Los días 22 y 23 de junio de 2016 organizamos en la Fundación Ramón Areces un simposio internacional sobre 'Materiales bidimensionales: explorando los límites de la física y la ingeniería'. En colaboración con el Massachusetts Institute of Technology (MIT), científicos de este prestigioso centro de investigación mostraron las propiedades únicas de materiales como el grafeno, de solo un átomo de espesor, y al mismo tiempo más resistente que el acero y mucho más ligero.

1. Recent advances in graphene research
Outline
l Graphene and Majorana particles
l Graphene as an anharmonic membrane
l Gauge fields in graphene
P. San Jose, CSIC
R. Aguado, CSIC,
J. Lado,INL, Braga
J. Frrnandez_Roissier, INL,Braga
A. L. Vázquez de Parga, UAM
R. Miranda, Imdea Nano
F. Calleja; Imdea Nano
H. Ochoa, CSIC
M. Garnica, Imdea Nano
S. Barja, Imdea Nano
J. J. Navarrp, Imdea Nano
A. Black, Imdea Nano
M. M. Otrokov; DIPC
E. V. Chulkov, DIPC
A. Arnau, DIPC
M. I. Katsnelson (Nijmegen)
J. Gonzalez (CSIC)
P. San-Jose (CSIC)
V. Parente (Imdea)
B. Amorim (Braga)
R. Roldan (CSIC)
L. Chirolli (Imdea)
P. Le Doussal (Paris)
B. Horowitz (Beersheva)
K. Wiese (Paris)
C. Gomez-Navarro (UAM)
J. Gomez (UAM)
G. Lopez-Polin (UAM)
F. Perez-Murano (UAM)
E. Khestanova (Manchester)
I. V. Grigorieva (Manchester)
A. K. Geim (Manchester)
M. A. H. Vozmediano (CSIC)
M. P. López Sancho (CSIC)
Madrid, June 22nd, 2016
Materiales bidimensionales:
explorando los límites de la ciencia y
la ingeniería

2. Phys. Usp. 44, 131 (2001)
𝑡 𝑡 𝑡 𝑡 𝑡
−𝑡 −𝑡 −𝑡 −𝑡 −𝑡
+Δ +Δ +Δ +Δ
−Δ −Δ −Δ −Δ −Δ
+Δ
electrons
holes
𝐻 = 𝑡
𝑛=1
𝑛=𝑁−1
𝑐 𝑛+1
†
𝑐 𝑛 + 𝑐 𝑛
†
𝑐 𝑛+1 + Δ
𝑛=1
𝑛=𝑁−1
𝑐 𝑛+1
†
𝑐 𝑛
†
+ 𝑐 𝑛 𝑐 𝑛+1
The Kitaev model
𝐻±
𝜙
=
0 𝑡 ± Δ + 𝑡 ∓ Δ 𝑒 𝑖𝜙
𝑡 ± Δ + 𝑡 ∓ Δ 𝑒−𝑖𝜙
0
𝑡 + Δ 𝑡 − Δ 𝑡 + Δ 𝑡 − Δ 𝑡 + Δ
𝑡 − Δ 𝑡 + Δ 𝑡 − Δ 𝑡 + Δ 𝑡 − Δ
𝛾𝑛 =
𝑐 𝑛
†
+ 𝑐 𝑛
2
, 𝛾𝑛 =
𝑐 𝑛
†
− 𝑐 𝑛
2𝑖
𝜖 𝜙 = ± 2 𝑡2 + Δ2 + 2 𝑡2 − Δ2 cos 𝜙
𝐻± = 2𝑖 𝑡 ± Δ
𝑛=1
𝑛=
𝑁−1
2
𝛾2𝑛−1 𝛾2𝑛 + 2𝑖 𝑡 ∓ Δ
𝑛=1
𝑛=
𝑁−1
2
𝛾2𝑛 𝛾2𝑛+1
One dimensional spinless superconductor

3. Phys. Rev. Lett. 105, 077001(2010)
Phys. Rev. Lett. 105, 177002 (2010).
Realization of the Kitaev model
l One dimensional system
l Strong spin-orbit coupling
l Magnetic field
l Superconductivity

