Pseudospintronics: a new electronic
device scheme in graphene systems
− 연필심 위의 물리학
Hongki Min
Seoul National University, May 16, 2012
Department of Physics and Astronomy, Seoul National University, Korea
Motivation
Text book examples vs research problems
• How can we overcome the barrier efficiently
between the text book and real research problems?
Text book
examples
Research
problems
What is graphene?
Graphene is 2D honeycomb lattice of carbon atoms
Adapted from Wikipedia
Graphene (2D)
nm 246.0anm 246.0a
Graphite (3D)
How to make graphene?
Scotch tape method
Other methods: epitaxial growth, chemical vapor deposition, …
Adapted from Scientific American Adapted from NIST
Why graphene?
Graphene has high electron mobility
Y. Zhang et al,
Nature 438, 201 (2005)
100 times faster than
conventional semiconductors
at room temperature
Graphene is described by relativistic wavefunction
P. R. Wallace,
Phys. Rev. 71, 622 (1947)
Relativistic wavefunction
in a low energy system
Why graphene?
Bilayer graphene is a band gap tunable semiconductor
E. McCann,
Phys. Rev. B 74, 161403 (2006)
Band gap opening
by an external electric field
Graphene can be used as touch screen and flexible display
S. Bae et al,
Nat. Nanotech. 5, 574 (2010)
Atomically thin and conducting
Why graphene?
C. Lee et al,
Science 321, 385 (2008)
Graphene is 200 times stronger than steel
NEMS application
A. Balandin et al,
Nano Lett. 8, 902 (2008)
Graphene is an excellent thermal conductor
Heat spreader
1. Electronic structure
― Tight-binding model
Outline
2. Transport and optical properties
― Boltzmann transport theory
3. Electron-electron interactions
― Mean-field theory
4. Conclusion and future work
― New electronic device scheme
Electronic structure
− Tight-binding model
• What is the model Hamiltonian for graphene?
• What is the effective theory near the Fermi energy?
• What is the effect of stacking sequence?
Hamiltonian
)()()()( kkkk nnn EH
wave vector k
energy band nE
band index n
Time independent Schrödinger equation
a
α β α β
t t εα εβ
Tight-binding model
Example: 1D chain composed of two kinds of atoms
||2.0 t
ka
E/|t|
2
0
-2 -π π 0
)(
)(*
k
k
f
fH
2cos2)( )2/()2/( ka
teetf aikaikk
α β
K
K
M
σ , σ*
π, π*
Ener
gy
(eV
)
K M K
30
-20
20
10
0
-10
Monolayer graphene
Tight-binding model
321)(RkRkRk
k
iii
eeetf
α
β t1R
2R 3R
0)(
)(0*
k
k
f
fH
α β
)(0 yx ikkv
Expand around K
β β
: sp2
*, : pz *,
6C=1s22s22p2
pvE 0)( p
Monolayer graphene has a linear dispersion at low energies
Effective theory around K
Monolayer graphene
ka E
(eV
)
σp
0
*
00
0vvH
α β
01
10x
0
0
i
iy
in-plane velocity 0vyx ipp
kp
Pseudospin, chirality and Berry phase
Monolayer graphene
σB H
BE BE
kxa
E (
eV)
kya
pB 0vσp
0
*
00
0vvH
α β pseudospin
chirality
B σ B σ
Bilayer graphene
Bilayer graphene is composed of a pair of coupled graphene
A 35.3d
A 46.2a
000
00
00
000
0
*
0
0
*
0
v
vt
tv
v
H
Bilayer graphene is composed of a pair of coupled graphene
4-band model around K
Electronic structure of bilayer graphene
ka E
