Graphene

Handout 11 Energy Bands in Graphene: Tight Binding and the Nearly Free Electron Approach Energy

In this lecture you will learn:

• The tight binding method (contd…) • The π-bands in graphene

FBZ

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Graphene and Carbon Nanotubes: Basics y

3a

• Graphene is a two dimensional single atomic layer of carbon atoms arranged in a Honeycomb lattice (which is not a Bravais lattice) • The underlying Bravais lattice is shown by the location of the black dots and is a hexagonal lattice • There are two carbon atoms per primitive cell, A and B (shown in blue and red colors, respectively)

a

A

a = 2.46 A

a

x

B

v a1 v a2 v ⎛ 3 1 ⎞ a1 = ⎜⎜ xˆ + yˆ ⎟⎟ a 2 ⎠ ⎝ 2

⎛ 3 v 1 ⎞ a2 = ⎜⎜ xˆ − yˆ ⎟⎟ a 2 ⎠ ⎝ 2

• Graphene can be rolled into tubes that are called carbon nanotubes (CNTs)

CNT

CNT


1

Graphene: Sp2 Hybridization y

Sp2 hybridization in carbon:

r r r 1 1 1 φ2 s (r ) + φ2 px (r ) + φ2 py (r ) 3 6 2 r r r r 1 1 1 ϕ 2 (r ) = φ2 s (r ) + φ2 px (r ) − φ2 py (r ) 3 6 2 r r r 1 2 ϕ 3 (r ) = φ2 s (r ) − φ2 px (r ) 3 3 r

ϕ1(r ) =

x

• All carbon atoms are all sp2 hybridized (one 2s orbital together with the 2px and the 2py orbitals generate three sp2 orbitals) • All sp2 orbitals form σ-bonds with the sp2 orbitals of the neigboring carbon atoms • The bonding orbital associated with each σ-bond is occupied by two electrons (spin-up and spin-down) • There is one electron per carbon atom left in the 2pz orbital


Graphene: 2pz orbitals

y

π-bonding: • Each carbon atom contributes one 2pz-orbital that participates in bonding

x

⇒ Each primitive cell contributes two 2pz-orbitals that participate in bonding • The 2pz orbital stick out of the plane of the chain and form π-bonds with neigboring 2pz orbitals • The π-bonding results in energy bands (π-bands) that we will study via tight binding


2

Graphene: Some Useful Vectors y

a = 2.46 A

3a

Basis vectors:

y

a

r r d1 d 2 B

A

a

x A

r a d1 = − xˆ 2 3

x

B

v a1

r a d2 = xˆ 2 3

v a2

Nearest neighbor vectors:

r n2

r a n1 = xˆ 3

r n1

r a ⎛ 1 3 ⎞ ⎜ − xˆ + n2 = yˆ ⎟ 3 ⎜⎝ 2 2 ⎟⎠

B

rA n3

These will be useful for writing the final solution in a compact form

r a ⎛ 1 3 ⎞ ⎜ − xˆ − n3 = yˆ ⎟ 3 ⎜⎝ 2 2 ⎟⎠ ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Graphene: Tight Binding Solution

y

a = 2.46 A 3a

• Each basis atom contributes one 2pz-orbital that participates in bonding

x

⇒ Each primitive cell contributes two 2pz-orbitals that participate in bonding

r

φ pzA (r ) ↔

r

φ pzB (r ) ↔

Ep

A

B

a

Ep a

One can thenr write the trial tight-binding solution for wavevector k as: r

r

[

r r r r r r r r r r r r r e i k . Rm ψ kv (r ) = ∑ c pzA k e ik . d 1φ pzA r − Rm − d1 + c pzB k e ik . d 2 φ pzB r − Rm − d 2 N m

()

(

)

()

(

)]


3

Graphene: Tight Binding Solution

y

a = 2.46 A

3a

Plug the solution into the Schrodinger equation:

()

r r r Hˆ ψ kr (r ) = E k ψ kr (r )

x

And then, one by one, multiply by from the left by the bra’s corresponding to every orbital in one primitive cell to generate as many equations as the number of orbitals per primitive cell

B

A

a a

Step 1: r Multiply the equation with φ pzA (r ) and: • keep the energy matrix elements for orbitals that are nearest neighbors, and • assume that the orbitals on different atoms are orthogonal

)

(

r r r r r r r r v v E p c pzA (k ) − Vppπ e i k . n1 + e i k . n2 + e i k . n3 c pzB (k ) = E k c pzA k

()

()

Notice that the final result can be written in terms of the nearest neighbor vectors ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Graphene: Tight Binding Solution y

a = 2.46 A

3a Step 2: r Multiply the equation with φ pzB (r ) and: • keep the energy matrix elements for orbitals that are nearest neighbors, and • assume that the orbitals on different atoms are orthogonal

)

(

x A

B

a a

r r r r r r r r v v E p c pzB (k ) − Vppπ e − i k . n1 + e − i k . n2 + e − i k . n3 c pzA (k ) = E k c pzB k

()

()

Notice that the final result can be written in terms of the nearest neighbor vectors


4

Graphene: Tight Binding Solution y

a = 2.46 A

Write the equations obtained in a matrix form:

⎡ Ep r ⎢ * ⎢⎣ − Vppπ f k

()

()

r − Vppπ f k ⎤ ⎥ Ep ⎥⎦

() r

Where the function f k

()

Solutions are:

x B

A

a is:

(

r r r r r r r f k = e i k .n1 + e i k .n2 + e i k .n3

()

3a

v v r ⎡c pzA (k )⎤ ⎡c pzA (k )⎤ v⎥=E k ⎢ v⎥ ⎢ ⎣c pzB (k )⎦ ⎣c pzB (k )⎦

()

)

a

()

r r E k = E p ± Vppπ f k

And the corresponding eigenvectors are:

v 1 ⎡c pzA (k )⎤ ⎤ 1 ⎡ r r ⎥ v⎥ = ⎢ * ⎢ f k f k − 2⎣ ⎦ ⎣c pzB (k )⎦ +

() ()

v ⎡c pzA (k )⎤ 1 ⎡ r1 r ⎤ v⎥ = ⎢ ⎢ * ⎥ 2 ⎣f k f k ⎦ ⎣c pzB (k )⎦ −

() ()


Graphene: π-Energy Bands Energy

ky K K’ M

()

()

r r E k = E p ± Vppπ f k K’ M

Γ

kx

FBZ K

K K’

FBZ

• Bandgaps open at the M-points between the first and the second bands • No bandgaps open at the K-points and the K’-points • In the homeworks you showed (using nearly free electron approach) that bandgaps can open at the K-points (or K’-points) if the inversion symmetry is broken


5

Graphene: π-Energy Bands

• Since graphene has two electrons per primitive cell contributing to π-bonding, the lower π-band will be completely filled at T ≈ 0K

3Vppπ

• The location of Fermi level near T ≈ 0K is shown by the dashed curve

In generating the plots I chose energy zero such that:

Ep = 0 And for graphene:

Vppπ = 3.0 eV ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Graphene: A Comparison of NFEA and TB

Scale normalized 2

h2 ⎛ 1 ⎞ ⎜ ⎟ 2m ⎝ a ⎠ and offset by Vo to :

To compare the nearly free electron approach (NFEA) to tight-binding (TB) I assumed the DC potential in NFEA to be:

Vo = E p − 3Vppπ And in graphene:

Vppπ = 3.0 eV ECE 407 – Spring 2009 – Farhan Rana – Cornell University

6

Graphene

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