8. Semiconductors
What does it mean semiconductor?
• the term semiconductor indicates that the electrical conductivity of such materials is lying in between
that one of metals and insulators
• good metals have at room temperature an electrical resistivity of about 10-7… 10-8Ωm, whereas good
insulators have values of about 1012Ωm; for semiconductors the resistivity varies typically between
104Ωm and 107Ωm
• semiconducting elements with a simple crystal lattice can be found in the 4th main group of the
periodic table, where mainly silicon and germanium (diamond-type semiconductors) are of industrial
importance; diamond (carbon) belongs rather to the insulators and lead to the metals; thin can appear
as metallic or semiconducting allotrope
• further semiconducting elements are: red phosphorus, baron, selenium and tellurium with quite
complicated crystal structures
• nomenclature for semiconductor compounds:
(1) compounds of chemical formula AB, where A is a trivalent element and B is a pentavalent
element, are called III-V (three-five) semiconductors; examples are gallium arsenide (GaAs)
and indium antimonite (InSb); GaAs of some interest for high-speed electronic devices
380
(2) where A is divalent and B is hexavalent, the compound is called II-VI semiconductor; examples
are zinc sulphide (ZnS) and cadmium sulphide (CdS)
(3) an example for a IV-IV compound is silicon carbide (SiC)
• at zero temperature semiconductors behave like insulators: the lower energy bands are completely
occupied with electrons while the bands above are completely empty so that no charge transport
occurs
• at finite temperatures there are electrons in the conduction band, which are thermally excited from the
valence band; both, the electrons in the conduction band as well as the holes in the valence band
contribute to electrical conductivity
• for technical applications, the deliberate addition of impurities and imperfections (doping) drastically
affect the electrical properties of semiconductors
• the doping of semiconductors allows to modify the physical properties of the material in accordance
with the technical needs
8.1. Intrinsic Crystalline Semiconductors
Band Gap and Optical Absorption
• in semiconductors the uppermost completely occupied energy band is called valence band, while the
empty band above is called conduction band
381
electron energy
overlap
EC
conduction
band
Fermi level
EV
metal
semiconductor
valence
band
insulator
EG= EC-EV
band gap
(forbidden
band)
band scheme for intrinsic conductivity in
semiconductors; at T = 0K the conductivity is
zero as all states of the valence band are filled
and all states of the conduction band are
vacant; if the temperature is increased, the
electrons are thermally excited and pass on
from the valence band to the conduction band,
where they become mobile; such carriers are
called intrinsic; the energy of the lower edge of
the conduction band is EC, the energy of the
upper edge of the valence band is EV; the
forbidden band (or energy gap) is EG=EC-EV
• the energy of the upper edge of the valence band is EV, the energy of the lower edge of the
conduction band is EC; the band gap EG=EC-EV in between these two bands is for many electronic
properties of great significance
• the band gap depends only weakly on temperature; from low temperatures upward the gap decreases
first quadratically and then linearly with temperature; the total decreases up to room temperature is
only about 10% and is caused by thermal expansion and effects of electron-phonon interaction
• if the maximum of the valence band (valence band edge) and the lowest minimum of the conduction
band (conduction band edge) occur at the Γ point (the origin of the first Brillouin zone), the gap is
called direct band gap and the semiconductor is called direct semiconductor
382
• if both extreme occur not at the same point, we have an indirect gap and the semiconductor is said
to be an indirect semiconductor
• band gap for some selected semiconductors at room temperature, zero temperature and type of gap
semiconductor
EG (T = 300K) [eV]
EG(T = 0K) [eV]
type of band gap
diamond
5.47
5.48
indirect
Si
1.12
1.17
indirect
Ge
0.66
0.75
indirect
GaP
2.26
2.32
indirect
GaAs
1.43
1.52
direct
InSb
0.18
0.24
direct
InP
1.35
1.42
direct
CdS
2.42
2.58
direct
after Hunklinger
• the electronic band structure of the technically important semiconductors gallium arsenide and
germanium are shown below; the graphs show calculations of the electronic band structure, the
results of which were fitted to the data of spectroscopic measurements
• silicon is the semiconductor of greatest technical importance, the electronic band structure of silicon
is very similar to that of germanium
383
4
EG
0
energy E [eV]
energy E [eV]
4
- 4
- 8
-12
- 4
- 8
-12
L
(a)
EG
0
Γ
X U,K
wavevector k
L
Γ
(b)
Γ
X U,K
wavevector k
Γ
after Hunklinger
electronic band structure of gallium arsenide (a) and germanium (b); the band gap is
shown in grey; (a) valence band maximum and conduction band minimum meet each
other at the Γ point, gallium arsenide is a direct semiconductor; (b) for the indirect
semiconductor germanium the valence band maximum occurs at the Γ point while the
lowest conduction band minimum occurs at L point
• as can be seen from the graph, GaAs is a direct semiconductor:
the valence band maximum and the
r
conduction band minimum occur at the same value of k
• for germanium the valence band maximum also occurs at the Γ point; however, the conduction band
minimum from across has not the smallest distance in energy to the valence band; the lowest
conduction band minimum occurs in the 111 direction at the L point
384
• optical absorption allows to measure the band gap in a very simple manner: a phonon is absorbed by
the crystal with the creation of an electron and a hole; the electron transits into the conduction band,
while in the valence band a hole remains (inter-band transition)
EC
ħωγ
EV
absorption α [cm-1]
energy E
• note: for a vacuum wave length λ = 500 nm (green light) we get ħωγ = 2.48 eV which is in the same
order of magnitude as for example the band gap of cadmium sulfide (CdS) at room temperature
EG = 2.42 eV!
