Part I Fundamentals

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Part I Fundamentals

Electron Theory : Matter Waves

Chap. 3 The Schrödinger Equation

Chap. 4 Solution of the Schrödinger Equation for Four Specific Problems

Chap. 5 Energy Bands in Crystals

Chap. 6 Electrons in a Crystal

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5. Energy Bands in Crystals

* For Free electron

const.

1/ 2

. k

x

= E

sin a cos

P a

a

α α

α +

5.1 1-D Zone Schemes

) 2 cos(

cos

cos α a = k

_x

a ≡ k

_x

a + π n

In a crystal

ka a cos cos α =

If P = 0,

2 / 1 2

2 m E

=  α

n a

k

a

_x

π

α = + 2

2 / 1 2

2

2 m E

a n k

_x

=  + π

}

,....

2 , 1 ,

0 ± ±

= n

E is a periodic function of with the periodicity of k_x 2

π

/a

2 2

2 k

^x

E =  m

(3)

, 1, 2, 3, ..., k a

x

= n π n = ± ± ±

If an electron propagates in a periodic potential we always observe discontinuities of the energies when has a maximum or a minimum from Fig.4.11

cos k a

_x

This discontinuities of the energies come from the fact that periodic

potential is not perfectly zero. In other words, it is not a perfect free electron.

a n k

_x

π

=

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We call the allowed bands, for the time being, the n-band or the m- band, and so forth. In later sections and particularly in semiconductor physics, we will call one of these

bands the valence band and the next higher one the conduction band.

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2 2

2 , 0, 1, 2, ....

2

^x

E h k n n

m a

 π 

=  +  = ± ±

 

2

n 0 yields

2

(parabola with 0 as origin);

2

^x

E h k

= = m

2 2

n 1 yields parabola with as origin ;

2

^x

E h k

m a a

π π

   

= − =   −      

2 2

specifically, for =0 follows 4

2

;

x

2

k E h

ma

= π

2 2

and for = follows 1

2

.

a 2

x

k E h

ma

π = π

Free electron bands (see Fig 5.6)

2 2

2

2 



 



 +

= k n a

E  m

_x

π

2 2

2 k^x E = m

2 2

2

2 



 



 −

= k a

E  m

_x

π

2 2 2

4 2

E

π

ma

=

2 2 2

12

E

π

ma

=

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5. Energy Bands in Crystals

5.2 One- and Two-Dimensional Brillouin Zones

1-d Brillouin Zone

The first Brillouin Zone (BZ) : π/a ≤ k_x≤ π/a : n-Band

The second Brillouin Zone (BZ):

π/a ≤ k_x≤ 2π/a, -π/a ≤ k_x≤-2π/a : m-band

- Individual branches in an extended zone

scheme (Fig. 5.5) can be shifted by 2π/a to left or to right.

Shift the branches of 2^nd BZ to the positive side of E- k_x diagram by 2π/a to the left, and likewise the left band by 2π/a to the right → The result is shown in Fig. 5.4

(a reduced zone scheme)

- The same can be done in 3^rd BZ and all BZ (because of the 2π/a periodicity) →

relevant information of all BZ can be contained in the 1^st BZ (a reduced zone scheme)

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Between the boundaries and , corresponds to the first electron band, which we arbitrarily labeled as n-band. This region in k-space between and is called the first

Brillouin zone (BZ). Accordingly the area between and , and also between

and , which corresponds to the m-band, is called the second Brillouin zone.

/ a π

− π / a

/ a π

− π / a / a

π 2 / π a − π / a

2 / π a

−

5.2 One-and Two-Dimensional Brillouin Zones

2-d Brillouin Zone

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5.2 One-and Two-Dimensional Brillouin Zones

2-d Brillouin Zone

Description for the movement of an electron in the potential of 2-d lattice

- Wave vector k = (k_x, k_y) : 2-d reciprocal lattice (Fig 5.7) - A 2-d field of allowed energy regions which

correspond to the allowed energy band → 2-d BZ - 1st zone in 2-d: the area enclosed by four “Bragg planes” having four shortest lattice vectors, G₁: bisectors on the lattice vectors

- For the following zone, construct the bisectors of the next shortest lattice vectors, G₂, G₃…

- For the zone of higher order the extended limiting lines of the zones of lower order are used as additional limiting lines.

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Example: in 2-d lattice, an electron travels at 45^o to k_x-axis, then the boundary of the BZ is reached for

“Usefulness of BZ”

- Energy bands of solids (discussed in later section)

- The behavior of electrons which travel in a specific direction in reciprocal space

a 2 k

_crit

= π

) 2 (

1

2 2 2

max

a m

E = π  k

_crit

= π a a m

E

₂

2 2 max

π 

this yields a maximal attainable energy of

=

If the boundary of a BZ is reached at

the largest energy of electrons moving parallel to k_x or k_y axis

- Once the maximal energy has been reached, the electron waves (those of the incident and the Bragg-reflected electrons) form standing waves (or equivalently, the electrons are reflected back into the BZ.)