4. Science 336, 1003 (2012)
Science 346, 602 (2014)
Phys. Rev. Lett. 109, 237003 (2012)
Experiments

5. arXiv:1511.05161
Quantum link between QDev in Denmark and QuTech in
Holland
Research collaboration
What do you do when you have two of the leading giants in the
same research field? – compete with each other? – fight each
other? – no, you start collaborating. The Center for Quantum
Devices, QDev at the Niels Bohr Institute at the University of
Copenhagen and QuTech at Delft University of Technology in
Holland have therefore entered into an international partnership
in the research of quantum technologies. The collaboration will
be celebrated with an official ceremony with the attendence of
ministers from both countries and the Dutch royal couple.
Recent developments

6. arXiv:1603.04069

7. Edge states: the Integer Quantum Hall Regime
Graphene
l 2D metal
l Excellent platform for QHE physics.
l Very weak spin-orbit coupling
l High degeneracy (spin and valley)
l No superconductivity

8. Phys. Rev. Lett. 98, 157003 (2007)
Phys. Rev. Lett. 110, 186805 (2013)
Edge modes: theory
Phys. Rev. Lett. 100, 096407 (2008)

10. Phase diagram of a Superconductor-graphene IQHE-Superconductor junction

11. Critical current,
and Fraunhofer
pattern for different
phases
Spectrum of the
SNS junction, as
measured by a
normal point
contact

12. Recent experiments
Science 352, 966 (2016)

13. 0 50 100 150 200
0.1
0.2
0.3
0.4
0.5
0.6
Δ 𝑠𝑐
Δ 𝐴𝐹
E 𝑐
0 50 100 150 200
0.2
0.4
0.6
0.8
1.0
t
t’
SC SCAF
Generic SC-AF edge
Flat band of midgap states
Confirmed by analytical
calculations. Also in 3D
Square lattice
Almost perfect nesting
3 2 1 1 2 3
k
0.4
0.2
0.2
0.4
E

14. 0 50 100 150 200
0.1
0.2
0.3
0.4
0.5
0.6
Δ 𝑠𝑐
Δ 𝐴𝐹
E 𝑐
0 50 100 150 200
0.2
0.4
0.6
0.8
1.0
t
t’
SC SCAF
Generic SC-AF edge
Flat band of midgap states
Confirmed by analytical
calculations. Also in 3D
Square lattice
Almost perfect nesting
3 2 1 1 2 3
k
0.4
0.2
0.2
0.4
E

15. GRAPHENE’S SUPERLATIVES
l Thinnest imaginable material
l largest surface area (~2,700 m2 per gram)
l strongest material ‘ever measured’ (theoretical limit)
l stiffest known material (stiffer than diamond)
l most stretchable crystal (up to 20% elastically)
l record thermal conductivity (outperforming diamond)
l highest current density at room T (106 times of copper)
l completely impermeable (even He atoms cannot squeeze
through)
l highest intrinsic mobility (100 times more than in Si)
l conducts electricity in the limit of no electrons
l lightest charge carriers (zero rest mass)
l longest mean free path at room T (micron range)

16. Bgraphene =22 eV Å-2 = 352 N/m
Bdiamond x d=52.4 N/m
T=300K
L=1Km
Why are there two dimensional crystals?
    






d
L
B
Tk
uLu B
log0

Thermal fluctuations:

17. Elastic properties of graphene
courtesy of M. M. Fogler

18. 𝐸𝑇
𝜅2 𝑞2
𝐸 ≃ 22 eVÅ−2
𝜅 ≃ 1 eV
𝑇 = 300K
𝑞−1 =
𝜅
𝐸𝑇
≃ 1.3 Å

19.      
      













