Modern Semiconductor Devices for Integrated Circuits Chapter 1. …bkict-ocw.knu.ac.kr/include/download.html?fn=559E036F732... · Modern Semiconductor Devices for Integrated Circuits

Chapter 1. Electrons and Holes in SemiconductorsModern Semiconductor Devices for Integrated Circuits

Allowed values of energy can be plotted vs. the propagation constant, k.

Since the periodicity of most lattices is different in various direction, the E-k diagram must be plotted for

the various crystal directions (complex).

Si, Ge, GaP, AlAs :

indirect band gap semiconductor

A transition must necessarily include and

interaction with the crystal so that crystal

momentum is conserved.

GaAs :

the minimum conduction band energy and

maximum valence band energy occur at the same

k-value.

direct band gap semiconductor

semiconductor lasers and other optical devices

-Direct bandgap: a minimum in the conduction band and a maximum in the

valence band for the same k value

-Indirect bandgap: a minimum in the conduction band and a maximum in the

valence band at a different k value

Direct and Indirect Semiconductor

0

i f

i f photon f

photon

E E hv

k k k k

where k

0

i f phonon

i f photon phonon f phonon

phonon

E E E

k k k k k k

where k


Extension to Three Dimensions

Electrons traveling in different

directions encounter different

potential patterns and therefore

different k-space boundaries.

The (100) plane of a face-centered

cubic crystal showing the [100] and

[111] directions.


The k-Space Diagrams of GaAs and Si

Plot the direction[100]

[111]


• A perfect semiconductor crystal with no impurities or lattice defects

• No charge carrier at 0 K

• EHP generation at higher temperature

For intrinsic material

in p n

• At a given temperature there is a certain concentration of electron-hole

pairs ni

• If a steady state carrier concentration is maintained, there must be

recombination of EHPs at the same rate at which they are generated.

i ir g

Intrinsic Material


• Doping is the most common technique for varying the conductivity of

semiconductors

• By doping, a crystal can be altered so that it has a predominance of

either electrons (N-type) or holes (P-type)

• Extrinsic: the equilibrium carrier concentrations n0 and p0 are different

from the intrinsic carrier concentration ni

[N-type][P-type]

Extrinsic Material


For the case of N-type doping

The energy level of the 5th electron is inside the energy gap but very close to the bottom of the conduction band. At RT, it gets enough energy to jump into the conduction band, becoming a free electron. It leaves behind positively charged donor atoms, which is immobile.

For the case of P-type doping

The acceptor atom introduces a localized energy level into the energy gap which is

close to the top of the valence band. As a consequence, an electron from the valence

band jumps onto this level, leaving behind a mobile hole and creating a negatively

charged immobile acceptor atom.

Effects of (a) N-type and (b) P-type doping in energy-band model presentation

(C.B., conduction band; V.B., valence band.)


donor level

acceptor level

column V impurities

column III impurities


Donors and Acceptors in the Band

Model

- Shallow levels: acceptor and donor levels with small ionization energies,

such as As, P, Sb, and B

- Deep levels: impurity levels with large ionization energies, such as Au, Cu, Pt..

Energy levels of donors and acceptors


Ionization Energy of Donor and Acceptor

(Binding Energy)• Estimated by modifying the theory of the ionization energy of a hydrogen atom.

• Energy required for electron in solid to make a transition from the donor level to the

conduction band and become (quasi) free.

r3e, m0

r2n=1n=1

n=2n=3

0

re, m*r3

r1

r1n=1

n=3

n=2

Si

= 11.7 for Sir

Hydrogen atom

in vacuum

n = , E = 0

n = 3, E3

n = 2, E2n = 1, E1

hydrogen atom

Donor atom

in Si atom

n = , Ed=Ecn = 3, Ed3

n = 2, Ed2n = 1, Ed1

Donor atom

2n

EE ionn

eVqm

EEEion 6.1332 220

4

01

, n=1, 2, 3,

eVm

m

EEEEE

n

r

dcion

)()(6.130

*2

0

0

1

Ec=Ed~6meVn=1, Ed1=Ed


No broken bond ↔ No electron in conduction band andno empty state in valence band

Comparison between Bonding and Energy Band Model

Eg: equal to the energy required to break a bond

Electron – hole pair (EHP) generation

Electrons and Holes

+-

Completely empty

Completely filled

Eg

Ec

Ev

+- E

At 0 K

No current flow

-

+-

broken bond

Ec

Ev

Eg

+

empty states in valence band

With excitation with thermal or optical energy

electron in conduction band

Broken bond ↔ Electron (in conduction band and)Hole (empty state in valence band)

- Electrons in conduction band

- Holes (empty state) in valence bandCarriers:

E

E

E


Water analogy

originally missing bond

+-

newly missing bond

+-

Ec

Ev

movement of empty state (movement of missing bond)

Movement of Hole

water flow

Ec

Ev

Ec

+-

e

Ev

Electrons in conduction band: moves like free

electron

partially

filled water in tubeelectrons in conduction band

electron flow

completely

filled water

in tube

Ec

Ev

Ev

Ec

+-

electrons in valence band

no water flowno electron flow

filled water with some bubble

bubble

flow

Ec

Ev

Ev

Ec

+-Holes (empty states) in valence band: moves like positively charged free

particle

empty state flow

E

E

E

E

E


- The electrons and holes in a crystal interact with a periodic coulomb field in the crystal.

