Original Lecture Notes Prepared by:

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Lecture 12 Band Theory:

E-K Diagram and Energy Gaps

Original Lecture Notes Prepared by:

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

Arab Academy for Science and Technology AAST – Cairo

Spring 2016

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Original Lecture Notes Prepared by:

Dr. Amr Bayoumi, Dr. Nadia Rafat

Solid State Electronics EC210, Spring 2016 Arab Academy for Science and Technology

AAST – Cairo,

Lecture 12:E-K Diagram and Band Gap

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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.

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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

using Kronig-Penney Model

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Source: Dr. M. Fedawy’s Lecture notes

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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].

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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

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

McGraw-Hill, 2005)

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E-k Diagram for Holes in Valence Band

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)

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E-k Diagram for Electrons in Conduction Band

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₊

(b)

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E-k Diagram: Emission of a Photon Due to Electron-Hole Recombination

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)

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E-k Diagram: Effective Mass

Reminder: Using E-K diagram:

𝑚

^∗

= 1 ℏ

²

𝑑

²

𝐸 𝑑𝑘

²

−1

= ℏ

²

𝑑

²

𝐸 𝑑𝑘

²

−1

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