Bloch Electrons in Semiconductors

A. Band Gap
B. Equations of motion
C. Intrinsic Carrier Concentration
D. Impurity conductivity
E. Thermoelectric Effects
F. Bloch Oscillation
 Energy Band Gap in a Periodic Potential
Recall the electrostatic potential energy in a crystalline solid along a line
passing through a line of atoms:

bare ions
solid

Along a line parallel to this but running between atoms, the divergences
of the periodic potential energy are softened.
A simple 1D model that captures the periodicity of such a potential is:
æ 2px ö
U ( x) = U 0 + U1 cosç ÷ U 0 < U1 < 0
U è a ø
x
U0

U1
Electron Wavefunctions in a Periodic Potential
Consider the following cases:

Wavefunctions are plane waves i ( kx -wt ) !2k 2

U1 = 0 and energy bands are parabolic: y = Ae E=
2m
U1 ¹ 0 Electrons wavelengths much larger than a, so wavefunctions and
k << pa energy bands are nearly the same as above

U1 ¹ 0 Electrons wavelengths approach a, so waves begin to be strongly

back-scattered :
k £ pa
y ± = Aei ( kx -wt ) ± Be -i ( kx -wt ) B< A
U1 ¹ 0 Electrons waves are strongly back-scattered (Bragg scattering) so
standing waves are formed:
k = pa
y ± = C [ei ( kx -wt ) ± e -i ( kx -wt ) ] = 1
2
[ ]
A eikx ± e - ikx e - iwt
 Bloch Theorem and Wavefunctions
Bloch theorem is one of the most important formal results in all of solid
state physics because it tells us the mathematical form of an electron
wavefunction in the presence of a periodic potential energy.
In independent-electron approximation, the time-independent
Schrodinger equation (SE) for an electron in a periodic potential is:
é "2 2 !ù
ê - Ñ + U ( r )úy = Ey
ë 2m û
where the potential energy is invariant under a lattice translation vector T :
U(r + T) = U(r) and T = ua + vb + wc

Bloch showed that the solutions to the SE are the product of a plane
wave and a function with the periodicity of the lattice:

𝜓k(r ) = uk(r)eik･r and uk(r + T) = uk(r)

 Bloch Wavefunctions
! ! !!
y k! (r ) = uk! (r )e ik × r

This result gives evidence to support the nearly-free electron approximation,

in which the periodic potential is assumed to have a very small effect on the
plane-wave character of a free electron wavefunction. It also explains why
the free-electron gas model is so successful for the simple metals!
 Band Gap
188
The band gap is the difference in energy between
qxd the lowest
8/13/04 4:23 PM point of the conduction band and the
Page 190
Vacant conduction band
highest point of the valence band. The lowest

Energy
point in the conduction band is called the Forbidden band E g

conduction band edge; the highest point in the Filled valence band
190
valence band is called the valence band edge. Figure 2 Band scheme for intrinsic conductivity in a semiconductor. At 0 K the conduct
zero because all states in the valence band are filled and all states in the conduction band a
Table 1 Energy gap between the valence
cant. Asand conduction
the temperature bands
is increased, electrons are thermally excited from the valence band
(i ! indirect gap; d ! conduction
direct gap)band, where they become mobile. Such carriers are called “intrinsic.”

1014 1014
Eg, eV Eg, eV
Crystal Gap 0K 300 K Crystal Gap 0K 300 K

Electron concentration in carriers per cm3 1013SiC(hex) — 1013

Electron concentration in carriers per cm3

Diamond i 5.4 i 3.0
Si i 1.17 1.11 Tc d 0.33 —
Ge i 0.744 0.66 HgTea d $0.30
Si
!Sn d 0.00 0.00 PbS dGe 0.286 0.34–0.37
InSb d 0.23 0.17 1012PbSe i 0.165 0.27 10
12

InAs d 0.43 0.36 PbTe i 0.190 0.29

InP d 1.42 1.27 CdS d 2.582 2.42
GaP i 2.32 2.25 CdSe d 1.840 1.74
GaAs d 1.52 1.43 1011CdTe d 1.607 1.44 10
11

GaSb d 0.81 0.68 SnTe d 0.3 0.18

AlSb i 1.65 1.6 Cu2O d 2.172 —

1010 1010
a
HgTe is a semimetal; the bands overlap.
 !Sn d 0.00 0.00 PbS
Transparent !Sn d 0.00 0.00 PbS d 0.286
region InSb d 0.23 0.17 PbSe
InSb d 0.23 0.17 PbSe i 0.165
InAs Direct and Indirect Band Gaps
d 0.43
InAs
InP
0.36
8 Semiconductor
Onset of indirect
d
Crystals d
PbTe
0.43
189 1.42
0.36
i
1.27
PbTe
0.190
CdS
Onset of direct
photon
InP 1.42 GaP 1.27
d photon transition i
CdS 2.32 d
2.25
2.582
CdSe
transition GaP i 2.32 2.25
Onset of direct
GaAstransition Band CdSe
d gaps d 1.840
CRYSTAL WITH DIRECT GAP CRYSTAL WITH INDIRECT GAP
can be 1.43
1.52 measured CdTe
by
Absorption GaAs d
Absorption 1.52
Eg + !" EGaSb
1.43 d CdTe 0.81 d
0.68 1.607
SnTe
!!g vert optical absorption. In a direct
GaSb
Photon energy !! d 0.81 0.68
Photon energy !!
AlSb i SnTe d
1.65 process
1.6 0.3 Cu2O
Transparent
(a) (b) absorption the
region AlSb i 1.65 1.6 Cu2O d 2.172
Figure 4 Optical absorption in pure insulators at absolute zero. In (a) the threshold determines
the energy gap as Eg # !!g. In (b) the optical absorption is weaker near the threshold: at
threshold of continuous optical
!! # Eg $ !" a photon is absorbed with the creation of three particles: a free aelectron,
Onset of indirect
absorptionthe
HgTe aisfree at bands
ωg measures
a semimetal; overlap. the
hole, and a phonon
Onset of directof energy !". Ina (b) the energy Evert marks the threshold for the creation of a
photon
free electron and a free hole, with no HgTe
phononisinvolved.
a semimetal; the bands
photon transition
Such a transition overlap.
energy band gap Eg = ℏωg.
is called vertical; it is
transition Onset of direct
similar to the direct transition in (a). These plots do not show absorption lines that sometimes are
seen lying just to the low energy side of the threshold. Such lines are due to the transition
creation of a
bound electron-hole !!pair,
g called
Photon energy !!
an exciton. E g + !" E vert
Photon energy !!
In the indirect absorption
(a) (b) process the minimum energy
threshold determines wavevector k c . Here a direct photon
Figure 4 Optical absorption in pure insulators at absolute zero. In (a) the substantial
the energy gap as" E # !! . In (b) the optical absorption is weaker near " the threshold: at gap of the band structure
!! # E $ !" a photon is substantial of three particles: k
wavevector . Here
aminimum electron, aa direct
gap photon
cannot transition
satisfy theand atholes
the ener
requirement
g g
g absorbed with the creation cfree
hole, and a phonon of energy !". In (b) the energy E marks the threshold for the creation of a
free
involves electrons
minimum gap cannot satisfy thephoton
it requirement of conservation of atwath
vert
free electron and a free hole, with no phonon involved. Such a transition because
is called vertical; wavevectors
is separated by aaresubstantial
negligible
similar to the direct transition in (a). These plots do not show absorption lines that sometimes are
because
seen lying just to the low energy side of thephoton lines are dueifto a
wavevectors
threshold. Such the phonon
are negligible
creation of wavevector
of a wavevectorat the K
If and
kc. energy frequency
a phonon
range of inte
of !
bound electron-hole pair,Conduction
called an exciton. Conduction
!
g
if a
band edge
phonon
Valence band edge
of
band edge
wavevector
" %
we K canand have wavevector!Kisand
frequency frequency
created in theΩproc
Valence band edge
we can have is created, then
" "
k(photon) " kc # K " 0 ; !
k k
0
(a)
–kc
k(photon)
0
(b)
" kkcc # K " 0 ; !" " Eg # !! ,
Figure 5 In (a) the lowestConduction
point of the conduction
as required by the conservation laws. The phon
band occurs at the same value of k as the highest
Conduction
 The work !" done on the electron by the electric field E in the time
interval !t is
192 Equations of Motion
!" ! "eEvg !t . (2)
k !" done
The on the electron
propagation speed by
of theelectron
an electric field E in
wavepacket in the
a time crystal can
periodic
We observe that
be calculated fromThetheworkenergy done
!"band on that
along the direction
electroninby the electric
reciprocal space: field
interval !t is!" ! (d"/dk)!k ! !vg !k , (3)
!" ! "eEv g !t . (2)
electron velocity: (1-D) v g = dw = 1 dE (3-D) vg(k) = ℏ-1𝝯kϵ(k)
hat using (1). On comparing (2) with (3) dk !" ! "eEvg !t .
dk we !have
" done on the electron by the electric field E in the time
The work 𝛿ϵ Wedone on thethat
observe electron
!k ! by the electric
"(eE/!)!t , field E in the time interval (4)
𝛿t is !" ! (d"/dk)!k ! !vg !k , (3)

(2)!"
whence !dk/dt
comparing !!"eEv
with (3)
"eE.weg !t ,. and since !" ! (d"/dk)!k(2)
have ! !vg !k ,
We may write (4) in On
terms of the external force Fweashave
we have using (1).
!k ! "(eE/!)!t , comparing
and (2)
ℏdk/dt with
= −eE (3)
= F , so we obtain
(4)
!" ! (d"/dk)!k ! !vg !k , dk !k ! (3)
"(eE/!)!t ,
t ! "eE. ! !F . (5)
write (4) (2)
mparing in terms of we
with (3) external force F
thehave dt as
whence !dk/dt ! "eE.
ThisThis is important
is an an
!k ! Werelation:
important may
"(eE/!)!t write
relation:
, in (4)
in
a a in terms
crystal
crystal of the
ℏdk/dt
!dk/dt is isexternal
equal
equal (4)
to toforce
thethe F as force
external
external
dk
force on the electron.
! !InF free . space d(mv)/dt is equal to the force. (5) We have
on the electron. In free dt space d(mv)/dt is equal to the force. We have not over-
"eE. not overthrown Newton’s second law of motion: the dk electron in the crystal
thrown Newton’s second
fromlaw
theof motion: the electron
! in!the F crystal
. is subject to
e (4) inisterms
subjectofto forces
the external forcecrystal
F as lattice as well asdtfrom external sources.
forces
portant from the
relation: incrystal lattice
a crystal as well
!dk/dt as from
is equal external
to the sources.
external force
 equation of motion ofwhere
an electron of group
the right-hand side of eachvequation
velocity in a constant
is the magnetic
Lorentz for
field B is With the group velocity v ! !"1grad k", the rate of change of
In a constant magnetic field B, the equation of motion is

dk e dk -1
(CGS) ! ! "cv $ B ;or dk ! " e2 #k" $ B(SI)
(CGS) ; with! v =!ℏ"ev
𝝯kϵ$ B dk(6)
(SI) !"
dt dt !c dt dt
,YC T63&7 !""" Z@63/@G [.2)776/)*3. 63+ /%& $*'*7*41 *5 A&(G) I@(562&.

where the right-hand sidewhereof now

eachboth anisof
sides
equation electron
theLorentz
the moves
equation in kon
refer
force space
tothe incoordinate
a
theelectron.
!"#$% !"""

