L2_Semiconductor_device_physics_f

Prévia do material em texto

Physics of Semiconductor Devices 
 PN-junction diodes
 Schottky Barrier Diodes
 Optical Devices
PN Junction (Diode)
 When N-type and P-type dopants are introduced side-by-side in 
a semiconductor, a PN junction or a diode is formed. 
PN Junction (Diode)
 When N-type and P-type semiconductors are combined we have a 
material with very large concentration of holes on one side and a 
very large concentration of electrons on the other side.
 We expect electrons to diffuse from n side to p side and holes to 
diffuse from p side to n side.
 Therefore, a diffusion current will flow from p side to n side.
 When electrons diffuse to p side, n side will have a net positive 
charge due to positively charged donor ions.
 When holes diffuse to ne side, p side will have a net negative 
charge due to negatively charged acceptor ions.
PN Junction (Diode)
 Electrons which diffuse to p side will be minority carriers.
 These minority carriers will recombine with majority carriers 
(holes) in p region. 
 Holes which diffuse to n side will be minority carriers.
 These holes will recombine with majority carriers in n side 
(electrons) in n region.
 There will be a net static positive charge on the n side and a net 
static negative charge on the p side.
 This will create an electric field.
 Electric field will drift electrons from p side to n side and holes 
from n side to p side.
 In equilibrium, these 2 currents will balance each other.
Depletion Region
 When Diffusion and Drift Mechanisms balance each other, a 
region depleted of free carriers containing charged ions is formed 
around the metallurgical junction.
 This region is called Space Charge Region or Depletion Region. 
 Region beyond Depletion Region are neutral 
 Neutral N and Neutral P Regions maintain characteristic of 
original n-type and p-type materials. 
PN Junction (Diode)
 Depletion Region has net negative charge 
on the p side and a net positive charge on 
the n side.
 Since both n type and p type materials 
were neutral initially, the complete pn
junction has to be neutral.
 Regions outside of the depletion region are 
neutral with fixed ions and free electrons 
and holes neutralizing each other.
 Therefore, charge on the n side of the 
depletion region has to be equal to the 
charge on the p side of the depletion 
region.
 Electric field is maximum at the 
metallurgical junction and zero at the ends 
of the depletion region. 
PN Junction (Diode)
 Consider a volume from edge of the depletion region on p side to a 
point x.
 Total Electric Flux leaving this volume is:
 By Gauss’ Law, this flux is proportional to total charge in the volume:
E jEgdS E A  
   

 0e A j p
E j
si si
q N A x xQE A
 

   
 
 0e D p
si
q NE x x

 

-
-
-
--
--
--
--
--
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Aj
E
-xp0 x
-
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-
-
-
PN Junction (Diode)
 Similar analysis for a volume from x to depletion region edge at n 
side will give the electric field strength at the n side.
 Depletion Region Charge on both sides is equal
 At metalurgical junction (x=0) electric 
field is continuous
 0e D n
si
q NE x x

 

0 0A j p D j nQ N A x Q N A x   
0 0
0 0
e A e D
p n
x xsi si
q N q NE x E x
   
   
 
 0e D j n
E j
si si
q N A x xQE A
 

   
 
Aj
E
xn0x
+
+
+
+
+
+
+
+
+
+
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+
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+
+
+
+
Current Flow Across Junction: Equilibrium
 At equilibrium, the drift current flowing in one direction cancels out 
the diffusion current flowing in the opposite direction, creating a net 
current of zero.
 The figure shows the charge profile of the PN junction.
ndiffndrift
pdiffpdrift
II
II
,,
,,


PN Junction (Diode)
 Energy band structures of p type material and n type material are 
different.
 Fermi level of a system in equilibrium will be constant.
 When the 2 materials are merged, Fermi level of pn junction will 
be constant.
 Therefore, energy bands will bend.
PN Junction (Diode)
 Energy level of conduction band on the p 
side is higher than energy 
level of conduction band on the n side. 
 There is a potential difference which 
favors motion of electrons from p side to 
n side.
 Therefore, a drift current will flow from 
p side to n side.
 Difference between conduction band 
minimum (or Valence Band Maximum) 
on n side and p side is the potential 
difference created by band bending.
cp cn vp vn e biE E E E q V   
Derivation of Builtin Potential using Einstein’s Relation
,drift p e p e p
dVI q pE q p
dx
    ,diff p e p
dpI q D
dx
 
2
1
p
n
px
p p
x p
dpdV D
p
  
    2 1 22 2( ) ( ) ln ln ln lnp p p p Ax x
ip p n p
D
D D p D NV x V x p p
np
N
  
    
2 1 2( ) ( ) ln
A D
bi
e i
N NkTV V x V x
q n
  
 If we equate these 2 currents and solve for the resulting differential 
equation, we can calculate the built in potential.
PN Junction (Diode)
 Built-in Potential is:
 Minority Carriers are electrons in p side and holes in n side.
 Minority Carrier concentrations are given by Mass Action Law just like 
any semiconductor.
 Minority Carrier concentrations may also be expressed in terms of Vbi.
 e- concentration at the neutral p region:
 h+ concentration at the neutral n region:
2 2ln ln
n p D A
bi
e i e i
n p N NkT kTV
q n q n
   
