SEMICONDUCTORS Semiconductors Semiconductor devices Electronic Properties Robert M Rose, Lawrence...
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Transcript of SEMICONDUCTORS Semiconductors Semiconductor devices Electronic Properties Robert M Rose, Lawrence...
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