With the group velocity v ! We!"1see from

grad direction
the vectornormal to the direction
cross-product
k", the rate of change of the wavevector is
in (7) that in
an electron moves in ofkthe gradient
space of the energy
in a direction normal ϵ, soto the di
dient of the energy that the the
", so that electron
electron moves moveson on a a su
dk e surface of dk
constant eenergy. The
(CGS) ! " 2 #energy.
k" $ B ; (SI)
The value of the projection kB2 #ofk" k$ on
! " B B (7) is co
dt !c dt !
value of the projection kB of k on B
motion. The motion in k space is on a plane normal to the d
is constant during the motion. The
the orbit is defined #$%&by !"""&'&
the intersection of *(0)/
this *5
plane with a s
where now both sides of the equation refer motion
to the in
$%&kA&(G)8.@(562&
space is in
coordinates onk aspace.
plane
energy. 63 &7&2/(*3 )3 6 .*7)+ @3+&( /%& )357@8
normal
We see from the vector cross-product &32& 63 to
*5 in (7)the
6''7)&+ direction
that of$%&
in a5)&7+9
G643&/)2 B, and field
magnetic
an electron moves in k space in a direction thenormal
'&()*+ orbit
*5 isthe
/%& *(0)/ ). defined
'(*'*(/)*367by /* the
/%& +&()>6/)>& *5 /%& 2(*..8.&2/)*367of the gra-
to direction
intersection of this plane with a
dient of the energy ", so that the electron 6(&6 moves
=)/% (&.'&2/ on/* a&3&(41
surface of constant
surface of constant energy.
energy. The value of the projection kB of k on B is constant during the
motion.! !"
The
! !$ in k space is on a plane normal to the direction of B, and
motion
the orbit!# !! "
is defined ! #!"""!#$
!# by the intersection of this plane with a surface of constant
 8 Semiconductor Crystals
2
using " #C(k ! G) # # 1. 8 Semiconductor Crystals
We examine the transfer of momentum between the electron and the lat-
Physical Derivation of Equations of Motion
Physical
Physical
Derivation
Physical of !k̇
Derivation
Derivation
of #!k̇F#F
tice when the state k of the electron is changed to k ! "k by the application
!k̇ # F
ofBloch
We We
consider consider
the
of an external force. We imaginethe
Bloch eigenfunction
eigenfunction belonging
an insulating!!kcrystal
k belonging totothethe
energy eigen-
energy
electrostatically eigen-
neutral
We
value
except "consider
value
forWe
and
a the
"k and
consider Bloch
wavevector
single eigenfunction
wavevector
the Bloch
electron k:
in the ofk belonging
state k 𝜓
k: eigenfunction to the
k belonging
an!otherwise toenergy
emptytheband.eigen-eigen-
energy
k
value
value ϵ"kk and
and wavevector
wavevector k: k:
We suppose that a weak external force is applied for a time interval such
! k # " C(k ! G) exp[i(k ! G) ! r] . (8)
that the total impulse ! given
k # toC(k
G
"
the !entire G) ! r]is. J # !F dt. If the(8)
crystal!system
G) exp[i(k
! k #G" C(k ! G) exp[i(k ! G) ! r] . (8)
conduction electron were freeG (m* # m), the total momentum imparted
The expectation value of the momentum of an electron in the Bloch state k is
to
theTheThe expectation
crystal system by
expectation value
the of
value ofthe
themomentum
impulse momentum
would appear ofan
of an electron
inelectron in the
the change
in of Bloch
the state
momentum
Bloch state ofis
k
kThe
is expectation value of the momentum of an electron in the Bloch state k is
the conduction Pel #electron:
(k#$i!%#k) # " !(k ! G)#C(k ! G)#2 # !(k ! " G#C(k ! G)#2) ,
G G
Pel # ##$i!%
Pel(k #k) ##"J"!(k
(k#$i!%#k) ! G)#C(k
G)#C(k ! G)
G)# #
# "ptot # "pel # !"k .
!(k ! #
# !(k
!(k !!"" G #C(k
G#C(k ! !
G)#22
G)#
2 (9)
) ,
2
(10)) ,
GG G G
(9) (9)
using
using "#C(k ! 2
The neutral crystal suffers no net interaction with the electric field, either
G)# # 1.
We examine
directly or indirectly the transfer
through the free of momentum
electron. between the electron and the lat-
2
usingusing
We "suppose
"#C(k
#C(k
tice !
when that
!
G)theG)#
#2 a weak
# #
state1. 1.
k ofexternal
the forceis is
electron applied to to
k! the crystal
by the in a time
If the conduction electron interacts withchanged
the periodic "k
potential application
of the crys-
We We
interval
of anexamine
such that
external
examine the the
the transfer
force.total
We impulse
transfer ofmomentum
imagine
of momentum
is
an J = ∫F
insulatingbetween
dt. We
crystal
between the
will electron
have
electrostatically
the electron andand
the the
lat- lat-
neutral
tal lattice, we must
tice except
when for
have k of the electron is changed to k ! "k by the application
theastate
tice when the state singlek ofelectron in the state
the electron k of an otherwise
is changed to k !empty "k by band.
the application
of an external force.that We imagine an insulating crystal electrostatically neutral
of an external force. WeJimagine . (11)
We suppose a weak
# "ptot # external force is applied for a time interval such
an"p lat ! "pelcrystal
insulating electrostatically neutral
except
thatfor
thea single electron
total impulse in the
given to state k of an
the entire otherwise
crystal systemempty !F dt. If the
is J #band.
except
From thefor a single
result
conduction
We (9) for
suppose
electron
pelawe
electron
that were
weak
in
havethe
free state #k m),
(m* force
external
of an
isthe
otherwise
total momentum
applied
empty imparted
band. to
for a time interval such
Wethe
that suppose
thecrystal that abyweak
total system
impulse given external
the impulse
to thewould force
entire is applied
appear
crystal in the
system foris aJoftime
change !F interval
#momentum of such
dt. If the
thatconduction
thethetotal impulse
"p
conduction el #
electron given
!"k
were!free
electron: "
to !G[(%
the #k #C(k
(m*entire m), crystal
! G)
the #2)system
total "k] .is J #imparted
!momentum !F dt. (12)
If
to the
 The change "plat in the Jlattice # "ptotmomentum
# "plat ! "presulting from the change of
el . (11)
state of the electron may be derived by an elementary physical consideration.
An From Thethe
electron change
resultof(9)
reflected momentum
by pthe
for el we ofhave
the electron
lattice transferswill be
momentum to the lattice. If an
incident electron with plane wave component of momentum !k is reflected
"pel # !"k ! " !G[(%k #C(k ! G)#2) ! "k] . (12)
with momentum !(k ! G), the lattice G acquires the momentum $!G, as re-
quired by the
because
The momentum
portion"pinlatconservation. The momentum transfer to theof lattice
because thechange
The change
portion∆plat the
in lattice momentum
the lattice momentum resulting from
resulting the
fromchange
the change of
when the ofstate
state
state of
thethe goes over
!kelectron
electron is to
may k!"k is by an elementary physical consideration.
be !derived
$ !C(k " G)!2 2 ! !k (14)
$ k !C(k
k " G)!
An electron reflected by the lattice transfers momentum ! !k to the lattice. (14)
If an
incident electron"p lat #
with $!"wave
plane k #C(k state
G[(%component ! G)#of
2
! "k] ,
momentum !k is (13)
reflected
of each individual component of the initial is reflected during the state
of each individual component ofG the initial state is reflected during the state
with momentum
change !k. !(k ! G), the lattice acquires the momentum $!G, as re-
change The!k. total momentum change is therefore
quired
The by momentum
total momentumconservation.
change is thereforeThe momentum transfer to the lattice
The total momentum change is therefore
when the state ! goes over to ! is
k k!"k
!pel " !plat # J # !!k , (15)
!pel " !plat # J # !!k , (15)
"plat # Eq.
exactly as for free electrons, G[(%
$! (10). " k #C(k
Thus ! G)#
from the definition
2
! "k] , of J, we have (13)
exactly Since
as fordJ/dt = F, we thusEq.
free electrons, have G
(10). Thus from the definition of J, we have
!dk /dt # F , (16)
!dk /dt # F , (16)
derived in (5) by a different method. A rigorous derivation of (16) by an en-
derived
tirelyindifferent
(5) by amethod
different method.
is given A rigorous
in Appendix E. derivation of (16) by an en-
tirely different method is given in Appendix E.
Holes
 Band Electron in E Field
Now we see that the external electric field causes a change in the k
vectors of all electrons: ! " "
! dk ! dk - eE
F =" = -eE =
dt dt !
E If the electrons are in a partially filled band, this
will break the symmetry of electron states in the
1st BZ and produce a net current. But if they are
in a filled band, even though all electrons change
k vectors, the symmetry remains, so J = 0.
kx When an electron reaches the 1st BZ edge (at k =
p/a) it immediately reappears at the opposite
- pa p
a edge (k = -p/a) and continues to increase its k .
v
As an electron’s k value increases, its velocity
kx increases, then decreases to zero and then
becomes negative when it reemerges at k = -p/a.
Thus, an AC current is predicted to result from a
DC field (Bloch oscillations).
 Properties of Holes
Holes are vacant orbits in a band whose properties are important in an
almost filled band.
A hole acts in applied electric and magnetic fields as if it has a positive
charge +e. The reason is given in five steps that follow:

