-
Metals, Semiconductors, and Insulators
Metals have free electrons and partially filled valence bands,
therefore they are highly conductive (a).
Semimetals have their highest band filled. This filled band,
however, overlaps with the next higher band, therefore they are
conductive but with slightly higher resistivity than normal metals
(b). Examples: arsenic, bismuth, and antimony.
Insulators have filled valence bands and empty conduction bands,
separated by a large band gap Eg(typically >4eV), they have high
resistivity (c ).
Semiconductors have similar band structure as insulators but
with a much smaller band gap. Some electrons can jump to the empty
conduction band by thermal or optical excitation (d). Eg=1.1 eV for
Si, 0.67 eV for Ge and 1.43 eV for GaAs
Every solid has its own characteristic energy band structure.In
order for a material to be conductive, both free electrons and
empty states must be available.
-
An energy band is a range of allowed electron energies.The
energy band in a metal is only partially filled with
electrons.Metals have overlapping valence and conduction bands
Metals
Conduction in Terms of Band
-
Drude Model of Electrical Conduction in Metals
Conduction of electrons in metals – A Classical Approach:In the
absence of an applied electric field (ξ) the electrons move in
random directions colliding with random impurities and/or lattice
imperfections in the crystal arising from thermal motion of ions
about their equilibrium positions. The frequency of
electron-lattice imperfection collisions can be described by a mean
free path λ -- the average distance an electron travels between
collisions. When an electric field is applied the electron drift
(on average) in the direction opposite to that of the field with
drift velocityThe drift velocity is much less than the effective
instantaneous speed (v) of the random motion
v
-
In copper while where
The drift speed can be calculated in terms of the applied
electric field ξ and of v and λWhen an electric field is applied to
an electron in the metal it experiences a force qξresulting in
acceleration (a)
Then the electron collides with a lattice imperfection and
changes its direction randomly. The mean time between collisions
is
The drift velocity is
If n is the number of conduction electrons per unit volume and J
is the current density
Combining with the definition of resistivity gives
1210 −−≈ scmv . 1810 −≈ scmv . Tkvm Be 23
21 2 =
emqa ξ=
vλτ =
vmq
mqav
ee ⋅⋅⋅
=⋅⋅
=⋅=λξτξτ
σξ νnqJ ==
vmqne ⋅⋅⋅
=λσ
2
ee mq
vmq τλμ ⋅=⋅⋅
=
q=1.6x10-19C
-
For an electron to become free to conduct, it must be promoted
into an empty available energy stateFor metals, these empty states
are adjacent to the filled statesGenerally, energy supplied by an
electric field is enough to stimulate electrons into an empty
state
States Filled with Electrons
Empty States
“Freedom”
Ele
ctro
n E
nerg
y
Distance
Energy Band
-
At T = 0, all levels in conduction band below the Fermi energy
EF are filledwith electrons, while all levels above EF are
empty.Electrons are free to move into “empty” states of conduction
band with only a small electric field E, leading to high electrical
conductivity!At T > 0, electrons have a probability to be
thermally “excited” from below the Fermi energy to above it.
Band Diagram: Metal
EF
EC
Conduction band(Partially Filled)
T > 0
Fermi “filling”function
Energy band to be “filled”
E = 0
-
Resistivity (ρ) in MetalsResistivity typically increases
linearly with temperature:
ρt = ρo + αTPhonons scatter electrons. Where ρo and α are
constants for an
specific materialImpurities tend to increase resistivity:
Impurities scatter electrons in metalsPlastic Deformation tends to
raise resistivity dislocations scatter electrons
The electrical conductivity is controlled by controlling the
number of charge carriers in the material (n) and the mobility or
“ease of movement” of the charge carriers (μ)
μρ
σ nq== 1
-
Temperature Dependence, MetalsThere are three contributions to
ρ:ρt due to phonons (thermal)ρi due to impuritiesρd due to
deformation
ρ = ρt + ρi+ ρd The number of electrons in the conduction band
does not vary with temperature.
All the observed temperature dependence of σ in metals arise
from changes in μ
-
Scattering by Impurities and PhononsScattering by Impurities and
Phonons
Thermal: Phonon scatteringProportional to temperature
Impurity or Composition scatteringIndependent of
temperatureProportional to impurity concentration
Solid SolutionTwo Phase
Deformation
Taot += ρρ
)1( iii cAc −=ρ
ββαα ρρρ VVt +=
determinedallyexperimentbemust=dρ
-
InsulatorE
lect
ron
Ene
rgy
“Conduction Band”Empty
“Valence Band”Filled with Electrons
“Forbidden” EnergyGap
Distance
The valence band and conduction band are separated by a large
(> 4eV) energy gap, which is a “forbidden” range of energies.