104
InSb
102
T = 77 K
0
(a)
wavevector k
100
(b)
0.2
0.4
0.6
0.8
after Hunklinger
energy ħωγ [eV]
optical absorption for direct semiconductors; (a) schematic diagram of the absorption
process; the fat blue arrow represents the transition with the lowest possible energy; at
higher photon energies deeper-seated electrons are excited; (b) optical absorption
coefficients α (I = I0e-αd, d thickness of crystal sample) of indium antimonite in a
logarithmic scale as a function of the energy of the irradiated photons
385
• in a direct absorption process the transition of rthe electron in the electronic band scheme appears
to be vertical as the momentum of the phonon h k γ is small compared to the typical momentum of a
crystal electron
• as the electron has to jump over the band gap, a photon can only be absorbed, if its energy exceeds
the minimum value (threshold energy)
hω γ = E C − E V = EG
• therefore, many semiconductors are transparent in the near infrared (NIR); note: for λ = 1.5µm we
get hω γ = 0.8 eV
• above the threshold energy, which is given by the band gap, the optical absorption increases rapidly
with increasing frequency, cf. the graph for indium antimonite; such a steep slope is typical for direct
semiconductors
• the indirect absorption process is a little more complicated
386
absorption α [cm-1]
energy E
E′C
EC
EV
103
T = 300K
T = 77K
101
Ge
0
(a)
wavevector k
km
10-1
0.6
(b)
0.7
0.8
0.9
energy ħωγ [eV]
1.0
after Hunklinger
optical absorption in the case of an indirect band gap: (a) schematic diagram of the absorption
processes; the energy of the conduction band minimum at the Γ point is denoted by E'C; the
transition with smallest possible energy (solid arrow) requires the assistance of a phonon; the
direct transition is shown by a dashed arrow; (b) optical absorption coefficient α of germanium
in a logarithmic scale as a function of the energy of the irradiated photons; the weaker indirect
absorption comes in before the direct absorption process starts
• the absorption comes in already at photon energies smaller than the energy difference of
conduction band and valence band at the Γ point E'C −E V
387
r
• if the lowest conduction band minimum occurs at k m ≠ 0, than a direct transition is impossible
because of the small value of the photon momentum; for reasons of conservation of energy and
quasi momentum the assistance of a phonon is required
r
r
• if ω q is the circular frequency of the phonon and q is the wavevector, we find for the generation of
an electron-hole pair with the smallest possible energy the following conditions
hω γ ± hω rq = EG
r
r
r
and hkγ ± hq = h km
r
r
• for the indirect absorption process we get hω << EG and kγ << km ,or, roughly speaking, the
photon provides the energy and the phonon the required momentum for the transition
r
q
• the probability of the occurrence of an indirect absorption process is much smaller than that of the
direct absorption as the electrons have to couple to the phonons; consequently, indirect processes
give rise to weak absorption effect as can be seen from experimental results above
• the absorption coefficient is rather small at the threshold energy (absorption edge) and is increasing
with the photon energy; obviously, the slope of the absorption graph further increases at the onset of
the direct processes
• note: the absorption edge varies with temperature as the band gap EG increases with decreasing
temperature
388
Effective Mass of Electrons and Holes
• the electrical properties of semiconductors are mostly determined by the electrons at the minimum
of the conduction band and the holes at the maximum of the valence band; therefore, the band
structure in these ranges of energy needs to be studied in more detail
• as known from elsewhere, the effective mass of electrons and holes is determined by the band
curvature; around the extremes of the band the curvature of the energy curves is approximately
constant and therewith also the dynamic effective mass m*, which is just there in good agreement
with the cyclotron mass
0.0
cyclotron resonance measurements at a
germanium sample at 4K and 23 GHz; for the
crystal orientation chosen here, all types of
charge carriers can be observed while varying
the magnitude of applied magnetic field; the
maxima caused by the electrons originate from
different extremal orbits in the conduction band;
the maxima marked with a star (*) are higher
harmonics of heavy holes
heavy mass holes
electrons
electrons
absorption
light mass holes
electrons
• for semiconductors the cyclotron resonance is a very useful method to determine the masses of
electrons and holes; as in semiconductors the skin depth in the microwave range is quite large,
