energy bands and charge carriers

Energy bands and charge carriersin semiconductors

Chapter 3Mr. Harriry (Elec. Eng.)

By: Amir Safaei2006

In the name of God

2

Outlines

3-1. Bonding Forces and Energy Bands in Solids 3-1-1. Bonding Forces in Solids 3-1-2. Energy Bands 3-1-3.

Metals, Semiconductors & Insulators 3-1-4. Direct & Indirect Semiconductors 3-1-5.

Variation of Energy Bands with Alloy Composition

3

Outlines

3-2. Carriers in Semiconductors 3-2-1. Electrons and Holes 3-2-2. Effective Mass 3-2-3. Intrinsic Material 3-2-4. Extrinsic Material 3-2-5. Electrons and Holes in

Quantum Wells

4

Outlines

3-3. Carriers Concentrations 3-3-1. The Fermi Level 3-3-2. Electron and Hole

Concentrations at Equilibrium

5

3-1. Bonding Forces & Energy Bands in Solids In Isolated Atoms In Solid Materials

3rd Band2nd Band

1st Band

Core

6

3-1-1. Bonding Forces in Solids

Na (Z=11) [Ne]3s1

Cl (Z=17) [Ne]3s1 3p5

Na+ Cl_

7


e_

Na+

8


9


Si<100>

10

3-1-2. Energy Bands Pauli Exclusion Principle

C (Z=6) 1s2 2s2 2p2

2 states for 1s level

2 states for 2s level

6 states for 2p level

For N atoms, there will be 2N, 2N, and 6N states of type 1s, 2s, and 2p, respectively.

11

3-1-2. Energy Bands

Atomic separation

Diamond lattice

spacing

En

erg

y

1s

2s

2p

Valence band

Conduction band

2p

2s

2s-2p

4N States

4N States

Eg

1s

12

3-1-3. Metals, Semiconductors & Insulators

For electrons to experience acceleration in an applied electric field, they must be able to move into new energy states. This implies there must be empty states (allowed energy states which are not already occupied by electrons) available to the electrons.

The diamond structure is such that the valence band is completely filled with electrons at 0ºK and the conduction band is empty. There can be no charge transport within the valence band, since no empty states are available into which electrons can move.

13


The difference bet-ween insulators and semiconductor mat-erials lies in the size of the band gap Eg, which is much small-er in semiconductors than in insulators.

Insulator Semiconductor

Filled

Filled

Empty

Empty

Eg

Eg

14


Metal

Filled

Partially Filled

Overlap

In metals the bands either overlap or are only partially filled. Thus electrons and empty energy states

Metal are intermixed with-

in the bands so that electrons can move freely under the infl-uence of an electric field.

15

3-1-4. Direct & Indirect Semiconductors A single electron is assumed to travel

through a perfectly periodic lattice. The wave function of the electron is assumed

to be in the form of a plane wave moving.

xjkxk

xexkUx ),()( x : Direction of propagation k : Propagation constant / Wave vector : The space-dependent wave function

for the electron

16

3-1-4. Direct & Indirect Semiconductors

U(kx,x): The function that modulates the

wave function according to the periodically of the lattice.

Since the periodicity of most lattice is different in various directions, the (E,k) diagram must be plotted for the various crystal directions, and the full relationship between E and k is a complex surface which should be visualized in there dimensions.

17


Eg=hνEg Et

k k

EE

Direct IndirectExample 3-1

18


Example 3-1: Assuming that U is constant in

for an essentially free electron, show that the x-component of the electron momentum in the crystal is given by

xx khP Example 3-2

),()( xkUx xk xjkxe

19


x

x

xjkxjk

x

khdxU

dxUkh

dxU

dxexj

heU

P

xx

2

2

2

2

)( Answer:

The result implies that (E,k) diagrams such as shown in previous figure can be considered plots of electron energy vs. momentum, with a scaling factor .

