Download Lecture 13B:

PHYS 342/555
Introduction to solid state physics
Instructor: Dr. Pengcheng Dai
Associate Professor of Physics
The University of Tennessee
(Room 407A, Nielsen, 974-1509)
(Office hours: TR 1:10AM-2:00 PM)
Lecture 13, room 304 Nielsen
Chapter 8: Semiconductor crystals
Lecture in pdf format will be available at:
http://www.phys.utk.edu
1
Semiconductor crystals
A solid with an energy gap will be nonconducting at T = 0
unless electric breakdown occurs or unless the AC field is
of such high frequency that =ω exceeds the energy gap.
However, when T ≠ 0 some electrons will be thermally
excited to unoccupied bands (conduction bands).
If the enegy gap Eg ≈ 0.25 eV, the fraction of electrons across
− E / 2k T
the gap is of order e g B ≈ 10−2 , and observable conductivity
will occur. These materials are semiconductors.
Dai/PHYS 342/555 Spring 2006
Chapter 8-2
The energy gap can be measured by optical absorption.
If the conduction band minimum occurs at the same point
in k -space as the valence band maximum, the energy gap
can be directly determined from the optical threshold. If the
minima and maxima occur at different points in k -space, then
for phonon must also participate in the process.
The energy can may also be deduced from the temperature
dependence of the intrinsic conductivity, varying as e
Dai/PHYS 342/555 Spring 2006
− E g / 2 k BT
.
Chapter 8-3
Equations of motion
The group velocity vg = dω / dk , ω = ε / =, so vg = = −1d ε / dk .
The work δε in interval δ t is δε = −eEvgδ t.
Note δε = (d ε / dk )δ k = =vgδ k .
G
G G
dk
=δ k = −eEδ t , Hence =
= −eE = F .
dt
The electron in the crystal is subject to forces from the crystal
lattice as well as from external sources.
G
G
e
F = −eE − v × B.
c
Dai/PHYS 342/555 Spring 2006
Chapter 8-4
Holes
Dai/PHYS 342/555 Spring 2006
Chapter 8-5
1. The total wavevector of the electrons in a filled band
G
G
G
is zero: ∑ k = 0 or kh = − ke .
2. The energy of the hole is opposite in sign to the energy
of the missing electron, because it takes more work to remove
an electron from a low orbital than from a high orbital.
G
G
ε h (kh ) = −ε e (ke ).
G G
3. vh = ve . The velocity of the hole is equal to the velocity of
the missing electron.
4. mh = − me .
G
G 1G G
dkh
5. =
= (eE + vh × B)
dt
c
Dai/PHYS 342/555 Spring 2006
Chapter 8-6
Dai/PHYS 342/555 Spring 2006
Chapter 8-7
Effective Mass
For U positive, an electron near the lower edge of the second
band has an energy: ε (k ) = ε c + (=k )2 /(2me )
Here k is the wavevector measured from the zone boundary,
and me denotes the effective mass of the electron near the
second band.
The group velocity vg = d ω / dk , ω = ε / =, so vg = = −1d ε / dk .
1 d 2ε 1 ⎛ d 2ε dk ⎞ 1 ⎛ d 2ε ⎞ ⎛ dk ⎞ 1 ⎛ d 2ε ⎞ ⎛ F ⎞
=
= ⎜ 2
⎟ = ⎜ 2 ⎟⎜ ⎟ = ⎜ 2 ⎟⎜ ⎟
= dkdt = ⎝ dk dt ⎠ = ⎝ dk ⎠ ⎝ dt ⎠ = ⎝ dk ⎠ ⎝ = ⎠
dt
1
1 ⎛ d 2ε ⎞
F = ma, then we have * = 2 ⎜ 2 ⎟ .
= ⎝ dk ⎠
m
dvg
Dai/PHYS 342/555 Spring 2006
Chapter 8-8
Physical interpretation of Effective Mass
Near the bottom of the lower band, ψ = eikx with momentum
=k ; m* ≈ m. Positive m* means that the band has upward
curvature (d 2ε / dk 2 > 0). A negative effective mass means that
on going from state k to state k + Δk , the momentum transfer
to the lattice from the electron is larger than the momentum
transfer from the applied force to the electron.
Dai/PHYS 342/555 Spring 2006
Chapter 8-9
Effective Mass in Semiconductors
The angular rotation frequency ω c of the current carriers
is:
eB
ω c = * , where m* is the effective mass.
mc
Dai/PHYS 342/555 Spring 2006
Chapter 8-10
Dai/PHYS 342/555 Spring 2006
Chapter 8-11
Intrinsic carrier concentration
In semiconductor physics μ is called Fermi level.