 








 







 









 








22222
2
222
2222
2
2
2
22
1
22
22222
hh
uu
h
u
h
urd
hh
uurdhrd
t
h
rdH
yx
xyyx
y
yy
x
xx
yx
yyxx




Two dimensional membranes
Out of plane displacements
lead to changes in area
h
L
L
h
L
2
2

Kinetic Bending Stretching
Shear
Two dimensional crystaline membranes are intrinsically anharmonic

20. Thermal expansion
 
 
2
24
22
1
2
qu
quq
q
q
q
q













Flexural phonon
Grüneisen parameter






 22
3
log
8 Y
kB




Thermal expansion
In plane strains change the frequency of out of plane modes
Negative thermal expansion coefficient
Low T
High T
Lattice constant
Bindingenergy
Thermal expansion

21. Substrate effects
Gapped flexural modes
Thermal expansion

22. Out of plane fluctuations
screen the in plane
elastic constants
2
221 hcuYcE 






















221
log


cuYc
T
TF
2
22
2
2
2
1




TY
u
F
Y 




23. Load 2
Experiments

24. C D
1
2
N
SiO2
Au
A B
0.5 1.0 1.5 2.0 2.5
0
4
8
z(nm)
x (µm)
Si
SiO2
Au
A
I
Ar+F

Si
SiO2
Au
2a
Graphene
Experiments
1500 2000 2500 3000
0
100
200
300
Counts
Raman Shift (cm
-1
)
<E2D>=33611 N/m
<E2D>=53852 N/m
C
0.9 1.2 1.5 1.8
300 400 500 600
0
10
20
30
Counts
E2D
(N/m)
E3D
(TPa)
0 25 50 75
0.0
0.2
0.4
0.6
F(N)
 (nm)
B
A
GD
0 2x10
13
4x10
13
6x10
13
200
300
400
500
600
0.0 0.3 0.6 0.9 1.2 1.5 1.8
0.6
0.9
1.2
1.5
1.8
E3D
(TPa)
E2D
(N/m)
Defects/cm
2
Defects (%)
0 2x10
12
4x10
12
6x10
12
3x10
13
4x10
13
0.6
1.2
1.8
0.00 0.05 0.10 0.15 1.0
FractureForce(N)
Defects/cm
2
Defects (%)
A
B

25. The self consistent screening approximation
J. Physique, 48, 1085 (1987)
4
2
q
YT
qd 


 
     
 
 
     
 
 
 
 
   





 
qpGqGqpqqdpI
qIb
b
qb
pqGqpPqqbqdq
qqGqG
T





222
2
0
0
22
2
1
0
1
28
1
31
2
2


=
=
=
+
+
+
 
   
358.0
821.0
,



 
u
u
qqq
qq






Power law divergences
Self consistent theory, valid in high dimensions
Agrees well with numerical simulaions

26. Vacancies and flexural modes
 
 

,
1
, 42
qq
qG


 


















0
log
0
2
24
2
22
h
a
n
hn
V
V





T-matrix approximation
infinite mass
vacancies
21
44











Vn
 

localization length
l Vacancies localize flexural
modes
l Long wavelength flexural
modes do not contribute to
the screening of the elastic
constants

27. 0 2x10
13
4x10
13
6x10
13
200
300
400
500
600
0.0 0.3 0.6 0.9 1.2 1.5 1.8
0.6
0.9
1.2
1.5
1.8
E3D
(TPa)
E2D
(N/m)
Defects/cm
2
Defects (%)
0 2x10
12
4x10
12
6x10
12
3x10
13
4x10
13
0.6
1.2
1.8
0.00 0.05 0.10 0.15 1.0
FractureForce(N)
Defects/cm
2
Defects (%)
A
B


















 VV ncn
R
KY
u
2
0
2
2
0
2
1
1
11


geometric factor
intrinsic localization length
percolation
1
0
0 nm10020



Fk


28. 20 30 40 50 60 70 80
0,00
0,05
0,10
0,15
Prestress(N/m)
Temperature (ºC)
Pristine
Irradiated
Thermal Expansion Coefficient:
· Pristine: -9.4 x 10 -6 K-1
· Irradiated (LD ~ 5.5 nm): -1 x 10 -6 K-
1
Graphene thermal expansion coefficient
20 40 60 80 100 120
0,05
0,10
0,15
0,20
Prestress(N/m)
Temperature (ºC)
Thermal Expansion Coefficient:
· Pristine: -6.2 x 10 -6 K-1
· Irradiated (LD ~ 5 nm): -1.1 x 10 -6 K-1
Membrane 1 Membrane 2
Pristine
Irradiated
LD : Mean distance between defects as measured by Raman