    
   
2 bi
e
V
kT qi
n A
D
np N e
N

 
2 bi
e
V
kT qi
p D
A
nn N e
N

 
PN Junction (Diode)
 Potential difference between potential of 
neutral p region and a point at p side of the 
depletion region:
 Potential at metallurgical junction (x=0) 
relative to potential of neutral p region
 Potential difference between metallurgical 
junction and a point at depletion region in n 
side:
   
0
2
0 02
p
x
e A e A
p p
si six
q N q N xx x d x x  
 
      
 
   
22
0
0 0
0 2 2
x
pe D e D e A
n n
si si si
xq N q N q Nxx x d x x  
  
         
 
  200 2
e A
p
si
q N x

 
0 0px x  
00 nx x 
PN Junction (Diode)
 Built-in Potential is the difference of potentials at xn0 and xp0.
 xn0 is given by
 xp0 is given by
 Depletion Region Width:
  2 2 2 20 0 0 0 01 12 2 2 2
e D e A e eA D
bi n n p A p D n
si si si D si A
q N q N q qN NV x x x N x N x
N N

   
            
   
 0
2 si bi A
n
e A D D
V Nx
q N N N
      
 0
2 si bi D
p
e A D A
V Nx
q N N N
      
 0 0
2 2si bi si biA D A D
D n p
e A D D A e A D
V VN N N NW x x
q N N N N q N N
               
Diode’s Three Operation Regions 
 In order to understand the operation of a diode, it is necessary to 
study its three operation regions: equilibrium, reverse bias, and 
forward bias.
Diode’s Three Operation Regions 
 When no external circuit is 
connected a pn junction, it is 
in equilibrium.
 When a voltage source is 
connected between terminals 
of a pn junction, equilibrium is 
disturbed.
 Fermi Level is no longer 
constant in the pn junction.
 Fermi Level in the p side and 
p side increase ordecrease 
relative to the other side 
depending on the applied 
external voltage.
Diode in Reverse Bias 
 When the N-type region of a diode is connected to a higher 
potential than the P-type region, the diode is under reverse bias
 Voltage source will inject electrons to p side and holes to n side.
 These extra carriers will 
recombine with majority carriers
 Number of ions in both sides 
will increase.
 Therefore, depletion region 
is even wider.
Diode in Reverse Bias 
 When the depletion region gets wider (more 
ions on each side), the potential difference 
across the depletion region is larger.
 When increase in depletion region potential 
is equal to the applied voltage, the battery is 
balanced by this extra voltage.
 Battery can not pull any charge.
 Therefore, the increase in internal voltage 
difference is equal to applied voltage.
 No current flows. 
Diode in Reverse Bias 
 Actually, electrons and holes are generated 
thermally in the depletion region. (Just like 
intrinsic semiconductor)
 When an electron hole pair is generated in 
the depletion region, internal electric field 
will drift electron to the n side and hole to 
the p side. 
 Therefore, there will be current flow in the 
reverse biased diode.
 This current is limited by the thermal 
generation rate of the electron-hole pairs.
 This current is a very small current.
 This current is called generation current.
Diode in Reverse Bias 
 VR is added to built-in potential
difference.
Hint: Vbi in all equilibrium equations are
replaced with Vbi+VR
 Electric field is stronger
 Depletion region is also wider.
 There is a very weak current flowing
across the diode.
 The current is mainly generation current.
 2 si bi R A D
D
e A D
V V N NW
q N N
     
 
Reverse Biased Diode’s Application: Voltage-
Dependent Capacitor
 The PN junction can be viewed as a capacitor. 
 Neutral regions are conductor plates and depletion region is a 
dielectric with no charge carriers.
Reverse Biased Diode’s Application: Voltage-
Dependent Capacitor
 Distance between conductor plates is depletion region width.
 By varying VR, the depletion width changes.
 Thus, capacitance value cahnges.
 PN junction is actually a voltage-dependent capacitor. 
Voltage-Dependent Capacitance
 The equation above is usually written in terms of equilibrium 
(VR=0) capacitance density Cj0'
0
0
1
1
2
j j
j
j R
bi
si e A D
j
A D bi
C C
C
A V
V
q N NC
N N V


  

 

   22
j sisi si e A D
j j j
D bi R A Dsi bi R A D
e A D
A q N NC A A
W V V N NV V N N
q N N
 

          
 
 
Voltage-Controlled Oscillator
 A very important application of a reverse-biased PN junction is 
VCO, in which an LC tank is used in an oscillator. By changing 
VR, we can change C, which also changes the oscillation 
frequency. 
LC
fres
1
2
1


