Handout 3 - Cornell University

Lecture 3
Electron and Hole Transport in Semiconductors
In this lecture you will learn:
• How electrons and holes move in semiconductors
• Thermal motion of electrons and holes
• Electric current via drift
• Electric current via diffusion
• Semiconductor resistors
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Thermal Motion of Electrons and Holes
In thermal equilibrium carriers (i.e. electrons or holes) are not standing still but
are moving around in the crystal lattice and undergoing collisions with:
• vibrating Silicon atoms
• with other electrons and holes
• with dopant atoms (donors or acceptors) and other impurity atoms
Mean time between collisions =  c
In between two successive collisions electrons (or
holes) move with an average velocity which is
called the thermal velocity = Vth
In pure Silicon,
 c  0.1 10 12 s  0.1 ps
v th  107 cm s
Brownian Motion
Mean distance traveled between collisions is
called the mean free path    v th  c
7
In pure Silicon,   10  0.1 10
12
 10- 6 cm  0.01m
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
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Drift: Motion of Electrons Under an Applied Electric Field
Silicon slab
E
L
E
+ V
-
V
L
• Force on an electron because of the electric field = Fn = -qE
• The electron moves in the direction opposite to the applied field with a
constant drift velocity equal to vdn
• The electron drift velocity vdn is proportional to the electric field strength
v dn   E
v dn    n E

cm2
V-s
• The constant n is called the electron mobility. It has units:
• In pure Silicon, n  1500
cm2
V-s
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Drift: Motion of Holes Under an Applied Electric Field
Silicon slab
E
L
E
+ V
-
V
L
• Force on a hole because of the electric field = Fp = qE
• The hole moves in the direction of the applied field with a constant drift
velocity equal to vdp
• The hole drift velocity vdp is proportional to the electric field strength
v dp  E
v dp   p E

2
• The constant p is called the hole mobility. It has units: cm
• In pure Silicon,  p  500
cm2
V-s
V-s
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
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Derivation of Expressions for Mobility
Electrons:
Force on an electron because of the electric field  Fn  qE
Acceleration of the electron  a 
Fn
qE

mn
mn
Since the mean time between collisions is  c , the acceleration lasts only for a
time period of  c before a collision completely destroys electron’s velocity
So in time  c electron’s velocity reaches a value  a  c  
This is the average drift velocity of the electron, i.e.
Comparing with v dn   n E we get,
n 
q c
mn
q c
E
mn
v dn  
q c
E
mn
Holes:
q c
Similarly for holes one gets,  p 
mp
Special note: Masses of electrons and holes (mn and mp) in Solids are not the
same as the mass of electrons in free space which equals 9.1 10  31 kg
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Mobility Vs Doping
Mobility (cm2/V-s)
More doping means more frequent collisions with Donor and acceptor atoms and
this lowers the carrier mobility
Dopant concentration (1/cm3)
Note: Doping in the above figure can either be n-type or p-type
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
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Drift Current Density of Electrons and Holes
Flux Density:
Flux density is the number of particles crossing a unit area surface per second
It has units cm-2-s-1
Unit area surface
Density: n
Velocity: vdn
Flux density: nvdn
Area
Time
Volume = 1 x (vdn x 1)
Electrons Drift Current Density:
Electron flux density from drift  n v dn
Electron drift current density J ndrift is,
Check directions
E
v dn
J ndrift
J ndrift  q  electron flux density
 q n v dn  q n  n E
Jndrift has units: Coulombs  Amps
cm2 - s
cm2
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Drift Current Density of Electrons and Holes
Holes Drift Current Density:
drift ,
The hole drift current density is J p
Check directions
J pdrift   q  hole flux 
E
v dp
J pdrift
  q p v dp  q p  p E
J pdrift has units: Coulombs  Amps
cm2 - s
cm2
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
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Conductivity and Resistivity
Total Drift Current Density:
drift
drift
The total drift current density J drift is the sum of J n
and J p
J drift  J ndrift  J pdrift


 q n n  p  p E
 E
The quantity  is the conductivity of the semiconductor:
  q n n  p  p 
Conductivity describes how much current flows when an electric field
is applied. Another related quantity is the resistivity  which is the
inverse of the conductivity,

1

Units of conductivity are: Ohm-1-cm-1 or -1-cm-1 or S-cm-1
Units of resistivity are: Ohm-cm or -cm or S-1-cm
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Example: A Semiconductor Resistor
Silicon slab
For a resistor we know that,
I
V
R
1
L
We also know that,
Area  A
I  J drift A
 E A
V
 A