29. Young modulus and induced strains
Young modulus measured by Raman
is two times larger than the one
measured by indentation
arXiv:1504.05521

30. arXiv:1504.05521
Recent experiments

31. Ripples in graphene
l Quenched (non thermal) ripples in suspended
samples
lLateral scale ~102
− 103
Å
l Vertical scale ~10Å
Instability due to the coupling to
low energy electron-hole pairs?
Also: wrinkles induced by absorbates,
non trivial fixed point?

32. Strong and non uniform, spatially varying
Spin-Orbit coupling in Pb-intercalated graphene leads to
the observation of sharp pseudo-Landau levels without a
external magnetic field
C. L. Kane and E. J. Mele, Quantum Spin Hall in Graphene, Phys. Rev. Lett. 95,
226801 (2005).
C. Weeks, J. Hu, J. Alicea, M. Franz, and R. Wu, Engineering a Robust Quantum Hall
State in Graphene via Adatom Deposition, Phys. Rev. X 1, 021001 (2011).
Experiments: B.
Özyilmaz, et al.,
Nature Comm. 5,
4875 (2014).

34. Vs= 1 V , It= 0.9 nA
4.6 K
Periodically Rippled Graphene on Ir(111)
Wavelength ~ 25.2 ± 0.4 Å
Corrug ~0.2 Å

35. Pb evaporated
on gr/Ir(111)
at 800K
Partial
Intercalation
of Pb below
graphene
Intercalation of Pb /Graphene/Ir(111)

36. Atomic arrangement in Pb-intercalated Gr/Ir(111)

37. Graphene/Pb/Ir(111):
Landau Levels without a magnetic field
ED
EF
4.6 K

38. Graphene/Pb/Ir(111) versus gr/Ir(111)
Large
Spin-Orbit
coupling
Small
Spin-Orbit
coupling
δISO/δx
pn

39. F. G., M. I. Katsnelson, A. K. Geim, Nature Phys. 6, 30 (2010)
Scaling of resonances
observed with STM
Bubbles and strains in graphene
Topography and
spectroscopy of bubbles
in graphene on Pt
Comparison of theory and
experiment

40. DFT calculations
l Lead shows SO
splittings of order 1 eV
l Lead and graphene
bands are strongly
hybridized near the
chemical potential

41. Effective Dirac-like Hamiltonian
𝐻 = 𝑣 𝐹Σ ∙ 𝑘 − 𝐴 ± 𝐴0 𝑠 𝑦
𝐴 = 𝐴 𝑥 𝑠 𝑦, 𝐴 𝑦 𝑠 𝑥 Non-abelian gauge potential
Scalar potential
Σ = ±𝜎 𝑥, 𝜎 𝑦
𝐴0
DFT (in blue) and tight-binding (in red) band structure
calculation for a distance between graphene and the Pb
adatoms of 2.7 Å, with spin-orbit coupling. The right panel
zooms into the Dirac point region.

42. Inhomogeneous SO texture
Smooth change of the
SO coupling

43. The non-uniform spatial variation of the S-O
coupling and related gauge fields leads to
electronic confinement and pseudo-Landau
levels….
… but associated to effective magnetic
fields with opposite sign for each in plane
spin polarization

44. Non trivial one dimensional channels at boundaries of 2D materials
l Non trivial edge modes are possible at SC-graphene
interfaces, when graphene is in the Integer Quantum Hall regime.
l Generic states between superconductors and 2D
antiferromagnets
l Intercalated Pb induces resonances in the density of states of
graphene
l A large, inhomogeneous, spin-orbit coupling is induced
l Spin-orbit coupling is a source of gauge fields
Giant enhancement of spin-orbit coupling in graphene
l Graphene is a highly anisotropic membrane.
l The elastic properties of graphene are sample dependent
Graphene and other 2D systems as elastic membranes

45. (interacting) Majoranas from way back: