Transcript Review-QM’s and Density of States
Review QM’s and Density of States
Last time, we used a quantum mechanical, kinetic model, and solved the Schrodinger Equation for an electron in a 1-D box. (x) = standing wave =
A
sin
n
L x
Extend to 3-D and use Bloch Function to impose periodicity to boundary conditions (x) = traveling wave =
k x
0, 2
L
, 4 2
L
,...
k y
0, 2
L k
, 4 2
L
exp ,...
k z
0, 2
L
, 4 2
L
,...
We showed that this satisfies periodicity.
r
a
e ik
a
Review QM’s and Density of States
Plug into H = E
k
to find energy. 2 2
k
2 2
m
2
m
k x
2
k y
2
k z
2 Since k, higher energy corresponds to larger k.
For k =k Fy =k Fz (i.e. at the Fermi Wave Vector), the Fermi Surface (surface of constant energy, wave vector, temperature) the is a shpere. This applies to valence band for Si, Ge, GaAs; and couduction band for GaAs.
2
F
2
m k F
2 < F , filled states > F , empty states
Fermi Surfaces in Real Materials
If k xF ≠k Fy ≠k Fz , then you get constant energy ellipsoids. Also, if k=0 is not lowest energy state, (e.g. for p z orbitals where k= is the lowest energy), you don’t get a constant energy sphere.
e.g. Silicon and Germanium CB energy min occurs at Si - X point, k= along <100> directions Ge - L point, k= along <111> directions
Review QM’s and Density of States
# of states in sphere =
volume
_
of volume
_
of
_
sphere
_1_
state
4 3 2
k F
3 3
L
# of electrons = 2x Number of states, because of spin degeneracy (2 electrons per state)
L
3
V N
3
V
2
k F
3
F
2 2
m k F
2
k F
F
2 2
m
Now we wish to calculate density of states by differentiating N with respect to energy
D
dN d
V
2 2 2
m
2 3 2 1 2
Review QM’s and Density of States
D
1 2 2
m e
* 2 3 2
E
E C
1 2
D
1 2 2
m e
* 2 3 2
E V
E
1 2 Number of electrons, n, can be calculated at a given T by integration.
n
B
.
E
.
D
f
d
Review QM’s and Density of States
n
N C
exp
E C
E F
kT
p
N V
exp
E F
E V
kT
N C
2
m e
2 *
kT
2 3 2
N V
2
m h
* 2
kT
2 3 2 Eff. DOS CB Eff. DOS VB We can now calculate the n,p product by multiplying the above equations.
np np
N C N V
exp
E g kT
n i
2 =(intrinsic carrier concentration) 2 For no doping and no electric fields
p
n i
N C N V
exp
E g
2
kT
Outline - Moving On
1. Finish up Si Crystal without doping 2. Talk about effective mass, m* 3. Talk about doping 4. Charge Conduction in semiconductors
Intrinsic Semiconductors
It is useful to define the “intrinsic Fermi-Level”, what you get for undoped materials. (E F (undoped) = E i , n=p=n i )
N C
exp
E C
E i
kT
N C
exp
E i
E V
kT
E i
kT E C
E V
kT E i
ln
N V N C
E i
2
E i
E C E C
E V E V
E i
kT kT
ln
N V N C
ln
N V N C
2
E i
E V
E C
E V
kT
ln
N V N C
Intrinsic Semiconductors
E i N V N C
E V
E g
2 2
m h
*
kT
2 2
m e
*
kT
2
kT
ln
N V N C
2 3 2 2 3 2
m h
*
m e
* Si GaAs
kT
2 ln
N C N V
-13 meV -7 meV 35 meV E g 1.12 eV 0.67 eV 1.42 eV The energy offset from the center of the band gap is in magnitude compared to the magnitude of the bandgap. REMEMBER: There are no states at Ei or EF. They are simply electrochemical potentials that give and average electron energy
Intrinsic Semiconductors
N V N C
2
m h
*
kT
2 2
m e
*
kT
2 2 3 2 2 3 2
m h
*
m e
*
E i
E V
E g
2 3 4
kT
ln
m
*
h m
*
e
For Si and Ge, m e * > m h *, so ln term < 0, E i < E g For GaAs, m e * < m h *, so ln term < 0, E i < E g
Effective Mass
The text book 3.2.4 derives m* from QM treatment of a wave packet.
m
* 2
d
2
E dk
(This expression can be derived quantum mechanically for a wavepacket with group velocity v g ) We can use this quantum mechanical results in Newtonian physics. (i.e. Newton’s Secone Law)
F
dk
2
d
2
E
2
m
*
dv g dt
m
*
a
Thus, electrons in crystals can be treated like “billiard balls” in a semi-classical sense, where crystal forces and QM
1
m
* 1 2
d
2
E dk
2 or
Effective Mass
1
m ij
* 1 2
d
2
dk i
dk j
So, we see that the effective mass is inversely related to the band curvature. Furthermore, the effective mass depends on space we are “looking”
m
*
de
m
1 *
m
* 2
m
* 3 1 3 In silicon, for example
m
*
de
m l
*
m t
* 1 3 Where m l direction of the ellispoids and m t * is the effective mass along the transverse direction of the ellipsoids
Effective Mass
Relative sizes of m l * and m t * are important (ultimately leading to anisotropy in the conductivity)
m l
*
m t
* max_
length width
_
of
_ _
of
_
ellipsoid ellipsoid
_ _
along
_
axis perpendicular
_
to
_
axis
2 1
m
* 1 2
d
2
E dk
2 or Again, we stress that effective mass is inversly proportional to 1
m ij
* 1 2
d
2
dk i
dk j
will have negative mass and accelerate in the direction opposite to what is expected purely from classical considerations.
Effective Mass
One way to measure the effective mass is cyclotron resonance v. crystallographic direction.
strength.
-We measure the absorption of radio frequency energy v. magnetic field
c
qB m
* Put the sample in a microwave resonance cavity at 40 K and adjust the rf frequency until it matches the cyclotron frequency. At this point we see a resonant peak in the energy absorption.
-Replace Si lattice atoms with another atom, particularly with an extra or deficient valency. (e.g. P, As, S, B in Si)
D
E C
E D
ε D is important because it tells you what fraction of the dopant atoms are going to be ionized at a given temperature. For P in Si, ε D = 45 meV, leading to 99.6% ionization at RT.
Then, the total electron concentration (for and n type dopant) is
n
n i
N D
Carrier Statistics in Semiconductors
If we look at the Fermi Level position as a function of temperature (for some sample), we see that all donor states are filled at T = 0 (n ≈ 0, no free carriers), E F = E D . At high temperatures, such that, n i >> N D , then n ≈ n i and E F ≈ E i . E F ranges between these limiting values at intermediate temperatures. (see fig) Np=n i 2 n = n i still holds, but one must substitute + N D + p = n i 2 /(n i + N D + ).
At room temperature n ≈ N D + For Silicon - n i = 10 10 cm -3 .
- N D = 10 13 → 10 16 cm -3 (lightly doped → heavily doped)
n
Carrier Statistics in Semiconductors
N C
exp
E C
E F
kT
N n D
10 16 10 19 exp
E C
E F
kT
This is 60meV/decade. Three decades x 60 meV/decade = 180 meV = E C is very close to the conduction band edge.
– E F . E F Similar to E F (T), let’s look at n(T)
Charge Conduction in
KE KE
1 2
m e
2
v th
3 2
kT
3-D (from Statistical Mechanics) Brownian Motion Applying a force to the particle directionalizes the net movement. The force is necessary since Brownian motion does not direct net current.
F
qE
m e
*
a
q < 0 for e , q > 0 for h + Constant Field leads to an acceleration of carriers scaled by m e *.
If this were strictly true, e ’s would accelerate without bound under a constant field. Obviously, this isn’t the case. Electrons are slowed by scattering events.
Scattering Processes in Semiconductors
1) Ionized Impurity Scattering 2) Phonon (lattice) Scattering 3) Neutral Impurity Scattering 4) Carrier-Carrier Scattering 5)Piezoelectric Scattering
Scattering Processes in Semiconductors 3) - Donors and acceptors under freez-out. - Low T only - Defect = polycrystalline Si 4) - e - h + scattering is insignificant due to low carrier concentration of one type or another - e - e or h + - h + don’t change mobility since collisions between these don’t change the total momentum of those carriers.
5) - GaAs: displacement of atoms → internal electric field, but very weak.
2) - collisions between carriers and thermally agitated lattice atoms. Acoustic ~
m
* 5 2
T
3 2 - Mobility decreases as temperature increases due to increased lattice vibration - Mobility decreases as effective mass increases.
Scattering Processes in Semiconductors 1) Coloumb attraction or repulsion between charge carriers and N D + or N A .
~
m
* 5 2
T
3 2 Due to all of these scattering processes, it is possible to define a mean free time, m , and a mean free path
l
m.
qE F
qE
Average directed velocity
v avg
.
_
drift
a
me
m e
*
a a
t a
v drift
q
m
,
e E q
m
,
e m e
*
m e
*
a
m v d
e e
m
*
e
e E
m
,
e
1
m
*
e
For solar cells (and most devices) high mobility is desirable since you must apply a smaller electric field to move carriers at a give velocity.
e -
Current Flow in Semiconductors
(cm-3)(cm/s) = e /cm -2 s
J flux
nv d
current
_
density
A/cm -2 = C/cm -2 s
J J J
qnv d qn
n
E E
conductivi ty
qn
n
n
n
For Holes Total
Current Flow in
qp
Semiconductors
h qn
p
qp
h
Resistance n-type 1
q
n n
1
p p
1
q
n N D n
n n N D
p p