The 1D chain of atoms with s-p hybridization (a model Silicon): see Exercise 19 3

The 1D chain of atoms with s-p hybridization (a model Silicon): see Exercise 19

3s and 3p bands in silicon as a function of interatomic distance.

In the exercise we vary the hoping energy t : decreasing the distance means increasing the hoping integral !

(Electronic band structure, wikipedia)

The 1D chain of atoms with s-p hybridization (a model Silicon): see Exercise 19

1) 2)

First question : chain of atom with one s-orbital per atom (exactly what was done in class)

Bloch theorem: _k

(x + a) = e

(x)

k

(x) = e

u

(x) with u

(x) periodic

(I) (II)

With only one type of s-type orbital per atom, the only possible Bloch state for a given k-vector is :

k

(x) = 1 p N

X

e

(x na)

For a given k-vector, the energy of this state is given by the Schrödinger equation

that we project on

h

|

h ^k | H ˆ | ^k i = " _k h ^k | ^k i = " _k

H ˆ |

i = "

|

i

to obtain :

h

| H ˆ |

i = 1 N

X

n=1

X

m=1

e

e

h

(x na) | H ˆ |

(x ma) i

= 1 N

X

n=1

X

m=1

⇣ e

"

+ e

t + e

t ⌘

= "

2 cos(ka)

that is :

"₀+ 2

4

e

Second question : chain of atom with one p-orbital per atom

With only one type of p-type orbital per atom, the only possible Bloch state for a given k-vector is :

k

(x) = 1 p N

X

n=1

e

(x na)

"_k =

h

| H ˆ |

i = 1 N

X

n=1

X

m=1

e

e

h

(x na) | H ˆ |

(x ma) i

= 1 N

X

n=1

X

m=1

⇣ e

"

+ e

t

+ e

t

⌘

= +"

+ 2 cos(ka)

e

Overlap of s- and p-band begins at:

"

+ 2 = "

2

= "

/2

+"

"

"₀ + 2

4

Question 3 : Now we put the s and p orbitals all-together => there are 2 atomic orbitals per atom (and per unit cell) => 2 possible basis Bloch state:

s

(x) = 1 p N

X

e

(x na)

p

(x) = 1 p N

X

e

(x na)

k

(x) = ↵

(x) +

(x)

Any Bloch state is a linear combination of these 2 basis Bloch states:

By projection of onto

h

| and h

|

↵H _ss + H _sp = "↵

↵H _ps + H _pp = "

H ˆ |

i = "

|

i

H

= h

| H ˆ |

i = "

(k) = "

2 cos(ka) (first question)

H

= h

| H ˆ |

i = "

(k) = +"

+ 2 cos(ka) (second question)

H_sp = h ^s|Hˆ| ^pi = 1 N

XN n=1

XN m=1

e ^iknae^ikmah ^ats (x na)|Hˆ| ^atp (x ma)i

= 1 N

XN

n=1

⇣e^ika + e ^ika( )⌘

= 2isin(ka)

sp coupling : no onsite energy and hoping change sign from right to left :

(H

")(H

") | H

|

= 0

(" " _s (k ))(" " _p (k)) = 4 ² sin ² (ka)

" _av = (" _s + " _p )/2 and " = " _p " _s

Solve :

Trick :

(" "

)(" "

) = (" "

✏/2)(" "

+ ✏/2)

= (" "

)

( ✏/2)

" = " _av ±

q

( ✏/2) ² + 4 ² sin ² (ka)

Note : when sp coupling goes to zero, one recovers the original uncoupled s and p bands.

+ -

H

= h

| H ˆ |

i = "

(k) = "

2 cos(ka) (first question)

H

= h

| H ˆ |

i = "

(k) = +"

+ 2 cos(ka) (second question)

✏

= ✏

+ ✏

2 = 0 ✏

2 = ✏

✏

2 = "

+ 2 cos(ka)

"(k) = ±

q

"

+ 4 "

cos(ka) + 4

Limit

/"

>> 1 "(k) = ± 2

s ⇣ "

⌘

+ "

cos(ka) + 1

= ± 2 ⇣

1 + 1 2

"

cos(ka) + · · · ⌘

= ± ⇣

2 + "

cos(ka) + · · · ⌘

Two bands centered on

± 2"

with a bandwidth

"

2

No sp-coupling With sp-coupling

"₀

"

= "0/2

With 2 electrons per atom, the s-band is filled => insulating for

< "

/2

> "

/2

For •  metallic in the case of no sp-coupling

•  insulating in the case of sp-coupling

Exercise 29 Localized vibrational modes around an impurity

We study the vibrational modes of a 1D chain of atoms. All springs have the same spring constant C. All atoms have the same mass M except the central atom that has a mass m.

a) In the case of a chain with all identical atoms with mass M and spring constants C, re- derive the equation of motion for any atom with index (n), calling u_n its displacement out of its equilibrium position. Plot the phonon modes energy dispersion relation in the

Brillouin zone for the perfect chain (q = phonon momentum).

M d

u

dt

= C (u

u

) C (u

u

) = C (u

+ u

2u

)

u

= u

e

e

!(q) =

r 4C

M | sin(qa/2) |

Use :

b) In the presence of the impurity, we try vibrational modes that are localized around the impurity of mass m, assuming displacements of the form:

where A is a constant and |n| the absolute value of the n-index.

- What is the equation of motion for an atom n with (n > 1) ?

-  Show that the "localized" solution satisfies this equation of motion provided that:

u

= A( 1)

e

e

M !

= C ⇣

2 + X + 1 X

⌘ with X = e

M d

u

dt

= C (u

+ u

2u

)

M !

( 1)

e

= C ⇣

( 1)

e

+( 1)

e

2( 1)

e

⌘

M ! ₀ ² = C ⇣

( 1) ¹ e ^↵ + ( 1) ¹ e ^↵ 2 ⌘

Plug in localized mode (with 0 < n-1 < n < n+1) :

-  Deduce that the energy of the localized vibrational mode is necessarily above the band of energy of the phonon modes of the 1D chain without impurities.

Indication: cosh(x) always larger than cosh(0)=1.

!

= 2C M

⇣ 1 + cosh(↵) ⌘

> 4C M

c) What is the equation of motion for the central impurity atom of mass m ? Using this equation of motion, obtain a second equation relating , m, C and X.

!

m d

u

dt

= C (u

+ u

2u

)

u

= A( 1)

e

e

m!

= C ⇣

e

e

2 ⌘

= ) m!

= 2C ⇣

1 + 1 X

⌘

cosh

d) Using the two expressions for derived in b) and c), calculate (M-m)/m as a function of X and show that the above solution is possible (not diverging) only if (m < M).

!

M !

= C ⇣

2 + X + 1 X

⌘ with X = e

m! ₀ ² = 2C ⇣

1 + 1 X

⌘

M m

m = 2 + X + 1/X 2 2/X

2 + 2/X = X 1/X

2 + 2/X = 1 2

X 1 X + 1

↵ > 1 = ) X = e

> 1 = ) (M m) > 0

Localized vibrations around defect

u

= A( 1)

e

e

2

= X 1 2

0