1. Crystal momentum: kh = −ke

The total wavevector of the electrons in a filled band is zero: Σk = 0,
where the sum is over all states in a Brillouin zone. This result follows
from the geometrical symmetry of the Brillouin zone: every
fundamental lattice type has symmetry under the inversion operation
r→ −r about any lattice point; it follows that the Brillouin zone of the
lattice also has inversion symmetry. If the band is filled all pairs of
orbitals k and −k are filled, and the total wavevector is zero.
If an electron is missing from an orbital of wavevector ke, the total
wavevector of the system is −ke and is attributed to the hole.
 2. Energy: !h(kh) ! "!e(k e) . (18)
ch08.qxd 8/13/04 4:23 PM Page 196
the zero ofThe
4:23 PM energy
energy
Page 195
ofofthe
thevalence
hole is opposite
band is atin the signtop to the of theenergy of the missing
band.
wer in theelectron,
band thebecause
missing itelectron
takes more workhigher
lies, the to remove the energy an electron
of from a
low orbital than from a high orbital.
stem. The energy of the hole is opposite in sign to the energy of
issing electron, because it 196 takes more work to remove an
8 Semiconductor Crystals 195
electron
a low orbital than from! a high orbital. Thus if the band is symmet-
!
e (k e ) ! !e ("k e ) ! "! h ("k e )!Conduction
"!h(kband h ). We construct in Fig. 8 a
Hole band constructed
Q with kh = –ke and
scheme to represent the properties of a hole. This hole band is a
!h(kh) = –!e(ke), to
l representation because it appears right side up. simulate dynamics
of a hole.
$" kh
ke vh !kve . (19)
h k
Electron removed k
elocity of the hole is equal to the velocity of the kmissing
E e
G
electron.
Valence band Valence band
Fig. 8 we see that #!h(kh) ! #!e(ke), so that vh(kh) !withve(kone e).
electron missing

re 7 Absorption of a photon of energy !" and negligible wavevector takes an electron from
the filled valence band to Q in the conduction band. If ke was the wavevector of the electron
sit are always
becomes symmetric
the wavevector under
of the electron at Q.the inversion
The total kof→
Figure
wavevector 8 The
the "k ifband
upper
valence the
half ofspin-orbit
the figure showsinteraction is simulates the dynamics of a ho
the hole band that
the absorption is "ke, and this is the wavevector we must ascribe to constructed
the hole ifbyweinversion
describe of the valence band in the origin. The wavevector and energy of the h
Even with spin-orbit interaction, bands arearealways
alence band as occupied by one hole. Thus kh ! "ke; the wavevector ofequal,
symmetric
but isopposite
the hole the same
if the crystal structure
in sign, to the wavevector and energy of the empty electron orbital in the
einversion
wavevector of operation. Without
the electron which remains ataG.center of symmetry,
For the entire lencethe
system band. but
We
total do with
not show
wavevector spin-orbit
the dispositioninteraction, the from the valence band at k e.
of the electron removed
 d scheme to represent the properties of a hole. This hole band is a
pful representation because it appears right side up.
3. Velocity: vh = ve v ! v . (19)
ch08.qxd 8/13/04h 4:23 ePM Page 196

The velocity of the hole is equal to the velocity of the missing electron.
e velocity of the hole is equal to the velocity of the missing electron.
m Fig. 8 we seeSince
that #!h(kh) ! #!e(ke), so that vh(kh) ! ve(k e).
196

4. Effective mass: mh = −me !

ands are always symmetric under the inversion k → "k Hole if the spin-orbit interaction is
band constructed
ed. Even with The effectiveinteraction,
spin-orbit mass is inversely
bands are always symmetric
with kh = –kif the crystal structure
e and
!h(kh) = –!e(ke), to
the inversionproportional to thea center
operation. Without curvature
of symmetry, but with
simulatespin-orbit
dynamics interaction, the
re symmetricd2ifϵ/dk
we 2compare of a hole. is reversed: !(k, ↑) !
, and forsubbands
the holefor which the spin direction
band kh
). See QTS, Chapter 9.
this has the opposite sign to k
that for an electron in the
ke
valence band. Valence band
with one
electron missing

Figure 8 The upper half of the figure shows the hole band that simulates the dynamics of a
constructed by inversion of the valence band in the origin. The wavevector and energy of the
are equal, but opposite in sign, to the wavevector and energy of the empty electron orbital in th
lence band. We do not show the disposition of the electron removed from the valence band at k
 is negative, so that mh is positive. Ex
F F
kx k
dkh D E G H1I D
E G
H
5. Equation
ch08.qxd
of motion:
8/13/04 4:23 PM Page 197
5. ! "C e(E #
c Jvh $ B) . C I (21)
dt B
K B J
A
K
The equation of motion for a hole is that of a particle ofApositive
This comes
charge e. from the equation of motion8 Semiconductor Crystals 197

! dke !
E1
!
(CGS) ! " !e(E # c vk e $ B) (22)
x Ex

E G
F
kx dt E FG (a) x
F
E G
kx
(b)
D H D H D H
C I
that applies
A
B toJKthe missing FigureB9Celectron
(a) At t I!J 0whenall stateswe aresubstitute
B
Cfilled except
I F!k
J
at thefor
h kethe
top of and vhth
band;
because A d!/dkx ! 0.K(b) An electric field Ex is applied in the "x direction.
for ve. The equation of motion for a hole
trons is in the #kx direction and all electrons
A
is that K of a particle of
make transitions together in t
positive charge e.ingThe positive
the hole to the statechargeE. (c) is
After consistent withthethe
a further interval electric
electrons move f
and the hole is now at D.
current carried (a)
by the valence (b)
band of Fig. 9: the (c)
current is carried by
the unpaired
Figure 9 (a) At t ! 0 allelectron in the
states are filled except orbital
F at the G:the velocity v is zero at F
top of the band; x
because d!/dkx ! 0. (b) An electric field Ex is applied in the "x direction. The force on the elec-
E together in the #kx direction, mov-
trons is in the #kx direction and all electrons make transitions
j " (!e)v(G) "(!e)[!v(E)] " ev(E) ,
ing the hole to the state E. (c) After a further interval the electrons move farther along in k space
and the hole is now at D.
(23)
e
ve j
which is Ejust the current of a positive charge
e
moving with the velocity as-
cribed to thee missing electron at E. The current is shown in Fig. 10.
ve je
vh Figure 10 Motion of electrons in the
h holes in the valence band in the elect
vh jh
Figure 10 Motion of electrons in the conduction band and
and electron drift velocities are in opposi
h holes in the valence band in the electric field E. The hole
jh electric
and electron drift velocities are in opposite directions, but their currents are in the same directio
electric currents are in the same direction, the direction of the
electric field.
electric field.
 Electrons and Holes
±N /2 ! ! ±N /2 ! ! !
A full band can be written as: å
i¹+ j
ki + k j = 0 and å ki = - k j = k - j
i¹+ j

So when the state +j is empty in a band, the band has effective wavevector k−j.
Now the current flow in the incomplete band under the influence of a field
E: ±N /2
! ±N /2
! !
å (- ev i ) = å (- ev i ) - ev j =0
i = ±1 i¹+ j

!
ev j
This shows that an incomplete band (state +j empty) behaves just like a
positive charge moving with the same velocity an electron would have in that
state. Thus the properties of all of the remaining electrons in the incomplete
band are equivalent to those of the vacant state j if the vacant state has:
a. A k-vector k−j
b. A velocity v+j We call this vacant state a positive “hole” (h+)
c. A positive charge +e
 Dynamics of Electrons and Holes
! !
dvh
If this hole is accelerated in an applied electric field: mh = +eE
dt
! !
dve
The corresponding equation for the electron is: me = -eE
dt
But earlier we deduced that the hole velocity is the
! !
same as that of the corresponding “missing” electron: vh = ve
! !
eE eE
So by equating the derivatives we find: =- mh = -me
mh me
However, note that near the top of a band the band curvature is negative,
so the effective electron mass is also negative. The corresponding hole
mass is then positive.
So the equation of motion of a “hole” in an electromagnetic
field is: !
! ! ! !
= e(E + vh ´ B )
dk h This explains why some
F ="
dt metals have positive RH.
 Band Effective Mass of an Electron
We can write the equation of motion of a Bloch electron in 1-D:
dv dv dk dv x dv g d æ 1 dE ö 1 d 2 E
ax = x = x x and = = çç ÷÷ =
dt dk x dt dk x dk x dk x è ! dk x ø ! dk x2

Also, from the acceleration theorem: dk x = Fx

dt !
This gives: 1 d 2 E æ Fx ö
ax = !2
2 ç ÷ or Fx = ax
! dk x è ! ø 2
d E
dk x2
With the analogy of F = ma, the band
effective mass is defined as: !2
m* º 2 or 1 1 d 2E
d E º 2
m * ! dk x2
dk x2
The effective mass depends on the electron’s energy and thus its location
in the band.
 ! "F ;
dvg 1 d2# !2 dvg
" 2 2 or F" 2 2 dt
. (27
dt ! dk d ##dk
Physical Meaning of Effective Mass
If we identify !2/(d2#/dk2) as a mass, then (27) assumes the form of Newton’s
second law. We define the effective mass m* by
For a free electron: 2 2 !2
! k m2* º =m
E= x
and
1 1 d #
. æ! ö
2
2m "
m* !2 dk2 çç ÷÷ (28
èmø
It generalize
It is easy to is easy to generalize thissolid
this to a 3-D to take account
to take of anofanisotropic
account electron en
an anisotropic
ergyenergy
electron surface, as for electrons
surface. We wouldinfind
Si orthat
Ge.m*
Weisintroduce the components
a second-order tensor of the
with 9reciprocal
components:effective mass tensor

! " ! "
1
2
1 d #k dv! 1
" 2 ; " F! , (29
m* !! ! dk! dk! dt m* !!

wherewhere
μ, 𝑣 are are Cartesian
!, "Cartesian coordinates.
coordinates.
The effective
Physicalmass concept is of
Interpretation useful because it
the Effective allows us to retain the
Mass
notion of a free-electron even when we have a periodic potential, as long
How can an electron of mass m when put into a crystal respond to applied
as we use m* to account for the effect of the lattice on the acceleration
fields as if the mass were m*? It is helpful to think of the process of Bragg re
of theflection
electron.
of electron waves in a lattice. Consider the weak interaction approxi
mation treated in Chapter 7. Near the bottom of the lower band the orbital is
represented quite adequately by a plane wave exp(ikx) with momentum !k
 Physical Meaning of Effective Mass

The effective mass is inversely proportional

to the curvature of the energy band.