If ε − μ k BT , the Fermi-Dirac distribution function
reduces to f e exp[( μ − ε ) / k BT ]. The energy of an
electron in the conduction band is ε k = Ec + = 2 k 2 / 2me .
Note the total number of electrons N :
V ⎛ 2mε ⎞
dN
1 ⎛ 2me ⎞
1/ 2
,
D
(
)
E
.
ε
ε
≡
=
−
(
c)
2 ⎜
2 ⎟
2 ⎜
2 ⎟
3π ⎝ = ⎠
d ε 2π ⎝ = ⎠
The concentration of electrons in the conduction band is
3/ 2
3/ 2
N=
⎛ me k BT ⎞
n = ∫ De (ε ) f e (ε )d ε = 2 ⎜
2 ⎟
Ec
⎝ 2π = ⎠
∞
3/ 2
exp[( μ − Ec ) / k BT ].
Dai/PHYS 342/555 Spring 2006
Chapter 8-12
The distribution function f h for holes is f h = 1 − f e .
1
≅ exp[(ε − μ ) / k BT ].
We have f h = 1 −
exp[(ε − μ ) / k BT ] + 1
If the holes near the top of the valence band behave as particles
with effective mass mh , the density of hole states is given
3/ 2
dN
1 ⎛ 2mh ⎞
1/ 2
Dh (ε ) ≡
= 2 ⎜ 2 ⎟ ( Ev − ε ) .
d ε 2π ⎝ = ⎠
The concentration of holes in the valence band is
⎛ mh k BT ⎞
p = ∫ Dh (ε ) f h (ε )d ε = 2 ⎜
2 ⎟
−∞
2
π
=
⎝
⎠
The energy gap Eg = Ec − Ev ,
Ev
3/ 2
exp[( Ev − μ ) / k BT ].
3
3/ 2
⎛ k BT ⎞
m
m
exp(− Eg / k BT ).
np = 4 ⎜
2 ⎟ ( e h)
⎝ 2π = ⎠
Dai/PHYS 342/555 Spring 2006
Chapter 8-13
In an intrinsic semiconductor, n = p, Eg = Ec − Ev ,
⎛ k BT ⎞
n = 2⎜
2 ⎟
π
=
2
⎝
⎠
3/ 2
( me mh )
3/ 4
exp[( Eg / 2k BT ].
Dai/PHYS 342/555 Spring 2006
Chapter 8-14
Intrinsic carrier mobility
The mobility is the magnitude of the drift velocity per unit
electric field: μ =| v | / E
The electric conductivity is the sum of the electron and hole
contributions. σ = (neμ e + peμ h )
Dai/PHYS 342/555 Spring 2006
Chapter 8-15
Impurity conductivity
Impurities that contribute to the carrier density of a semiconductor
are called donors if they supply additional electrons to the conduction
band, and acceptors if they supply additional holes to (i.e., capture
electrons from) the valence band.
The impurities are at N D fixed attractive centers of charge + e, per
unit volume, along with the same # of additional electrons. If the
impurity were not embedded in the semiconductor, but in empty
space, the binding energy of the electron would just be the first
ionization potential of the impurity ion, 9.81 eV for arsenic.
However, since the impurity is embedded in the medium of the pure
semiconductor, this binding energy is enormously reduced
(to 0.013 eV for arsenic in Ge).
Dai/PHYS 342/555 Spring 2006
Chapter 8-16
1. The field of the charge representing the impurity must be reduced
by the static dielectric constant ε (16 for Ge) of the semiconductor.
2. A electron moving in the medium of the semiconductor should
G
be described by the semiclassical relation, where =k is the electron
crystal momentum. The additional electron introduced by impurity
should be thought of as being in a superposition of conduction band
levels of the pure host material. The electron can minimize its energy
by using only levels near the bottom of the conduction band.
We have a particle of charge − e and mass m* , moving in free space
in the presence of an attractive center of charge e / ε .
Dai/PHYS 342/555 Spring 2006
Chapter 8-17
The radius of the first Bohr orbit, a0 = = 2 / me 2 , becomes
m
4
2
a
,
and
the
ground-state
binding
energy,
me
/
=
= 13.6 eV
ε
0
*
m
m* 1
becomes ε =
× 13.6 eV. r0 = 100 angstrom.
2
m ε
r0 =
Dai/PHYS 342/555 Spring 2006
Chapter 8-18
Consider the energy surface
2
2
⎛
+
k
k
k z2 ⎞
x
y
2
+
ε (k )== ⎜⎜
⎟⎟ , where mt and ml are transvers and longitudinal
2ml ⎠
⎝ 2mt
mass parameter. Use equation of motion to show that
ωc = eB /(ml mt )1/ 2 c when the static magnetic field B lies in the
x, y plane.
Dai/PHYS 342/555 Spring 2006
Chapter 8-19