E vs k diagram - INAOE - Pjmolina/2010_Part3-06-08_Bandas de e… · E vs k diagram m k m p E 2 2!2...

E vs k diagram

m

k

m

pE

22

222

The parameter k is called the crystal momentum and

is a parameter that results from applying Schrödinger

wave equation to a single-crystal lattice.

Electrons traveling in different directions encounter different potential patterns

and therefore different k-space boundaries. The E vs k diagrams are in general

a function of the k-space direction in a crystal.

111Curso propedéutico de Electrónica INAOE 2010 Dr. Joel Molina & Dra. Claudia Reyes

22

2

/*

dkEdm

*

22

2)(

e

Cm

kEkE

When the conduction band edge occurs at k = 0, we

can represent the band-structure as a simple parabola,

1

2

2

2

* 1

dk

Edme

where:

EC = minimum of the conduction band energy

me* = effective mass of an electron,

The effective mass of an electron in a band

with a given (E, k) relationship

Effective mass of carriers – electrons

mdk

Ed 2

2

2

Narrow parabola

small effective mass

GaAs me* = 0.063m0

Ge me* = 0.55m0

Si me* = 1.09m0


Curso propedéutico de Electrónica INAOE 2008 Dr. Joel Molina & Dr. Pedro Rosales

*

22

2h

Vm

kEE

Similarly, the energy momentum relation

for the valence band can be written…

where:

EV = maximum of the valence band energy

mh* = effective mass of a hole.

There are actually two bands near the top

of the valence band of different widths,

leading to heavy holes and light holes.

GaAs mhh* = 0.45m0 mlh* = 0.08m0

Si mhh* = 0.49m0 mlh* = 0.16m0

113

Effective mass of carriers – holes

A perfect semiconductor crystal with no

impurities or lattice defects is called an intrinsic

semiconductor. In such material there are no

charge carries at T = 0 K.

At higher temperatures EHPs are generated as

valence band electrons are thermally excited

across the bandgap to the conduction band.

If a steady state carrier concentration is

maintained, there must be recombination of

EHPs at the same rate at which they are

generated ri = gi


Intrinsic material

In addition to the intrinsic carriers generated thermally … When a group V or III

atom (As, B) is substituted into the Si lattice an electron is donated or accepted and

the semiconductor becomes n-type or p-type respectively.

In an extrinsic semiconductor at any

temperature the carriers concentration have

two contributions:

1. Thermal

2. Doping [ ND or NA ]

For a n-type semiconductor:

n = ND and p = ni2 / ND

For a p-type semiconductor:

p = NA and n = ni2 / NA

[ND]

[NA]

The crystal is extrinsic when the

doping is such that:

(n0, p0) ni


Extrinsic material

At 0K the extra electrons associated with the donor atoms are ‘fixed’ to

the donor sites at an energy level Ed.

As the temperature increases there is enough thermal energy to ionize

the donor atoms i.e. for an electron to make the transition into the

conduction band which is only an energy jump of Ed where Ed <<Eg.

To create holes in the valence band in a p-type semiconductor, electrons

need only an energy of Ea to reach the acceptor level where Ea <<Eg.


25 meV for Si

7 meV for GaAs

50 meV for Si, GaAsVt= kBT/q at 300 K = 26 meV

In reality, different dopants have different ionization levels and deep levels (|E| > 3kBT), which can be important, but this simple model gives the correct order of magnitude.

He

S

D Em

mE

0

*2

0

hydrogen

energy levels

donor levels

acceptor levels


Approximate energy required to excite the 5th electron of a donor atom into

the conduction band:

Bohr Model

Ionization level of dopants

Ionization level of dopants


He

S

D Em

mE

0

*2

0


A quantum well laser (QWL) is a laser diode in which the active region of the device is so narrow that quantum confinement occurs.

The wavelength of the light emitted by a QWL is determined by the width of the active region rather than just the bandgap of the

material from which it is constructed. This means that much shorter wavelengths can be obtained from QLW than from conventional

laser diodes using a particular semiconductor material. The efficiency of a QLW is also greater than a conventional laser diode due to

the stepwise form of its density of states function.

Electrons and holes in Quantum Wells

To obtain the carrier density per unit volume we must first calculate the

number of allowed states (including spin) per energy range per unit volume.

For electrons in the conduction band

where the E-k relation is of the form,

*

22

2 e

Cm

kEE

21

23

2

*24 E

h

mEN e

The density of states is given by:

See appendix IV of Streetman

*

22

2 h

Vm

kEE

Similarly, for holes in the valence band

where the E-k relation is of the form,

21

23

2

*24 E

h

mEN h

The density of states is given by:

23

*23

*23

*

hhlhh mmmAccounting for the

contribution from both

light and heavy holes


Density of states function

The probability that an electron occupies an electronic state with energy E

is given by the Fermi-Dirac distribution function:

TkEE BFEF

/exp1

1)(

The distribution of

electrons over a range of

allowed energy levels at

thermal equilibrium.