Near the bottom of a nearly-free electron band

m* is approximately constant, but it increases
dramatically near the inflection point and even
becomes negative near the zone edge.
 Metals, Insulators, and Semiconductors
• Metals are solids with incompletely filled energy bands
• Semiconductors and insulators have a completely filled or empty bands and
an energy gap separating the highest filled and lowest unfilled band.
Semiconductors have a small energy gap (Eg < 2.0 eV).
Quick quiz: Does this mean a divalent element will always be an insulator?
Answer: In 1-D, yes, but not necessarily in 2-D or 3-D! Bands along different
directions in k-space can overlap, so that electrons can partially occupy both
of the overlapping bands and thus form a metal.

Energy

Insulator Metal Semimetal Semiconductor Semiconductor

Figure 1 Schematic electron occupancy of allowed energy bands for an insulator, metal, semi-

But it is true that only crystals with an even number of valence electrons in a
metal, and semiconductor. The vertical extent of the boxes indicates the allowed energy regions;
the shaded areas indicate the regions filled with electrons. In a semimetal (such as bismuth) one

primitive cell can be insulators.

band is almost filled and another band is nearly empty at absolute zero, but a pure semiconduc-
tor (such as silicon) becomes an insulator at absolute zero. The left of the two semiconductors
shown is at a finite temperature, with carriers excited thermally. The other semiconductor is
electron-deficient because of impurities.
Carrier Concentrations

Metals, semimetals, semiconductors

and insulators can be categorized by
their carrier concentrations at room
temperature. They can also be
differentiated by the electrical
resistivity at room temperature.
Semiconductors usually have this value
in the range of 10-2 to 109 ohm-cm,
and dependent on temperature
strongly. At absolute zero a pure,
perfect crystal of most semiconductors
will be an insulator with a resistivity
above 1014 ohm-cm.
 This is the probability that a conduction electron orbital is occupi
fe # exp[(!!")/kBT] .
approximation valid when fe # 1.
IntrinsicThis
Carrier Concentrations
The energy of an electron in the conduction band is
is the probability that a conduction electron orbit
INTRINSIC
approximation validthe"k CARRIER
when 2 2 CONCENTRATION
#!1.
Efec % k /2me ,of intrinsic
For a semiconductor, we want to know concentration
$
carries as function of We The energy
temperature of an electron in gap
the conduction band is
want the T, in terms
concentrationof itsofband
intrinsic Ecarriers
g. as a functio
where Ec is the energy at the conduction band edge, as in Fig.
8/13/04 4:23 PM Page 206
The energy of an ture,
me isinthe
electron terms
theofconduction
themass
ineffective bandof gap.
an We
is do"kthe
electron.
band $ Ecalculation
Thus for
from2 (6.20)
2
c % ! k /2me ,
thesimple
density
edges.
at " is We first calculate in terms of the chemical potential !
electrons The density
PM Pageexcited
of electronband states
at per unit volume
ch08.qxd 8/13/04 4:23
where206
Ec isto the
the conduction
energy
at ϵ isthe Fermi 1level. at the temperature
conduction band T. In
edge,

206
physicsm!isisthe
suppose
e
at for
called
effective mass
De(")
2# ! ! "
2meAt3/2the temperatures
$ of2 an 2electron. (" ! EThus
" is the conduction band of a semiconductor that " ! !
c)
1/2 of in
. from (6.20)

the Fermi-Dirac Fermi-Dirac

distributiondistribution
function reduces for ϵ−μto>> kBT
reduces
! "
The206
concentration of electrons in thetoconduction band 1 is 2me 3/2 1/2
feD#e(") $ 2
exp[(!!")/k T] . (" ! E c) .
#
B2
3/2 2# !
! " 1 2m
!
The concentration of electrons in e the conduction band is
n! De(")fe(")d"
The concentration
! 2 2 of exp(!/k BT) is
electrons $
2$ !
This is the probability that a conduction electron orbital is o
E

#
c

!
# (" " E ) exp(""/k T)d" , " 1 2me
! 3/2
(38)
approximation valid when
!
f # 1.
n! D (")f
1/2e e e
(")d" ! 2
!2
exp(!/kBT) $
Ecc B 2$
The energy of an electron in the conduction band is
#
Ec !
(" " Ec)1/2 exp(""/kBT)d" ,
which integrates to give Ec
"k $ Ec % !2k2/2me ,
which integrates
m ktoTgive3/2
which integrates to give n!2 !
where Ec is the energy
e B
2$!2 at the " exp[(! " E )/k T] .
mekBconduction
c B
band edge, as in
(39)
n!2
me is the effective mass of an2$!
T 3/2
electron.
2 ! "
exp[(! " Ec)/kBT] .
Thus from (6.20) the de
 !" #$!%$
2-' &$'" ()*!&!0'/: %$2-1'6I 2"6 2"6 2%%'(&)-*
!)"!F'6 2%%'(&)-*8 C 2"6
( ! -"// " " J" H"'12&!0'/: 8 2"6 &$' 6'"*!
" " D%$2-1'6I3
!" #$!%$ /),+()*!&!0'/:
'(" >*%"#&"
D 5*#-: 2"6 .?"?:
4" 2 $).)1'"')7* *'.!%)"67%&)-8 2-'"'12&!0'
&$' - &$'"
E $ - !
IG !".%G !.0 #!
%
%$2-1'
- " %$2-1'6I
! / #G"
6'"*!&: J !)"!F'6 2%
.7*&
E ! " # " H( !
! -
" H ! " H " D