h

20


Si Ge GaAs AlAs Gap

1.11 1350 480 2.5E5 D 5.43

0.67 3900 1900 43 D 5.66

1.43 8500 400 4E8 Z 5.65

2.16 180 0.1 Z 5.66

2.26 300 150 1 Z 5.45

Eg(eV) n p Lattice Å

Properties of semiconductor materials

21

3-1-5. Variation of Energy Bands with Alloy Composition

X

L

1.43eVk

E

0.3eV

AlxGa1-xAs

2.16eV

AlAsGaAsX

E

1.4

2.0

1.8

1.6

2.2

2.4

2.6

2.8

3.0

0 0.2 0.4 0.6 0.8 1

X

L

X

L

22

3-2. Carriers in Semiconductors

Ec

Ev

Eg

0ºK3ºK2ºK4ºK5ºK1ºK6ºK7ºK8ºK9ºK10ºK11ºK12ºK13ºK14ºK300ºK

15ºK16ºK17ºK18ºK19ºK20ºK

Electron Hole PairE H P

23

3-2-1. Electrons and Holes

N

iiVqJ 0)(

k

Ekj-kj

j` j

j

N

ii VqVqJ )()(

0

J jVq) ( jV)q(

24

3-2-2. Effective Mass

The electrons in a crystal are not free, but instead interact with the periodic potential of the lattice.

In applying the usual equations of electrodynamics to charge carriers in a solid, we must use altered values of particle mass. We named it Effective Mass.

25


Example 3-2: Find the (E,k) relationship for a free electron and relate it to the electron mass.

E

k

26


khmvp

222

2

22

1

2

1k

m

h

m

pmvE

Answer: From Example 3-1, the electron

momentum is:

m

h

dk

Ed 2

2

2

27


Answer (Continue): Most energy bands are close to

parabolic at their minima (for conduction bands) or maxima (for valence bands).

EC

EV

28

3-2-2. Effective Mass The effective mass of an electron in a band

with a given (E,k) relationship is given by

2

2

2*

dkEd

hm

X

L

k

E

1.43eV

) ()( or** LXmm

Remember that in GaAs:

29

3-2-2. Effective Mass At k=0, the (E,k) relationship near the

minimum is usually parabolic:

gEkm

hE 2

*

2

2 In a parabolic band, is constant.

So, effective mass is constant.

Effective mass is a tensor quantity.

2

2

dk

Ed

2

2

2*

dkEd

hm

30


EV

EC

02

2

dk

Ed

02

2

dk

Ed

0* m

0* m2

2

2*

dkEd

hm

Ge Si GaAs

† m0 is the free electron rest mass.

Table 3-1. Effective mass values for Ge, Si and GaAs.

mn

*

mp

*

055.0 m 01.1 m 0067.0 m

037.0 m 056.0 m 048.0 m

31

3-2-3. Intrinsic Material

A perfect semiconductor crystal with

no impurities or lattice defects is

called an Intrinsic semiconductor.

In such material there are no charge

carriers at 0ºK, since the valence

band is filled with electrons and the

conduction band is empty.

32

3-2-3. Intrinsic Material

SiEgh+

e-

n=p=ni

33

3-2-3. Intrinsic Material If we denote the generation rate of EHPs

as and the recombination rate

as equilibrium requires that:

)(Tgi

)( 3scmEHPri

ii gr Each of these rates is temperature depe-

ndent. For example, increases

when the temperature is raised.

)( 3scmEHPgi

iirri gnpnr 200

34

3-2-4. Extrinsic Material

In addition to the intrinsic carriers generated thermally, it is possible to create carriers in semiconductors by purposely introducing impurities into the crystal. This process, called doping, is the most common technique for varying the conductivity of semiconductors.

When a crystal is doped such that the equilibrium carrier concentrations n0 and p0

are different from the intrinsic carrier concentration ni , the material is said to be

extrinsic.

35


0ºK3ºK2ºK4ºK5ºK1ºK6ºK7ºK8ºK9ºK10ºK11ºK12ºK13ºK14ºK50ºK15ºK16ºK17ºK18ºK19ºK20ºK

Ec

Ev

Ed

Donor

V

P

As

Sb

36


0ºK3ºK2ºK4ºK5ºK1ºK6ºK7ºK8ºK9ºK10ºK11ºK12ºK13ºK14ºK50ºK15ºK16ºK17ºK18ºK19ºK20ºK

Ec

Ev

Ea

Acceptor

ш

B

Al

Ga

In

37


h+

Al

e- Sb

Si

38


We can calculate the binding energy by using the Bohr model results, consider-ing the loosely bound electron as ranging about the tightly bound “core” electrons in a hydrogen-like orbit.

rKnhK

mqE 022

4

4, 1;2

39


Example 3-3: Calculate the approximate donor binding energy for Ge(εr=16, mn

*=0.12m0).

40


eVJ

h

qmE

r

n

0064.01002.1

)1063.6()161085.8(8

)106.1)(1011.9(12.0

)(8

21

234212

41931

220

4*

Answer:

Thus the energy to excite the donor electron from n=1 state to the free state (n=∞) is ≈6meV.