Diode in Forward Bias
 When the N-type region of a diode is at a lower potential than the 
P-type region, the diode is in forward bias.
 The depletion width is shortened and the built-in electric field 
decreased.
Diode in Forward Bias 
 A voltage source will inject extra electrons to n side and extra holes 
to p side.
 These extra carriers will be attracted by the static ionic charge in 
the depletion (space charge) region. 
 When electrons flow to the depletion region of n side, these 
electrons will balance some of the positively charged ions in the n 
side of the depletion region. 
 Therefore, the net positive charge in the n side will be smaller.
 When holes flow to the depletion region of p side, these holes will 
balance some of the negatively charged ions in the p side of the 
depletion region. 
 Therefore, the net negative charge in the p side will be smaller.
 Depletion region will be narrower.
 Decrease in net static charge will decrease internal electric field 
and internal voltage across the depletion region.
Diode in Forward Bias 
 Decrease in the potential difference between the 2 sides will
balance the applied voltage field.
 When internal potential difference is equal to the applied voltage, 
the system will be balanced.
 However, the decrease in internal electric field will result in a 
smaller drift current.
 Since drift current is reduced, diffusion current is unbalanced.
 Net diffusion current starts flowing.
 Electrons diffusing to p region and holes diffusing to n region 
increase minority carrier concentrations. These minority carriers 
move in the neutral regions by diffusion.
 They recombine with majority carriers.
 The external power supply provide carriers to replace the majority 
carriers that recombined with minority carriers. 
Diode in Forward Bias 
 In steady state, the rate of carrier diffusion is equal to rate of carrier 
recombination. Thus, concentrations don’t change with time.
 However, system is not in equilibrium due to constant injection of extra 
carriers by the voltage source.
 Carrier concentrations change as a function of distance from 
metallurgical junction.
Product of minority and majority carriers is not 
equal to ni2.
    2p n ip x p x n
Diode in Forward Bias 
 Excess carrier concentration at a point x is defined as the difference 
between the non equilibrium carrier concentration and the equilibrium 
concentration:
 np0, nn0, pp0 and pn0 are majority and minority carrier concentrations in 
equilibrium (no bias applied).
    0n n np x p x p       0p p pn x n x n  
    0n n nn x n x n       0p p pp x p x p  
Diode in Forward Bias 
 In steady state, concentration of excess minority carriers and 
spatial distribution of the excess minority carriers are related by:
 Minority carrier lifetime is the average time an excess minority 
carrier can exist before recombining.
 Minority carrier lifetime remains constant for the extrinsic p-type 
semiconductor under low injection.
   
2
2 0
n
p n
p
p x
D p x
x


  
  
 2
2 0
p
n p
n
n x
D n x
x



  

Diode in Forward Bias 
 Solutions to these differential equations are 
 Concentration of excess minority carriers is maximum at the edge 
of the depletion region and decays exponentially as carriers diffuse 
deep into the neutral regions.
 Ln and Lp are diffusion lengths of electrons and holes. This is the 
average distance a minority 
carrier can travel before recombining.
 It is related to diffusion constant
and lifetime.
     p p p
x x
D L
n n n n np x p x e p x e
  
 
 
     n n
x xD Ln
p p p p pn x n x e n x e
  
 
   
Diode in Forward Bias 
 Recombination removes majority carriers as well as minority 
carriers. 
 If majority carriers are not replaced, depletion region will get wider 
and electric field will restore its value and diffusion will stop.
 This is what happens when the depletion region is first formed.
 The destroyed majority carriers are supplied by the external power 
supply. 
 That is the reason why diffusion process goes on.
Diode in Forward Bias 
 There are 2 types of currents flowing in the forward biased PN 
Junction; diffusion current and recombination current. 
 Recombination current is the current due to majority carriers 
recombining with minority carriers. 
 On the p side, electrons flow from depletion regionn into the depths 
of p region (from depletion region towards the battery). As they 
move they recombine with majority carrier holes. 
 Battery supplies holes to replace the majority carriers.Therefore, 
holes are flowing into the p region in the opposite direction (from 
battery towards the depletion region).
 Since the 2 carriers with opposite charges are moving in opposite 
directions , the 2 currents add up.
 A similar process is happening in the n side as well.
ELECTRON AND HOLE COMPONENTS OF 
CURRENT IN A FORWARD-BIASED P-N JUNCTION
 Injected minority hole current is higher on the n side than electron 
current on the p side because n doping is lower than p doping.
FORWARD BIAS
 In equilibrium, ratio of concentrations of electrons and holes on each
side of the junction are given by:
 Potential barrier is lowered by qeVf under forward bias
 Effective barrier voltage is
,bi eff bi fV V V 
e biq V
n D kT
p p
n N e
n n
 
e biq V
p A kT
n n
p N e
p p
 
FORWARD BIAS
 You may substitute the effective internal potential difference into the
equtions in the previous slide to calculate the non equilibrium minority
carrier concentrations.
If you want to know the actual reasoning behind this assumption, details are provided in 
the next slide which is optional
 Minority carrier concentrations at depletion region edges are:
  0
bieff f fbi
e e e e
V V VV
kT q kT q kT q kT q
p p D D pn x N e N e e n e
 
   
  0
f fbi
e e e
V VV
kT q kT q kT q
n n A np x N e e p e

 
Minority Carrier Concentration (Optional)
 pn junction is not in 
equilibrium when bias is 
applied. 
 Therefore, Fermi Level is 
not constant.
 2 Quasi Fermi Levels formed.
 The 2 quasi Fermi levels are seperated by qeVf in the depletion region. 
 EFp determines hole concentrations on both sides, EFn determines
electron concentration on both sides. 
 EFp approaches EFn in the n region since concentration of excess holes
approaches 0.
 EFn approaches EFp in the p region since concentration of excess
electrons approaches 0.
 Regions where the 2 quasi Fermi levels are equal are under
equilibrium.
Fn Fp e fE E q V   cp cn vp vn e bi fE E E E q V V    
Minority Carrier Concentration (Optional)
 At the edges of the depletion region, concentration of minority carriers
can be calculated using this information:
 Therefore, replacing Vbi in equilibrium equations with effective internal
voltage gives the correct result.
 