Impurity Carrier Concentrations

"C # "C ! "C ! C C"//
2CI$).)1'"')7*
4" ()*!&!0' $?@!?N 2%
*'.!%)"67%&)-8
% 6'"*!&: G &$' "'1
9' %).('"*2&'6 9: 2" 'K72//: /2-1' - $ -(
" I$
%$2-1'
&7 -
% %
"43".
H% ! !! . $G"
#! ! C!7 H*''
/' !
"ED ! "
D # ?@3AI3
" "D 9'#" EH !$"-DH! % -1)0'-"*
-%).('"*2&'6
H
H%)"6!&!)" H "9:
H 0
2"&$'9%
$?@!?N 'K72//:
()*!&!)"/2-1')+ ()*!
",!13 D !
L'"%'8 &$' +)//)#!"1 "'7&-2/!&: D
I M= * /1)'(") =.+3%./.&*'.,# !7 ?N;" JO *=I
For,)-a &:(!%2/
&$' doped!.(7-!&:
,'-.! /'0'/semiconductor,
! , !" 2
%)"%'"&-2&!)"* weH?Ewant
$).)1'"')7*/:,)- ,!13
JJ0)""KK
?O -?N?@3AI3
J?E Hto
6)('6
&:(!%2/ %. know
$ -
"%"&'),#=
!.(7-!&:
JO H L'"%'8
*'.!%)"67%&)->
&7 %
I8 !" #$!%$.#"43".&$'
the &$'
'(" +)//)#!"1
%)"%'"&-2&!)"*
H !"6!5 42F5"'7&-2/!&:
concentration
! .
&,#-1&'.,# 0 #! !
H?E/' ?O
5*#- J?E!of
&*#%. ,#%6
6)::'#: 3
conductive
# ! "carries 0!672/ /),+
&$'
6)")-* &,#-1&'.>.'6
'("
,'-.! >*%"#&"
2"6 /'0'/ 5*#-:
2%%'(&)-* ='"+=
!".?"?:
!,)%%7(2&!)"
6) 2")& /),+
$).)1'"')7*/:
!"+/7'"%'.,#.N"- -,#
6)(
)"' 2")&$
D # $ !as function
! 6) ")& of temperature T and impurity energy levels
0!672/ 6)")-* 2"6
" 2%%'(&)-*
"! !"+/7'"%' )"' 2")&$'-8 &$' $?@!?M%
C
)+ &$' 6)")-* 9: '/'%&-)"* H#CI 2"6 )+ &$' JJ0)""KK
)+ 2%%'(&)-*
6)")-* /,)
9: "%"&'),#=
9: '/'%&-)"*
%"4*+3%"
$)/'*" H$ I.#!* H#
'("
2(
1!0'" &,#-1&'.,#
C!I. Q2"6&)7?R
2)+) 5*#-
2%%'(&)-*
7I 7I
&+ &*#
9: ,#
ST $)/'
*'
Ed and Ea. Neutrality %#
$!-$"%&'()**% D
! ! $5*#-:
9: condition demands ! in which
(#'("
1))6 !"2((-)P!.2&!)"
#$!%$ /),+ !"#
1))6 2((-)P!.2&!)" "
>*%"#&"
HD 9: ! " +#),',-./*
C.?"?: &0*/1'0, '&
%,; -,3.#$ %">"%=? 0,) '(.= =.+3%" &*=
! M=
"? !" #$!%$(E $*-/1)'(")
%
% ! ! =.+3%./.&*'.,# ;" *==1+"
"? '(*' " '(""
#C # ""CEC # "C E &? ! 'P($! " %" # # &? ! 'P($! " %" ! !
! " ! ! ! !'# ! " " ! $?@!?Q "!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
2% !'
$?@!?N 2%
C C C , C C HC %/),+ C , I-,#,)=:
and &,#-1&'.>.'6 ( ='"+=
* - $ - .,#.N"-
! - "3 #.?"?
" '(*' -
7 !! !
H
#! !!7I %"!! ;"
2" H H
E *!/1)'(")
"4C),$?@!?Q 49% 5$?@!?N ! #9%
E " "? E ! 7I ST "?
"D # "#DD # # "DED&?!!"'P($$
D ! ,
" $D %"! !'"D /,)!# #M= "4*+3%"
"#
D C# "2(
D &? !. Q
C
3=.+3%./.&*'.,#
! 7?R
'P($$ " $D&+ !'*' *==1+"
"
TII
6 U9 '(*'
'(.= '.
%,; -,3.#$
&,#-1&'.>.'6.?"??O ;.'(%">"%=?
='"+= 0,)
278?7C/),+ '(.=.,#.N"-
=.+3%" -,#,)=:
&*=": #.. 278?7L9 .= )
.?"? '(*
?NE JO " *9 ,#" (*=
,)- &:(!%2/ !.(7-!&: %)"%'"&-2&!)"* /,)For "Dthe
H?E
"4*+3%" #% case
J?E "2( %.
D. !
Q" of
7?RI8
DI) ! a7I#$!%$
!" 7I
pure
&+ &$'
ST
n-type
*' !"6!5
TII U9 '(.=
0!672/E 6)")-* ( * - $ - ! - "
C – ED = Ed
2"6 2%%'(&)-* 6) ")&%,; !"+/7'"%'
!"'& H )"' 2")&$'-8
#9$:#&&'0, H.0: &$' 1"# )%%7(2&!)"
(0,7/(1'0, #*#(
H
!
semiconductor,
-,3.#$
,)-
%">"%=? 0,)
&:(!%2/ with
!.(7-!&: N
'(.= D >>
+=.+3%" ni,&*=":7278?7L9
%)"%'"&-2&!)"*
.=
H?E
)+ &$' 6)")-* 9: '/'%&-)"* H#CI 2"6 .?"? )+ 7/(10:&
2%%'(&)-* ( *9: $)/'*
H 7
*9-,#" H$DI !* 1!0'" &) 2
!
;.'( % (0,1)',&
278?7C (*=I 1"# .0**0<',- *'+'1',-
1))6 2((-)P!.2&!)" 9: 0!672/(*- 6)")-*
!$1"# -H2"6 " 7 % "43&".
! -H2%%'(&)-* 6) H "!*0<
")& .0 #!!
!"+/7'
>?H>. 1#+$#:)1/:# ! '& &0 1"
)+ &$' 6)")-* 9: '/'%&-)"* 7 H# C I 2"6 )+ 2%%
#C # "EC # "C &? ! 'P($!C " !,.?"? (!'*"?-278?7C
%"! ;.'( !H '(" 7 !*9 ,#" (*= $?@!?Q 2% !
1))6!.'( 2((-)P!.2&!)" ("%3
! 7@% "43&". ,/ 9:
$7 ! ! H
278?C *9:
! ! .0 #! /'
! . 0 " 5" "
&*#
"? 5##3 !# $# 7% $?@!?Q
4
"D # #ED # #D &? ! 'P($$, " $D %"! !' ! E #.. 9% ! "
!.'( Since*
(#'("
C- # H "7
("%3 ,/!# "
278?C
C G7 % "43&".C. &?
*9: ! !
!./ 0'P($!
&*# .5" C! " !
"43)"=="-
! / , %" ! !'
>.*
"(!- "// #"
G
$!" .00 #!! /'
H "
1"#,G0,#.EG !")& !#/ # 0 !! / 'P($$ " $ %"! !'"
"
"(!-D #
!.'( '(""//("%3$
#
#" D $ "
D
.&?
,/##############
278?C *9:
! ."0 &*# ,5" "43)"=="-
D >.*
', 6."%- #
"$7 !2 ! !
', 6."%- "G & .G !!#/3 #6 #
. 0 !! /
#.. !
"(!-"// #" $" H " -
( *-H "
(* "
- !! /
.%)"%'"&-25
!"#$ %&$'$ UP(/2"2&!)" )+ &$' ")&2&!)" %)..)"/: 7*'6
', >,
+)- 1"'&
%2--!'-
6."%- :#-'0,C
2"6 7 /% " <"#:#
.- !!.(7-!&:
! G ) &/..'('#,1*%
(!- G *):
7 % ")+ &$' (!-
!"#$ %&$'$ UP(/2"2&!)" ")&2&!)" %)..)"/: "// 7*'6 +)- %2--!'- 2
 the absence ofexp[("
anfconcentration
electron. 1 p. Therelated to the electron 1 distribution function fe b
$
equilibrium h !" 1" !)/kWe B
of
T] have
,
holes distribution
isT]maintained ! function
byan" black-body f h for holes is
photon radia
exp[("
related to the electron distribution " !)/k B &
the 1
absence exp[(! of ")/k
electron.
function fe by fh ! 1 "1 fe, because a hole is B T] We& 1
have (40
1 generate electron-hole pairs at a rate A(T)
provided The(!
the ") f#
distribution
"absence h! kofB1T.
$function
an exp[("f"
" electron.
exp[(" for
h We
" !)/k
!)/k holes
have T]
T] &,is 1related
!
exp[(! to the
" electron
")/k T] & distribution
1 1
BB
combination fh ! 1 "
reaction B
e ! h " photon. (40) !The
function
If the holesfe near
by the top of the 1 valence band behave
1 as exp[("
particles
" !)/k B with1
T] & exp
provided (! $
" fexp[("
")!# 1" k "T.!)/kBT] , !
effective mass mh, the densityBexp[("
h
of hole " !)/k states
B T] & is
1 givenexp[(! by" ")/k$BT] exp[("
& 1 dn/dt
" !)/kB" T](40),
A(T) # B(T
provided If (!
the" holes
") # knear T. the top of the valence band behave as particles with
$ Bexp[(" " !)/kBT] , 3%2provided
! "
provided (! " ") # kBT.
If the holes
effective massnear mDh, (") thedensity
the top 1 of2m of thehhole In equilibrium
valence
states band
is1/2given by "as0, particles
behave
dn/dt dp/dt " 0,
withwhenb
provided h ! (E v " If ")the ,
holes near the top of the (41)valence
effective mass (! , the kBT. 2$of
") #density
mh"
at ϵ ishole !2 states is given the by product of the electron a
2
The density of hole states Because
! "
effective
2mh band behave
3 2 mass mh, the density of hole states is gi
If the holes near the top of the 1 valence %
1/2 as particles with
Dh(") ! (E , (41
! " independent of impurity concentration 3%2at a
2m
1 hole2 states 3%2 " ")
where E# iseffective
the energy mass at mhthe Dhvalence
(") ! ofband edge. Proceeding
h 2 is given vby1/2
as in (38) we obtain
! "
, the density 2$ 2 ! (E v " ") , 1 2m (41)
h
2$2 !of a3%small proportion ofDh(")a suitable
! 2 impurity (Ev "

#
2
2$ !

! ! ""
2m
1 mhvalence
kThis
2
BT
Ec 3/2
The
where concentration
where E#pis! E is the
the energy p
energy atof holes
at
D
the the
(") in
!
valence the
valence band band
h
(Eband
edge.
result
edge. isProceeding
")is
1/2
v "Proceeding important
, as in as(38)
in in (38)
practice—we we obtain
(41)obtain
we ca
#
Dh(")fh(")d" ! 22$ ! 2 whereexp[(E
h 2 2
E# is the " !)/k T]
c energyBat the valence band edge. P (42)
2$!
##
tionmn k!3/2Tp in 3/2 an impure crystal, sometimes

! ! " "
"! Ec

#
Ec m k T
! "
where Ep# is the!energy at (")f
the valence band h B
edge. Proceeding asE !)/k
in"(38) we T]obtain m(42) hkBT(42
3/2
! p
or the concentration p"!of"! D D
(")f (")d" (")d"
holes in the valence
h hh h ! 2 ! 2 h
duction
B
band. of
2 exp[(E exp[(E
suitable
p! c "impurities.
c
c !)/kT]
Dh(")fh(")d" ! 2
B B Such a reduc
2$! 2$!2

#
2$!2
m
! k T
"
E 3/2 "!
We the
multiply together BIn anexp[(E
pintrinsic semiconductor the nu
p ! pthe Dhexpressions forh n and to obtain the equilibrium
c

for for concentration of (")fh(")d"

holes in ! 2 valence
the band. " !)/k T] (42)
the concentration p of holes in the valence
for band.
the concentration c B
p of equilibrium
holes in the valence b
elation,We multiply
with
We We the
multiply
together
energy
together
"!the
gap Ethe
expressions
! E
g expressions
c " E v
for
2$!
number
as n2 and
in
for Fig.
n of pholes,
and
to obtain
18,p to obtain
the
because thethe thermal exci
equilibrium
relation multiply together the expressions We
for multiply together the
n and p to obtain the equilibrium expressions for n a
for the concentration p of holes in the valence
hole band.
in thewith valence band.
relation, with the energy gap E3g ! Ec " v as in Fig.
Erelation, 18, gapThusEg ! Efrom (43)
! !" "
relation, with the energy k T gap E ! E " E as in Fig.the energy
18, c " Ev as in
We multiply together B the expressions g 3/2c for nv and p to obtain the equilibrium
np ! 4 (m c3 h note
m ) exp("E intrinsic g/kBT) and , . Eg " Ec k#T Ev3, (43)
! "
relation, with the energy 2kBTE !
gap
2$! Ec3" E3/2 v as in Fig. 18,

!! " "
g B
np ! 4 k
2B T (m cm h ) exp("E g /k B T) .
np ! 4 (mcmh(43) )3/2 exp(
3/2
np !2$! 4 kBT 23 (mcmh) exp("Eg/kBT) . 2$!k T 3/2 (43 2
This useful result The
This useful
s 2.10 $ 10 cm 19 result
This$useful
is 2.10
doesproduct
1019 cm result"6
not np
"6 does not involve
, 2.89 $
, does
involve
2.89 $
!of
10
4np 2$!
cm
the
is
not10involve
26
Fermi
constant
26 2$! "6the Fermi
2
(mcmh)3/2at
,
cm"6the
and
, and
level
6.55
isFermi
a
exp("E
This level
2.10
given
6.55
$
!.g/k
useful !.
10
$level
At
$1910
10
300 K the value
temperature.
BT) n. i " pi " 2
12 At does
result
cm
cm
300
12"6
!.
"6 not
At
,
K the
300
, 2.89
cm "6for
! the
value
involve
2"!
K10the
,$for 26
"
B of np
(43) (memh
2 of
the
actual
cm
the value
np lev
Fermi
, and
actual
"6
of 6.5
np
bandband This useful
structures of result
Si, Ge, does
and not
GaAs, involve the
respectively. Fermi level !. At 300 K the value of np
2.10 $ 1019
is structures of19cm Si, "6Ge,
"6 and $
, 2.89 GaAs,10
26
26 respectively.
cm
"6
"6band
, and structures
6.5512$of10 Si,
"6 Ge,
12
cm and"6 GaAs, respectivel
, for the actua
 duction
mall of suitable
proportion
number of aholes,
of impurities.
suitable impurity
because
$ Such
exp[(" the a reduction
tothermal
increase n, say,is must
excitation called compensation.
decrease
of an electron
independent p. of impurity
leaves behin
concen
" !)/k BT] ,
esult isIn
ntration important
holeof theinvalence
anelectrons
inintrinsic practice—we band.can
in semiconductor
the conduction Thus reduce
the
band isthe
from totalwe
number
(43) carrier
ofhave, ofconcentra-
electrons
aletting istheequal
small proportion to athe
of
subscriptsuitab
id
! p innote
anFor provided
impure (!
crystal, " ") #
sometimes k T.
B enormously, by the controlled This result is important
intro-leaves in pract
number of holes,
intrinsic because
carriers, the
n andthermal
p , excitation of an electron behind a
nhole
of suitable
n! #!intrinsic
in theD impurities.
and
E effective mass 2$
E
If the holes1 near
Such
g
! a2 Thus
e(")fe(")d"band.
valence
" i E
! "
c #
reduction
mh, the
i
3/2E
from
v

