. Physics, 4 (2003) 835, ed. S. G. Pandali, Transworld Res. Network S. M. Dubiel and J. Cieslak, Europhys. Lett., 53 (2001) 383 J. Cieslak and S. M. Dubiel, Acta Phys. Pol. A, 91 (1997) 1131 K. Mibu et al., Hyp. Inter.(c), 3 (1998) 405 K. Mibu et al., J. Phys. Soc. Jpn., 67 (1998) 2633 K. Mibu and T. Shinjo, J. Phys. D; Appl. Phys., 35 (2002) 2359 K. Mibu et al., Phys. Rev. Lett., 89 (2002)