Reading and manipulating valley quantum
states in Graphene
Atindra Nath Pal
Arindam Ghosh
Department of Physics
Indian Institute of Science
Atin Pal et al. ACS Nano 5, 2075 (2011)
Atin Pal and Arindam Ghosh PRL 102, 126805 (2009)
Atin Pal, Vidya Kochat & Arindam Ghosh PRL 109, 196601 (2012)
Vidya Kochat
Layout
• A brief introduction to Graphene – The valleys
• Uniqueness in the structure of graphene – Valleys and new effects
in quantum transport
• Graphene as an electronic component
• Valley manipulation with disorder and gate
• Valley reading: Mesoscopic conductance fluctuations in Graphene
• Graphene on crystalline substrates: Manipulating valleys at atomic
scales
• Conclusions
Graphene
Graphene excitement
 Electronics of Graphene
 Backbone of post-Silicon nanoelectronics, Flexible
 Higher mobility, speed, robustness, miniaturization
 > 100 GHz transistors (Can be upto 1.4 THz)
 Electrical sensor for toxic gas
 Novel Physics – Astrophysics, Spintronics… more?
 Strongest material known – Electromechanical sensing
 Bio-compatibility: Bio sensing, DNA sequencing
 Transparent – Application in solar cells
What is different in Graphene?
Existence of valleys
Single layer graphene: Sublattice symmetry
A
B
0

 i v F 
  x  i y
 x  i  y   A 
 A 
   E  
0
 B 
  B 
 
 v F ( . k )   E 
Pseudospin
Single layer graphene: Valleys
E
Valleys
K
K’
 
 v F ( . k )   E 
 
 v F ( . k )    E  
 
  v F  .k

0

k

 s
 v F  .k 
0
 Es
Implications to Random Matrix Theory and
universality class
Suzuura & Ando, PRL (2002)
Removed
Effective spin rotation
symmetry broken
Wigner-Dyson orthogonal
symmetry class
Preserved
Effective spin rotation
symmetry preserved
Wigner-Dyson symplectic
symmetry class
Valley
symmetry
Valley-phenomenology in graphene
• Valleytronics
Valley-based electronics, equivalent to SPIN (generation and
detection of valley state)
• Valley Hall Effect
Analogous to Spin Hall effect (Berry phase supported
topological transport)
• Valley-based quantum computation
Example: Zero and One states are valley singlet and triplets in
double quantum dot structures
Phenomenology
Berry phase
Observed
 A 


B

s  
'
  B 
 ' 

 A 
Half-integer integer
Quantum Hall effect
Absence of backscattering
Antilocalization
Klein Tunneling
Valley Hall Effect
Valley Physics
Nontrivial universality class
Universality of mesoscopic fluctuations?
Magnetism
Time reversal symmetry
Edges , magnetic
impurities,
adatoms, ripples…
Graphene: An active electrical component
The Graphene field-effect transistor
Au contact pads
VBG
Exfoliation of Graphene
Typical HOPG (highly
oriented pyrolitic graphite )
surface prior to exfoliation
The Graphene field-effect transistor
3
 (k)
Au contact pads
2
1
VBG
0
-40
0
VBG(V)
40
Effect of valleys on quantum transport in graphene
Disorder in graphene
 Atomic scale defects: Grain boundaries,
topological defects, edges, vacancies…
1.
2.
Source of short range scattering
Removes valley degeneracy
 Charged impurity
Long range scattering
Substrate traps, ion drift, free charges
Does not affect valley degeneracy
Linear variation of conductivity
Graphene
1.5
Conductivity (mS)
1.
2.
3.
4.
Silicon oxide
1.0
Doped silicon
0.5
-3
-2
-1
0
12
1
2
n2D (10 /cm )
2
Valley symmetry: Quantum transport
Isospin singlet
Broken valley symmetry
Isospin singlet
Isospin triplet
Presence of Valley symmetry
2
2e D
Quantum correction to
 
