ka a

: 1st Brillouin Zone

“Extended” “Reduced”

Weak binding model

Weak binding model

Point : extrema in energy band

𝐸 𝐸ℏ 𝑘 𝑘

2𝑚∗

: Free electron like behavior

𝐸 𝐸ℏ 𝑘

2𝑚∗

𝑘 : k at the bottom of band

𝑚∗ : effective mass at the bottom of band

𝑘 : k at the top of band

𝑚∗: effective mass at the top of band

𝑁 𝐸 ∝ 𝐸 / near the bottom of the band𝑁 𝐸 ∝ 𝐸 𝐸 / near the top of the band

(Nearly free electron model)

The first Brillouin zones

The k values in the Brillouin zone depend on the crystal symmetry. The first Brillouin zone = unit cell of reciprocal lattice

ex) FCC crystal has BCC Brillouin zone. Greek and Latin letters indicate the directions and zone faces.

Simple cubic

FCC

BCCHexagonal

Density of states in a band

𝑁 𝐸 ∝ 𝐸

Near the bottom of the band

Near the top of the band

E

( )N E

𝑁 𝐸 ∝ 𝐸 𝐸 /

𝑁 𝐸1

4𝜋2𝑚∗

ℏ 𝐸

𝑁 𝐸1

4𝜋2𝑚∗

ℏ 𝐸 𝐸 /

Density of states in a band• Metal

• Semiconductor or Insulator

E

( )N E

FE E

( )N E

or

Partially occupied Bands overlap

E

( )N E

△ 𝐸 𝑘𝑇 E

( )N E

△ 𝐸 𝑘𝑇

Small gap Large gap

Different band representationE

𝑘𝜋𝑎

𝜋𝑎

0

E

𝑥

𝑁 𝐸

E

𝑘

Equal energy surfaces𝑘

Different band representations

E vs k: Dispersion relation for electron waves that is useful in describing the transport

Free electron like behavior1. 𝐸 ∝ 𝑘 (parabolic E-k)

2. 𝑁 𝐸 ∝ 𝐸

3. Spherical equal energy surfaces

E vs 𝑥: Flat band diagram that emphasizes the non-localized nature of the band states

𝑁 𝐸 vs E: Variation of the DOS within a band important when describing a variety of electron transport processes, optical excitations, etc.

x

Electron velocity

𝑣𝜕𝜔𝜕𝑘

1ℏ

𝜕𝐸𝜕𝑘

𝐸ℏ 𝑘𝑚∗

Solution: 𝐸 𝐴𝑘 𝐵* This is satisfied only for an energy band extremum at 𝑘 0

𝑣ℏ𝑘𝑚∗

𝑚∗ ℏ𝜕 𝐸/𝜕𝑘

𝜕𝐸𝜕𝑘 𝑘

𝜕 𝐸𝜕𝑘

E

𝑘

Velocity is zero at band extrema. In free electron system, group velocity

goes to infinite or increase as k increases.

In thermal equilibrium, there are equal number of electron-occupied states with positive velocity and negative velocity.

Therefore, in completely filled band, no net charge transport under an electric field (ex: insulator, no conductivity of valence electrons)

Electron velocity

𝑘

𝑘

𝐸 𝐸

𝑣 𝑣

+ +- -

𝑣𝜕𝜔𝜕𝑘

1ℏ

𝜕𝐸𝜕𝑘

: proportionality factor between force and acceleration (ex: 𝐹 𝑚𝑎)𝐹𝑑𝑡 𝑑𝑝 ℏ𝑑𝑘 p: momentum

𝐹 ℏ𝑑𝑘𝑑𝑡

𝑑𝑘𝑑𝑡

𝐹ℏ

: For positive F, k of all occupied states is shifted toward positive kvalues. So, there is an unbalance between occupied positive-velocity states and occupied negative-velocity states under electric field, leading to net transport.

𝐸

𝑣

+-

𝑘

𝑘

𝐸

𝑣

+-

𝑘

𝑘

𝐹 0 𝐹 0

Balanced velocitiesNo net transport

Unbalanced velocitiesNet transport

Electron velocity

Effective mass

𝑚∗ ℏ𝑑 𝐸/𝑑𝑘

The effective mass of an electron is the reciprocal of the curvature of the E vs. k plot.

For 3-D,𝑑𝐯𝑑𝑡

1ℏ ∇ 𝐅 ⋅ ∇ 𝐸 𝐤 ,

1𝑚∗

1ℏ

𝜕 𝐸𝜕𝑘 𝜕𝑘

𝑑𝑣𝑑𝑡

1ℏ

𝑑𝑑𝑡

𝑑𝐸𝑑𝑘

1ℏ

𝑑𝑑𝑘

𝑑𝑘𝑑𝑡

𝑑𝐸𝑑𝑘

In one dimension with periodic potential barriers

∵ 𝐹 ℏ𝑑𝑘𝑑𝑡

𝐹ℏ

𝑑 𝐸/𝑑𝑘𝑑𝑣𝑑𝑡 𝑚∗ 𝑑𝑣

𝑑𝑡

in one dimension

cf) For free electrons, 𝑚∗ 𝑚

Effective massE

k

②③

①

𝑚∗ 𝑚∗ 𝑚∗

: Greater the curvature, smaller the effective mass

E

k

k

E

k

k

*m *m

a

+

--

+

-

+

For a free electron,

In a crystal, electrons at the extrema have effective mass.

at the bottom of a bandat the top of a band

A negative mass implies that the induced acceleration is on the opposite direction to the force that caused it (Result of Bragg reflection)

𝑚∗ 𝑚, 𝑣 ℏ𝑘𝑚

𝑚∗

𝑚∗

In a semiconductor or insulator

Holes

Similar to “bubbles (holes) in water (electrons)”

E

𝑘

𝐸

𝑘

C.B

V.B

electrons m∗, 𝑞

electrons m∗, 𝑞

Holes(+m*, +q)

𝐸𝑘

𝐸

𝑘

: missing electrons in a nearly filled band with a positive effective mass and a positive charge.

: In the presence of an electric field, electrons in the bottom of the conduction band and holes at the top of the valence band move in the opposite directions in real space (same sign mass but different sign charge), whereas electrons and holes both at the top of the valence band move in the same direction (different sign mass cancels different sign charge)

Move in opposite directions under electric field