mostly one single resonance is observed for a given mass
0.1
0.2
0.3
magnetic field B [T]
0.4
after Hunklinger
389
• for cyclotron resonance experiments microwaves are used: from ωC= eB/mel (mel is the rest mass of
the electron) we get for a magnetic field of B = 1T a resonance frequency of fC= 23GHz
• as can be seen from the experiment, a number of resonances appear if the magnetic field is tuned;
these resonances can be attributed to electrons and holes in different energy bands
• in the experiment shown here, the germanium crystal was oriented in such a way, that all different
effective masses can be observed simultaneously
• in order to get significant signals, the electrons have to run though the orbit several times without
any collision, i.e. ωCτ >> 1 must be fulfilled
• as at room temperature the mean time between collision τ is about 10-13s, one can get useful results
only for very clean samples at low temperatures
• however, in this case the small number of charge carriers is a problem; trick: by irradiation with light
of energy higher than the band gap, the number of charge carriers in the sample can be increased
artificially
• choosing the strength and the direction of the magnetic field in a suitable manner, the masses of the
different electrons and holes can be measured
• for semiconductors with a direct band gap (for example GaAs) the conduction band at the Γ point
determines essentially
the energy is practically independent
r the electronic properties; as at this point
∗
of the direction in k space, we find only one effective mass mn for electrons
390
• the dispersion relation takes the simple form
h 2k 2
En = EC + ∗
2mn
(note: we use here the same indices “n” and “p” as already used elsewhere in order to distinguish
between electrons and holes)
• effective mass of electrons at the conduction band minimum; the indices “L” and “T” denote
longitudinal or transversal, respectively (see below)
semiconductor
m∗n /m
mn,∗ L /m
m∗n, T /m
GaAs
0.067
-
-
GaSb
0.047
-
-
InSb
0.015
-
-
InP
0.073
-
-
Si
-
0.98
0.19
Ge
-
1.64
0.082
• note: in most cases the effective mass of electrons is much smaller than the mass of the free
electrons; the interaction of the electrons apparently results in a reduction of mass
391
• for semiconductors with an indirect band gap, such as silicon or germanium, the band structure is
more complex; as already mentioned, for germanium the conduction band minimum is parallel to the
111 direction close to the L points and for silicon one finds the minima parallel to the 100 direction
close to the X points
r
close to the minimum the E(k) surface is not isotropic but has an ellipsoidal shape
• in the reduced zone scheme the surfaces of constant energy of the conduction band in germanium
are halves of ellipsoids located of the L points with the major axes parallel to the 111 directions
surfaces of constant energy (halves of ellipsoids)
of the conduction electrons in germanium shown
in the reduced zone scheme (first Brillouin zone);
the conduction band minima appear parallel to the
111 directions at the L points, which are the
centres of the ellipsoids in the periodic zone
scheme
after Hunklinger
392
• in the periodic zone scheme the energy surfaces are complete ellipsoids on which the electrons are
circulating at cyclotron resonance measurements; depending on the direction of the magnetic field
different extremal orbits contribute to the signal
if the cyclotron resonance is measured at several tilt angles of the sample in the magnetic field,
the shape of the energy surfaces can be investigated
• because of the ellipsoidal shape, the effective mass is determined by two quantities, the
longitudinal mass m∗n,Land the transversal mass m∗n,T; therefore, the energy surface close to the
minimum is given by the relation
2

 k12 + k 22
k
3
,
En = EC + h 
+
 2m∗n, T 2m∗n, L 


where the origin of the coordinate system is the corresponding minimum
• for germanium the direction of k3 corresponds to the 111 direction and for silicon it corresponds to
the 100 direction
2
• as the radiuses of curvature are rather different along the different directions, also the corresponding
effective masses are different (for silicon and germanium see table above)
• note: in the graph of cyclotron resonance shown above the four maxima, which can be attributed to
the electrons, are caused by extremal orbits on different ellipsoids as the sample was not aligned
along one of the preferred directions
393
• note: from the graph of the energy band structure for GaAs and Ge shown elsewhere we can see,
that for those semiconductors two energy bands overlap each other in the valence band maximum;