 
0
vn Fn e fvn Fp e f e fvn Fn
E E q VE E q V q VE E
kT kT kT kT kT
n n v v v np x N e N e N e e p e
  
   
 
 
0
Fp e f cpFn cp Fp cp e f e fE q V EE E E E q V q V
kT kT kT kT kT
p p C C C pn x N e N e N e e n e
   
    
 Excess minority carrier concentrations at edge of depletion region are:
 Excess carriers recombine as they
move away from the depletion edge
 The spatial distribution of excess
carriers is an exponential function
Calculating Junction Current From Excess Minority 
Carrier Distributions
    0 0 1
e fq V
kT
n n n n n np x p x p p e
 
      
    0 0 1
e fq V
kT
p p p p p pn x n x n n e
 
        
  0 1
ne f
p
x xq V
LkT
n np x p e e
 
    
  0 1
pe f
n
x xq V
LkT
p pn x n e e
 
    
 Diffusion current is proportional to spatial derivative of excess minority 
carriers.
 h+ diffusion current at the edge of the depletion region on the n side:
 e- diffusion current at the edge of the depletion region on the p side:
Calculating Junction Current From Excess Minority 
Carrier Distributions
     0 1
e fq V
e p n kT
p n e p n n
p
q D p
J x q D p x e
x L

        
     0 1
e fq V
e n p kT
n p e n p p
n
q D n
J x q D n x e
x L

         
 Diffusion is maximum at the depletion region edge and decreases as 
we move away from depletion region.
 On the other hand, recombination current is 0 at the depletion region
edge (no minority carriers recombined yet) and increases as we move
away from the depletion region.
 Sum of diffusion and recombination currents are constant throughout
the diode.
 Therefore, diffusion current at the edge of depletion region is equal to
the total current.
Calculating Junction Current From Excess Minority 
Carrier Distributions
 Electron and hole diffusion currents are continuous and constant in 
depletion region.
 Total diffusion current is
Calculating Junction Current From Excess Minority 
Carrier Distributions
0 0 1 1
e f e fq V q V
e p n e n p kT kT
s
p n
q D p q D n
J e J e
L L
    
                
2 2
1 1
e f e fq V q V
e p i e n i kT kT
s
D p A n
q D n q D nJ e J e
N L N L
    
                
   p n p pJ x J x     n n n pJ x J x 
           p n n n p p p p p n n pJ J x J x J x J x J x J x        
ALTERNATIVE WAY OF CALCULATING JUNCTION 
CURRENT (OPTIONAL)
 Average time for charge that diffuses to the other side of the 
depletion region to recombine is minority carrier life time.
 If you divide total excess minority carrier charge by minority carrier 
lifetime, this will give the recombination current.
  00 1 1
ne f e f
p
n
x xq V q V
Lpn e p ne kT kT
p n n
j p p px
Q q L pqJ x p e e dx e
A   
    
             

  00 1 1
pp e f e f
n
x xx q V q V
np e n pLe kT kT
n p p
j n n n
Q q L nqJ x n e e dx e
A   


   
              

ALTERNATIVE WAY OF CALCULATING JUNCTION 
CURRENT (OPTIONAL)
 Recombination currents far away from depletion region are equal 
to diffusion currents at the depletion region edges.
 Sum of recombination currents is the total current.
0 0 1 1
e f e fq V q V
e p n e n p kT kT
s
p n
q L p q L n
J e J e
 
    
                
2
n n nD L 
2
p p pD L 
0 0 1 1
e f e fq V q V
e p n e n p kT kT
s
p n
q D p q D n
J e J e
L L
    
                
Temperature Dependence of Diode Current
 ni2 has a very strong dependence on temperature. 
 Even though Ln and Lp have temperature dependence as well, 
we can ignore their temperature dependence.
 Temperature dependence of Eg is negligble.
)(2
pD
p
nA
n
is LN
D
LN
DAqnI 
3/2 exp
2
g
i
E
n T
kT


2 ( )pns i
e A n D p
kTI Aqn
q N L N L
 
  3/2n T T 
4
24.73 10( ) 1.166
636g
E T T
T
 

  3/2p T T 
2.5
gE
kT
sI CT e

 2.5 e f g
q V E
kT
DI CT e


Temperature Dependence of Diode Voltage
 If ID is constant
1.5 2.5
2
12.5 1
e f g e f gq V E q V E
ekT kT
D f
qI CT e CT V e
T kT T kT
                     
2.5f g
f
e e
V E kV
T T q T q
   

2.5
2
2.5 1
e f gq V E
e f ge kT
D f
q V EqI V CT e
T T kT T kT
              
2
1 2.5 e f ge
D f
D
q V EqI V
I T T kT T kT
       
 Typically, Eg =1.12V, kT/qe=26mV and Vf is around 0.8V for Si
0.8 1.12 0.06465 1.28
300f
mVV KT
    

