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P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors1

Effective Mass

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

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

P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors

Effective Mass – an example

Find the (E,k) relationship for a free electron and relate it to the electron mass.

khmvp

222

2

22

1

2

1k

m

h

m

pmvE

m

h

dk

Ed 2

2

2

E

k

The electron momentum is:

2

2

2

dk

Ed

hm

3

Effective MassMost energy bands are close to parabolic at their minima (for

conduction bands) or maxima (for valence bands).

EC

EV

3

• 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:


Effective Mass

• At k=0, the (E,k) relationship near the minimum is usually parabolic:

2

2

dk

Ed

2

2

2*

dkij

Eijdij

hm

gEkm

hkE 2

*

2

2)(

In a parabolic band, is constant. So, effective mass is constant.

In most semiconductors the effective mass is a tensor quantity.

Effective Mass

EV

EC

02

2

dk

Ed

02

2

dk

Ed

0* m

0* m2

2

2*

dkEd

hm

Effective mass Ge Si GaAs

† m0 is the free electron rest mass.

Table: 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


Crystalline structure of Si

• Si N=14 4 bonds, IV-th column of the periodic table

► Diamond lattice, lattice constant a=0.543 nm

► Each atom has 4 nearest neighbor

real 3D

simplified 2D

undoped or intrinsic semiconductor


Intrinsic Semiconductor

a) Energy level diagrams showing the excitation of an electron from the valence band to

the conduction band. The resultant free electron can freely move under the

application of electric field.

b) Equal electron & hole concentrations in an intrinsic semiconductor created by the

thermal excitation of electrons across the band gap

-123 JK 1038.1 Bk


For T > 0K, some VB electrons get enough thermal energy to be excited (through Eg) up to the CB.

Consequently, the semiconductor material will have some electrons in the previously empty CB and some unoccupied states in the previously full VB


Semiconductors: Doping

– When silicon is doped with phosphorous, it becomes a n-type semiconductor, in which an electrical current is carried by negatively charged electrons

– When silicon is doped with boron, it becomes a p-type semiconductor, in which an electrical current is carried by positively charged holes


C

V

Eg

Vacancy

Holes

Energy bandModel

BondModel


Likewise, acceptor levels can bethermically settled with VBelectrons, by there generatingholes.

Donors and acceptors

At 0K, the donor energy level is filled with electrons and too little thermicalenergy is needed in order to excite these electrons up to the CB. So, between 50-100K, electrons are virtually “donated” to the CB.


Donor and Acceptor levels with temperature


Extrinsic Semiconductors (n-type)

Extrinsic semiconductor (p-type)


• Doping: the process to create carriers in semiconductors by purposely introducing impurities into the crystal

- There are two types of doped semiconductors, n-type and p-type.

• Extrinsic semiconductors: the materials that have a characteristic of n0 p0 ni when they are doped

• n-type semiconductors:

- A “donor” impurity from column V (P, As, Sb; donor) introduces an donor energy level (Ed) near the bottom of CB Ec( within the band gap)

- At 50K, all of the electrons in Ed (filled with electrons at 0K) are “donated” to CB

- (e-: majority carrier, h+: minority carrier)

Extrinsic Material

00 , pnn i

n-type semiconductor Energy Band Model

Charge Carriers in doped Semiconductors


• p-type semiconductor (e-: minority carrier, h+:majority carrier):

- An “acceptor” impurity from column III (B, Al, Ga, In; acceptor) introduces an acceptor energy level (Ea) near the top of VB Ev( within the band gap)

- At 50K, all of the energy states in Ea (empty at 0K) “accept” electrons from the VB, leaving behind holes in the VB.

-

Extrinsic Material (continued)

00 , nnp i

• Bond Model:- An As atom (column V; 5 valence electrons) in Si lattice has 4

valence electrons to complete the covalent bonds with the 4 neighboring Si atoms, plus one extra electron; the fifth valence electron is loosely bound to As atom A small amount of thermal E enables this extra electron to overcome its Coulombic binding to the impurity atom extra electron is donated to the lattice

- The column III impurity B has only 3 valence electrons to contribute to the covalent bonding, leaving one bond incomplete (hole) With a small amount of thermal E, this incomplete bond is transferred to other atoms as the bonding electrons exchange positions (electron hopping)

p-type Energy Band Model

Bond Model

Charge Carriers in Semiconductors


5 valance dopant: donor (As, P, Sb)

• Electron: majority carrier• Hole: minority carrier

conduction band

valance band

n-type semiconductor


3 valance dopant: acceptor (B, Ga, In)

• Electron: minority carrier• Hole: majority carrier

conduction band

valance band

p-type semiconductor


n-Type Semiconductor

a) Donor level in an n-type semiconductor.

b) The ionization of donor impurities creates an increased electron

concentration distribution.


p-Type Semiconductor

a) Acceptor level in an p-type semiconductor.

b) The ionization of acceptor impurities creates an increased hole

concentration distribution


Intrinsic & Extrinsic Materials

• Intrinsic material: A perfect material with no impurities.

