Fariborz ParhizgarInstitute for research in Fundamental Science

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Intro.

•Transistors (From Ge to C)• Importance of Graphene

Model

•monolayer Graphene•Bilayer Graphene

conclusion

•Bilayer: a good condidatefor FET

2

GeC

3

Smallsize

Highmobility

2 importantProperties

that atransistorsmust have

4

5

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Wallace Phys. Rev. 104, 666Geim et. al. Nat. Mat. 6, 183

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� Carbon: smallest atom in group 4

sp2 orbitals which makes σ-bondpz Orbital which makes π-bond

2 2 2:1 2 2C S S P

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Goodcondidate to

maketransistors

Crystalupto nm

sizes

Highmobility

~ 105

cm2/Vs

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MonolayerGraphene

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� Honeycomb lattice is not a Bravious Lattice

� Hoping of PZ Orbitals

† . .ij i jij

H t C C H C< >

= +å

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� Taking Fourier Transform:

1 2

( )

*( ),

( ) [1 ]

pA A

B Bp

ik a ik a

E f kH

f k E

f k e e

y yy y

g × ×

æ öæ ö æ öç ÷=ç ÷ ç ÷ç ÷è ø è øè ø

= + +r rr r

r

r

r

1 2 2 1( ) 3 2cos 2cos 2cos ( )pE k E k a k a k a a= ± + × + × + -r r rr r r r

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.,( ) iik ri

ic k c ea a=å

Two independent K-Point

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� Expanding the function f(k):

� massless Dirac Fermions

�Where is the Pauli matrix for pseudospins

( ) ( )'F x y

Kf k v k ik

Kì

= ± íî

H =

ˆ.k

H ks=år

r

s

( ) FE k v k=610 / / 300Fv m s c: :

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� Absence of e-backscattering:

Castro Neto et. Al. Rev. Mod. Phys. 81,109

ˆ.k

H ks=år

r

( ) ( )( )

( ) ( )( )

( )

1 11( )2 2

1 1( )

' '2 21

( )2

x y x y

x y x y

x y

i k x k y i k x k yI i i

i q x k y i q x k yII i i

i k x k yIII i

rr e ese se

a br e es e s e

tr ese

j p j

q p q

j

y

y

y

+ - +

-

+ - +

-

+

æ ö æ ö= +ç ÷ ç ÷

è ø è øæ ö æ ö

= +ç ÷ ç ÷è ø è øæ ö

= ç ÷è ø

10sgn( ), ' sgn( ), tan ( / )y xs E s E V k qq -= = - =

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� Quasi particles can not be trappedyour PC will not turn off

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BilayerGraphene

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� Most possible config.

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† †, , , , 1, , 2, ,

† †4 1, , 2, , 4 2, , 1, ,

†3 1, , 2, ,

( . .) ( . .)

( . .) ( . .)

( . .)

m i m j j jij m j

i j i jij m ij m

i jij m

H t a b H C t a a H C

tv a b H C tv a b H C

tv b b H C

s s s ss s

s s s ss s

s ss

^< >

< > < >

< >

= - + - +

- + - +

- +

å å

å å

å

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T.Chakraborty et. Al. advance in phys. April 2010

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Fourier transform:*

4* *

† 4 3*

4*

4 3

( ) ( )( ) ( ) ( )

( ) ( )( ) ( )

( ) ( ) ( )k

f k t v f kf k v f k v f k

H k kt v f k f k

v f k v f k f k

^

^

æ öDç ÷

Dç ÷= Y Yç ÷-Dç ÷ç ÷-Dè ø

å

† † † † †1, , 1, , 2, , 2, ,( ) ( , , , )k k k kk a b a bs s s sY =

3.

1( ) ( )iik d

Fi

f k t e v q q=

= Få ;

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� Dispersion Relation:

� Independent of

2 22 2 2 2 2 2 2 4( ) / 4

2 4t Vv p v p V t te ^

^ ^= + + - + +

( )kF

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We go to2-bandmodel

LowEnergy is

preferable

Smaller is(usually)simpler

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In low energy limit ( ):

0

/ 2 0 00 / 2 0 0

0 / 2 00 0 0 / 2

V tV

Kt V

V

^

^

æ öç ÷ç ÷=ç ÷-ç ÷

-è ø

4*

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4

4

0 0( ) 0 00 0

0 0

i i

i

i i

i i

pe v pef k v pe

Kv pe pe

v pe pe

f f

f

f f

f f

-

-

-

æ öç ÷ç ÷= ç ÷ç ÷ç ÷è ø

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3

0 0 0 00 0 00 0 0 00 0 0

i

i

v peK

v pe

f

f

=

-

æ öç ÷ç ÷ç ÷ç ÷è ø

0 1 2H K K K= + +

te ^<<

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using degenerate 2nd order perturbation:

McCann et. al. PRL 96, 086805

Nilsson et. al. PRB 78, 045405

20 0 0 0 2 0 0 1 1 1 1 0

0

1' [ ] ' 'l P HP l l P K K P l l P K P PK P lE K

l l= + +-

† 2 †

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1 00 ( ) 010 12 2( ) 0 0low

VH vm

p pp p

æ ö æ ö æ ö-= + +ç ÷ ç ÷ ç ÷-è øè ø è ø

25

T.Chakraborty et. Al. advance in phys. April 2010

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Experimental evidence of the gap:

Ohta et. al. Science 313, 951

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� Low energy band with the gap:

at momentum:

2 22 2 2 2 2 2 2 4( ) / 4

2 4t Vv p v p V t te ^

^ ^= + + - + +

2 2

t VUV t

^

^

=+

%

2 222

2 2 2

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V tVpv V t

^

^

+=

+

.2V eV:

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Gate VoltageChange the

Fermienergy.

Using Dualgated BLG:

2 tuningparameter

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BilayerFET

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Digital Electronics

Pseudospintronics

Terahertz Technology

Infrared nanophotonics

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SLG isGapless

On/offratio ~ 5

We needGap

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Uniaxial Strain

Substrate Interaction

Lateral Confinement

Biased BLG

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Advantages

•Gap~Hundereds of meV•High On/Off ratio

Disadvantages

•Low mobility•Undesirable shape

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Gap upto .3eV

On/off ratio: (100-2000)

Nano lett. 2010,10,715

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� Thanks for yourattention