(eV
)
yx ipp
βb αt αb βt
αb
βb αt βt
t
200
20
02
002
0
*
0
0
*
0
Uv
vUt
tUv
vU
H
Band structure control by an external electric field
4-band model around K
Band gap opening in bilayer graphene
yx ipp
βb αt αb βt
+U/2
−U/2
ka E
(eV
)
U=0.0 eV
U=0.2 eV
Bilayer graphene
Source Drain SiO2
gate
SiO2
gate
Band gap opening in bilayer graphene
Band structure control by an external electric field
Tunable energy gap semiconductor
Possible application to a switch device
eEextd (eV)
Egap
(eV
)
0.3
0 1 2 3
0.2
0.1
Min, Sahu, Banerjee, and MacDonald, Phys. Rev. B 75, 155115 (2007)
000
00
00
000
0
*
0
0
*
0
v
vt
tv
v
H
Bilayer graphene is composed of a pair of coupled graphene
4-band model around K
αb
βb αt βt
t
Effective theory of bilayer graphene
ka E
(eV
)
yx ipp
αb βt βb αt
Bilayer graphene has a parabolic dispersion at low energies
Effective theory around K
m
pE
2)(
2
p
σn p
)(
20
0
2
12
2
2
2*
m
p
mH
Effective theory of bilayer graphene
ka E
(eV
)
αb βt
)2sin,2(cos)(2 pppn
yx ipp
)/(tan 1
xy ppp2
02vtm
Pseudospin, chirality and Berry phase
kxa
E (
eV)
kya
Effective theory of bilayer graphene
cx kk
cy kk
)2sin,2(cos)(2 pppn σn p
)(
20
0
2
12
2
2
2*
m
p
mH
αb βt pseudospin
chirality
kxa
E (
eV)
kya
Effective theory of bilayer graphene
cx kk
cy kk
xσm
p
p
p
mH
20
0
2
1 2
2
2
m
pE
2)(
2
p
Pseudospin, chirality and Berry phase
)sin,(cos)( pppn JJJ
)(0
00
*
0 pnσ
J
J
J
J
J pH
“↑” “↓”
J
J pE 0)( p
Monolayer graphene : 2D chiral system with chirality J=1
Bilayer graphene : 2D chiral system with chirality J=2
Pseudospin and Chirality
Effective theory of a general chiral system
Pseudospin: Two-valued quantum degrees of freedom
Example: sublattice, layer
01
10x
0
0
i
iy
yx ipp
)/(tan 1
xy ppp
Chirality: Projection of pseudospin to momentum direction
McCann & Fal’ko , Phys. Rev. Lett. 96, 086805 (2006)
Min & MacDonald, Phys. Rev. B 77, 155416 (2008)
Three distinct stacking arrangements, labeled A,B,C
Multilayer stacking
α
β
A α β
C
α β
B
Electronic structure of multilayer
Stacking sequence and energy band structure A
C
B
ABC trilayer (chirality=3)
ka
E (
eV)
E~p3
ka
ABA trilayer (chirality=1,2)
E~p2 E~p
E (
eV)
0
0~
3
3*
effH
0
0
0
0~
2
2**
effH
pi
yx peipp
Min and MacDonald, Phys. Rev. B 77, 155416 (2008)
Electronic structure of multilayer
Stacking sequence and energy band structure A
C
B
ABCA (chirality=4)
ka
E (
eV)
E~p4
ka
ABAB (chirality=2,2)
E~p2 E~p2
E (
eV)
0
0~
4
4*
effH
0
0
0
0~
2
2*
2
2*
effH
Min and MacDonald, Phys. Rev. B 77, 155416 (2008)
pi
yx peipp
Electronic structure
Important lessons
Low-energy band structure is
described by a set of chiral systems.
High-energy band structure follows that of monolayer graphene.
Electronic structure strongly depends on the stacking sequences.
Electronic structure engineering by stacking sequences
ka
E~p2 E~p
E (
eV)
0
0
0
0~
2
2**
effH
Ex: ABA trilayer (J=1,2)
Min and MacDonald, Phys. Rev. B 77, 155416 (2008)
Transport and optical properties
− Boltzmann transport theory
• What is the effect of impurities?
