LCAO Introduction - Solid State Physics - Lecture Slides, Slides of Solid State Physics

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Introduction to theTightbinding (LCAO)

Method

Tightbinding:

1 Dimensional Model

-^ Consider an

Infinite Linear Chain

of identical atoms, with

1 s-orbital

valence e

-^ per atom

& interatomic spacing =

a

-^ Approximation:

Only Nearest-Neighbor interactions.

(Interactions between atoms further apart than

a^ are^ ~ 0

This model is called the

“Monatomic Chain”.

Each atom has s electron orbitals only! Near-neighbor interaction only means that the

s^ orbital on site

n^ interacts with the

s^ orbitals on sites

n – 1^ &^

n + 1^ only!

n =^ Atomic Label

a^   

n =^ -

-^ The localized atomic orbitals on each site for this

Monatomic

Linear Chain

of atoms look qualitatively like this:

n = -^

-^ -^ -^

^ a^  ^3

The spherically symmetric

s^ orbitals on each site overlap

slightly with those of their neighbors, as shown. Thisallows the electron on site

n^ to interact with its

nearest-neighbors on sites^ n – 1

&^ n + 1

The True Hamiltonian in the solid is

:

H = (p)

2 /(2m)o^

+ V(x),

with^ V(x) = V(x + a)

-^ Instead

,^ approximate

it as^ H

^ ∑ Hn^

(n) +^ at

∑ U(n,nn,n

where,^ H

(n)^ ^ Atomic Hamiltonianat

for atom

n.

U(n,n)

^ Interaction Energy

between atoms

n^ &^ n.

Use the assumption of

Only Nearest-neighbor Interactions

:

^ U(n,n

) = 0^ unless

n^ = n -

or^ n^ = n +

-^ With this assumption, the

Approximate Hamiltonian

is:

H^ ^ ∑

[H^ (n) + U(n,n -1) + U(n,n + 1)]n at^

-^ Dirac notation:

Ek

^  Ψ |H| k^

Ψ k

(This Matrix Element is shorthand for a spatial integral!)

-^ Using the assumptions for

H^ &^ Ψ k

(x)^ already listed:

^ E^ =k^

 Ψ | ∑ Hkn^

(n) | Ψ atk +^  Ψ |[ ∑ k

U(n,n-1) + U(n,n-1)]|n

Ψ k

also note that

H(n)| ψ at

^ =^ ε | ψ nn 

-^ The LCAO assumption is

:^ | Ψ ^ k

ikna ∑ en

| ψ n

-^ Assume

orthogonality of the atomic orbitals:^ ^  ψ n

| ψ ^ =^ n^ δ (= 1, n = nn,n^

;^ = 0, n^

^ n)

-^ The Nearest-neighbor interaction

assumption:

^ There is nearest-neighbor overlap energy only!

( α^ =^ constant)

 ψ |U(n,nn

^ 1)| ψ n

^ ^

-^ α ;^ (n

= n,^ &^ n

= n^ ^ 1)

 ψ |U(n,nn

^ 1)| ψ n

^ = 0,^ otherwise

It can be shown that for

α^ > 0, this^ must be negative!

-^ As a student exercise, show that the

“energy band”

of this model is:

E=^ ε^ - 2 α k

cos(ka)^

or^ E

=^ ε^ - 2 α k^

• 4 α^ sin

2 [(½)ka]

-^ A trig identity was used to get last form.

ε^ &^ α^ are usually taken as parameters

in the theory, instead of actually calculating them from the atomic

ψ n

^ The^ “Bandstructure”

for this monatomic chain with nearest-neighbor interactions only looks like (assuming

2 α^ <^ ε^ ):^

(E^ E-T^ k^

ε^ + 2 α ) It’s interesting to

note that: The form^

E=^ ε^ - 2 α k

cos(ka)^ is similar to Krönig-Penney modelresults in the linear approximationfor the messy transcendentalfunction! There, we got:^ E

= A - Bcos(ka)k^ where A^ &^ B^ were constants.

E^ T^4 α

The True Hamiltonian in the solid is

H = (p)

2 /(2m)o^

+ V(x),

with^ V(x) = V(x + a)

-^ Instead

,^ approximate it

(with^ γ^ = A

or =B)^ as H^ ^ ∑γ n

H( γ ,n) +at^

∑ U( γ n, γ n

γ ,n; γ ,n

where,^ H

( γ ,n)^ ^ at Atomic Hamiltonian

for atom

γ^ in cell^

n.

U( γ ,n; γ ,n

)^ ^ Interaction Energy

between atom of type

γ

in cell^ n^

& atom of type

γ^ ^ in cell

n.

Use the assumption of

only nearest-neighbor interactions

The only non-zero

U( γ ,n; γ 

,n)

are^ U(A,n;B,n-1) = U(B,n;A,n+1)

^ U(n,n-1)

^ U(n,n+1)

-^ With this assumption, the

Approximate Hamiltonian

is:

H^ ^ ∑^ γ n

H^ ( γ ,n) + at

∑ [U(n,n -1) + U(n,n + 1)]n^

H^ ^ ∑γ n

H( γ ,n) +at^

∑ [U(n,n -1) + U(n,n + 1)]n

-^ Goal:

Calculate the bandstructure

Eby solving the Schrödinger Equation:k^ H Ψ (x) = E^ Ψ kk

(x)k

-^ Use the LCAO (Tightbinding) Assumptions:^ 1. H

is as above. 2. Solutions to the atomic Schrödinger Equation are known:^ H(at

γ ,n) ψ (x) = E γ n^

ψ (x) γ n γ n^

3.^ In our simple case of

1 s-orbital/atom

:

E=^ ε =AnA^

the energy of the atomic e

-^ on atom

A

E=^ ε =BnB^

the energy of the atomic e

-^ on atom

B

4.^ ψ (x) γ n^

is very localized near cell

n

5.^ The Crucial

(LCAO

)^ assumption is:ikna Ψ (x)^ ^ ∑ e^ kn^

∑^ C ψγ n γγ

(x)

That is, the Bloch Functions are linear combinations of atomic orbitals!