IV Characteristic of PN Junction
 We deriived diode eqn for fwd bias, but it is valid for reverse 
bias as well. 
 Current is relatively constant in reverse bias region.
)1(exp 
T
D
SD V
VII
Reverse Breakdown
 When a large reverse bias voltage is applied, breakdown 
occurs and an enormous current flows through the diode.
Zener Breakdown
 Valence band of p side and conduction band of n side are 
very close to Fermi level at equilibrium dueto heavy doping
 When heavily doped junction is reverse biased, valance 
band of p side is at a higher potential than conduction band 
of n side
Zener Breakdown
 The depletion region is very narrow due to heavy doping 
levels
 Electrons can tunnel through the narrow energy barrier 
(potential barrier) 
 Large currents will flow through reverse biased junction.
Impact Ionization
 An (primary) electron gains kinetic energy in the electric 
field of the depletion region
 Electron hits the crystal and gives its kinetic energy to 
create an (secondary) electron–hole pair by impact 
ionization
 The primary electron losing most of its kinetic energy in 
the process
 A single ionizing collision 
by an incoming electron 
in the depletion region of 
the junction creates a 
chain if ionizations.
Zener vs. Avalanche Breakdown
 Zener breakdown is a result of the large electric field inside the 
depletion region that breaks electrons or holes off their covalent 
bonds.
 Avalanche breakdown is a result of electrons or holes colliding 
with the fixed ions inside the depletion region. 
PhotoDiodes
 If a semiconductor is illuminated with light, electrons in the 
semiconductor will absorb photons.
 Only those photons with energy larger than bandgap energy will 
be absorbed and covalent bonds will be broken (electron hole 
pairs will be created).

 Silicon bandgap corresponds to 1.1µm (infrared).
 Therefore, visible light can be detected by silicon.
 This is similar to the creation of electron hole pairs with heat.
 However, these electron hole pairs are excess carriers. 
 Excess carriers mean: 
2
innp 
g
ph
ph E
chE 

PhotoDiodes
PhotoDiodes
 Normally, these excess carriers will recombine.
 However, if there is an electric field, created electrons and hole will 
be seperated.
 Built in electric field in a photodiode will drift electrons from p side to 
n side and holes from n side to p side.
 Electric field exists only in depletion region, 
 Excess electron-hole pairs created in depletion region will be drifted 
by electric field.
 Also those electrons and hole that can diffuse to the depletion region 
before recombining will be drifted by electric field.
 Electrons and holes created in regions close to depletion region 
(within 1 diffusion length) will be be able to diffuse to depletion 
region.
 Other electron-hole pairs will recombine.
PhotoDiodes
Excess electron-hole pairs 
created in depletion region 
will be drifted by electric 
field.
Electrons and holes created in 
regions close to depletion 
region (within 1 diffusion 
length) will be be able to 
diffuse to depletion region.
PhotoDiodes
 Therefore, a photo generated current will flow from n region to p region in 
a photodiode.
 Diode equation under illumination is given by:
 Photodiodes are generally reverse biased and current will flow from n 
side to p side.
 G is generation rate of excess carriers, it is proportional to light intensity
 Aj is junction area
 WD is depletion region length
 Lp and Ln are diffusion lengths of holes and electrons in n and p regions 
respectively (minority carrier diffusion lengths) 
1
DqV
kT
D s phI I e I
 
   
 