• What is the response to light?
• How can we identify the stacking sequence?
Model
• Relaxation time approximation
t
tf
dt
dtf
dt
tdf
),(),(),( kk
k
kk
• Non-equilibrium distribution function
)(),()2(
)( kvkk
J tfd
ed
Boltzmann transport theory
k
kk
)(),( 0ftf
),()2(
tfd
dnd
kk
Ek
)( edt
d
k
)(0 kf
relaxation time
equilibrium
distribution function
electrical conductivity
Model
DOS Diffusion constant in 2D Fermi velocity
FFFvs vgge 22 v
2
1)(
Fk
F
1v
2
2v
impimp
F
Vn
Boltzmann transport theory
• Electrical conductivity Ekvkk
J
)(),()2(
)( tfd
ed
• Relaxation time
)(21 2
Fimpimp
F
vVn
Fermi’s Golden rule
DOS
Density dependence of conductivity
2
2v~
impimp
F
Vn
constant~impV
2~ Fkn
1~v J
FF k
n (1012 cm-2)
σ (
e2/h
)
Short range scatterers
Short-range scatterers
imp
J
n
n 1
~
impn
n0
~ at high n
For J-chiral system
A→J=1, AB→J=2, ABC→J=3, ABA→J=1,2
1~J
~n0
~nJ-1
Min et al. Phys. Rev. B 83, 195117 (2011)
Transmittance of graphene
Optical conductivity measurements
Nair et al, Science 320, 1308 (2008)
2
)(2
1)(
cT
)(1)( TR
Optical conductivity of bilayer graphene E
(eV
)
ka
Optical conductivity Bilayer graphene
σ(ω
) /σ
uni
h
euni
2
2
(eV)
High frequency limit
At high frequencies, each decoupled layer gives σuni, thus
optical conductivity approaches to Nσuni.
h
euni
2
2
Optical conductivity of bilayer graphene E
(eV
)
ka
Optical conductivity Bilayer graphene
σ(ω
) /σ
uni
h
euni
2
2
(eV)
Intermediate frequency regime (t=0.3~0.4 eV)
At intermediate frequencies, optical conductivity shows
characteristic peaks depending on the stacking sequence.
h
euni
2
2
Optical conductivity of bilayer graphene E
(eV
)
ka
Optical conductivity Bilayer graphene
σ(ω
) /σ
uni
h
euni
2
2
(eV)
Low frequency limit
At low frequencies, optical conductivity approaches to Nσuni.
h
euni
2
2
Optical conductivity of graphene multilayers
Low frequency limit
• Monolayer graphene (J=1) h
euni
2
2
• J-chiral system uniJ J
• Chirality sum is the number of layers
uni
iiJ N
At low frequencies, optical conductivity approaches to Nσuni.
Optical conductivity of ABC trilayer E
(eV
)
ka
Optical conductivity ABC trilayer
Min and MacDonald, Phys. Rev. Lett. 103, 067402 (2009)
Optical conductivity and stacking sequence A
C
B
σ(ω
) /σ
uni
(eV)
h
euni
2
2
Optical conductivity of ABA trilayer
Optical conductivity ABA trilayer
Min and MacDonald, Phys. Rev. Lett. 103, 067402 (2009)
A
C
B
σ(ω
) /σ
uni
ka
Optical conductivity measurements provide a useful way
to identify the number of layers and stacking sequences.
h
euni
2
2
E (
eV)
(eV)
Optical conductivity and stacking sequence
Min and MacDonald, PRL 103, 067402 (2009)
Comparison with experiments
Mak et al, PRL 104, 176404 (2010)
σ(ω
) /σ
uni
(eV)
ABAB
ABCA
Optical conductivity and stacking sequence
Electron-electron interactions
− Mean-field theory
• What is the effect of electron-electron interaction?
