This document summarizes quantum oscillations in strongly correlated electron systems, with a focus on applications to cuprate superconductors. Key points include: (1) quantum oscillations can be used to measure the Fermi surface and quasiparticle properties; (2) in cuprates, oscillations provide evidence of long-lived quasiparticles and Fermi surface reconstruction near optimal doping; and (3) while oscillations support a small reconstructed Fermi surface in underdoped cuprates, details like the number of pockets are still controversial.

1. Quantum oscillations in strongly
correlated electron systems
S.R. Julian
Department of Physics, University of Toronto
•Measuring the Fermi surface:
•Basic theory: Onsager relation
•examples
•Effect of scattering
•Measuring the quasiparticle mass
•Spin dependent effects
•Application to the cuprates

2. Electron motion is quantized in a
magnetic field.
Fermi surface
Landau tubes
k
∂
∂
=
ε

1
v
B
q
F ×
= v
•Quasiparticles follow lines of constant
energy
• orbit area in k-space is quantized

3. k-space quantization: (semi)-
classical derivation

4. Landau tubes
B

5. ε
n(ε)
εF
k
ε
Landau tubes

6. The Onsager relation.

7. Lifshitz-Kosevich Theory
ext
2
A
e
F
π

=
;
2
sin
*
1
~
0
1
2
/
3






+
∝ ∑
∞
=
−
φ
π
B
F
p
p
m
M
p
B1
Applied field B
ε
n(ε)

8. Harmonic content of dHvA signals

9. The Fermi surface of Sr2RuO4
C. Bergemann

10. The Fermi surface of
Sr2RuO4
Mackenzie et al., PRL 1996
B

11. Electronic structure:
Bergemann, Adv. Phys, 2003

12. ARPES Fermi surface:
Damascelli et al., Yokoya et al., etc….

13. Warping patterns on the Fermi surface

14. Data:amplitude vs field
Bergemann, Adv. Phys, 2003

15. Amplitude vs field and angle

16. Allowed warping patterns on the Fermi
surface
Bergemann et al., PRL 85 (2000) 2662.

17. Impurity and temperature damping
of dHvA signals

18. Impurity scattering: broadening of Landau levels
o
c l
r
e /
π
−

19. Applying the Dingle factor to the
surface state in a topological
insulator:
Xiong et al., arXiv:1101.1315:
- Quantum oscillations arise from surface
state.
- Analysis of Dingle factor yields surface
mobility 2,800 cm2/Vs, compared with
about 50 in the bulk
o
c l
r
e /
π
−

20. Effect of temperature: Fermi function

21. Derivation of conventional T dependence in dHvA
1
1
/
)
(
)
(
2
+
∝ −
−
∫ kT
e
e
d
N c
i
µ
ε
ω
µ
ε
π
ε 
ε
n(ε)
c
kT
X
X
X
R
T
ω
π 
/
2
sinh
2
where =
=
*
m
eB
c =
ω

22. Mackenzie et al., PRL 1996

24. Spin splitting:
A. McCollam, Ph D thesis, 2004

25. CeRu2Ge2: a ferromagnet with a “small” Fermi
surface
King and Lonzarich,
Physica B 171 (1991) 161.

26. Backprojection of dHvA frequencies:
Pauli susceptibility is invisible
( )






+
∂
∂
−
=





 +
∂
∂
−
+
=






φ
π
π
π
B
B
F
B
F
B
B
F
B
B
F
B
B
F /
2
sin
/
)
(
2
sin
)
(
2
sin 0
0
0
0 

27. McCollam et al., PRL, Physica B 2006

28. Quantum
oscillations in
cuprates

29. Shubnikov-de Hass
oscillations in ortho-II
ordered YBCO
Doiron-Leyraud et al., Nature 2007

30. dHvA oscillations in
ortho-II ordered YBCO
Jaudet et al.,
Cond-mat, 26 Nov 07.

31. Quantum oscillations in the
specific heat:
Riggs et al., Nature Physics, 2011.
F = 531 T
B dependence of C(T) -> sample
is superconducting at high field

32. How Many Pockets,
and where?
Doiron-Leyraud et al., Nature 2007,
deBoef et al., Nature 2007

33. How many pockets, and where?
Hossain et al., Nature Physics 2008.

34. Observation of hole
pocket in
underdoped YBCO?
Sebastian et
al., Nature,
2008

35. How many pockets, and where?
Audouard et al., PRL 2009: The
main frequency is composed of at
least three components.

36. How many pockets, and where?
Sebastian et al., PRB2010: similarly sized
electron and hole pockets, with different
corrugations. Electron pocket disappears at
xc.

37. How many pockets and where?
…
Sebastian et al., Nature
Communications, 2011
Harmonic analysis suggests
only one pocket, probably
nodal, split by bi-layer
splitting?

38. Carrier density from FS size:
Yelland et al., 2008

39. What causes the large Fermi
surface to reconstruct?
Chakravarty and Kee, 2007

40. Spin-splitting of the Fermi surface
Sebastian et al., PRL 2009, no
spin-zeroes. This means SDW is
the symmetry breaking field that
gives small Fermi pockets

41. More spin-
splitting …
Ramshaw et al., Nature Physics
2011: there are spin zeroes, but
they occur at different angles for
the different oscillatory
components

42. More spin splitting, more how
many pockets and where …
Sebastian et al., Nature
Communications, 2011: spin
zeroes do exist.

43. Doping Dependence:
Quantum oscillations in Y124
Yelland et al., PRL, 2007; Bangura et al, PRL 2007.

44. Doping dependence:
Singleton et al., PRL 2010:
-Higher pulsed field magnets
allow more disordered samples
to be studied
-m* and F vs. doping

45. Quantum oscillations
in the cuprates
Hussey et al.,
Nature 2003

46. SdH oscillations in overdoped Tl-
cuprate
Vignolle et al., Nature 2008

47. Overdoped thallium cuprate ….

48. The main issue is reconstruction of the Fermi
surface near optimal doping in the cuprates

49. Conclusions
• Quantum oscillations are a powerful probe of
quasiparticle properties, giving Fermi surface size, and
quasiparticle scattering rates and effective masses
• Impurities are both a problem (you need pure crystals)
and a blessing (you know you are looking at the cleanest
part of your sample)
• In the cuprates, quantum oscillations have established
the existence of long-lived quasiparticles, with
conventional spins, on both the under and over-doped
sides of the phase diagram, and strongly support that the
Fermi surface reconstructs at optimal doping. But the
details of the Fermi surface on the underdoped side are
still controversial.
Thanks: NSERC, CIFAR