SEMICONDUCTORS

Semiconductors

Semiconductor devices

Electronic PropertiesRobert M Rose, Lawrence A Shepart, John Wulff

Wiley Eastern Limited, New Delhi (1987)

Energy gap in solids

In the free electron theory a constant potential was assumed inside the solid

In reality the presence of the positive ion cores gives rise to a varying potential field

The travelling electron wave interacts with this periodic potential (for a crystalline solid)

The electron wave can be Bragg diffracted

Bragg diffraction from a 1D solid

n = 2d n = 2d Sin 1D =90o

...

3

2d d, 2d,λCritical ,

... ,

d

d ,

dkCritical

3,

2

The Velocity of electrons for the above values of k are zero These values of k and the corresponding E are forbidden in the solid The waveform of the electron wave is two standing waves The standing waves have a periodic variation in amplitude and hence the

electron probability density in the crystal The potential energy of the electron becomes a function of its position (cannot be assumed to be constant (and zero) as was done in the

free electron model)

Sin d

nk

m

khE

2

22

8

k →

E →

Band gap

d

d

2

d

d

2

The magnitude of the Energy gap between two bands is the difference in the potential energy of two electron locations

d

k →

E →

K.E of the electron increasingDecreasing velocity of the electronve effective mass (m*) of the electron

Within a band

Effective energy gap → Forbidden gap → Band gap

k →

E →

d

E →

d

2

[100] [110]

k →

Effective gap

o90Sin dk

Sin d

nk

o45Sin dk

The effective gap for all directions of motion is called the forbidden gap There is no forbidden gap if the maximum of a band for one direction of motion is higher than the minimum for the higher band for another direction of motion this happens if the potential energy of the electron is not a strong function of the position in the crystal

Energy band diagram: METALS

Monovalent metals

Divalent metals

Monovalent metals: Ag, Cu, Au → 1 e in the outermost orbital outermost energy band is only half filled

Divalent metals: Mg, Be → overlapping conduction and valence bands they conduct even if the valence band is full

Trivalent metals: Al → similar to monovalent metals!!! outermost energy band is only half filled !!!

Energy band diagram: SEMICONDUCTORS

2-3 eV

Elements of the 4th column (C, Si, Ge, Sn, Pb) → valence band full but no overlap of valence and conduction bands

Diamond → PE as strong function of the position in the crystal Band gap is 5.4 eV

Down the 4th column the outermost orbital is farther away from the nucleusand less bound the electron is less strong a function of the positionin the crystal reducing band gap down the column

Energy band diagram: INSULATORS

> 3 eV

P(E) →

E →

1

FE

0

EgEg/2

5.0

Intrinsic semiconductors

At zero K very high field strengths (~ 1010 V/m) are required to move anelectron from the top of the valence band to the bottom of the conduction band

Thermal excitation is an easier route

T > 0 K

kT

EEEP

Fexp1

1)(

2

gF

EEE

kT

EEP g

2exp)(

eV E

EESilicon

g

SiliconF 55.02

eV kT 026.0 1

kT

EE F

kT

ENn g

e 2exp

ne → Number of electrons promotedacross the gap (= no. of holes in the valence band)

N → Number of electrons available at the top of the valance band

for excitation

Unity in denominator can be ignored

Under applied field the electrons (thermally excited into the conduction band) can move using the vacant sites in the conduction band Holes move in the opposite direction in the valence band The conductivity of a semiconductor depends on the concentration of these charge carriers (ne & nh)

Similar to drift velocity of electrons under an applied field in metals in semiconductors the concept of mobility is used to calculate conductivity

Conduction in an intrinsic semiconductor

sVmmV

sm

gradient field

velocity driftMobility //

/

/ 2

Mobility of electrons and holes in Si & Ge (at room temperature)

Species Mobility (m2 / V / s)

Si Ge

Electrons 0.14 0.39

Holes 0.05 0.19

hhee enen

Conductivity as a function of temperature

kT

E)( e N g

he 2exp

kT

EC g

2exp

kT

EC g

2ln 1

Ln()

→1/T (/K) →

k

Eg

2

hhee enen

Extrinsic semiconductors

The addition of doping elements significantly increases the conductivity of a semiconductor

Doping of Si V column element (P, As, Sb) → the extra unbonded electron is practically free (with a radius of motion of ~ 80 Å) Energy level near the conduction band

n- type semiconductor III column element (Al, Ga, In) → the extra electron for bonding

supplied by a neighbouring Si atom → leaves a hole in Si. Energy level near the valence band

p- type semiconductor

Eg

Donor level

n-type Ionization Energy→ Energy required to promote an electron from the Donor level to conduction band

EIonization < Eg

even at RT large fraction of the donor electrons are exited into the conduction band

Electrons in the conduction band are the majority charge carriers

The fraction of the donor level electrons excited into the conduction band is much larger than the number of electrons excited from the valence band

Law of mass action: (ne)conduction band x (nh)valence band = Constant

The number of holes is very small in an n-type semiconductor

Number of electrons ≠ Number of holes

FE EIonization

Acceptor level

Eg

p-type

At zero K the holes are bound to the dopant atom As T↑ the holes gain thermal energy and break away from the dopant atom

available for conduction The level of the bound holes are called the acceptor level (which can accept

and electron) and acceptor level is close to the valance band Holes are the majority charge carriers Intrinsically excited electrons are small in number Number of electrons ≠ Number of holes

EIonizationFE

Ionization energies for dopants in Si & Ge (eV)

Type Element In Si In Ge

n-type

P 0.044 0.012

As 0.049 0.013

Sb 0.039 0.010

p-type

B 0.045 0.010

Al 0.057 0.010

Ga 0.065 0.011

In 0.16 0.011

(

/ Ohm

/ K

)→

1/T (/K) →

0.02 0.04 0.06 0.08 0.1

410

310

210

110

010

Intrinsic

Exhaustion

Extrinsic

Exponentialfunction

Slope can be usedfor the calculationof EIonization

10 K50 K

+ve slope due to Temperature dependent

mobility term

All dopant atoms have been excitedk

Eg

2 slope

Semiconductor device chose the flat region where the conductivity does

not change much with temperature Thermistor (for measuring temperature) maximum sensitivity is

required