Electrons must be promoted across the energy gap to conduct, but
the energy gap is large. Energy gap º Eg
-
Band Diagram: Insulator
At T = 0, lower valence band is filled with electrons and upper
conduction band is empty, leading to zero conductivity.Fermi energy
EF is at midpoint of large energy gap (2-10 eV) between conduction
and valence bands.At T > 0, electrons are usually NOT thermally
“excited” from valence to conduction band, leading to zero
conductivity.
EF
EC
EV
Conduction band(Empty)
Valence band(Filled)
Egap
T > 0
-
Conduction in Ionic Materials (Insulators)Conduction by
electrons (Electronic Conduction): In a ceramic, all the outer
(valence) electrons are involved in ionic or covalent bonds and
thus they are restricted to an ambit of one or two atoms.
If Eg is the energy gap, the fraction of electrons in the
conduction band is:Tk
E
B
g
e 2−
A good insulator will have a band gap >>5eV and
kBT~0.025eV at room temperature
As a result of thermal excitation, the fraction of electrons in
the conduction band is
~e-200 or 10-80.There are other ways of changing the electrical
conductivity in the ceramic which have a far greater effect than
temperature.
•Doping with an element whose valence is different from the atom
it replaces. The doping levels in an insulator are generally
greater than the ones used in semiconductors. Turning it around,
material purity is important in making a good insulator.
•If the valence of an ion can be variable (like iron), “hoping”
of conduction can occur, also known as “polaron” conduction.
Transition elements.
•Transition elements: Empty or partially filled d or f orbitals
can overlap providing a conduction network throughout the
solid.
-
Conduction by Ions: ionic conductionIt often occurs by movement
of entire ions, since the energy gap is too large for electrons to
enter the conduction band.
The mobility of the ions (charge carriers) is given by:
Where q is the electronic charge ; D is the diffusion
coefficient ; kB is Boltzmann’s constant, T is the absolute
temperature and Z is the valence of the ion.
The mobility of the ions is many orders of magnitude lower than
the mobility of the electrons, hence the conductivity is very
small:
TkDqZ
B ...
=μ
μσ ... qZn=Example:Suppose that the electrical conductivity of
MgO is determined primarily by the diffusion of Mg2+ ions. Estimate
the mobility of Mg2+ ions and calculate the electrical conductivity
of MgO at 1800oC.Data: Diffusion Constant of Mg in MgO =
0.0249cm2/s ; lattice parameter of MgO a=0.396x10-7cm ; Activation
Energy for the Diffusion of Mg2+ in MgO = 79,000cal/mol ;
kB=1.987cal/K=k-mol; For MgO Z=2/ion; q=1.6x10-19C;
kB=1.38x10-23J/K-mol
-
First, we need to calculate the diffusion coefficient D
⎟⎟⎠
⎞⎜⎜⎝
⎛+−
−=⎟
⎠⎞
⎜⎝⎛ −=
KKxmolcalmolcal
scm
kTQDD Do )(/.
/exp.exp27318009871
79000023902
D=1.119x10-10cm2/s
Next, we need to find the mobility
sJcmCCioncarriers
TkDqZ
B ...
))(.().)(.)(/(
... 29
23
1019
10121273180010381
101110612 −−
−−
×=+×
××==μ
C ~ Amp . sec ; J ~ Amp . sec .Volt μ=1.12x10-9 cm2/V.s
-
MgO has the NaCl structure (with 4 Mg2+and 4O2- per cell)
Thus, the Mg2+ ions per cubic cm is:
32237
2
1046103960
4 cmionscmcellionsMgn /.
).(/
×=×
= −+
sVcmcmC
nZq
....
).)(.)()(.(
3
26
91922
109422
10121106121046
−
−−
×=
×××==
σ
μσ
C ~ Amp.sec ; V ~ Amp.Ω σ = 2.294 x 10-5 (Ω.cm)-1
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Example:
The soda silicate glass of composition 20%Na2O-80%SiO2 and a
density of approximately 2.4g.cm-3 has a conductivity of 8.25x10-6
(Ω-m)-1 at 150oC. If the conduction occurs by the diffusion of Na+
ions, what is their drift mobility?
