Mobility 2
The average momentum is proportional to the applied force, which is qE. The
electrons, on an average, collide in time n (called momentum relaxation time), so
the momentum they achieve before reaching steady state is given as qnE
The average drift velocity of electrons is then given as
q n
  *   * E  n E
mn
mn
p
Note: Velocity in different directions
can be different even though the
field acting is the same
mn* is the conductivity effective mass for electrons, which is the harmonic
mean of the band structure effective masses. Note that this is different from
the density-of-states effective mass, which is the geometric mean. n is the
electron mobility
1 1 1
2  in Si (since there are 6 equivalent
    minima, and effective masses in three
Example:
*
mn 3  ml mt  directions are m , m , m )
l
t
t
Slide # 1
Conductivity and DOS effective masses
J
q 2n  m
1 1 1
1
1 








 

m
3  m1 m2 m3 
• J is given as:
m
m* is the conductivity effective mass
• For semiconductors with a single minimum (direct bandgap
materials), the current density will be different in different
directions if the effective masses are different. However, for
indirect bandgap materials, the current density can be
isotropic even if the effective masses are not same.
• For semiconductors with elliptical or cylindrical symmetry,
the effective mass is same along the shorter axes
• As a reminder, the density of states effective mass is given
13



*
as: m  m1 m2 m3
DOS

E,

Slide #
Conductivity effective mass
qE  m
• The average velocity of the electrons2 is v 
m
q n m
• The current density is given as J 
E

Jx 
q 2n  m
Ey J z 
m1
m2
q m
• The mobility is given as   
m

Ex
Jy 
q 2n  m

m
q 2n  m

m3
Ez
• When there are gc equivalent conduction band minima,
and total electron density, n, the electron density at each
minimum is n/gc
• For Si, with 6 equivalent minima, the current density in
2
q
n m  2
any direction is:
2
2 
Ji 
6
       Ei
 m1 m2 m3 
i  x, y , z
Slide #
Equivalent energy minima in Si
Slide #
Mobility 3
(1) Current caused due to motion of only electrons in applied electric field:
qn  tS
Q
j

S  t
S t
nq 2 n
j  qn  
E  nq n E
*
mn
From Ohm’s Law:
j  E
q
S
vt
nq 2 n
n 
 qn n
*
mn
(only due to
electrons)
(2) Total current due to both electrons and holes:
j  qn  n  qp  p  nq  n  pq  p E   E
Note for holes,
p 
q p
m
*
p
E  pE
  qn  n  qp  p
(  p is the hole mobility)
Slide # 5
Electron and hole mobility vs. bandgap
• The electron and hole mobilities vary inversely with the
bandgaps of the semiconductors
Slide #
Types of mobilities
• Conductivity mobility
This mobility relates current density to the electric field and is given
as: J   c qvE
• Hall mobility: Measured from Hall measurement by
application of magnetic field
  r c
H
where r is called the Hall scattering factor,
and given as
r  n
2
n
2
Depending on the scattering mechanism, r can be significantly more than one.
Slide # 7
Lorentz force and Hall effect
F  qv  B
Jx
q
B
nq
nq e E x
q
B
nq
VH
 qE y  q
w
VH
LVH
LVH
VH
e 



wE x B wVx B wRx IB BI  sheet
Slide # 8