Condensed Matter Physics
At Low Dimensions
Philip Kim
Department of Physics
Columbia University
Condensed Matter Physics
~1023 electrons
~1023 ions
SP2 Carbon: 0-Dimension to 3-Dimension
p
Benzene
molecule
0D
Fullerenes (C60)
s
Atomic orbital sp2
1D
Carbon Nanotubes
2D
Graphene
3D
Graphite
Electronic Band Structure of Graphene
Band structure of graphene (Wallace 1947)
ˆ  t
H
E(k2D)
AB

An
E
Bm  c.c
( n,m)

E  vF k 
empty
filled
ky
kx'
K’
K
x
kx
ky'
kx
ky
2D Brillouin Zone
Zero effective mass particles
moving with a constant speed
vF = c/300
Single Layer Graphene: Pseudo Spinor
Graphene Lattice Structures
Spinor Representation
‘A’ sublattice: pz orbitals
A
‘B’ sublattice : pz orbitals
B
Superposition:
[ A  eiq B ] Cs
X
Two inequivalent lattice sites!
Spin
q
Pseudo spin
Dirac Fermions in Graphene : “Helicity”
ˆ  t
H
E

An
E
Bm  c.c
( n,m)
momentum
pseudo spin
E
ky
K’
ky
kx
kx
K
ky
kx
Effective Dirac Equations
H eff   v
 0 k x  ik y 


F  k x  ik y 0 
G. Semenoff PRL (1984)
 
 vF s  k
k
= eik.r
1
2
1 
 
iq
e
 
qk = tan-1(ky / kx)
k
Hall (1879)
Hall Effect
Rxx = Vxx / I
Rxy = Vxy / I = B/en
B
I
+
-
+
Vxx
+
+
Rxy
Quantum Hall Effect:
Klitzing (1980)
-
Rxx
Vxy
Quantized Cyclotron Orbit
s
s
Quantum Hall Effect in Graphene (2005)
E
kx'
Quantization:
1 __
_
Rxy = 4 (n + ) e
2 h
-1
2
ky'
spin (2) X pseudo-spin (2)
pseudo-spin rotation
Klein Tunneling (1928)
Step Potential problem
V>m: transmission via negative energy states
V
T 1
0
x
T e
V
0
V
0
x
x

p m 2
E
Klein result:
barrier sharpness
~Compton wavelength
Klein Tunneling and Pseudo spin
Chiral tunneling in graphene pn junctions
Katsnelson et al. (2006)
VBG > 0
VTG < 0
graphene
n
2
Magnetic field modulation of FP
n
p
1 mm
+1
Gosc (e2/h)
p
n
0
p
B (T)
electrode
-1
VBG < 0
VTG > 0
0
1
20 nm
Young et al. (2009)
2
|n2| (1012 cm-2)
3
4
Spin ½ and Electron Interaction
“Triplet”

Exchange Interaction:
1
e2
 X     dr dr 
 i* (r ) i (r) *j (r) j (r )  si , s j
2 i, j
| r  r |

Pseudo Spin
“Singlet”