A
V
L
L
1  2   R 
+
V
-
I
2
L
L

 A
A
where  
1


 q n n  p  p

Lessons:
• Knowing electron and hole densities and mobilities, one can calculate the
electrical resistance of semiconductors
• n-doping or p-doping can be used to change the conductivity of
semiconductors by orders of magnitudes
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
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Diffusion
Diffusion of ink in a glass beaker
Why does diffusion happen?
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Diffusion and Diffusivity
There is another mechanism by which current flows in semiconductors …….
• Suppose the electron density inside a semiconductor is not uniform in space,
as shown below
n(x)
electron flux in +x direction
electron flux in -x direction
slope 
d n x 
dx
x
• Since the electrons move about randomly in all directions (Brownian motion),
as time goes on more electrons will move from regions of higher electron
density to regions of lower electron density than the electrons that move from
regions of lower electron density to regions of higher electron density
d n x 
dx
d n x 
  Dn
dx
• Net electron flux density in +x direction  
• The constant Dn is called the diffusivity of electrons (units: cm2-s-1 )
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
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Diffusion Current Density
Electrons Diffusion Current Density:
d n x 
Electron flux density from diffusion   Dn
dx
Check directions
n x 
Electron diffusion current density J ndiff is,
x
J ndiff  q  electron flux 
 q Dn
d n x 
dx
elec. flux
J ndiff
Holes Diffusion Current Density:
d p x 
Hole flux density from diffusion   Dp
dx
Check directions
p x 
diff
Hole diffusion current density J p
is,
diff
J p   q  hole flux

 q Dp

d p x 
dx
x
hole flux
J pdiff
J ndiff and J pdiff has units Coulombs  Amps
cm2 - s
cm2
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Einstein Relations
Einstein worked on other things besides the theory of relativity……..
• We introduced two material constants related to carrier transport:
1) mobility
2) Diffusivity
•Both are connected with the transport of carriers (electrons or holes)
•It turns out that their values are related by the Einstein relationships
Einstein Relation for Electrons:
Dn
n

KT
q
Example:
2
In pure Silicon, n  1500 cm
 p  500 cm V - s
Einstein Relation for Holes:
Dp
p

KT
q
V-s
2
This implies,
Dn  37.5 cm2 s
Dp  12.5 cm2 s
• K is the Boltzman constant and its value is: 1.38x10-23 Joules  K
•
K T has a value equal to 0.0258 Volts at room temperature (at 300 oK)
q
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
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Total Electron and Hole Current Densities
Total electron and hole current densities is the sum of drift and diffusive
components
Electrons:
J n  x   J ndrift  x   J ndiff  x 
 q n x  n E  x   q Dn
d n x 
dx
Holes:
J p  x   J pdrift  x   J pdiff  x 
 q p  x   p E  x   q Dp
d p x 
dx
Electric currents are driven by electric fields and also by carrier density
gradients
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Thermal Equilibrium - I
There cannot be any net electron current or net hole current in thermal
equilibrium ……… what does this imply ??
Consider electrons first:
J n  x   Jndrift  x   J ndiff  x   0
 q no  x  n E  x   q Dn
1
d no  x 
0
dx
can also be written as:
1
d logno  x 
q

E x 
dx
KT
Since the electric field is minus the gradient of the potential: E  x   
We have: d logno  x   q d  x 
dx
KT
dx
d  x 
dx
q  x 
The solution of the above differential equation is: no  x   constant  e KT
But what is that “constant” in the above equation ???
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
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Thermal Equilibrium - II
q  x 
We have: no  x   constant  e KT
Note: one can only measure potential differences and not the absolute values
of potentials
Convention: The potential of pure intrinsic Silicon is used as the reference
value and assumed to be equal to zero.
q x 
So for intrinsic Silicon, no  x   constant  e KT
 constant
But we already know that in intrinsic Silicon, no  x   ni
So it must be that, constant  ni
q  x 
And we get the final answer, no  x   ni e KT
Consider Holes Now:
One can repeat the above analysis for holes and obtain: po  x   ni e

q x 
KT
Check: no  x  po  x   ni2
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
Potential of Doped Semiconductors
What are the values of potentials in N-doped and P-doped semiconductors ??
N-doped Semiconductors (doping density is Nd ):
The potential in n-doped semiconductors is denoted by: n
no  x   Nd
 Nd  ni e
 n 
q n  x 
KT
N 
KT
log d 
q
 ni 
Example:
Suppose,
Nd  1017 cm- 3 and ni  1010 cm-3
 n 
N 
KT
log d    0.4 Volts
q
 ni 
P-doped Semiconductors (doping density is Na ):
The potential in n-doped semiconductors is denoted by: p
po  x   Na
 Na  ni e
 p  

q p  x 
KT
N 
KT
log a 
q
 ni 
Example:
Suppose,
Na  1017 cm- 3 and ni  1010 cm-3
 p  
N 
KT
log a    0.4 Volts
q
 ni 
ECE 315 – Spring 2005 – Farhan Rana – Cornell University
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ECE 315 – Spring 2005 – Farhan Rana – Cornell University
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