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2 sin a θ = n λ , n = 1, 2, 3,...

2 sin a n 2 k θ = π

crit

.

k n sin a

π

= θ

(Bragg Reflection)

In previous pages we have seen an electron traveling in a

specific direction. When we consider energies with all directions, overlapping of allowed energy bands occur

At k_crit, the transmission of an electron beam through the lattice is prevented, and then the incident and the Bragg-reflected electron wave form a standing wave.

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Crystals have symmetrical properties. Therefore, a crystal can be described as an accumulation of “unit cells”. There are two kinds of unit cell, “primitive unit cell”

means a smallest possible cell, and “conventional unit cell” is a nonprimitive unit cell having the advantage that the symmetry can be better recognized,

Wigner-Seitz Cell is a special type of primitive unit cell that shows the cubic symmetry of the cubic cells

5.3-4 3-D Brillouin Zone and Wigner-Seitz Cells

W-S cell construction: bisects the vectors from a given atom to its nearest neighbors and place a plane perpendicular to these vectors at the bisecting points.

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1 1 2 2 3 3,

n n n

= + +

R t t t

1 1 2 2 3 3

2 (π h h h ),

= + +

G b b b

1 1

1 2

1 3

1, 0, 0.

⋅ =

⋅ = b t b t b t

t₁, t₂, t₃ : Primitive (or fundamental) vectors

1, 2, 3 : reciprocal lattice vectors b b b

5.5 Translation Vectors and the Reciprocal Lattice

5. Energy Bands in Crystals

Kronecker-Delta symbol

m n for 0 and

m n for 1

where _nm _nm

nm m

n

≠

=

•

δ δ

δ , t

b

Translation

vectors

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1

= const.

2

×

3

.

b t t

1

⋅ =

1

const.

1

⋅ × =

2 3

1.

b t t t t

1 2 3

const.= 1 .

⋅ × t t t

2 3

1

1 2 3

= × .

∴ ⋅ ×

t t b t t t

3 1

2

1 2 3

= × ,

⋅ × t t b t t t

1 2

3

1 2 3

= × .

⋅ × t t b t t t

1

( ),

2

= a − + +

t i j l

1

(111) 2

= a

t

₂

(111),

2

= a

t

₃

(111).

2

= a t

Calculation for the reciprocal lattice of a BCC crystal Real crystal

a: lattice constant , t₁, t₂, t₃: primitive lattice vectors_,

i, j, l : unit vectors in the x, y, z coordinate system (see Fig. 5.14(b))

Abbreviated,

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2 2 2

2 3

2

1 1 1 ( ) (2 2 )

4 4 4

1 1 1

( )

2

a a a

a

× = − = + + + − + = +

−

= +

i j l

t t i j l l i j j l

j l

3 3 3

1 2 3

( ) (0 ) (0 1 1) .

4 4 2

a a a

⋅ × = − + + ⋅ + + = + + =

t t t i j l j l

2

1 3

( )

2 1 ( ),

2 a

a a

= j l+ = +

b j l

1

1 (011).

= a

b ₂ 1

(101).

= a

b ₃ 1

(110).

= a b

The end points of the reciprocal lattice vectors of a bcc crystal are at the center of the edges of a cube. This means that reciprocal lattice of the bcc structure are identical to the lattice points in a real lattice of the fcc structure

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′ = + .

k k G

2

k

( ) .

2 E h

′

= m k + G

2 2

2 ,

2

E h x

m a



π



=  i G+ 

2

( )

2 2 2 2

2

E h x Cx

m a



π



=   i ≡

2 2 2 2

2

2 2

, 2

h h

C m a ma

π π

 

=   =

( )

2 .

a

= −

π

+

G i l

( ) ( )

2 2

2

2 2

2 1

1 1 2 2

h x

E C x

m a a

C x C x x

π π

 

=  − +  =  − − 

 

=  − + = − +

i i l i l

0 2

1 1 .

x E C

= → =

5.6 Free Electron Bands for BCC

In Fig 5.17, three important directions [100] from (origin) to point H : [110] from to N :

[111] from to P :

∆ Σ

Λ



2



²



2



2



2

2

Γ Γ Γ

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5.6 Free Electron Bands for FCC

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5.7 Band Structures for Some Metals and Semiconductors

Band structure of actual solids: Figs. 5.21-24

(results of extensive, computer-aided calculations)

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5.8 Curves and Planes of Equal Energy

E(k) curves

Fig 5.25: curves of equal energy for free electrons Fig 5.26: near boundary of BZ- deviation from a circular form (2-d)

Fig 5.27: 3-d BZ for Cu