Transcript conductivity effective mass
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 n E The average drift velocity of electrons is then given as
p m n
*
q
n m n
*
E
n E
Note: Velocity in different directions can be different even though the field acting is the same
m
*
n
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 Example: 1
m n
* 1 3 1
m l
2
m t
in Si (since there are 6 equivalent minima, and effective masses in three directions are m l , m t , m t )
Slide # 1
Conductivity and DOS effective masses
• • J is given as: m * is the
J
q
2
n m
m E
, 1
m
1 3 1
m
1
conductivity effective mass
1
m
2 1
m
3
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
as:
m DOS
m
1
m
2
m
3 * 1 3 is given
Slide #
Conductivity effective mass
• The average velocity of the electrons is • The current density is given as
J
q
2
n m
J
q
2
n
m J y
2
n
m J z
2
n m
3
v m
E qE m
m
• When there are g c
q m
2
E
y q
m
q
m E z m
equivalent conduction band minima, and total electron density, n , the electron density at each minimum is n/g c • For Si, with 6 equivalent minima, the current density in any direction is:
J i
q
2
n
6
m
2
m
1 2
m
2 2
m
3
E i 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:
j j
S qn
Q
t
qn
t
S nq
2
S
t n E m n
*
nq
n E
From Ohm’s Law:
j
E
S
n
v
t
nq
2
n m n
* (2) Total current due to both electrons and holes:
q
qn
n j
qn
n
qp
p
nq
n
pq
p
E
E
(only due to electrons)
qn
n
qp
p
Note for holes,
p
q
p m
*
p E
p E
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
H
r
c
where r is called the Hall scattering factor, and given as 2
r
n
2
n
Depending on the scattering mechanism, r can be significantly more than one.
Slide # 7
Lorentz force and Hall effect
F
qv
B
q
q J x
B nq nq
e E x nq
B
qE y
q V H w
e
V H wE x B
LV H wV x B
LV H wR x IB
BI V
H sheet
Slide # 8