Pseudospintronics: a new electronic

device scheme in graphene systems

− 연필심 위의 물리학

Hongki Min

hmin@snu.ac.kr

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