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