HARMONICALLY
MODULATED STRUCTURES
S. M. Dubiel*
Faculty of Physics and Computer Science,
AGH University of Science and Technology,
PL-30-059 Krakow, Poland
*e-mail:
dubiel@novell.ftj.agh.edu.pl
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 spindensity is modulated. If both densities are modulated we speak about the coexistence 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 TaS3 and NbSe3 , 2D layered transition-metal
dichalcogenides such as TaS2, VS2, or NbSe2, 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, TSF of 123 K, SDWs in chromium are
transversely polarized i.e. the wave vector, q , is perpendicular to the
polarization vector, p. Below TSF 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
Fermi Surface of chromium
3D
3D
3D
2D
SIMULATED SPECTRA
SDWs can be described by a sinusoidal function or a series of odd
harmonics,

SDW   H 2i 1 sin[( 2i  1)
i 1
CDW 

I
0
sin    I 2i sin( 2i   )
i 1
and CDWs can be described by a series of even harmonics, where   q · r and
 is a phase shift. H2i-1 and I2i 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, 57Fe 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 119Sn has prooved to be useful. In the following, all spectra were
simulated and/or recorded on 119Sn.
INCOMMENSURTATE CDWs and SDWs
• CDW = I0· sin  – effect of I0
• SDW = H1· sin  – effect of H1
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 = I0 · sin  + I2 · sin (2+) - Effect of I2>0 and 
• 119Sn simulated spectra and underlying
distributions of the charge-density for I2
> 0 and  = 0o, (a) and (b), respectively,
and for  = 90o (c) and (d). I0 = 0.5.
J. Cieslak and S. M. Dubiel, Nucl. Instr.
Metyh. Phys. Res. B, 101 (1995) 295
INCOMMENSURTATE SDWs
• SDW = H1 · sin  + H3 · sin 3 - Effect of H3 and its sign
• Simulated spectra for (a) H3 > 0 and (b) H3 < 0 with various amplitudes of H3 shown,
and underlying distributions of the spin-density. H1 = 60.
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
SINGLE-CRYSTAL CHROMIUM
• First ME determination of H3 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.
S. M. Dubiel and G. LeCaer, Europhys. Lett., 4 (1987) 487; S. M. Dubiel et al., Phys. Rev. B, 53 (1996) 268
POLYCRYSTALLINE CHROMIUM
• 119Sn 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 spinand 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
H3 < 0.
S. M. Dubiel and J. Cieslak, Phys. Rev. B, 51 (1995) 9341
INFLUENCE OF VANADIUM
4.2 4.2
K K
295 K
cm
• Spectra recorded at 4.2 K
(left) and 295 K (right) on
single-crystal samples of CrVx
with (a) x = 0, (b) x =0.5, (c) x
=2.5 and (d) x =5. The
quenching effect of V is
clearly seen.
S. M. Dubiel, J. Cieslak and F. E. Wagner, Phys. Rev. B, 53 (1996) 268
IMPLANTED CHROMIUM
Hf. Field [T]
25
20
15
10
5
0
10
15
20
25
30
35
40
<d> [nm]
• Right: (a) CEMS spectrum recorded at RT on a single-crystal chromium implanted with 119Sn 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 119Sn ions by diffusion.
Left: Hyperfine field vs. average implantation depth, <d>; 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.
S. M. Dubiel et al., Phys. Rev., 63 (2001) 060406(R), J. Cieslak et al., J. Alloys Comp., 442 (2007) 235
CONCLUSIONS
Harmonically modulated structures (SDWs and CDWs) can be
studied in detail with 119Sn-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) 287202