Data: Atomic masses of Na, O and Si are 23, 16 and 28.1
respectively
Solution:
We can calculate the drift mobility (μ) of the Na+ ions from the
conductivity expression
ii qn μσ ××=Where ni is the concentration of Na+ ions in the
structure.
20%Na2O-80%SiO2 can be written as(Na2O)0.2-(SiO2)0.8 . Its mass
can be calculated as:
14860
162128118016123220−=
+×++×=
molgMM
At
At
..))().((.))()((.
The number of (Na2O)0.2-(SiO2)0.8 units per unit volume can be
found from the density
-
3802202
22
1
1233
10392
486010023642
−
−
−−
−×=
×=
×=
cmunitsSiOONanmolg
molxcmgM
NnAt
A
.. )()(...
).()..(ρ
The concentration of Na+ ions (ni) can be obtained from the
concentration of (Na2O)0.2-(SiO2)0.8 units
32122 101831039221801220
220 −×=××⎥⎦
⎤⎢⎣
⎡+×++×
×= cmni ..)(.)(.
.
And μi
11214
362119
116
10621
101018631060110258
−−−
−−
−−−
×=
××××Ω×
=×
=
sVmmC
mnq
i
ii
.).().(
).(
μ
σμ
This is a very small mobility compared to semiconductors and
metals
-
Electrical BreakdownAt a certain voltage gradient (field) an
insulator will break down.
There is a catastrophic flow of electrons and the insulator is
fragmented.
Breakdown is microstructure controlled rather than bonding
controlled.
The presence of heterogeneities in an insulator reduces its
breakdown field strength from its theoretical maximum of ~109Vm-1
to practical values of 107V.m-1
-
Energy Bands in SemiconductorsEnergy Levels and Energy Gap in a
Pure Semiconductor.The energy gap is < 2 eV. Energy gap º Eg
Semiconductors have resistivities in between those of metals and
insulators.Elemental semiconductors (Si, Ge) are perfectly
covalent; by symmetry electrons shared between two atoms are to be
found with equal probability in each atom.Compound semiconductors
(GaAs, CdSe) always have some degree of ionicity. In III-V
compounds, eg. Ga+3As+5, the five-valent As atoms retains slightly
more charge than is necessary to compensate for the positive As+5
charge of the ion core, while the charge of Ga+3 is not entirely
compensated. Sharing of electrons occurs still less fairly between
the ions Cd+2 and Se+6 in the II-VI compund CdSe.
Elec
tron E
nerg
y
“Conduction Band” (Nearly) Empty – Free electrons
“Valence Band” (Nearly) Filled with Electrons – Bonding
electrons
“Forbidden” Energy Gap
-
Semiconductor MaterialsSemiconductor Bandgap Energy EG
(eV)Carbon (Diamond) 5.47Silicon 1.12 Germanium 0.66Tin
0.082Gallium Arsenide 1.42Indium Phosphide 1.35Silicon Carbide
3.00Cadmium Selenide 1.70Boron Nitride 7.50 Aluminum Nitride
6.20Gallium Nitride 3.40Indium Nitride 1.90
IIIA IV A V A V IA
10.8115
BBo ro n
12.011156
CC a rb o n
14.00677
NN itro g e n
15.99948
OO xyg e n
IIB
26.981513
AlA lum inum
28.08614
SiSilic o n
30.973815
PPho sp ho rus
32.06416
SSulfur
65.3730
ZnZinc
69.7231
G aG a llium
72.5932
G eG e rm a nium
74.92233
AsA rse nic
78.9634
SeSe le nium
112.4048
C dC a d m ium
114.8249
InInd ium
118.6950
SnTin
121.7551
SbA ntim o ny
127.6052
TeTe llurium
200.5980
HgM e rc ury
204.3781
TiTha llium
207.1982
PbLe a d
208.98083
BiBism uth
(210)84
PoPo lo nium
Portion of the Periodic Table Including the Most Important
Semiconductor Elements
-
Band Diagram: Semiconductor with No Doping
At T = 0, lower valence band is filled with electrons and upper
conduction band is empty, leading to zero conductivity.Fermi energy
EF is at midpoint of small energy gap ( 0, electrons thermally
“excited” from valence to conduction band, leading to measurable
conductivity.
EFECEV
Conduction band(Partially Filled)
Valence band(Partially Empty)
T > 0
-
Semi-conductors (intrinsic - ideal)Perfectly crystalline (no
perturbations in the periodic lattice).Perfectly pure – no foreign
atoms and no surface effectsAt higher temperatures, e.g., room
temperature (T @ 300 K), some electrons are thermally excited from
the valence band into the conduction band where they are free to
move.“Holes” are left behind in the valence band. These holes
behave like mobilepositive charges.