 ph j p D nI GA L W L  
Light Emitting Diode
 When electrons and holes recombine, electrons emit their 
energy.
 Forward current of a diode is a recombination current.
 Therefore, electrons and holes recombine all the time when 
current flows in a diode.
 Emitted energy might be in the form of a photon (light) and/or a 
phonon (vibration/heat).
 If bandgap energy of a semiconductor is larger than energy of 
photons in visible spectrum, visible photons might be emitted.
 Silicon bandgap energy is less than energy of red photons. 
(corresponds to infrared spectrum) 
 Moreover, portion of the energy is emitted as phonons. 
 Therefore, LEDs are not built with silicon.
Metal – Semiconductor Junction
 The whole semiconductor pn junction structure is a single crystal 
with different doping profiles on two sides.
 It is sufficient to raise electrons to conduction band for electrons 
to flow from one side to the other.
 When a metal and a semiconductor are fused to form a junction, 
two sides of the junction are completely different materials.
 Unlike a pn junction formed between identical materials, a metal 
semiconductor junction will not act as a continuous crystal.
 Electrons or holes that reach the surface of the semiconductor 
crystal have to break free from the crystal completely before they 
can flow to the metal and vice versa. 
 Since the two sides of the junction are different materials, their 
energy band levels must be expressed relative to a common 
reference energy level. 
Metal – Semiconductor Junction
 Assume the semiconductor and metal are placed in vacuum.
 If a certain energy is provided to an electron in the conduction 
band of the metal or the semiconductor, the electron might break 
free of the material and it will be released to the vacuum.
 If a free electron in the vacuum is captured by the metal or the 
semiconductor this energy will be released by the electron.
 Common sense suggests only electrons at the surface of the 
material can break free of the material.
 Energy level an electron at the surface 
of a semiconductor or metal has to reach
to leave the semiconductor or the metal 
is called the vacuum level.
Metal – Semiconductor Junction
 Metal Work Function (φm) is the electric potential difference 
between vacuum level and metal Fermi Level.
 Fermi Level of a metal is the same as conduction and valence 
bands.
 Electron Affinity (χ) is the the electric potentail difference 
between vacuum level and conduction band of a semiconductor.
 We also define Semiconductor Work Function (φs) as the 
electric potential difference between 
vacuum level and Fermi level of the 
semiconductor.
Metal – Semiconductor Junction
 If an electron in the semiconductor surface is to be captured by the 
metal:
 First of all, an electron to leave the semiconductor must be a freely 
moving electron at the surface (not an electron stuck in a covalent 
bond). In other words, it must be in the conduction band and 
physically it must be at the surface of the material.
 Electron will be excited from the conduction band to vacuum level. 
 Therefore, its potential will increase by . 
Metal – Semiconductor Junction
 When the electron in vacuum is captured by the metal surface, it
falls from vacuum level to the Fermi Level of the metal.
 Therefore, its potential will decrease by . 
 Difference between final and initial electrical potentials of the 
electron is 
 Electron in the metal first jumps to vacuum level to break free and it 
falls to semiconductor conduction level when it is captured.
 Change in its electric potential will be
 Typically, position of energy bands in the metal 
and semiconductor will be different.
 Electron will end up at a lower or higher 
energy level in the end.
m
   vac m vac mV V       
   vac vac m mV V       
Metal – Semiconductor Junction
 Since all interactions between the 2 materials use vacuum level as 
an intermediate step, all energy levels in metal and semiconductor 
are expressed relative to the vacuum level.
 The potential difference between vacuum level and the energy 
bands in the material is a property of the material.
 Putting a material next to another material will not change its 
material properties.
 When a metal and a semiconductor are fused,
their surfaces will become a junction.
 At the surface of the semiconductor and 
the metal, the positionof conduction bands
relative to vacuum level will be constant.
Metal – N-Type Semiconductor Junction
 If a junction is formed between a semiconductor and a metal, its 
behavior will depend on the relative band locations with respect 
to vacuum level and each other.
 First consider the case where the Metal work function (φm) is 
larger than electron affinity of semiconductor. 
“Metal – N” Schottky Diode
 When φm> φs, Fermi Level of the semiconductor is above Fermi 
Level of metal.
 When metal and semiconductor are fused, Fermi level will be 
constant in the whole system in equilibrium.
 Since the conduction and valence bands are pinned at the 
junction surface, semiconductor bands will bend. 
 Since difference between conduction band and Fermi Level 
increases, the electron concentration in conduction band has to 
be reduced in vicinity of the junction.
 These electrons diffuse to the metal.
 As free electrons in the conduction band are gone, static 
positively charged ions are left behind in the vicinity of the 
junction.
 Therefore, a depletion region is formed.
“Metal – N” Schottky Diode
 Metal Fermi Level is at a lower potential than semiconductor 
conduction band at the surface.
 Thus, there will be potential barrier between the semiconductor 
and metal. It will stop electron movement from metal to 
semiconductor.
 This potential barrier is called Schottky Barrier and it depends 
only on the type of materials. It is independent of doping 
concentration. 
 Larger work function difference will increase the Schottky barrier 
further.
0B m s   
“Metal – N” Schottky Diode
 Conduction band at the junction surface is at a higher level than 
the conduction band at the neutral regions due to band bending.
 Thus, there is a built-in potential barrier in the semiconductor 
side as well.
 Therefore, electrons cannot move from semiconductor to the 
metal either.
 An electron in the conduction band of the semiconductor will be 
stopped by this potential barrier.
 This barrier is the amount of bending in conduction band.
 Difference between the Schottky Barrier and Vbi is the difference 
between Fermi level and conduction band in the semiconductor.
0bi m s B nV       
ln ln Dn C F
C C
NnE E kT kT
N N
      