Note!! The C’s^ are unknown γ

-^ Student exercise to show that these simplify to:

0 = ( ε A^

• E^ )C^ k^ A

+ 2 μ cos(ka)C

, (3)B^

and

0 = 2 μ cos(ka)C

+ ( ε - EA^ B^

)C^ ,^ k^ B

-^ ε ,^ ε AB

,^ μ^ are usually taken as parameters in the theory, instead of (^) computing them from the atomic

ψγ n

-^ (3) & (4) are linear, homogeneous algebraic equations for

C^ &^ C^ A^ B

^2 ^2 determinant of coefficients = 0

-^ This gives:

( ε - EA^

)( ε - Ek B^

) - 4^ μ k

2 [cos(ka)]

^ A quadratic equation

for^ E!k^

^ 2 solutions:

a^ “valence” band

& a^ “conduction” band

-^ Results:^ “Bandstructure” of the Diatomic Linear Chain

(2 bands):

E(k) = (½)(

ε +^ ε A^ B

)^ [(¼)(

ε -^ ε )A^ B

{cos(ka)}

2

-^ This gives a

k = 0^ bandgap of E = E^ (0) - EG +^

(0) = 2[(¼)(-

ε -^ ε )A^ B^

2 2 + 4 μ ]

½

-^ For simplicity, plot in the case

ε -^ ε )A^ B

2 &^ ε B^

>^ ε A

^ Expand the

½^ [ ….]part of E(k)^ & keep the lowest order term

^ E^ (k)+

^ ε + A[cos(ka)]B^

2 ,^ E

(k)^ ^ ε -^ A

• A[cos(ka)]

2

E^ (0)^ ^ G^

ε –^ ε + 2AA^ B^

,^ where^ A

(^2)  (4 μ )/| ε -^ ε |A^ B^

Tightbinding Method:

3 Dimensional Model

-^ Model:

Consider a monatomic solid, 3d, with only nearest-neighbor interactions. Hamiltonian:

H = (p)

2 /(2m) + V(r)o^ V(r) = crystal potential,

with the full lattice symmetry & periodicity.

-^ Assume

(R,R^ = lattice sites):^ H^

^ ∑ HR^

(R) +at

∑ U(R,RR,R

H(R)^ at^

Atomic Hamiltonian

for atom at

R

U(R,R)

^ Interaction Potential

between atoms at

R & R

Near-neighbor interactions only! ^ U(R,R

) = 0^ unless

R & R

are nearest-neighbors

-^ Goal:

Calculate the bandstructure

Eby solving the Schrödinger Equation:k^

H Ψ (r) = Ek^

Ψ (r)kk^

-^ Use the LCAO (Tightbinding) Assumptions:^ 1. H

is as on previous page. 2. Solutions to the atomic Schrödinger Equation are known: H(R) ψ (R) = Eat n

ψ (R),^ nn

n^ = Orbital Label (

s, p, d,..),

E=^ Atomic energy of the en

-^ in orbital

n

3.^ ψ (R)n

is very localized around

R

4.^ The Crucial

(LCAO

)^ assumption is:

Ψ (r) =k^

ik ∑ eR

R ∑ b^ ψ n^ n

(r-R)^ n

(bto be determined)n^

ψ (R):^ The atomic functions are orthogonal for differentn^

n^ &^ R

That is, the Bloch Functions are linear combinations of atomic orbitals!

-^ Manipulate

(several pages of algebra)

to get:

(E^ - E^ k^ m

)b^ +^ ∑ m^

∑ enR^0

ikR^ γ (R)bmn^

= 0 ,^ n^

(I)

where:^ γ mn

(R)^ ^  ψ

|U(0,R)|m ψ ^ ^ “Overlap Energy Integral” n

-^ The^ γ

(R)^ are analogous to themn

α^ &^ μ^ in the 1d models. They are

similar to

V, etc. in real materials, discussed next! The integrals aress σ horrendous to do for real atomic

ψ!^ In practice, they are treated asm

parameters to fit to experimental data.• Equation (I)

:^ Is a system of

N^ homogeneous, linear, algebraic

equations for the coefficients

b.^ N =^ number of atomic orbitals.n

-^ Equation (I)

for^ N atomic states ^ The solution is obtained by taking an

N^ ^ N^ determinant! This results in

N bands^ which have their roots in the atomic orbitals!

-^ If the^ γ

(R)^ are “small”, each band can be thought of asmn

E^ ~ E^ + kk^ n^

dependent corrections That is, the bands are

~^ atomic levels + corrections

-^ Equation (I)

: A system of homogeneous, linear, algebraic eqtns for the

bn

-^ N atomic states

^ Solve an

N^ ^ N^ determinant!

^ N bands

Note:^ We’ve implicitly assumed

1 atom/unit cell.

If there are

n atoms/unit cell

, we get^ nN

equations &

nN bands!

-^ Artificial Special Case #1:

One^ s^ level per atom  1 (s-like) band

-^ Artificial Special Case #2:

Three^ p

levels per atom

^ 3 (p-like) bands

-^ Artificial Special Case #3:

One^ s^ and three

p^ levels per atom &

(^3) spbonding

^ 4 bands^ NOTE that

For n atoms /unit cell, multiply by n to get the number of bands!