41


When a ш-V material is doped with Si or Ge, from column IV, these impurities are called amphoteric.

In Si, the intrinsic carrier concentration ni is about 1010cm-3 at

room tempera-ture. If we dope Si with 1015 Sb Atoms/cm3, the conduction electron concentration changes by five order of magnitude.

42

3-2-5. Electrons and Holes in Quantum Wells

One of most useful applications of MBE or OMVPE growth of multilayer compou-nd semiconductors is the fact that a continuous single crystal can be grown in which adjacent layer have different band gaps.

A consequence of confining electrons and holes in a very thin layer is that

43


these particles behave according to the particle in a potential well problem.

GaAs

Al0.3Ga0.7AsAl0.3Ga0.7As

50Å

E1

Eh

1.43eV

1.85eV

0.28eV

0.14eV

1.43eV

44


Instead of having the continuum of states

as described by , modified for

effective mass and finite barrier height.

Similarly, the states in the valence band

available for holes are restricted to

discrete levels in the quantum well.

2

222

2mL

hnEn

45


An electron on one of the discrete condu-ction band states (E1) can make a transition

to an empty discrete valance band state in the GaAs quantum well (such as Eh), giving

off a photon of energy Eg+E1+Eh, greater

than the GaAs band gap.

46

3-3. Carriers Concentrations In calculating semiconductor electrical

pro-perties and analyzing device behavior, it is often necessary to know the number of charge carriers per cm3 in the material. The majority carrier concentration is usually obvious in heavily doped material, since one majority carrier is obtained for each impurity atom (for the standard doping impurities).

The concentration of minority carriers is not obvious, however, nor is the temperature dependence of the carrier concentration.

47

3-3-1. The Fermi Level Electrons in solids obey Fermi-Dirac statistics. In the development of this type of statistics:

Indistinguishability of the electrons Their wave nature Pauli exclusion principle

must be considered. The distribution of electrons over a range of

these statistical arguments is that the distrib-ution of electrons over a range of allowed energy levels at thermal equilibrium is

48

3-3-1. The Fermi Level

kTfEE

eEf )(

1

1)(

k : Boltzmann’s constant

f(E) : Fermi-Dirac distribution function

Ef : Fermi level

49


2

1

11

1

1

1)( )(

kTfEfE

eEf f

Ef

f(E)

1

1/2

E

T=0ºKT1>0ºKT2>T1

50


Ev

Ec

Ef

E

f(E)01/21

≈≈

f(Ec

)f(Ec

)

[1-f(Ec)]

Intrinsicn-typep-type

51

3-3-2. Electron and Hole Concentrations at Equilibrium

CE

dEENEfn )()(0

The concentration of electrons in the conduction band is

N(E)dE : is the density of states (cm-3) in the energy range dE.

The result of the integration is the same as that obtained if we repres-ent all of the distributed electron states in the conduction band edge EC. )(0 CC EfNn

52


EC

EV

Ef

E

Holes

Electrons

Intrinsicn-typep-type

N(E)[1-f(E)]

N(E)f(E)

53


kTFECE

kTFECE

ee

Ef C

)(

)(

1

1)(

kTFECE

eNn C

)(

0

23

) 2

(22

*

h

kTmN nC

54


)](1[0 VV EfNp

kTVEFE

kTFEVE

ee

Ef V

)(

)(

1

11)(1

kTVEFE

eNp V

)(

0

23

) 2

(22

*

h

kTmN pV

55


kTvEiE

eNp Vi

)(

kTiEcE

eNn Ci

)(

kT

gEkT

vEcE

eNNeNNpn vcvc

)(

00

kTgE

eNNpn vcii

kTgE

eNNn vci2

2

00 inpn

kTFEiE

enp i

)(

0

kTiEFE

enn i

)(

0

56


Example 3-4: A Si sample is doped with 1017 As Atom/cm3. What is the equilibrium hole concentra-tion p0 at 300°K? Where is EF relative to Ei?

57


3317

20

0

2

0 1025.210

1025.2

cmn

np i

Answer: Since Nd»ni, we can approximate

n0=Nd and

kTiEFE

enn i

)(

0

eVn

nkTEE

iiF 407.0

105.1

10ln0259.0ln

10

170

58


Answer (Continue) :

Ev

Ec

EF

Ei1.1eV0.407eV

59

References:

Solid State Electronic Devices Ben G. Streetman, third edition

Modular Series on Solid State Devices, Volume I: Semiconductor Fundamentals

Robert F. Pierret

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