!2density of3/2
,
2methe top of the valence band
is called
exp(!/k
(43) BT)
wecompensation.
$ have,
hole states is given letting
behave
by the
duction
as particles
tion n ! p in an impure crystal,
subscript i de-
with
of suitable impurities. Su
an intrinsic semiconductor the number
! " ! "
kBT of electrons is equal(38) to the
c
note intrinsic and E " E # E ,
#
3/4
er of holes, because
!
the
g
thermal
c 2
ni1/2" pi "excitation
v (m2m emhh) 3%2 exp(#
2 of1an electron leaves
Eg/2k
In
behind
anBT)
a
.
intrinsic semiconduc(4
(" " Ec) exp(""/k DB2"!T)d" ,
h(") ! (Ev "number 1/2
") , of holes, because the (41)th

! "
2 2
n the valence band. Thus from (43)
E kBT we have, letting
3/2 2$ ! the subscript de-valence band. Thus
ithe
hole in
c
3/4
ntrinsic andToThe
obtain n#
Ethe i"
Eg "intrinsic Epv,i "
Fermi 2 μ,concentration
level
carrier (m mh) nexp(#
2we starte from = p
depends , Eg/2kexponentially
so BT) . on (45)
Eg#/2k
2"! i i
grates to givewhere E# is the energy at the valence band edge. Proceeding as in (38)g we obtain
c c note
v intrinsic and E " E EB
where Eg is the energy gap. We set (39) equal to (42) to obtain, for the Fer
! ! ""# ! " !
The m k Tk T3/2 3/2 E mhkBT 3/2 kBT
levelnnintrinsic carrier concentration
3/4 of the depends exponentially on E /2k T,
c
as
" measured
p
i ! 2i " e
2 B B
pfrom (m the
m
D top
)
(")f exp(#
(")d" E valence
/2k
2 T) . band,exp[(E
(39)c (45) n
T] " p " 2 (42)
2 exp[(! h Ech)/kBT] . g
! e " ! B " !)/kB i g i B
where Eg is the energy gap. We set (39) equal to (42) to obtain, for the Fermi2"!
2
2$! 2"!
2 "! 2$!
3/2
level
roblem for the
asis measured
solved for from
n when
exp(2!/k
concentration
the top
is p of T)
holes" (m
in
of theItvalence
known.B is the
h
useful
/m e)
valence
band,
to
exp(E
band.
calculateg/kThe
B
the
;
T) intrinsic carrier conc (4
he intrinsic carrier concentration
! depends exponentially
then,We multiply together the expressions for n and on E g /2k B T,
p to
where Eobtain the equilibrium
g is the energy gap. We
mEconcentration
g is the energy
of holes p. The distribution
gap. We set (39) !equal "gap
1
2 E
function
3
!!4Ek3/2
tog (42) to ln f
Tobtain,
B" (m
h for holes
/methe
for ) .18, is
Fermi (4
relation, with the energy
exp(2!/k T) " (mEg/m ) Ev as h/k
exp(E in Fig.
level
T) ; as measured from the
(46)top
the electron distribution functionBfe by fh !h 1 "
s measured from the top of the valence band, e fe, because
c g B a hole is

! "
e of an electron. We have
E3/2g4!k4BTkBT (m
1 3 3
!" ln c(m ) /mexp("E
3/2
e) .
exp(2!/k
(47) BT
np2! m hh g/kBT) . (43)
exp(2!/k 1 BT) " (mh/me) exp(E 2$! 1g/kBT) ;
2 (46) 1
fh ! 1 " ! !"2E
exp[("useful
"!!)/k 1 T] &31 exp[(! " ")/k & 1 level !.(40)
BT] Fermi
If mThis
h = m e , " result
then
2
BE ! does
g μ =
4 kE
Bg
T/2not
ln (minvolve
andh/m
the
e) . the
Fermi At (47)
level is in the 300 K theofvalue
middle the of np
is"2.10
$ exp[("
forbidden $BT]
gap.
!)/k 1019 26 "6 12
, cm , 2.89 $ 10 cm , and 6.55 $ 10 cm , for the actual
"6 "6

band structures of Si, Ge, and GaAs, respectively.

! " ") # kBT. We have nowhere assumed in the derivation that the material is intrinsic:
holes near the top ofholds
the result the valence bandionization
for impurity behave asasparticles
well. Thewith
only assumption made is
 Impurity Conductivity
Certain impurities and imperfections drastically affect the electrical
properties of a semiconductor. The addition of boron to silicon in the
proportion of 1 boron atom to 105 silicon atoms increases the
conductivity of pure at room temperature by a factor of 103. The
deliberate addition of impurities to a semiconductor is called doping.
We consider the effect of impurities in silicon and germanium. These
elements crystallize in the diamond structure. Each atom forms four
covalent bonds, one with each of its nearest neighbors, corresponding to
the chemical valence four. If an impurity atom of valence five, such as
phosphorus, arsenic, or antimony, is substituted in the lattice in place of
a normal atom, there will be one valence electron from the impurity
atom left over after the four covalent bonds are established with the
nearest neighbors, that is, after the impurity atom has been
accommodated in the structure with as little disturbance as possible.
Impurity atoms that can give up an electron are called donors.
 '% '() ;%"'/"<<: %$ '() ;%"#<;'/%" =!"#5
A3 ;!" =) 3))" $&%: '() )M>)&/:)"'!,

Donor States
#)3;&/>'/%" %$ '() #%"%& =2 ! (2#&%1)"/; :
%&#)& %$ :!1"/'<#) %$ '() /%"/0!'/%" )")&12
/:><&/'/)3. 3<;( !3 9. A3 !"# @=. 3(%<,# (!
The impurity atoms of valence five such as P, As, and Sb are called donors
D()" >&)3)"' /" '() 3!:) 3):/;%"#<;'%& (
because they donate electrons to the conduction band+J5H
## /" B!=,) in order to complete
3(%D. (%D)-)&. '(!' '() -
the covalent bonds with neighbor atoms, leaving electrons
'% #%"%&5 D=1> inR3#!-?
the band.
35 4'"!*3-,$*

4*8'")&!* !"#$ %&$'$ Y<!,/'!'/-) >%3/'/%" %$ '() 1&%<"# 3'!') ,)

The extra!"#$ %&$' () *$
electron moves in the %'#%'('-&)&!3- 35 &8'e/ϵr
coulomb potential
'() :/"/:<:
'K'*& 35
the) impurity
,3-3% B(Fion,
of ;%"#<;'/%"
%$ '()
)-, )- )*
=!"# #E !"# '() :
(!+!*3- +)&&!*'1 <8' /)+'-*'95!/' #83(#83%$( )&3" !( !-*3%#3%)&', !- &8' +)&&!*'
where ϵ in a covalent crystal is the static dielectric
Z<!"'/'/)3 constant of'()
## !"# #! !&) the/%"/0!'/%"
medium. )")&1/)3 %$ '(
(!+!*3- )&3"1 <8' 5!5&8 /)+'-*' '+'*&%3- 35 &8' #83(#83%$( )&3" !( -3& %'T$!%
)-, !( &8$( 3-+0 7')A+0 :3$-,1 <8' :!-,!-? '-'%?0 *)- :' '(&!")&', :0 &%')
 m by the effective mass me to obtain
We estimate the ionization energy of the donor impurity. The Bohr theory

" #
4
of the hydrogen atom may be modified to take einto
me account m
the
13.6Bohr e dielectric
We estimate the ionization energy(CGS) of the donor
Ed ! impurity.
2 2
! The 2 m
theory
eV ;
constant
of theofhydrogen
the mediumatom and
may the effectiveto
be modified mass
takeofinto
2!an!account
electron in dielectric
! the the periodic
4
potential of the
constant crystal.
of the The ionization
medium energy of
and the effective atomic
mass hydrogen
of an electronisin#ethe m$2!2
in CGS and #e
periodic
4
m/2(4"!
potential of the
2 as the
in SI. The
0!) crystal.
donor energy
ionization ionization energyhydrogen
of atomic of the semico
is
In−ethe
4m/2ℏ 2 . In the semiconductor
semiconductor with
with dielectric The Bohr !constant
dielectric
constant radius ofϵ we
we replace theereplace
2ground
e2 state
by e2/! and
m bybythe
e2/ϵ and m bymass
effective the effective mass4"!
me to obtain me0! to2/me 2
obtainin SI. Thus the Bohr radius of th

e4me
" # " #
4
13.6 me !!2 0.53!e me
(CGS) Ed ! 2 2 ! eV (CGS)
; and ad ! (SI)2 E
!d ! Å ;2 (51)
2! ! !2 m mee m /m
2(4"!!
e 0!)