- They surf over the periodic potential of the crystal, and therefore mn and mp are not the same as

the free electron mass, m0.

Effective Mass

n

p

qAcceleration electrons

m

qAcceleration holes

m

The solution becomes the plane wave as;

The electron wave function is the solution of the three-dimensional Schrödinger wave equation

ErVm

)(2

2

0

2

EmkwheretrkjA

02)],(exp[

2 0

2

20,

mE

, or constant0)( rV

For free electron,


2 0

2

2[ ( ) ] 0,

mV r E

0)( rV

0*2

2

2 Em

by neglecting and introducing new mass m* called “effective mass”)(rV

the periodic crystal potential

EmkwheretrkjA

*2)],(exp[

The solution looks like plane wave with m* ! (mn for electron and mp for hole)

By adopting effective mass concept, the carriers in solids can be treated as almost free carriers.

The calculation of effective mass must take into account the shape of the energy bands

in three-dimensional k-space.

Assuming the E-k relationship has spherical symmetry, an electric field, ,

would accelerate an electron wave packet with

*2*2

222

m

p

m

kE

2

2

2

*

1

dk

Ed

m

kp

2

2

2,

dk

Edq

m

qaonAccelerati

n

22

2

/ dkEdmassEffective

k

E

For electrons in the crystal,


For particles moving in a crystal, there are an internal force in addition to an externally applied force.

particleofmassrestmdt

vdmamFFF exttotal 000int ,

for free particle,,0int F

amFF totalext

0

m*: new directly related to the external force, assuming int 0F

correspond to ( ) 0V r

amFFF exttotal

*int

In a crystal,

glass

marble

vacuum

fast dropslow drop

water

: gravitational force

: viscosity of the liquid

e, m0e, m*

solid

in vacuum in semiconductor

qamFF totalext 0 int int* ,total extF F F m a q neglecting F

extF

intF

E E


Electrons in valence band

The mass calculated by will have a negative value

Valence band e- with (-) charge & (-) mass moves in an electric

field in the same direction as h+ with (+) charge & (+) mass

Selecting holes as the valence-band carriers

(the minimum kinetic-energy position of holes at the peak)

2

2 2*

/m

d E dk

2

2 2*

/

km

d E d

For parabolic energy band, the electron mass is inversely related to the curvature

of the (E, k) relationship

The curvature of the band determines the

electron effective mass, m*.


Sharper large light mass 2 2/d E dkWider small heavy mass 2 2/d E dk

For a band centered at k=0, the E-k relationship near the minimum is usually parabolic:

22

C*E = k E

2m

2

2 2*

/

km

d E d

(a) E–k diagram and (b) spherical constant-energy surface for GaAs

2 2/d E dk negative negative effective mass

E-k relationship for parabolic band

with isotropic effective mass in 3-D:

mx* = my* = mz* = m*

22 2 2

22 2

2 2 2

2 2 2

( )2 *

12 * 2 * 2 *

x y z

yx x

E k k km

kk k

m E m E m E

Equation for sphere in k space

The radius of the sphere stands for

energy and the surface of the

sphere is same energy, which is

called constant energy surface.


The real E-k diagram of Si is more complicated (indirect semiconductor).

The bottom of EC and top of EV appear for different values of k.

(b) ellipsoidal constant-energy surfaces in

the conduction band. There are 6

equivalent minima along [100] direction

Read subsection 1.5.2;How to measure

the effective mass?

(a) E–k diagram of Si

large light hole2 2/d E dk

small heavy hole2 2/d E dk

E-k relationship for parabolic

band with anisotropic effective

mass in 3-D: mx* ≠ my* ≠ mz*.

In Si, mx* = my* ≠ mz*.

The constant energy surface is not

sphere, but ellipsoid.

22 22

* * *

22 2

2 2 2* * *

22 2

( )2

12 2 2

yx z

x y z

yx x

x y z

kk kE

m m m

kk k

m E m E m E

Equation for ellipsoid in k space

Documents

Modern Semiconductor Devices for Integrated Circuits Chapter 1. …bkict-ocw.knu.ac.kr/include/download.html?fn=559E036F732... · Modern Semiconductor Devices for Integrated Circuits