Lecture Notes Prepared by:

(1)

Lecture 9

Band Theory:

E-K Diagram, Energy Gaps and Effective Mass

Lecture Notes Prepared by:

Dr. Amr Bayoumi, Dr. Nadia Rafat Solid State Electronics EC210

Arab Academy for Science and Technology AAST – Cairo

Spring 2015

1

(2)

Lecture Notes Prepared by:

Dr. Amr Bayoumi, Dr. Nadia Rafat

Solid State Electronics EC210, Spring 2015 Arab Academy for Science and Technology

AAST – Cairo,

Lecture 9:E-K Diagram, Band Gap,

Effective Mass 2

From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw-Hill, 2005)

These PowerPoint color

diagrams can only be used by instructors if the 3^rd Edition has been adopted for his/her course. Permission is given to individuals who have

purchased a copy of the third edition with CD-ROM

Electronic Materials and

Devices to use these slides in seminar, symposium and conference presentations provided that the book title, author and © McGraw-Hill are displayed under each diagram.

(3)

AAST – Cairo,

Effective Mass 3

Pages

• Kasap:

– P.355 (Kronig Penny)

– P.303-304, p. 454-455 (Effective Mass)

Lecture 8: Band Theory: Kronig-Penny Model and Effective Mass

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Fig 4.52

E-K Diagram

usingKronig-Penney Model

4

Source: Dr. M. Fedawy’s Lecture notes

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AAST – Cairo,

Effective Mass 5

Energy Gap (Bandgaps, Eg)

Fig 4.54

 a E

k₃[11]

E

k₁^[10]

Band Band Energy gap

Energy gap Band

Band First

Brillouin Zone Second Brillouin Zone

Second

Brillouin Zone

First

Brillouin Zone

a



The E-k behavior for the electron along different directions in the two dimensional crystal. The energy gap along [10] is at ^/a whereas it is at /a along [11].

From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (©

McGraw-Hill, 2005)

(6)

AAST – Cairo,

Effective Mass 6

Energy Gaps (Eg)

Fig 4.55

(a) Metal: For the electron in a metal there is no apparent energy gap

because the 2nd BZ (Brillouin Zone) along [10] overlaps the 1st BZ along [11]. Bands overlap the energy gaps. Thus the electron can always find any energy by changing its direction.

(b) Semiconductor or insulator: For the electron in a semiconductor there is an energy gap arising from the overlap of the energy gaps along [10] and [11] directions. The electron can never have an energy within this energy gap, Eg.

Energy gap

[11]

[10] Overlapped

energy gaps Energy gap =E

g

1st BZ band 2nd BZ

band

(b) Semiconductor and insulator

Energy gap

[10] [11] Bands overlap energy gaps Energy gap

1st BZ band

2nd BZ

&

1st BZ overlapped band 2nd BZ

band

(a) Metal

McGraw-Hill, 2005)

(7)

AAST – Cairo,

Effective Mass 7

E-

Fig 5.52

(a) In a full valence band there is no net contribution to the current.

There are equal numbers of electrons (e.g. at b and b') with

opposite momenta. (b) If there is an empty state (hole) at b at the top of the band then the electron at b' contributes to the current.

Ev

VB E

-k k

b' b (a)

VB E

-k k

b' (b)

From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap (© McGraw- Hill, 2005)

(8)

AAST – Cairo,

Effective Mass 8

Fig 5.49

The E-k diagram of a direct bandgap semiconductor such as GaAs. The E-k curve consists of many discrete points each point corresponding to a possible state, wavefunction y k(x), that is allowed to exist in the crystal. The points are so close that we normally draw the E-k relationship as a continuous curve.

In the energy range Ev to Ec there are no points (yk(x) solutions).

E_k

/a k

-/a

E_c

E_v CB

VB

E_c

E_v

The E-k Diagram The Energy Band Diagram

Empty _k

Occupied_k h+

e^- E_g

e^-

h⁺

h

VB h

CB

McGraw-Hill, 2005)

(9)

AAST – Cairo,

Effective Mass 9

Fig 5.51

(a) In the absence of a field, over a long time, average of all k values is zero, there is no net momentum in any one particular direction. (b) In the presence of a field E in the -x direction, the electron accelerates in the +x direction increasing its k value along x until it is scattered to a random k value. Over a long time, average of all k values is along the +x direction. Thus the electron drifts along +x.