CB electrons and VB holes can move around (carriers).
At edges of band the kinetic energy of the carriers is nearly
zero. The electron energy increases upwards. The hole energy
increases downwards.
-
Si Si Si Si Si Si Si
Si Si Si Si Si Si Si
Si Si Si Si Si Si Si
free electron
free hole
Si Si Si Si Si Si Si
Si Si Si Si Si Si Si
Si Si Si Si Si Si Si
positive ion core
valence electron
-
Semiconductors in Group IVCarbonSiliconGermaniumTinEach has 4
valence Electrons.Covalent bond
-
Generation of Free Electrons and HolesIn an intrinsic
semiconductor, the number of free electrons equals the number of
holes.Thermal : The concentration of free electrons and holes
increases with increasing temperature.Thermal : At a fixed
temperature, an intrinsic semiconductor with a large energy gap has
smaller free electron and hole concentrations than a semiconductor
with a small energy gap.Optical: Light can also generate free
electrons and holes in a semiconductor.Optical: The energy of the
photons (hν) must equal or exceed the energy gap of the
semiconductor (Eg) . If hν > Eg , a photon can be absorbed,
creating a free electron and a free hole.This absorption process
underlies the operation of photoconductive light detectors,
photodiodes, photovoltaic (solar) cells, and solid state camera
“chips”.
-
UV 100-400 nm 12.4-3.10 eVViolet 400-425 nm 3.10-2.92 eVBlue
425-492 nm 2.92-2.52 eVGreen 492-575 nm 2.52-2.15 eVYellow 575-585
nm 2.15-2.12 eVOrange 585-647 nm 2.12-1.92 eVRed 647-700 nm
1.92-1.77 eVNear IR 10,000-700 nm 1.77-0.12 eV
RedRed
OrangeOrange
YellowYellow GreenGreen
BlueBlue
VioletViolet
-
Photoconductivity
Eg ω � Eg
Conductivity is dependent on the intensity of the incident
electromagnetic radiation
E = hν = hc/λ, c = λ(m)ν(sec -1)
hν ≥ Eg
Band Gaps: Si - 1.11 eV (Infra red)Ge 0.66 eV (Infra red)GaAs
1.42 eV (Visible red)ZnSe 2.70 eV (Visible yellow)SiC 2.86 eV
(Visible blue)GaN 3.40eV (Blue)AlN 6.20eV (Blue-UV)BN 7.50eV
(UV)
Total conductivity σ = σe + σh = nqμe + pqμhFor intrinsic
semiconductors: n = p & σ = nq(μe + μh)
-
As T ↑ then ni ↑As Eg↑ then ni ↓
What is the detailed form of these dependencies?We will use
analogies to chemical reactions. The electron-hole formation can be
though of as a chemical reaction……..Similar to the chemical
reaction………
Question: How many electrons and holes are there in an intrinsic
semiconductor in thermal equilibrium? Define:no equilibrium (free)
electron concentration in conduction band [cm-3]po equilibrium hole
concentration in valence band [cm-3]Certainly in intrinsic
semiconductor: no = po = nini intrinsic carrier concentration
[cm-3]
+− +⇔ hebond−+ +⇔ )(OHHOH2
iOO npn ==
⎟⎠⎞
⎜⎝⎛−≈=
−+
kTE
OHOHHK exp
][]][[
2
The Law-of-Mass-Action relates concentration of reactants and
reaction products. For water……Where E is the energy released or
consumed during the reaction………….
This is a thermally activated process, where the rate of the
reaction is limited by the need to overcome an energy barrier
(activation energy).
-
By analogy, for electron-hole formation:
Where [bonds] is the concentration of unbroken bonds and Eg is
the activation energy
In general, relative few bonds are broken to form an
electron-hole and therefore the number of bonds are approximately
constant.
2iOO npn =×
⎟⎟⎠
⎞⎜⎜⎝
⎛−≈=
kTE
bondspnK goo exp
][]][[
tcons[bonds] ,pn[bonds] oo
tan=>>
⎟⎟⎠
⎞⎜⎜⎝
⎛−≈
kTE
pn goo exp
⎟⎟⎠
⎞⎜⎜⎝
⎛−≈
kTE
n gi 2exp
Two important results:
1)……………………..
2)………………………………………….……..