Ed is
as the the ionization
donor ionizationenergy
energyand ad the
of the Bohrapplication
The radius of the
semiconductor. ofdonor.
impurity state theory
2 2
The Bohr radius of the ground plicated
state by
of the anisotropic
hydrogen is
To obtain a general impression of the impurity levels we use me ≃ 0.1! $ effective
me mass
inmCGS
for orof
!2/me2 in in
4"!0electrons SI.germanium
Thus the Bohr
and m dielectric
radius constant
of the donor is has the more important e
e ≃ 0.2 m in silicon. The static dielectric
constant ϵ is 15.8 for Ge and 11.7 forit enters as the
Si. Then, we square,
obtain whereas the effective

" #
2
!!2 0.53! To obtain a general 4"!!0!
impression of(52)
the i
(CGS) ad ! ! Å ; (SI) a !
mee2 meE/m
d = 5 meV and ad = 80 Å for Ge;
for electrons in germanium
d
e2 me ! 0.2
meand
Ed = 20 meV and ad = 30
constant Å for Si.
is given in Table 4. The ionization e
The application of impurity state13.6 theory to germanium
eV. For germaniumand thesilicon
donor is com-
ionization
plicated by the anisotropic effective mass of the conduction electrons. But the
reduced with respect to hydrogen by th
dielectric constant has the more important effect on the donor energy because
 /:><&/'/)3. 3<;( !3 9. A3 !"# @=. 3(%<,# (!-) '() 3!:) /%"/0!'/%" )"
D()" >&)3)"' /" '() 3!:) 3):/;%"#<;'%& (%3'5 B() )M>)&/:)"'!, -
Acceptor States
## /" B!=,) +J5H 3(%D. (%D)-)&. '(!' '() -!,<)3 -!&2 3%:)D(!' $&%:
D=1> R3#!-? 35 4'"!*3-,$*&3%( P=S
'% #%"%&5

35 Y<!,/'!'/-)
!"#$ %&$'$
%'#%'('-&)&!3- 35 &8' 'K'*& ) ,3-3% >%3/'/%"
B(F )-,%$)- '())**'#&3%
1&%<"# 3'!')
B*F,)-),3
!- )%$ #%"%&3 !"# !;;)>'%&3
Trivalent impurities such as %$ B, '()
Al,;%"#<;'/%"
Ga, and In are#Ecalled
!"# '()acceptors because
*'95!/' #83(#83%$( )&3"'()!( :/"/:<:
!-*3%#3%)&', =!"#
!- &8' +)&&!*' )& &8' (!&':!M/:<:
35 ) %$ '() -!,)";) =!"#
they accept electrons
Z<!"'/'/)3from the
## !"# valence
#! !&) band )")&1/)3
'() /%"/0!'/%" in order to #%"%&
%$ '() complete the &)3>);'
!"# !;;)>'%&.
)+'-*' '+'*&%3- 35 &8' #83(#83%$( )&3" !( -3& %'T$!%', 53% :3-,!-?
:3$-,1covalent bonds'-'%?0
<8' :!-,!-? with neighbor atoms, leaving
*)- :' '(&!")&', holes in&8'
:0 &%')&!-? the(0(&'"
band.
',,', !- ) ,!'+'*&%!* "',!$"1 <8' *)(' 35 )- )**'#&3% B*F *)- :' ,'9
'-*'9&8%'' :3%3- )**'#&( )- '+'*&%3- 5%3" &8' (!+!*3- +)&&!*'1 <8' 83+'
 icmobility
ity Mobility
is defined to be positive
!e ! e"e/me ;
for both
!h ! e"h/mh ,
is defined to be positive for both electrons and holes, alth
electrons and holes, although
(50)

ervelocities
mobility where " is the collision time.
are
is the
drift velocities opposite
magnitude
are Carrier
inopposite
aofgiven
the drift field.
in Mobility
By
avelocity
given writing
offield.
a charge
!e By or !carrier
h with per
The mobilities depend on temperature as a modest power law.writing
The tem- ! e or ! h
for the
ectric
scripts electron
field:
The the or
forperature
mobility ishole
electron mobility
the magnitude
dependence or
of thehole weof thecandrift
mobility
conductivity avoid any
we
invelocity
the confusion
can
of
intrinsic avoidwill
a charge
region be-
any be confusio
carrier per
the chemical
unit potential
dominated
electric by theand
field: ! as
! the
exponential
!v mobility.
dependence
!/E . exp("Eg/2kBT) of the carrier con-
(48)
en ! as thecentration,
chemical Eq. (45). potential and as the mobility.
ectrical conductivity is the sum of the electron and hole contributions:
obility isThe
The electrical electrical
defined Tableto3 be conductivity
conductivity
gives isfor
experimental
positive theisvalues
theofelectrons
sum
both sum
of
the
2
the of electron
mobility the electron
at room
and holes, and and
holehole
temperature.
although contribut
The # ! (ne!ein# ape!
mobility in SI units is
h) , field. By writing ! or !
expressed in m /V-s and is 10 of the (49)with
mobility in
"4
contributions:
rift velocities are opposite given
practical units. For most substances the values quoted are limited by thehscat- e
pts for the electron or hole
#!
mobility
(ne! we e# can
pe! h) , any
avoid confusion
nd p are the concentrations of electrons and holes. In Chapter
tering of carriers by thermal phonons. The hole mobilities typically the be-
are6smaller
!tyasofnthe than the potential
chemical electron mobilities
and asbecause
the of the occurrence of band degeneracy at
mobility.
ere aand
charge
p
Thethe q
are
drift was
the found
velocity to
concentrations be v ! q"E/m, whence
of electrons and holes. In Chapter
valence bandof a charge
edge at the zoneq was found
center, to bemaking
thereby v = q𝜏E/m,
possible whence
interband
e electrical conductivity is the sum ofthethemobility
electron and hole contributions:
t velocity of !ae !charge
scattering e"processes
e/m e q; was
that !found
reduce
h ! e" h to
/m h be , v ! q"E/m,
considerably.
where whence
𝜏 is the collision (50)time.
# ! (ne!e # pe!at h) room
, temperature, in cm2/V-s (49)
the collision time. !e ! e"e/me ;
Table 3 Carrier mobilities
!h ! e"h/mh ,
and p are
nobilities the concentrations
depend of electrons and holes.law. In Chapter 6 the
Crystal on Electrons
temperature Holesas a modest Crystal power Electrons The tem- Holes
ere " isofthe
locity
dependence collision
a charge
of the q was time.
found to beinv !
conductivity theq"E/m, intrinsic whence region will be
Diamond 1800 1200 GaAs 8000 300
The mobilities
by the exponential
Si depend
!e !dependence
1350/m ;
e" e e
on temperature
exp("E
480 !h ! e"GaSb gh/2k
/mhBT) as a modest
, of the5000 power
carrier con- 1000(50)law. The
Ge 3600 1800 PbS 550 600
ature
Eq. (45).dependence
InSb
of
800
the conductivity
450 PbSe
in the 1020
intrinsic 930
region wi
"3isgives
the collision
experimentaltime.30000 values of dependence
the
450 mobility PbTe atexp("E room 2500 temperature.
minated byInPthe exponential
InAs
g /2k B T) of
1000the carrier
eitymobilities
in SI unitsdepend
is on
expressed
4500 temperature
in m
100
2
/V-s as
and aAgCl
modest
is 10 "4 power50law. The —
of the mobility intem-
tration,
e Eq.
dependence
AlAs(45).
of the
280
conductivity
—
in
KBr (100 K)
the intrinsic
100
region will
—
nits. For most AlSb substances 900 the values 400 quoted SiC are limited100 by the scat- 10–20 be
Table
ted by 3 gives
the experimental
exponential dependence valuesexp("E of gthe/2k mobility
T) of the at room
carrier con- tempera
arriers by thermal phonons. The hole mobilities typically are smaller B
 plicated the
ction (43) requires and np theproduct
equations toare
be solved
constant by at
numerical
a given methods.
tempera- However, the
of mass willaction (43) requires the np product to be constant at a given temp
excess of donors Thermoelectric
increase the electron Effect
concentration and de-
e hole concentration; the sum n ! p will increase.the
ture. An excess of donors will increase Theelectron concentration and
conductivity
4:23 PM Page 215
ase as The pcrease
if thethe
n !thermoelectric hole
mobilities concentration;
effect is the
are the sum
directasconversion
equal, n of p will increase.
! temperature
8 Semiconductor
in Fig. 22. The conduct
differences
Crystals 215
will increase
to electric voltage as nand if the versa
! p vice mobilitiesvia are equal, as in Fig. The
a thermocouple. 22. term
:23 PM Page 215
Ec is the "thermoelectric
energy effect" encompasses
at the conduction
THERMOELECTRIC band edge.three
EFFECTS We referseparately
the energyidentified
to effects:
mi level thebecause
Seebeck effect, conductors
different THERMOELECTRIC
Peltier effect, and Thomson
in contact EFFECTS
have theeffect.
same The Seebeck and
Fermi
he
derenergy fluxeffects
a Peltier that accompanies
semiconductor aremaintained
different themanifestations
charge flux is of
at a constant the same physical
temperature anprocess;
while Crystals
Consider a semiconductor maintained8 atSemiconductor a constant temperature whil
eld drivestextbooks jmay
through refer
it an# to!this
electric3 process
currentas)Ethe Peltier–Seebeck
density jq. current effect.
If the current is. If the curre
electric n(E
U "field c drives kBT)(#!
$ 2through ite an. electric (55)
density
8 Semiconductor Crystals j q 215
nly by electrons,
The charge
carried the
flux
only charge
is, assumingflux
by conduction
electrons,is
onlythe electrons
charge involved,
flux is refer the energy to
e E
e Peltier
c is the energy at the
coefficient ! is defined by jU " !jq; or the band edge. We energy carried
Fermi level because jq # n("e)("!
different e)E #j ne!
conductors , where
E contact
ein μe ne!
have isthe ,to (54)
theEsame
electron mobility.
Fermi
E c is the
charge. energy
For at
electrons,the conduction band q# n("e)("!
edge. We e)E
refer the
# energy
e

isThe
ermi the energy
level
Then flux
because
the
electron
where
that accompanies
different
energy
mobility.
!
! e is
"
flux
the
#(E
conductors
due
The #
to the
average
electron
! $
the
k
charge
in contact
3 electrons
energy
mobility.
T)/e
isflux
have
The
is
the same Fermi
transported
average by an
energy elec-
transported
(56) by an e
The energy flux that accompaniese c the charge
2 B flux is
erred to the Fermi
tron j level
is referred" n(E !,to#the !$ Fermi
3
level !,
k T)(#! )E . (55)
U c 2 B e
negative because jUthe energy
" n(Ec # ! flux
$ 2 k3is
3
opposite
BT)(#! e)E . to the charge flux.(55)
For
(Ec "carries
each electron kBT , (Ec " !) ! 2 kBT ,
!) ! 2energy 3
The Peltier coefficient ! is defined by j " !jq; or the energy carried
he Peltier coefficient ! is defined by jU " !jUq; or the energy carried
Thej Peltier
nit charge. For coefficient
electrons, ⫪ is defined by3 j = ⫪j ; or the energy carried per unit
hE ;
q " pe!
nit charge. For electrons, jU " p(! # Ev $ 2kBUT)!hE q , (57)
charge. So, 3
Ev is the energy at the ! "
!evalence #(E
" #(Ec band
e # 3! $
# ! $edge.
c 2 kBT)/e
Thus
2 kBT)/e , for electrons (56) (56)

snegative
negativebecause
because !the
the energy
" (!
henergy Ev $flux
#flux
3
is opposite
is2kopposite
BT)/e forto
, to the the charge
charge
holes flux. For
(58)
flux. For
 Applications of Thermoelectric Effect

where S is the Seebeck coefficient (also known as thermopower).