For E=EF then F(E)=0.5The Fermi energy is the energy for which the

probability of occupation by an electron is exactly ½


Fermi-Dirac distribution function

Curso propedéutico de Electrónica INAOE 2008 Dr. Joel Molina & Dr. Pedro Rosales

TkEE

BFBFeEFTkEE

3

TkEE

BFBFeEFTkEE

13

The Fermi distribution function is simplified

for an electron in the conduction band since,

and for a hole in the valence band since,

For Si at 300 K:

ni=pi 1010 cm-3

Density of available states at Ev and Ec:

1019 cm-3

Because of the relatively large density of states in

each band, small changes in f(E) can result in

significant changes in carrier concentration.

122

The concentration of

the electrons in the

conduction band is:

Ec

dEENEfn )()(

Ec

dEENEfp )()](1[

The concentration of

the holes in the

valence band is:

The distribution of electrons in the conduction band is given by the density of allowed

quantum states times the probability that a state is occupied by an electron.


toptop EE

dEEFENdEEnn00

taking the bottom of the conduction band to be E=0

The electron density in the conduction band is given by,

Taking the previous simplified expression for F(E),

dETk

EEE

h

mn

B

Fe

exp2

40

2

*

21

23


Electron concentration

dxexTk

ETk

h

mn x

B

FB

e

0

2

*

21

23

23

exp2

4

Let x = E / kT

Tk

E

h

Tkmn

B

FBe exp2

2

23

2

*

The effective density of states in the conduction band, NC



For Si (300 K) NC = 2.8 x 1019 cm-3

For GaAs (300 K) NC = 4.7 x 1017 cm-3

Taking the bottom of the conduction band as EC rather than E=0,

Tk

EENn

B

FCC exp



Where:

2/3

2

*22

h

TkmNc Be

For Si (300 K) NV = 1.04 x 1019 cm-3

For GaAs (300 K) NV = 7 x 1018 cm-3

Tk

EENp

B

VFV exp


Hole concentration

Similarly for holes in the valence band,

2/3

2

*22

h

TkmNv Bh

Where:

Tk

ENNnnp

B

g

VCi exp2

Tk

ENNn

B

g

VCi2

exp

This expression is independent of EF and is valid for extrinsic (doped) semiconductors too.

For Si (300 K) ni = 9.65 x 109 cm-3

For GaAs (300 K) ni = 2.25 x 106 cm-3


Law of Mass Action

Tk

EEnn

B

iF

iexp

Tk

EEnp

B

Fii exp

The product np is then,

Tk

EEn

Tk

EEnnp

B

iFi

B

Fii expexp

2

innp

So the law of mass action holds also for extrinsic semiconductors.


Law of Mass Action. Intrinsic Semiconductors

Calculation of extrinsic Fermi level

Tk

EENn

B

FCC exp

Tk

EENN

B

FCCD exp

D

CBFC

N

NTkEE ln

At 300 K there is usually enough thermal energy to completely ionize the

dopant atoms, so for n-type semiconductor n = ND (donor concentration)

So as the concentration of donor atoms increases the Fermi

level moves closer to the bottom of the conduction band


So as the concentration of acceptor atoms increases the Fermi

level moves closer to the top of the valence band

A

VBVF

N

NTkEE ln

Tk

EENN

B

VFVA exp

Tk

EENp

B

VFV exp

Similarly for p-type semiconductors, p = NA (acceptor concentration)…



It is often useful to express the carrier density in terms of the intrinsic

carrier concentration and the intrinsic Fermi level…

Similarly for holes,

Tk

EE

Tk

EENn

B

Fi

B

iCC expexp

Tk

EEnp

B

Fii exp

For electrons,



Tk

EEnn

B

iFi exp

Intrinsic carrier densities in

Si, Ge and GaAs as a

function of the reciprocal of

temperature.


Intrinsic carrier densities ni as a function of temperature

Tk

ENNn

B

g

VCi2

exp

For Si (300 K) ni = 9.65 x 109 cm-3

For GaAs (300 K) ni = 2.25 x 106 cm-3

Eg (Ge) = 0.66 eV

Eg (Si) = 1.12 eV

Eg (GaAs) = 1.42 eV

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 200 400 600 800 1000 1200

Temperature (K)

En

erg

y B

an

dg

ap

(eV

)

Eg (GaAs) = 1.42 eV

Eg (Si) = 1.12 eV

Eg (Ge) = 0.66 eV

The temperature dependence of the energy bandgap, Eg, has been

experimentally determined yielding the following expression for Eg as a

function of the temperature, T:

where Eg(0), a and b are the

fitting parameters. These fitting

parameters are listed for

germanium, silicon and gallium

arsenide in Table.


Energy gap Eg as a function of temperature

http://ece-www.colorado.edu/%7Ebart/book/book/chapter2/ch2_3.htm

At low temperatures the thermal

energy is insufficient to ionize all

donor atoms so n < ND

At higher temperatures the thermal

energy is sufficient to ionize all

donor atoms so n = ND

At some temperature the intrinsic

carrier density becomes

comparable to the donor

concentration and beyond this point

the semiconductor becomes

intrinsic.


Electron density n0 as a function of temperature

E vs k diagram - INAOE - Pjmolina/2010_Part3-06-08_Bandas de e… · E vs k diagram m k m p E 2 2!2...

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