• Extrinsic material: donor or acceptor type semiconductors.

• Majority carriers: electrons in n-type or holes in p-type.

• Minority carriers: holes in n-type or electrons in p-type.

• The operation of semiconductor devices is essentially based on the injection and extraction of minority carriers.

)2

exp(Tk

Enpn

B

g

i

2

inpn

n,p & ni are the electron, hole, & intrinsic concentrations respectively. Eg is the gap energy, T is temperature.


Calculation of carrier concentration

dWWfWgpvW

v )(1)(0

dWWfWgn

cW

c )()(

possible energy states

occupation probability

concentrations

electrons

holes

FD statistics:

kT

WWWf

Fexp1

1)(


T

TETE gg

2

)0()(

As the crystal temperature rises, the crystal expands and the gapenergy gets lower.

By submitting to pressure, the crystal is compressed and the gapenergy rises.

Varshni Equation

Temperature Effects

S.M.Sze, “Physics of Semiconductor Devices”, 2nd Edition, JohnWiley&Sons, 1981

Bandgaps of silicon, germanium and gallium arsenide


Intrinsic Semiconductor

Extrinsic Semiconductor

Donor impurities – provide extra electrons to conduction(type n)

Acceptor impurities – provide exceedingholes to conduction(type p)

inpn

Doping of SemiconductorsDoping of Semiconductors

• We can estimate the BINDING ENERGY of the surplus electron or hole provided by each

dopant in a rather simple way

* In the case where the dopant is a DONOR the loosely-bound additional electron orbits

a core with a net POSITIVE charge

The donor is thus essentially similar to a HYDROGEN ATOM and the binding

energy of the surplus electron can be computed from the expression for the

IONIZATION ENERGY of the hydrogen atom

)2.34(eV1.0)(8 22

4*

h

em

or

e

• THE IONIZATION ENERGY OF A HYDROGEN ATOM IS GIVEN AS

(SUBJECT_14)

• FOR A DONOR ELECTRON IN SILICON THIS EXPRESSION IS MODIFIED

BY INCLUDING THE EFFECTIVE ELECTRON MASS (me* = 1.18 mo) AND

THE DIELECTRIC CONSTANT (r = 11.8) IN SILICON

)1.34(eV6.138 22

4

h

me

o

A GROUP V DOPANT IN A SILICON LATTICE

CAN BE VIEWED AS A PSEUDO HYDROGEN ATOM!

e-

+

Doping of Semiconductors

• A similar argument to that above may also be made for an ACCEPTOR in silicon but now

the pseudo hydrogen atom consists of a POSITIVE HOLE that orbits a NEGATIVE core

* This gives a SIMILAR estimate for the binding energy of the HOLE

* The ACTUAL donor and acceptor binding energies measured in experiment agree

reasonably WELL with these SIMPLE estimates

• A GROUP III DOPANT IN A SILICON LATTICE CAN ALSO

BE VIEWED AS A PSEUDO HYDROGEN ATOM

• IN THIS CASE HOWEVER THE DOPANT CORE APPEARS

NEGATIVELY CHARGED AND IS ORBITED BY A HOLE

P As Sb B Al Ga In

Si .045 .049 .039 .045 .057 .065 .016

Ge .012 .013 .010 .010 .010 .011 .011

BINDING ENERGY FOR DONORS (eV) BINDING ENERGY FOR ACCEPTORS (eV)

B

e+

B

e+

Si:As

Si:B


Concept of Fermi-Dirac distribution function

Distribution of electrons over a range of allowed energy levels at thermal equilibrium is given by :

f(E) = 1

1 + e(E – EF) / kT

K – Boltzman’s constant = 8.62 x 10-5 eV / K= 1.38 x 10 -23 J / K

f(E) gives the probability of findiing a electron in a particular energy E.

effective mass - folk.uio.nofolk.uio.no/ravi/cutn/semiphy/6.l7_intrinsic-extrinsic.pdf · • the...

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