E

k

CB

-k

k₊ k-

Lattice scattering

k_av = 0

-x x

(a)

k_av> 0

-x x

E

k

CB

-k

E

k_1-

Lattice scattering

k₁₊k₂₊k₃₊

From Principles of (b)

McGraw-Hill, 2005)

(10)

Effective Mass

10

F = q e = m

_o

a where

m

_o

is the electron mass

F

_ext

= (-q) E F

_ext

+ F

_int

= m

_o

a F

_ext

= m

_n

*a

where

m_n* is the electron effective mass

In vacuum In semiconductor

q

(11)

AAST – Cairo,

Effective Mass 11

Effective Mass

Fig 4.19 From Principles of Electronic Materials and

McGraw-Hill, 2005) e-

Fext

VACUUM

x Fext

me

a =

(a) Ex

Fint

CRYSTAL

x Fext

me* a =

(b)

(a) An external force F_ext applied to an electron in vacuum results in an acceleration a_vac = F_ext / m_e . (b) An external force Fext applied to an

electron in a crystal results in an acceleration a_cryst= F_ext / m_e*. ( Ex is the electric field.)

Ex

(12)

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Effective Mass 12

Example of Effective Mass

From Principles of Electronic Materials and Devices, Third

McGraw-Hill, 2005)

(13)

AAST – Cairo,

Effective Mass 13

Lecture Notes Prepared by:

Lecture 9

Band Theory:

E-K Diagram, Energy Gaps and Effective Mass

Lecture Notes Prepared by:

Dr. Amr Bayoumi, Dr. Nadia Rafat Solid State Electronics EC210

Arab Academy for Science and Technology AAST – Cairo

Spring 2015

Pages

• Kasap:

– P.355 (Kronig Penny)

– P.303-304, p. 454-455 (Effective Mass)

E-K Diagram

usingKronig-Penney Model

Energy Gap (Bandgaps, Eg)

Energy Gaps (Eg)

E-

Effective Mass

F = q e = m

a where

m

is the electron mass

F

= (-q) E F

+ F

= m

a F

= m

*a

Effective Mass

Example of Effective Mass

Effective Mass

Group Velocity defined as the velocity of the wavefunction of the electrons (analogous to speed of sinusoidal wave ):

𝑣

= 𝑑𝑥

𝑑𝑡 = 𝑑ω

𝑑𝑘 , 𝑑𝑥 = 𝑣

𝑑𝑡 𝜔 = 𝐸

ℏ → 𝑣𝑔 = 1 ℏ

𝑑𝐸

𝑑𝑘 → 𝑑𝐸 = 𝑣𝑔

𝑑𝑘

𝑑𝐸 = 𝐹

𝑑𝑥 = 𝐹

𝑣𝑔 𝑑𝑡 → 𝐹

= 1 𝑣

𝑑𝐸

𝑑𝑡 → 𝐹

= ℏ 𝑑𝑘 𝑑𝑡

Effective Mass (2)

Acceleration:

𝑎 = 𝑑𝑣

𝑑𝑡 = 𝑑 𝑑𝑡

1 ℏ

𝑑𝐸

𝑑𝑘 = 1 ℏ

𝑑 𝑑𝑘

𝑑𝐸

𝑑𝑡 = 1 ℏ

𝑑 𝑑𝑘

𝑑𝐸 𝑑𝑘

𝑑𝑘 𝑑𝑡

𝑎 = 1 ℏ

𝑑

𝐸 𝑑𝑘

𝑑𝑘

𝑑𝑡 = 1 ℏ

𝑑

𝐸

𝑑𝑘

ℏ 𝑑𝑘

𝑑𝑡 = 1 ℏ

𝑑

𝐸

𝑑𝑘

𝐹

Using 𝐹

= 𝑚

𝑎

𝑚