conductivity

2
d q
 ( 2 )
(C  C  C  C )
x
2
y
z
0
Weak localization correction in
Graphene
2
 
2e D

2
d q
 ( 2 )
(C  C  C  C )
x
2
y
2

e   B 
B



 ( B )   (0)  
F

F
 
 B  2B
    B 
i
 

z
0



B
  2F 


 B  2 B 
z 


 
Short range scattering
C , C , C  Gapped
x
y
z
C  Contribute s
0
Negative MR: Localization
Long range scattering
C C
x
y
 C  C  Contribute
z
0
Positive MR: Anti-Localization
PRL (2009): Savchenko Group
Effect of valleys on mesoscopic fluctuations in graphene?
Universal Conductance Fluctuations
In a regular disordered metal
L
Bi film (Birge group, 1990)
 Aperiodic yet reproducible fluctuation of conductance with magnetic field,
Fermi Energy and disorder configuration
 For L < L: G  e2/h
 Quantum interference effect, same physics as weak localization
 Independent of material properties, device geometry: UNIVERSAL
Conductance fluctuations at low temperatures
G (e /h)
2
12
2
 (k)
3
10mK
11
500mK
1
1K
4.5K
10
0
-20
-40
0
VBG(V)
40
-19
VBG (V)
DG  e2/h  Universal conductance fluctuations
-18
Density dependence of conductance fluctuations
10 mK
B=0
8
10
-3
10
-4
10
-5
DG /<G>
2
4
2
G (mS)
6
2
0
-40
0
VBG(V)
40
-60
-30
0
30
VBG (V)
Need to find Conductance variance in single phase coherent box
60
Evaluating phase coherent conductance
fluctuations in Graphene
L
W
L
Classical
superposition
D G 
G
2
2
D G 
2

1
N box

2
G
N box 
LW
L
,
G  
G (mS)
6
4
2
10
-3
10
-4
10
-5
D /
2
0
2
DEVICE 1
T = 10mK
B=0
6
L (m)
2
DG ((e /h) )
1.5
1.0
0.5
4
2
0.0
L
Li
-40
0
2
VBG (V)
2
0
-60
-40
-20
0
VBG (V)
20
40
40
L (m)
G (mS)
DEVICE 2
2.0
1.5
1.0
0.5
2.0
1.5
L
Li
1.0
0.5
0.8
0.6
2
2
2
G ((e /h) )
1.0
0.4
0.2
0.0
-80
-40
0
VBG (V)
40
80
Valley symmetry: UCF
Universal Conductance
fluctuations
 G 2
Graphene
 N CD
 G 2
2 DEG
Number of gapless diffuson and
Cooperon modes
N CD  4
Low density: Valley symmetry preserved
N CD  1
High density: Valley symmetry destroyed
Implications to Random Matrix Theory and
universality class
Suzuura & Ando, PRL (2002)
Removed
Short range scattering
Effective spin rotation
symmetry broken
Wigner-Dyson orthogonal
symmetry class
Intervalley scattering by atomically sharp defects
Valley
symmetry
Preserved
Long range scattering
Effective spin rotation
symmetry preserved
Long range Coulomb potential from trapped charges
Wigner-Dyson symplectic
symmetry class
4
330mK
10mK
2

DG /DG
Temperature dependence
01
-70
0
70
-70
4.5K
1K
0
70
-70
VBG (V)
0
70
-70
 Factor of FOUR enhancement in UCF near the Dirac Point
 Possible evidence of density dependent crossover in universality class
0
70
BINARY HYBRIDS
GRAPHENE ON BN (INSULATOR)
GRAPHENE/BN BINARY HYBRIDS
VERTICALLY ALIGNED OVERLAY
Dr. Srijit Goswami
Paritosh Karnatak
GRAPHENE
EL9
Tape
Glass
h-BN (exfoliated)
GRAPHENE on h-BN
Aligner
Si/SiO2
GRAPHENE/BN
GRAPHENE-hBN HYBRIDS
ULTRA-HIGH MOBILITY
0.8
h-BN
SiO2
DOPED SILICON
Resistance (k)
Graphene
300 K
77 K
4.2 K
0.6
2
300K ~ 12000 cm /Vs
0.4
0.2
0.0
-2
2
4.2K ~ 30000 cm /Vs
-1
1
0
12
-2
density (10 cm )
GRAPHENE/BN
2
1/Rxy = gsgv(n+1/2)e2/h
= 2x2 (n+1/2)e2/h
GRAPHENE-hBN HYBRIDS
QUANTUM HALL EFFECT
n = 0, 1, 2,…
300
Rxx ()
200
Vg = -30 V
100
2
4
2
B (T)
6
8
B = 12 T
T = 4,2 K
10
12
1.0
0.5
2
Rxx (k)
LIFTING OF 4-FOLD
DEGENERACY
Ryx (h/e )
0
0
0.0
-1
0
-5
-6
-20
-4
1
-0.5
-3
-10
-2
2
0
10
Vg (V)
GRAPHENE/BN
3
4
20
5
-1.0
6
30
Summary
• A new effect of valley quantum state on the quantum
transport in graphene revealed
• The valley states are extremely sensitive to nature of
scattering of charge in graphene
• The degeneracy of the valley and singlet states can be
tuned with external electric field
• Universal conductance fluctuations can act as a readout of
the valley states
• Single layer graphene shows a density dependent crossover
in it universality class , along with a exact factor of four
change in its conductance fluctuation magnitude
• Valley degeneracy can be tuned with other means as well,
such as external periodic potential from the substrate
THANK YOU