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Quantum oscillations page 1 - M2 ICFP – Electronic properties of solids (Fabrice Bert)
Measurement of the Fermi
surface by quantum
oscillations
Quantum oscillations page 2 - M2 ICFP – Electronic properties of solids (Fabrice Bert)
Origin of the quantum oscillations
Effect of a magnetic field B on the trajectory of free electrons :
B
e-
Real space Reciprocal space:
Circular orbital on the FS
Period on this orbital
= 2/ wc related to the area of the
orbital
wc probes the FS geometry
Fermi
surface kx
ky
Quantum oscillations page 3 - M2 ICFP – Electronic properties of solids (Fabrice Bert)
),,( OBxOA p Aep
E
m
p
m
eBxp
m
p zyx
222
222
=energy of a 1D harmonic oscillator along x and free electron along z (Landau tube)
can be rewritten as an harmonic oscillator:
Effect of a magnetic field B on the trajectory of free electrons :
Quantum oscillations page 4 - M2 ICFP – Electronic properties of solids (Fabrice Bert)
kx
ky
surface de
Fermi
kx
ky
surface de
Fermi
n=0
n=1
n=2
n=3
B
kx
kz
n=0 n=1
n=2
B
ky
Without B With B
tubes de
Landau
n=3
kz
E
EF
kz
E
hwc
EF
Quantum oscillations page 5 - M2 ICFP – Electronic properties of solids (Fabrice Bert)
B increases
Special value of B for
which there’s a large
intersection of the Landau
tube and Fermi surface
When field B increases :
The section A(n) of the tubes increases because :
They cross the Fermi Surface at different position: m
eBc w
B
Quantum oscillations page 6 - M2 ICFP – Electronic properties of solids (Fabrice Bert)
Large intersection when the section of the nth Landau tube
corresponds to an extremal section area of the FS
therefore for a field
-> oscillation with 1/B of the density of state at the Fermi level
with the period:
~1/B
kz
E
hwc
EF
Quantum oscillations page 7 - M2 ICFP – Electronic properties of solids (Fabrice Bert)
Generalization to metals with arbitrary Fermi surface
*
CR
ccm
eB
m
eB ww
Where A(E) = area of the section of the FS by a plane
perpendicular to B in reciprocal space. Quantized as:
l Close to 1/2
Fermi
surface
Quantum oscillations page 8 - M2 ICFP – Electronic properties of solids (Fabrice Bert)
Some Fermi Surfaces
Quantum oscillations page 9 - M2 ICFP – Electronic properties of solids (Fabrice Bert)
Magnetization (de Haas-Van Alphen)
Peltier
Effect
Thermo
Electric
effect
Thermal
conductivity
Sound
attenuation
dT/dH
Resistivity
(Shubnikov-de Hass )
Some quantum oscillations
1/H
1/H
1/H
1/H
H
H
1/H
Quantum oscillations page 10 - M2 ICFP – Electronic properties of solids (Fabrice Bert)
Magneto resistivity
for a given Fermi surface
Quantum oscillations page 11 - M2 ICFP – Electronic properties of solids (Fabrice Bert)
B
Magneto resistivity
for a given Fermi surface
Quantum oscillations page 12 - M2 ICFP – Electronic properties of solids (Fabrice Bert)
Alkali metals example of potassium M(H) de Haas-Van Alphen
Spherical FS
-> 1 frequency
Quantum oscillations page 13 - M2 ICFP – Electronic properties of solids (Fabrice Bert)
To detect small deviation to the sphere, the crystal is rotated in a
static field: relative variation are as low as 10-4 :
Alkali metals example of potassium M(H) de Haas-Van Alphen
Quantum oscillations page 14 - M2 ICFP – Electronic properties of solids (Fabrice Bert)
More complex Fermi surface
Transition metals ex Copper: M(H) de Haas-Van Alphen
B <111>
2 extremal sections along 111
belly
neck
Quantum oscillations page 15 - M2 ICFP – Electronic properties of solids (Fabrice Bert)
along 110 :
Transition metals ex Copper :M(H) de Haas-Van Alphen
Quantum oscillations page 16 - M2 ICFP – Electronic properties of solids (Fabrice Bert)
Sr2RuO4
3 FS :
1 Hole FS « a »
2 electrons FS « b,g »
Quantum oscillations page 17 - M2 ICFP – Electronic properties of solids (Fabrice Bert)
Sr2RuO4 :M(H) de Haas-Van Alphen
Magnetization vs 1/B : Fourier transform in 1/B :
Quantum oscillations page 18 - M2 ICFP – Electronic properties of solids (Fabrice Bert)
Sr2RuO4 :oscillations vs angle
Quantum oscillations page 19 - M2 ICFP – Electronic properties of solids (Fabrice Bert)
Effects of temperature and disorder
kz
E
hwc
EF
kz
E
hwc
EF
h/t
kBT
Ideal case Effect of disorder: Broadening of the Landau
levels as h/t
t = scattering time
effect of
temperature: Broadening of the
Fermi level as kBT Oscillations are detectable only if
Low temperature weak disorder
Otherwise :
Quantum oscillations page 20 - M2 ICFP – Electronic properties of solids (Fabrice Bert)
g
B
FRRRMorR SDT 2sin
Extremal surface of FS
Cyclotron mass
Mean free path and
scattering time
Effective mass of the
electron (band curvature)
For clean enough sample and low enough temperature, unvaluable
informations can be measured
RD exponential term : if sample is not highly clean, not detectable
Mean free path
Mean free path
Effects of temperature and disorder
Quantum oscillations page 21 - M2 ICFP – Electronic properties of solids (Fabrice Bert)
Measurements in pulsed fields
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