The equilibrium np product in a semiconductor at a certain
temperature is a constant specific to the semiconductor.
-
Effect of Temperature on Intrinsic SemiconductivityThe
concentration of electrons with sufficient thermal energy to enter
the conduction band (and thus creating the same concentration of
holes in the valence band) ni is given by
⎟⎟⎠
⎞⎜⎜⎝
⎛ Δ−≈
TkEn
Bi exp
For intrinsic semiconductor, the energy is half way across the
gap, so that
⎟⎟⎠
⎞⎜⎜⎝
⎛ −≈
TkE
nB
gi 2
exp
Since the electrical conductivity σ is proportional to the
concentration of electrical charge carriers, then
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
TkE
B
gO 2
expσσ
-
Example
Calculate the number of Si atoms per cubic meter. The density of
silicon is 2.33g.cm-3and its atomic mass is 28.03g.mol-1.
Then, calculate the electrical resistivity of intrinsic silicon
at 300K. For Si at 300K ni=1.5x1016carriers.m-3, q=1.60x10-19C,
μe=0.135m2(V.s)-1 and μh=0.048m2.(V.s)-1
Solution
32810005 −−×=×= matomsSiA
NnSi
SiASi ..
ρ
( )myresistivit
mqn hei−Ω×==
−Ω×=+××= −−
3
13
10282
1043850
.)(.
ρ
μμσ
-
ExampleThe electrical resistivity of pure silicon is 2.3x103Ω-m
at room temperature (27oC ~ 300K). Calculate its electrical
conductivity at 200oC (473K). Assume that the Eg of Si is 1.1eV ;
kB =8.62x10-5eV/K
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
TkE
CB
g
2exp.σ ⎟⎟
⎠
⎞⎜⎜⎝
⎛ −=
)(exp.
4732473 Bg
kE
Cσ ⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
)(exp.
3002300 Bg
kE
Cσ
13300473
300
473
5300
473
300
473
04123851032
12385
7777
4731
3001
10628211
4731
3001
230024732
30024732
−
−
Ω=×
==
=⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎠⎞
⎜⎝⎛ −
×=⎟
⎠⎞
⎜⎝⎛ −=+
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−
−=
).(.)(.
)(
.ln
).(.
)()(ln
)()(exp
m
eVk
Ek
Ek
E
kE
kE
B
g
B
g
B
g
B
g
B
g
σσ
σσ
σσ
σσ
-
Example: For germanium at 25oC estimate (a) the number of charge
carriers, (b) the fraction of total electrons in the valence band
that are excited into the conduction
band and (c) the constant A in the expression when E=Eg/2
Data: Ge has a diamond cubic structure with 8 atoms per cell and
valence of 4 ; a=0.56575nm ; Eg for Ge = 0.67eV ; μe = 3900cm2/V.s
; μh = 1900cm2/V.s ; ρ = 43Ω-cm ; kB=8.63x10-5eV/K
eVKeVTkCT
B
o
0514025273106382225
5 .))(/.)(( =+×==
−
313
19 10521900390010610230
cmelectrons
qn
he
×=+×
=+
= − .)(..
)( μμσ
There are 2.5x1013 electrons/cm3 and 2.5x1013 holes/cm3 helping
to conduct a charge in germanium at room temperature.
(a) Number of carriers
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
TkE
AnB
g
2exp
-
b) the fraction of total electrons in the valence band that are
excited into the conduction band
The total number of electrons in the valence band of germanium
is :
371056575048
).()/)(/(
cmxatomselectronsvalencecellatoms
electronsValence −−=−
32310771 cmelectronselectronsvalenceTotal /. ×=−−
1023
13
3
3
10411107711052 −×=××
=−−−−
=− ...
//cmelectronsvalenceTotalcmelectronsexcitednumberexcitedFraction
(c) the constant A
319
05140670
13
2
101411052 cmcarriersee
nATk
E
B
g/..
..
−
⎟⎠⎞
⎜⎝⎛ −
⎟⎟⎠
⎞⎜⎜⎝
⎛ −×=
×==
-
Direct and Indirect SemiconductorsThe real band structure in 3D
is calculated with various numerical methods, plotted as E vs k. k
is called wave vectorFor electron transition, both E and p (k) must
be conserved.
momentum is pkp =
A semiconductor is indirect if the …do not have the same k
valueDirect semiconductors are suitable for making light-emitting
devices, whereas the indirect semiconductors are not.
A semiconductor is direct if the maximum of the conduction band
and the minimum of the valence band has the same k value