Schottky Effect**
 Ideally, Schottky barrier is constant.
 In reality, the electric field directed into a metal repels electron 
from the metal surface and the metal work function is reduced. 
 Electric field in depletion region in a metal semiconductor 
junction repels electrons from metal surface and lowers effective 
metal work function, so the Schottky Barrier is lowered.
 As the electric field increases, the Schottky Barrier will be 
lowered further.
 Barrier lowering is a weak 
function of the electric field.
 Effective Schottky Barrier is
lower than the Ideal Schottky 
barrier.
0Bn B m s     
Schottky Effect**
 Schottky barrier is the force that stops diffusion of electrons in 
the metal from diffusing to the semiconductor.
 Schottky Barrier height is a weak function of Electric field. 
 However, number of electrons crossing from metal to 
semiconductor is an exponential function of barrier height.
 Therefore, changes in electric field will yield large changes 
electron flow from metal to semiconductor (current flowing 
semiconductor to metal).
 This is especially important
under reverse bias.
“Metal – N” Schottky Diode Under Bias
 Equilibrium is disturbed when bias voltage is applied to the 
junction
Reverse Bias
 When the metal is connected to a more negative voltage than the 
semiconductor, Fermi level of the semiconductor will be reduced 
compared to Fermi level of the metal. 
 Therefore, the built-in potential on the semiconductor side will 
increase. 
 Less electrons can diffuse to semiconductor surface. Thus, less 
electrons can move to the metal.
 Ideally, Shottky barrier is unchanged since it depends on the 
difference between work function and electron affinity. 
 In reality, increasing reverse bias increases the electric field 
which reduces the Schottky Barrier.
 As reverse bias increases, number of electrons that flow from 
metal to semiconductor increases exponentially.
 Therefore, Reverse Leakage of a Schottky Diode increases 
exponentially with reverse bias.
Forward Bias
 When the metal is connected to a more positive voltage than the 
semiconductor, Fermi level of the semiconductor will increase 
compared to Fermi level of the metal. 
 Therefore, the built-in potential on the semiconductor side will 
decrease. 
 Electrons in conduction band of the semiconductor will face a 
lowered potential barrier. More electrons can diffuse to the 
surface of the semiconductor. Therefore, more electrons can flow 
into the metal.
 Schottky Barrier prevents flow of metal electrons to 
semiconductor.
 Shottky Barrier depends on the conduction band levels at the 
surface only. Ideally, it is constant. 
 In reality, Schottky Barrier increases compared to equilibrium 
level since the electric filed in the depletion region decreases 
with forward bias. 
 Therefore, electrons will move from semiconductor to the metal 
only.
Forward Bias
Metal – P-Type Semiconductor Junction
 Consider a junction is formed between a metal and a P-Type 
semiconductor when .
 Bands in semiconductor will bend up when equilibrium is reach 
since Fermi Level has to be constant.
 Concentration of holes in the valance band at the surface will 
have to decrease. Therefore, electrons will flow from metal to 
semiconductor to make the Fermi levels equal.
m s 
Metal – P-Type Semiconductor Junction
 Holes in the metal and semiconductor have to go down to the 
valence band level at the surface of the semiconductor to move 
to the other side.
 Holes in the metal face a Schottky Potential Barrier:
 Holes in the semiconductor face a built-in potential barrier equal 
to the difference between valence band level in neutral region 
and valence band level at the surface:
 It is equal to
 Ev is valence level in neutral region.
 0 g gB vac m vac m
e e
E EV V q q    
          
0bi s m B nV       
n F VE E  
ln lnV Vn
A
N NkT kT
p N
  
Metal – P-Type Semiconductor Junction
 Electrons in the metal have to jump to the conduction band level 
at the surface of the semiconductor to move to the other side.
 They face a potential barrier: 
 Notice that barrier for holes in the semiconductor and electrons 
in the metal increase and decrease together.
 Schottky Barrier for metal
holes is ideally constant.
 It changes with applied 
voltage due to Schottky
Effect.
   0Bn vac m vac n m n m F VV V E E                   
Reverse Bias
 When the metal is connected to a more positive voltage than the 
semiconductor, Fermi level of the semiconductor will be 
increased compared to Fermi level of the metal. 
 The built-in potential on the semiconductor side and the barrier 
stopping the metal electrons will increase. 
 Less holes can move to the surface of the semiconductor and 
from surface to the metal and less electrons can move from 
metal to semiconductor.
 Increasing reverse bias increases the electric field in the 
depletion region which reduces the Schottky Barrier.
 As reverse bias increases, number of holes that flow from metal 
to semiconductor increases exponentially.
 Therefore, Reverse Leakage of the Schottky Diode increases 
exponentially with reverse bias.
Forward Bias
 When the metal is connected to a more negative voltage than the 
semiconductor, Fermi levelof the semiconductor will increase 
compared to Fermi level of the metal. 
 Built-in potential for holes in the semiconductor decreases. More 
holes will reach the semiconductor surface and flow to the metal.
 Potential barrier for metal electrons will also decrease and more 
electrons will flow from metal to the semiconductor. 
 Schottky barrier preventing flow of metal holes into the 
semiconductor increases slightly due to the lower electric field.
 Therefore, current will flow from semiconductor to the metal only.
Schottky Diode Current
 Current in Schottky Diode is due to motion of majority carriers.
 It is given by:
 A* is called the effective Richardson constant
 Current equation is similar to the pn junction current equation.
 However, reverse saturation current is an exponential function of 
Schottky Barrier Voltage.
 Small changes in Schottky Barrier Voltage due to the depletion 
region electric field will cause exponential effects on the reverse 
saturation 
* 2 1 1
e Bn e a e aq q V q V
kT kT kT
SD STJ A T e e J e
     
      
   