temperature-sensing thermocouple thermoelectric cooler

K-type thermocouple (chromel-alumel) in

the standard thermocouple measurement
configuration. The measured voltage V can
be used to calculate temperature Tsense,
provided that temperature Tref is known.
 Bloch oscillations
The one-dimensional equation of motion for an electron with wave vector
k in a constant electric field E is:

Suppose the dispersion relation for a given band is

The group velocity v of the electron is given by

and the electron position x can be computed as a function of time:

This shows that the electron oscillates in real space. The angular
frequency of the oscillations is given by

Bloch oscillations are not routinely observed because the electrons in a

periodic system undergo collisions with ions in the lattice much too
frequently, on the time scale of t (»10-14 s).
&-" 7$'' 4/'&$2" !in steady state. " !$*#'"
" ;+"')"&$.&!<*$2,"&+.
"##'$.") " /##/!+&" " &/
"+,)3.&+/,
"H1 &/ !=
" /,"+! $,/&-"%
$##'+") <#
,/
7 ?-+.- 03+')!
;+"') 3# +, &-" +,)3.&+/,
<*$2,"&+. +, &-"
&$.&! !=#'$.")
+!)+%".&+/,
$##'+") +! ,/%*$'
/##/!+&" /,"
&/ &-"$,/&-"%
.3%%",&9 <#
7$'' (c)
4/'&$2" !7 &-" .$, &/ 0" /,"
*"$!3%") </#",
$.&'( ./*#",!$&") 0( &-" I/%",&J ;/%." /, $, "'".&%/,6
&$.&! #'$.") /##/!+&" &$.&!
7$''
&/ #'$.")
?-+.-
/," $,/&-"% /##/!+&"
$'!/ $.&!
4/'&$2" !7+, .$, 0"
<#"%#",)+.3'$%*"$!3%")$,/&-"%
&/ </#",
! <#"
$,)
# )+%".&+/,9Figure 14 The
A-" I/%",&J
4"'/.+&( ! $ cross-section
+!K
;/%." standard
7$'' 4/'&$2" Hall Effect
geometry
/, $, "'".&%/,
!
is placed in a magnetic
7
*/4+,2
.$, 7$''
0"
for
!-/?,
field
&-"
the $!.-"*$&+.$''(
+, &-"
!-/?, 4/'&$2"
*"$!3%")
B
Hall effect:
)+%".&+/,
!
!.-"*$&+.$''(
z , as 4/'&$2"
7$''
</#",
7
?+&-
.$, a rod-shaped
in (a). An!electric
0"+, *"$!3%")
+,
.+%.3+&
B+29 CDE9F9
B+29
?-+.-
specimen
CDE9F9
*"$!3%"*",&6
field
G+,."
</#",
G+,."
Ex applied
03+')! 3# +, +,
!-/?, !.-"*$&+.$''(
electrodes causes an electric current $.&'(
!-/?,
&-"
+, 7$''
B+29
density
!.-"*$&+.$''(
4/'&$2"
CDE9F9 G+,."
jx to flow down ! 7
7+, B+29
?-+.-
*"$!3%") CDE9F9
03+')!
the I/%",&J +, &-"
rod. The;/%."
G+,."
3#$0!"
drift
To# separately
% ! " &#! ! !! # determine
" & ! the carrier concentration./*#",!$&") n and 0(the &-"mobility μ
negatively-charged
#
&-" 7$'' 4/'&$2"
electrons !7 &-"$.&'(
immediately?-+.-7$'' 4/'&$2"
./*#",!$&")
03+')!
after 3# !
the electric +,70( ?-+.-
&-" &-"
field!$*#'" 03+')!
isI/%",&J 3#
;/%."
+, &-"
applied as sho+,
#
appearing$ ' " &in
! # the @ #conductivity σ = isneμ, one )+%".&+/,9
# )+%".&+/,9
measures A-"
both I/%",&J
the&-" ;/%." /, /,;/%."
$, "'""'"/
#&!deflection # in the "y./*#",!$&")
$.&'( direction #&-"
0($.&'(
caused by./*#",!$&")
the magnetic
I/%",&J A-"
;/%." I/%",&J
!CDE#F! 0(
field.
/, $,conductivity
;/%."
I/%",&J
Electrons
"'".&%/,6 $,
accumula
?-+.-
of 4"'/.+&( +!K
! $ +!K
7"%"and !7the
! # (the +!rod
L)Hall and # a)+%".&+/,9
&-"effect.
!/1.$''") positive
The ion
7$''Lorentz
A-"
;+"')9 excess #is4"'/.+&(
force
I/%",&J
:!!3*+,2 established
on;/%."
&-$& an !.3%%",&
)+%".&+/,9
&-" on
$,+!the
electron
$/,A-" opposite
moving
I/%",&J
"'".&%/,
.$%1 facethe
in
;/%."
*/4+,2 until,
/, as"'".
+,x$,&-" in$
verse0(electric field (Hall ! field)
+!K just cancels the !Lorentz
"+!K&#! force
! !! due " &to! the magnetic fie
direction with velocity vx is: $
%+") "H.'3!+4"'( "'".&%/,! 4"'/.+&(
<*1&(#" !"*+./,)3.&/% 4"'/.+&(
/% *"&$'=6
% ?" ! $-$4"
%## ! " &#! ! !!## " &! ##
+$ # "$!) ,! # "* &! $ !CDE#M!
%# ! " &#! ! !!# " &!%# ! "
#&! &#!'
#&! $ ' "
$ !&&!!
" !!### #"
# @&@!###
#&! $ ' " &! # #7"%"
@ # !#&!
Assuming
7"%" !##( ' "L)
(that
!77
$! the
L) #
&!+!+!#current # is carried
&-"@!/1.$''")
!/1.$''") 7$'' ;+"
;+"
eB!&-" 7$''
(SI) Ey !%+")
"""H.'3!+4"'(
%+") c!Ex !
exclusively by"electrons,
"H.'3!+4"'( 0(m
0( Ex .we have
"'".&%/,!
"'".&%/,! <*1&(#"
<*1&(#"
7"%" ! # ( !7L) +! &-" !/1.$''")
7"%" ! # ( !7$''
7L) +! ;+"')9&-" :!!3*+,2
!/1.$''") &-$&!"*+
7$'' &-"
;+"'
%+") "H.'3!+4"'( 0( "'".&%/,!
+$+$#<*1&(#"
#"$!)
"$!),!
%+") "H.'3!+4"'( !"*+./,)3.&/%
,! #
0(#"'".&%/,!
"*&!
"* /% *"&$'=6
&!$$ <*1&(#" !"*+.?
The quantity
+$ #defined
"$!) ,! #by"*and
&! $+$Hall coefficient
# "$!) ,! # "* (R &!
) is
$ defined by H

Ey
RH ! = −(ne)-1.
jxB
The sign of RH gives the type of carrier.
is called the Hall coefficient. To evaluate it on our simple model
2
./01 !"#121 ne !Ex/m
G.-"*$&+. and
!"&13# ;/% obtain
$ 7$'' ";;".& *"$!3%"*",&9 !K *$2,"&+. +,)3.&+/,O
-K .3%%",& &-%/32- &-" !$*#'"O !7K 7$'' 4/'&$2"9 A-" )$!-") '+,"! $%" &-" #$&-! ?-+.-
 (b) the radius of the ground state orbit. (c) At what minimum donor concentration
Problems
will appreciable overlap effects between the orbits of adjacent impurity atoms
occur? This overlap tends to produce an impurity band—a band of energy levels
1. which permit
Impurity conductivity
orbits. Indium presumably
antimonideby hasa hopping
Eg = 0.23mechanism in which
eV; dielectric electrons
constant ϵ
move
= 18;from one impurity
electron site to
effective a neighboring
mass ionized
me = 0.015 m. impurity site.(a) the donor
Calculate
ionization of
2. Ionization energy;
donors.(b) In
thea radius of semiconductor
particular the ground state thereorbit.
are 10(c)
13 At what 3
donors/cm
minimum
with donorenergy
an ionization concentration
Ed of 1 meVwill
andappreciable overlap
an effective mass 0.01effects betweenthe
m. (a) Estimate
the orbits of
concentration adjacent impurity
of conduction electrons atatoms
4 K. (b)occur?
What isThis overlap
the value of thetends to
Hall coeff-
produce
icent? an no
Assume impurity
acceptor band—a band of
atoms are present andenergy
that Eg #levels
kBT. which permit
conductivity presumably by a hopping mechanism in which electrons
3. Hall
moveeffect
fromwith
one two carrier
impurity sitetypes. Assuming concentration
to a neighboring n, p; relaxation
ionized impurity site. times
"e, "h; and masses me, mh, show that the Hall coefficient in the drift velocity approxi-
Hall effect
2. mation is with two carrier types. Assuming concentration n, p; relaxation
times 𝜏e, 𝜏h; and masses me, mh, show that the Hall coefficient in the drift
velocity approximation is p ! nb2
1
(CGS) RH " ec ! 2
,
(p $ nb)
where b = μe/μh is the mobility ratio. In the derivation neglect terms of
orderbB"2. #In
where e /#SI
h iswe
the drop
mobilitytheratio.
c. Hint:
In theIn the presence
derivation neglect of a longitudinal
terms of order B2. In
electric
SI we dropfield,
the find the Intransverse
c. Hint: the presenceelectric field such that
of a longitudinal thefield,
electric transverse
find the
current vanishes.
transverse Thesuch
electric field algebra maytransverse
that the seem tedious,
currentbut the result
vanishes. is worth
The algebra may
the trouble.
seem tedious, Neglect
but the (ω c𝜏) in comparison with ωc𝜏.
result 2 is worth the trouble. Use (6.64), but for two carrier

types; neglect ($c")2 in comparison with $c".

3. A semiconductor with a band gap energy Eg of 1eV and equal hole and
electron effective masses me* = mh* = m0 (m0 is free electron mass) is p-
doped with an acceptor concentration of p = 10-18 cm-3. The acceptor
energy level is located 0.2 eV above the valence band edge of the
material.
a) Show that intrinsic conduction in this material is negligible at 300 K.
b) Calculate the conductivity σ of the material at room temperature (300
K), given a hole mobility of μp = 100 cm2/Vs at 300 K.
c) Plot the logarithm of the hole concentration, ln p, versus reciprocal
temperature 1/T for the temperature range 100 to 1000 K.