• How can we treat the electron-electron interaction?
• Is there a pseudospin version of ferromagnetism?
Electron-electron interaction
Interaction-induced ordered states
• Magnetism
• Superconductivity
• Exciton-condensation
–,↑ –,↓
+
–
+
–
+
–
...
Mean-field theory
Weiss molecular-field approximation
ji
jiijJH,2
1SS
iii SSS
jjiiji SSSSSS
jijijiji SSSSSSSS
jiijji SSSSSS
i
i
MF
i
MFH SΒ j
jij
MF
i J SΒ
Weiss molecular-field approximation
Mean-field theory
0MF
iΒ
cTT
0iS
cTT
0MF
iΒ
0iS
–
–
–
–
–
–
–
–
–
Hartree-Fock approximation
Mean-field theory
Interacting electrons Non-interacting electrons
in an effective potential
Pseudospin
Bilayer graphene (chirality J=2)
“↑”=αb “↓”=βt
αb
βb αt βt
Two-valued quantum degrees of freedom
Is there spontaneous charge polarization in the presence
of electron-electron interactions?
Ferromagnetism means spontaneous spin polarization.
Pseudospin magnetism
)(20
0
2
12
2
2
2*
pnσ
m
p
mH
)2sin,2(cos)(2 pppn
Pseudospin magnetism
yx ipp
)/(tan 1
xy ppp
In-plane pseudospin direction
Min et al. Phys. Rev. B 77, 041407(R) (2008)
σkB )(00H
char
ge
po
lari
zati
on
Normalized gate voltage
cx kk
cy kk
yx ,
Pseudospin magnetism in bilayer graphene
No electron-electron interaction
+ + +
− − −
− − −
+ + +
)2sin,2(cos~
)0,2sin,2(cos2
)(22
0 kkkB m
k
In-plane pseudospin direction
cx kk
cy kk
yx ,
Min et al. Phys. Rev. B 77, 041407(R) (2008)
),2sin,2cos()(eff zBBB kkkB
Vortex formation in momentum space
Spontaneous charge transfer between the two layers
σkB )(effMFH
Pseudospin magnetism in bilayer graphene
Electron-electron interactions are turned on
char
ge
po
lari
zati
on
Normalized gate voltage
+ + +
− − −
− − −
+ + +
In-plane pseudospin direction
ABC stacked multilayers are excellent candidates. Ex) AB, ABC, ABCA, ABCAB,…
Pseudospin magnetism in multilayer graphene
Pseudospin magnetism is more stable for larger chirality.
J=1
J=3
J=2
J=4
)sin,(cos~, JJyx
A
C
B
Conclusion and future work
− New electronic device scheme
• How can we use the ordered states in devices?
• Are there other interaction-induced ordered states?
• Ongoing work
Dirac-like chiral wavefunction
New ordered states in graphene systems
Electron-electron interaction +
Energy band structure and stacking sequences
Search for ordered states in graphene systems
Pseudospin magnetism in coupled graphene bilayers
Interplay between chirality, e-e interaction and disorder
Disorder −
High-quality sample with low disorder
Single particle behavior
Collective behavior of many particles
Prospects
Classical/semi-classical phenomena
Quantum phenomena in macroscopic scale
Current electronic device scheme
Future electronic device scheme
Collective behavior of many electrons
Collective electron device scheme
Example: Giant magnetoresistance (GMR)
Magnetic field control of ordered states
• Collective behavior of
many electrons
Collective electron device scheme
Example: Pseudospin magnetism
Pseudospintronics!
• Can be switched with a
small gate voltage change
using much less power
• Can exhibit a pseudospin
version of GMR and spin-transfer torque
• Electrical control of ordered states
char
ge
po
lari
zati
on
Normalized gate voltage
Conclusion
Text book examples vs research problems
• Ask
Text book
examples
Research
problems
• Seek
• Knock
: Questions
: Motivation
: Doors
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