Schottky Diode vs. PN Junction
 Magnitudes of the reverse-saturation current densities and the 
switching characteristics are different.
 The current in a pn junction is due to diffusion of minority carriers 
while the current in a Schottky barrier diode is due to thermionic 
emission of majority carriers over a potential barrier.
 Ideal reverse-saturation current density of the Schottky Barrier 
junction is orders of magnitude larger than that of the ideal pn
junction diode.
 Since , forward-bias current of the Schottky is much 
larger than pn junction for the same applied voltage.
 Thus, effective turn-on voltage of the Schottky diode is much less 
than that of the pn junction diode.
ST SpnJ J
Ohmic Contact
 We need metal semiconductor contacts that act as resistors 
(ideally short circuits) to connect semiconductor devices to other 
circuits.
 There are 2 ways to make an Ohmic contacts:
 Non rectifying junction
 Barrier Tunneling
Non Rectifying “Metal – N” Contact
 When , Fermi Level of the semiconductor is below Fermi 
Level of metal. 
 When metal and semiconductor are fused, Fermi level will be 
constant in the whole system in equilibrium.
 Bands in the semiconductor will bend up.
 If as well, 
 Conduction band of the semiconductor is below conduction band of 
metal. There is NO Schottky Barrier.
 Conduction band of semiconductor is below fermi level at the surface. 
 Conduction band will be filled with electrons at the surface.
m 
m s 
Non Rectifying “Metal – N” Contact
 Conduction band is higher than Fermi level in neutral semiconductor. 
 Difference between conduction band of semiconductor and 
conduction band of metal is 
 When semiconductor is connected to a higher potential, Fermi Level 
in semiconductor will be increased relative to Fermi Level in the 
metal. The potential barrier is slightly higher.
 When metal is connected to a higher potential, Fermi Level in metal 
will be increased relative to Fermi Level in the semiconductor. The 
potential barrier is slightly lower.
ln Ce n Cs F
D
Nq E E kT N
      
Non Rectifying “Metal – N” Contact
 Potential difference between metal and semiconductor will be 
very low for a low resistance Ohmic Contact.
 When metal is at a higher potential than metal, electric field is 
applied from metal to semiconductor. 
 Electrons flowing from semiconductor to metal go down this 
potential barrier, so they move freely.
Non Rectifying “Metal – N” Contact
 When metal is at a lower potential than metal, electric field is 
applied from semiconductor to metal. 
 Electrons flowing from metal to semiconductor will face the 
potential barrier which is slightly lower than unbiased value.
 Electrons can overcome this potential barrier if the potential 
barrier is low.
 If semiconductor is heavily doped, is very low.
 If semiconductor is lightly doped, is higher.
 Metal contacts are connected to heavily doped semiconductors. 
e nq 
e nq 
Non Rectifying “Metal – P” Contact
 When , Fermi Level of the semiconductor is above Fermi 
Level of metal. 
 When metal and semiconductor are fused, Fermi level will be 
constant in the whole system in equilibrium.
 Bands in the semiconductor will bend down.
 If as well, 
 Valence band of the semiconductor is above valence band of metal. 
There is NO Schottky Barrier.
 Valence band of semiconductor is above fermi level at the surface. 
 Valence band will be filled with hole at the surface.
m g eE q  
m s 
Non Rectifying “Metal – P” Contact
 Valence band is lower than Fermi level in neutral semiconductor. 
 Difference between valence band of semiconductor and Fermi Level 
of metal is 
 When semiconductor is connected to a higher potential, Fermi Level 
in semiconductor will be increased relative to Fermi Level in the 
metal. The potential barrier is slightly lower.
 When metal is connected to a higher potential, Fermi Level in metal 
will be increased relative to Fermi Level in the semiconductor. The 
potential barrier is slightly higher.
ln Ve n V F
A
Nq E E kT N
      
Non Rectifying “Metal – P” Contact
 When metal is at a lower potential than semiconductor, electric 
field is applied from semiconductor to metal. 
 Holes flowing from semiconductor to metal go up this potential 
barrier, so they move freely.
Non Rectifying “Metal – N” Contact
 When metal is at a higher potential than metal, electric field is 
applied from metal to semiconductor. 
 Holes flowing from metal to semiconductor will face the potential 
barrier which is slightly lower than unbiased value.
 Electrons can overcome this potential barrier if the potential 
barrier is low.
 If semiconductor is heavily doped, is very low.
 If semiconductor is lightly doped, is higher.
 Metal contacts are connected to heavily doped semiconductors. 
e nq 
e nq 
Tunneling Type Contact
 The space charge width in a rectifying metal–semiconductor 
contact is inversely proportional to the square root of the 
semiconductor doping. 
 Thus, as the doping concentration increases, the probability of 
tunneling through the barrier increases
 For these types of barrier widths, tunneling may become the 
dominant current mechanism.
Ohmic Contact in P-Type Si
 Electron Affinity for Si is 4.01 eV.
 Typical metals used as interconnects in chip manufacturing 
processes have work functions larger than this value.
 Work function for Al is 4.28 eV and W is 4.55 eV.
 These metals and P-Type Si will have a negative Schottky 
Barrier for holes.
 We can build non-rectifying junctions.
 We usually use dope Si heavily to reduce the potential barrier.
 Current can flow both ways freely. 
 We obtain an Ohmic Contact.
Ohmic Contact in N-Type Si
 Typical metals used as interconnects in chip manufacturing 
processes and N-Type Si will have a positive Schottky Barrier 
for electrons.
 We cannot build non-rectifying junctions.
 We usually use degenerate silicon (extremely heavily deoped 
Si, ND>NC).
 Depletion region will be very narrow and carriers will tunnel 
through the Schottky Barrier.