5. Energy Bands in Crystals

Part I Fundamentals

Electron Theory : Matter Waves

Chap. 1 Introduction

Chap. 2 The Wave-Particle Duality Chap. 3 The Schördinger Equation

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

Chap. 5 Energy Bands in Crystals Chap. 6 Electrons in a Crystal

Electromagnetic Theory : Maxwell Equations

Chap. 4 Light Waves

(Electrons in Solids, 3^rd Ed., R. H. Bube)

5. Energy Bands in Crystals

5.1 One-Dimensional Zone Schemes

Energy E vs momentum of the electrons p (or k) For free electrons, the wave number in 1-dim

2 2

2 k

E = h m k

= const .E

ka

a a

P sin a + cos α = cos α

α

cos α a cos = ka

2 / 1 2

2 m E

= h α

) 2 cos(

cos

cos α a = k

a ≡ k

a + n π

π

α a = k

a + n 2

2

2

2 m E

n a k

= h

+ π

: more general form in 1-dim

,....

2 , 1 ,

0 ± ±

n =

Combining with

}

In a crystal

5. Energy Bands in Crystals

2 / 1 2

2

2 m E

n a k

= h

+ π

,..., 3 , 2 1,

, = ± ± ±

= n n

a

k

π

n a k

_{= ⋅} π

1 cos k

a = ±

If an electron propagates in a periodic potential, discontinuities of the electron energies are observed when cosk_x

a

E is a periodic function of with the periodicity of

2 π / a

5.1 One-Dimensional Zone Schemes

or

At these singularities, a deviation from the parabolic E vs k_xoccurs and the branches of the individual parabolas merge into the neighboring ones (see Fig.5.3)

5. Energy Bands in Crystals

The electrons in a crystal behave like free electrons for most k_x value except k_x → nπ/a

periodic zone scheme (see Fig 5.3)

reduced zone scheme (see Fig 5.4) π/a ≤ k_x≤ π/a

5.1 One-Dimensional Zone Schemes

extended zone scheme (see Fig 5.5) the deviations from the free electron parabola at the critical points k_x = n∙_π/a are particularly easy to identify.

free electron bands (see Fig 5.6)

free electrons in a reduced zone scheme from

5. Energy Bands in Crystals

,....

2 , 1 0,

, 2 )

2 (

2

2

+ = ± ± ±

= n

n a m k

E _h

π

5.1 One-Dimensional Zone Schemes

2 / 1 2

2

2 mE

n a k_x

= h +

π

5. Energy Bands in Crystals

origin) as

0

( 2

,

0

2

with parabola

m k E

n = = h

2 2 2

2 2 2 2 2

1 2

, For

4 2

, 0 For

origin) as

2

( 2 )

2 (

, 1

E ma k a

E ma k

with a parabola

k a E m

n

x x

x

h h h

π π

π

π π

=

=

=

=

−

=

−

=

,....

2 , 1 0,

, 2 )

2 (

2

2

+ = ± ± ±

= n

n a m k

E _h

π

By inserting different n-values, one can calculate the shape of branches of the free electron bands

5.1 One-Dimensional Zone Schemes

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)

5. Energy Bands in Crystals

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.

5.2 One- and Two-Dimensional Brillouin Zones

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

the BZ is reached,

according to Fig 5.8, for

The first four BZ shown in Fig 5.8

“Usefulness of BZ”

- energy bands of solids (discussed in later section)

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

5. Energy Bands in Crystals

a 2 k

₌ π

) 2 (

1

2 2 2

max

a m

E ₌ π h

k_{crit =}

π

E a

m

2 2 max

₌ π h

this yields with (4.8) 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

5.2 One- and Two-Dimensional Brillouin Zones

5. Energy Bands in Crystals

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

- Overlapping of energy bands: bands are drawn in different directions in k-space (Fig 5.9) :

the consequence of

a 2 k

₌ π

k

₌ π a

and

5.2 One- and Two-Dimensional Brillouin Zones

5. Energy Bands in Crystals

A different illustration of the occurrence of critical energies at which a reflection of the electron wave takes place :

Bragg relation

Since λ = 2π/k

For a perpendicular incidence, θ = 90^o, If θ = 45^o,

For increasing electron energies, a critical k-value is finally reached for which

“reflection” of the electron wave at the lattice plane occurs. At , the transmission of electron beam through the lattice is prevented.

1,2,3,...

, sin

2 a θ = n λ n =

n k a sin θ ² π

2 = θ

π sin n a

k

=

k

₌ π a a 2

k

₌ π

k

5.2 One- and Two-Dimensional Brillouin Zones

5. Energy Bands in Crystals

5.3 Three-Dimensional Brillouin Zones

- In previous section, it was shown that at the boundaries of the zones the electron waves are Bragg-reflected by the crystal.

- The wave vector, |k| = 2π/λ, was seen to have the unit of reciprocal length and thus is defined in the reciprocal lattice.

- The construction of 3-d Brillouin zones for two important crystal structures of face centered cubic (FCC) and body centered cubic (BCC) : important features in common with “Wigner- Seitz cells”

5. Energy Bands in Crystals

5.4 Wigner - Seitz Cells

Crystals have symmetrical properties - An accumulation of “unit cell”

- Smallest possible cell “primitive cell”

(consist of 1 atom)

- BCC, FCC : conventional non-primitive unit cells

- Wigner-Seitz cell : a special type of

primitive unit cell that shows the cubic symmetry of cubic 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. For BCC (Fig 5.11) & FCC (Fig. 5. 13)

5. Energy Bands in Crystals

5.4 Wigner - Seitz Cells

- The atomic arrangement of FCC:

corners and faces of cube,

or center points of the edges and the center of the cell (Fig 5.12)

-The W-S cell for FCC shown in Fig 5.13

5. Energy Bands in Crystals

5.5 Translation Vectors and the Reciprocal Lattice

Fundamental vectors or primitive vectors : t₁, t₂, t₃ Translation vectors, R : combination of primitive vectors

where n₁, n₂, and n₃ are integers.

