Set of supplementary notes.

94 CHAPTER 6. EXPERIMENTAL PROBES OF THE BAND STRUCTURE
666 A. P. Mackenzie and Y. Maeno: Superconductivity of Sr 2 RuO 4 and the physics of spin-triplet pairing
X-ray Spectrum
effect, in which the magnetization is studied. 20 Typical
raw data obtained using a field modulation method to
In this expression, ck z /2, where c is the he
body-centered tetragonal unit cell, and
study 2 M/B 2 are shown in Fig. 9. The modulation muthal angle of k in the (k x ,k y ) plane. T
field amplitude has been ‘‘tuned’’ to suppress the amplitude
of the otherwise dominant low-frequency Sr oscilla-
Fermi wave vector is given by k 00 (A e /)
tion, using a method described by Shoenberg 2
RuO
(1984). 4
is the cross-sectional area of the cylindri
surface sheet. Symmetry places constraints on
The Fourier transform of such data contains three fundamental
components, labeled , , and , each corre-
that exist in the expansion for a given sheet. F
which are centered in the Brillouin zone, k
sponding to a closed and approximately cylindrical sheet
only for divisible by 4. For , which runs
of the 0 Fermi surface (Fig. 10). By60
taking data at a closely
2 θ (deg)
zone corners, k
spaced series of angles for rotations about the (100) and
is nonzero only for even
visible by 4, or for odd and mod42. Pe
(110) directions, it has been possible to build an extremely
detailed Oscillation picture of Spectrum the Fermi-surface topogra-
Electronic
Quantum full fitStructure
of the dHvA data to this expansion
phy of Sr 2 RuO 4 . Even the out-of-plane dispersion is
volved a generalization of earlier theoretical
known, with a k resolution for the sheet of one part in
of dHvA amplitudes in nearly two-dimens
10 5 of the Brillouin zone.
materials] led to the Fermi-surface data sum
The standard way to describe out-of-plane dispersion Table I and Fig. 11 (Bergemann et al., 2
in low-dimensional metals is through hopping integrals 2002). 21
(t ). However, this involves making assumptions about It is important to note that the deviations
the shape of the Fermi surface that are too simple for fectly two-dimensional, nondispersing cylinde
Sr 2 RuO 4 . Instead, Bergemann et al. (2000) parametrized
the corrugation of each cylinder through an ex-
mation is adequate. For out-of-plane prope
so that for many properties, a two-dimension
pansion of the local Fermi wave vector:
ever, accurate knowledge of the dispersion is c
0 30
as we shall see, this is likely to be importan
QO frequency (kT)
cos
k F ,
,
k cos
0
mod40 standing key aspects of the superconductivi
. pect of the experimental Fermi surface is
sin mod42
Figure 6.7: Analogy between even quantum oscillations and x-ray Sr 2 RuO diffraction 4 to a higher inaccuracy Sr 2 RuOthan 4 : can be r
Obtaining the lattice structure from an x-ray diffraction experiment (2.1) tained
involves
from band-structure
solving an inverse
calculations.
problem, in which the spatial frequencies produced by plausible lattice structures are compared
against those measured as peaks in the diffraction pattern. Similarly, by analysing the periodicities
in bulk properties such as magnetisation (top panel) or electrical resistivity in magnetic
field sweeps and matching them against those expected from numerical TABLE I. Detailed calculations, Fermi-surface we can topography pa
determine the electronic structure of metals. Advances in crystalSr growth 2 RuO 4 . The
andwarping the availability
parameters kof
are given
10 7 m 1 . Entries symbolized by a dash are forbi
very high magnetic fields have allowed this technique to be applied body-centered in some tetragonal of the most Brillouin-zone complex
materials currently studied in condensed matter research, including Bergemannthe et al. high (2002). temperature
symm
superconducting cuprates and ferro-pnictides.
Fermi-surface k 00 k 40 k 01 k 02 k 21
sheet
Intensity
QO amplitude
experiments require
FIG. 10. A typical dHvA spectrum for Sr 2 RuO 4 (from Mackenzie,
Ikeda, et al., 1998). Both fundamental and harmonic
peaks can be seen. The split peak is due to the more pronounced
corrugation of that Fermi-surface sheet (see Fig. 11
below).
20 See Mackenzie et al. (1996a, 1996b); Yoshida, Settai, et al.
(1998); Yoshida et al. (1999); Bergemann et al. (2000, 2001).
Crystal Structure
FIG. 9. Typical raw
from the high-quali
Sr 2 RuO 4 that are n
(from Bergemann e
304 10 - 0.31 1.3
622 45 3.8 small -
753 small small 0.53 - s
• High purity samples: the electronic mean free path must be long enough to allow the
electrons to complete roughly one cyclotron orbit before scattering.
21 In constructing Fig. 11, dHvA has been com
probes such as angular magnetoresistance
(Yoshida, Mukai, et al., 1998; Ohmichi, Adachi, et
obtain the cross-sectional shapes. Angle-resolved
sion spectroscopy (ARPES) ought also to be an i
the determination of cross-sectional shape. As
Appendix B, ARPES on Sr 2 RuO 4 has had a check
but the recent work of Damascelli et al. (2000)
agreement with a 2D cut through Fig. 11.
• High magnetic field: high magnetic fields make the cyclotron orbits tighter, which equally
helps to fulfil the mean free path condition.
• Low temperature: The density of states oscillations are smeared out, when the Fermi
surface itself is smeared by thermal broadening of the Fermi-Dirac distribution. Typically,
Rev. Mod. Phys., Vol. 75, No. 2, April 2003