c1   c2 
“Quantum Hall bilayer”
“Valley spin”
…
SU(4) Quantum Hall Ferromagnet in Graphene
K’
K’
< SU(4)
Magnetic Wave Function
E
X
ky
q
kx
Spin
Degree of freedom:
Spin (1/2), Valleys
Under magnetic fields:
pseudospin = valley spin
Valley spin
K
K
K’
K’
Yang, Das Sarma and MacDonal, PRB (2006);
Possible SU(4) Quantum Hall Ferromagnetism at the Neutrality
FerroMagnetic
Anti FerroMagnetic
Kekule Distortion
Charge Density Wave
Spin & pseudo spins: many body physics in graphene
Dean et al. Nature Physics (2011)
5 mm
Mobility > 300,000 cm2/Vsec
• SU(4) hierarchical Fractional Quantum Hall Effect
• Spin and Pseudospin Ferromagnetic Quantum Hall Effect
• Spin Skyrmion and Valley Skyrmions
Phase Transitions Among Fractional Quantum Hall States
Bilayer Graphene
Encapsulated with top & bottom gate
sxx (S)
mobility > 106 cm/Vsec
Phase Transitions in Lowest Landau Levels
Bilayer graphene: Fractional Quantum Hall effect
70
12000
Rxx ()
8000
6000
4000
50
2/3
40
4/3
1
30
5/3
7/3
2
20
8/3
3
2000
0
60
Rxx
Rxy
0.5
1.0
Vg (Volts)
Maher*, Wang* et al. submitted
10/3
11/3 4
1.5
10
2.0
0
Rxy (k)
B=18T
SiGate
20
mK = +40V
10000
E-field tunable FQHE
2/3 5/3 8/3
1
2
3
4
5
6
Rxx (k)
Assembly of Various 2D Systems
Charge Transfer Bechgaard Salt
graphene
Metal-Chalcogenide
Bi2Sr2CaCu2O8-x
C
(TMTSF)2PF6
hexa-BN
X
Lead Halide Layered Organic
M
B
X
N
M = Ta, Nb, Mo, W, Eu …
X = S, Se, Te, …
Semiconducting materials: WSe2, NbS2, MoS2, …
Complex-metallic compounds : TaSe2, TaS2, …
Magnetic materials: EuS2, EuSe2 ,…
Superconducting: NbSe2, Bi2Sr2CaCu2O8-x, ZrNCl,…
A
C
A
B
A
Andreev Reflections – between NbSe2 & Graphene
Efetov et al. (2014)
Superconductivity and QHE
NbSe2
graphene
Tc = 7 K
Hc2 = 4.5 T
5 mm
Andreev Reflection btw graphene/NbSe2
Andreev Reflection
(dI/dV)/(dI/dV(10K))
1.08
Tomasch Oscillations
1.06
1.5 K
1.04
2.5 K
3.5 K
4.5 K
5.5 K
6.5 K
6.8 K
7.0 K
7.2 K
7.5 K
1.02
1.00
0.98
0.96
-40
-20
0
VSD(mV)
20
40
Andreev Reflection into QH edge
states are more efficient!
Atomically Thin vdW p-n junction
C. Lee et al, submitted
Vertical & Lateral Channels
- Al contact to MoS2 for electron injection
- Pd contact to WSe2 for hole injection
Gate
Tunable
Diode Characteristic
Lateral
Transport
in Channels
Lateral and vertical electron band alignment
-4
10
-5
WSe2
-6
MoS2
10
10
Vds = 0.5 V
Ids (A)
-7
10
-8
10
-9
10
-10
10
-11
Forward
10
-12
10
-60
-40
-20
0
Vg (V)
20
40
60
Interlayer recombination by inelastic tunneling process
Graphene Materials and Applications
Flexible/Transparent
Electrodes/Touch Panels
Printable
Inks
Transparent
Electrodes
Semiconductors
Ultrafast Transistors,
RFIC,
Photo/Bio/Gas Sensors
Large-Scale
CVD Graphene
+
Graphene
Nanoplatelet
Composites
Conductive Ink,
EMI shields
Gas
Barriers
Gas barriers fo Displays,
Solar Cells
Heat
Dissipation
Energy
Electrodes
Composites
LED Lights, BLU
ECU, PC …
Super Cap./Solar Cells
Secondary Batteries
Fuel Cells
Images: Royal Swedish Academy
Cars,
Aerospace
Appliations
Courtesy: B. H. Hong
Conclusions
Relativistic QM: High Energy Physics
CERN Electro-Positron Collider
Quasi Relativistic QM: Low Energy Physics
Kim Lab @ Columbia in City of New York
. Equation:
Dirac
Majorana Equation:
??
Acknowledgement
Amelia Barreiro
Chul-ho Lee (jointly with Nuckolls group)
Jean-Damien Pillet
Current Members
Students/postodcs
Jayakanth
RavichandranCollaborating
Yue Zhao
Cory Dean, Inanc Meric, Lei Wang,
Adam
Wei
Tsen (jointly with
Pasupathy
Mitsuhide
Takekoshi
Sebastian
Sorgenfrei,group)
Kevin Knox, Nayung
Andrea
Young
Jung,
Seok
Ju
Kang,
Jun
Yan, Yanwen Tan,
Dmitri Efetov
Dmitri Efetov
Kevin Knox
Fereshte
Ghahari
Fereshte Ghahari
Patrick Maher
Young-Jun Yu (jointly with GRL, POSTECH)
Vikram Deshpande (jointly with Hone group)
Paul Cadden-Zimansky (Columbia Frontier of Science Fellow)
Chenguang Lu (jointly with Hone and Herman
Patrick Maher
Carlos Forsythe
Giselle Elbaz (jointly with Brus group)
Collaborators
Austin
Cheng
Horst
Stormer,
Aron Pinczuk, Tony Heinz, Abhay
Pasupathy,
Latha Venkataraman
Frank Zhao
Louis
Brus,
George
Xiaomeng
LiuFlynn, Colin Nuckolls,
Jim Hone, Ken Shepard, Louis Campos, Rick Osgood
T. Taniguchi, K, Watanabe
Andre Geim, Kostya Novoselov, Sanka Das Sarma
Past Members
Melinda Han (Ph.D. 2010, Frontier of Science Fellow, Columbia University)
Meninder S. Purewal (Ph.D. 2008)
Josh Small (Ph.D. 2006)
Yuanbo Zhang (Ph.D. 2006, Professor, Fundan University)
Yuri Zuev (Ph.D. 2011, IBM Fishkill)
Kirill Bolotin (Assistant Professor, Department of Physics, Vanderbilt University)
Byung Hee Hong (Associate Professor, Department of Chemistry, Seoul National University)
Kim
Pablo Jarillo-Herrero (Assistant Professor, Department of Physics, MIT)
Keunsoo Kim (Assistant Professor, Department of Physics, Sejong University)
Namdong Kim (Research Scientist, POSTECH)
Barbaros Oezyilmaz (Assistant Professor, Department of Physics, National University of Singapore)
Collaborations: Brus, Dean, Heinz, Hone, Nuckolls, Shepard
Funding:
group and friends (2011)