Three vectors for the reciprocal lattice: b₁, b₂, b₃ a translation vector for the reciprocal lattice, G

where h₁,h₂, and h₃ is integer

3 3 2

2

1

t t t

R = n

+ n + n

) (

2

b

b

b

G = π h

+ h + h

) 2 (

1

i j l

t = a − + +

m n

m

n

nm nm m

n

≠

=

=

=

=

for 0

and

for 1 where

,

δ δ

δ t

b

5. Energy Bands in Crystals

The relation between real and reciprocal lattices By definition,

. 0

, 0

, 1

3 2

=

•

=

•

=

• t b

t b

t b

1 1

1

1 Kronecker-Delta symbol

}

3 2

1

. t t

b = const × b

• t

= const . t

• t

× t

= 1

3 2

1

1 t t

t • ×

= const

3 2

1

3 1 2

t t

t

t b t

×

•

= ×

3 2 1

1 3

2

t t t

t b t

×

•

= ×

3 2

1

2 1

3

t t t

t b t

×

•

= ×

5.5 Translation Vectors and the Reciprocal Lattice

5. Energy Bands in Crystals

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,

) 2 (

1

i j l

t = a − + +

) 11 1 2 (

1

= a

t ( 1 1 1 )

2

2

= a

t ( 11 1 )

3

2

= a t

) 2 (

) 2 2

4 (

) 4 (

1 1

1

1 1

4 1

2 2

2 2

3 2

l j l

j

j i l l j i k

j i

t t

+

= +

=

+

− + + +

=

−

−

=

×

a a

a a

5.5 Translation Vectors and the Reciprocal Lattice

5. Energy Bands in Crystals

(continued)

) 2 1 1 0 4 ( )

0 ( ) 4 (

3 3

3 3

2 1

a a

a − + + • + + = + + =

=

×

• t t i j l j l

t

3 2 1

3 1 2

t t t

t b t

×

•

= ×

1 ( ),

2 ) 2 (

3 2

1

j l

l j

b + = +

= a a

a

) 011 1 (

1

= a

b 1 ( 101 )

2

= a

b 1 ( 110 )

3

= a b

BCC (reciprocal lattice) FCC (real lattice)

1st Brillouin zone for BCC Wigner-Seitz cell for FCC Vice versa

5.5 Translation Vectors and the Reciprocal Lattice

Periodicity of E(k) → all information of electron contained in the 1st Brillouin Zone (BZ)

E_k' for k' for outside 1^st BZ → E_k with in 1^st BZ with a suitable translation vector G

“Energy bands are not alike in different directions in k-space”

for the demonstration, “free electron band” is used (Fig 5.6 ).

In 3-D, from (5.7)

G k

k

= +

5. Energy Bands in Crystals

5.6 Free electron Bands

G

k )

2 (

2

'

= +

E

h m

,....

2 , 1 0,

, 2 )

2 (

2 2

±

±

±

= +

= n

n a m k

E _h

π

(5.7)

5. Energy Bands in Crystals

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

[111] from to P:

Fig 5.18 calculated by using the following equation

Γ Γ Γ

Δ Σ

Λ

G

k ) 2 (

2

'

= +

E

h m

5.6 Free electron Bands

5. Energy Bands in Crystals

band calculation for BCC

Γ − H [ 100 ] direction

k

k

≡

For this direction (5.35) becomes

2 2

2 )

2 ( i + G

= x

a E _h m π

Where x may take values between 0 and 1. to start with, let G = 0, then where

this curve is labeled (000) in Fig 5. 18 since h₁,h₂,h₃ = 0,0,0 for G=0 between 0 and 2π/a (boundary of BZ)

2 2

2 2

) ( 2 )

2 ( x Cx

a

E = _h m π i ≡

2 2 2

2

2

2 )

2 m ( a ma

C _{= h} π _{= h} π

5.6 Free electron Bands

5. Energy Bands in Crystals

For the case of h₁,h₂,h₃ = 0,-1,0

combined (5.36) and (5.38)

) 2 2

( ]

1 )

1 [(

] )

1 (

[ )]

2 ( [ 2

2

2 2

2 2

2

+

−

= +

−

=

−

−

= +

−

=

x x

C x

C

x a C

a x

E _h m π i π i l i l

C E

x

C E

x

1 1

for and

2 0

=

→

=

=

→

=

) 0 1 0 (

The band labeled in Fig 5.18 obtained.

Similarly, For FCC, see Figs. 5.19 & 5. 20

) 2 (

l i

G = − +

a

π

For

5.6 Free electron Bands

- Band structure of actual solids : Figs. 5.21~5.24

(results of extensive, computer-aided

calculations)

- Directions in k-space [100] :

[110] : [111]:

5.7 Band Structures for Some Metals and Semiconductors

5. Energy Bands in Crystals

− X Γ

− K Γ

− L Γ

Band diagram for aluminum

- parabola-shaped band: free- electron like

5. Energy Bands in Crystals

Band diagram for copper

-Lower half of the diagram closely spaced and flat running bands (due to 3d-bands of Cu)

Band diagram for silicon - band gap : near 0~ 1eV →

“semiconductor properties”

5.7 Band Structures for Some Metals and Semiconductors

5. Energy Bands in Crystals

-Band diagram gallium arsenide : so called III – IV semiconductor Important for “optoelectonic devices”

5.7 Band Structures for Some Metals and Semiconductors

5. Energy Bands in Crystals

5.8 Curves and Planes of Equal Energy

Energy vs. wave vector, k

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