due to the different curvature of the two bands the corresponding holes have different masses and
are referred as heavy mass holes or light mass holes, respectively
• note: in the graph of the cyclotron resonance shown above one can find both types of holes
• note: in the graph of the energy band structure a third band appears; the valence band maximum of
this band is lowered by an energy ∆ compared to the two other two maxima; the charge carriers of
this band are called split-holes with the effective mass m∗∆
• this lowering of energy is due the spin-orbit interaction (interaction of the electron spin with motion of
electron which causes a split energy levels)
• for the vast majority of semiconductors in a rough approximation the valence bands can be
considered to be spherically close to the Γ point; consequently, one effective mass per band is
sufficient in order to characterise the valence bands
• table of effective masses of the different types of holes as well as the energy of the spin-orbit splitting
for various semiconductors; the indices “H” and “L” indicate the heavy mass and the light mass holes,
respectively (see below)
394
m∗p,H /m
m∗p,L /m
m∗∆ /m
∆ [eV]
GaAs
0.45
0.082
0.17
0.34
GaSb
0.3
0.06
0.14
0.80
InSb
0.39
0.021
0.11
0.82
InP
0.4
0.078
0.15
0.11
Si
0.49
0.16
0.23
0.044
Ge
0.28
0.044
0.075
0.29
semiconductor
after Hunklinger
Charge Carrier Density
• in semiconductors electrons as well as holes contribute to the charge transport; therefore, we get
the electrical conductivity the more general expression
σ = e(nµn + pµp) ,
where n and p are the densities of the free electrons or holes, respectively and µn and µp the
corresponding mobility's
• the contribution of the low mass holes dominates the total contribution of the holes to the charge
transport
• the density of charge carriers depends strongly on temperature, therefore also the conductivity
varies strongly with temperature
395
• the intrinsic conductivity can be observed at very clean semiconducting samples: thermally
excited electron are transferred into the conduction band while a hole in the valence band is left
• in calculating the charge carrier concentration it is not necessary to distinguish between the direct
and indirect semiconductors as in the thermal equilibrium the particular way of excitation does not
play any role
• the concentration of electrons n in the conduction band can be calculated from and integration of the
product of the density of states and the corresponding probability of occupancy f(E,T)
• as the Fermi-Dirac distribution decays rapidly with increasing energy, only the states close to the
lower edge of the conduction band matter; therefore, the energy of the upper band edge can be
substituted by infinity
• a similar argument holds for the calculation of the concentration of holes p
∞
EV
EC
−∞
n = ∫ DC (E)f(E, T)dE and p = ∫ D V (E)[1 − f(E, T)]dE,
where the probability of the holes is given by 1-f(E,T) as they originate from unoccupied electron
states
• as close to the extrema of the bands the dispersion relations have a parabolic shape, we can use for
the densities of states DC and DV the expressions derived for the free-electron gas; however, the
∗
∗
effective masses mn and mp for the different bands have to be taken into account:
396
DC (E) =
D V (E) =
3
∗ 2
n
2 3
E − EC
3
∗ 2
p
2 3
E V − E for E < E V
2(m )
2π h
2(m )
for E > E C
2π h
• note, that conduction band and valence band are separated by the band gap EG = EC- EV; there are
no states in the energy range EV < E < EC
• as we will show elsewhere, in the case of the intrinsic conduction the Fermi energy is positioned
about in the middle of the energy gap; this means, that the distances of the Fermi energy to the band
edges |EC- EF| and |EV- EF| are much greater than the thermal energy kBT at room temperature
• this allows us to simplify the integrals for the carrier concentration as we can approximate the FermiDirac distribution quite well as follows:
f (E,T ) =
1
e
(E −EF )/k B T
1 − f (E,T ) =
e
+1
1
(EF −E )/k B T
≅ e −(E −EF )/kB T
+1
≅ e −(EF −E )/kB T
for E > EF
for E < EF
as only the decaying slopes of the Fermi-Dirac function extend into the bands, the states available
there are rather poorly occupied and the classical Boltzmann distribution can be applied for the
description of the charge carriers in semiconductors
397
• therefore, the electrons in the conduction band and the holes in the valence band move largely
like atoms in classical gases and can be described by means of the Kinetic gas theory
• note however, that this is valid as long as the Fermi level is sufficiently far away from the band
edge; as we will show later this approximation of non-degeneracy can be used for intrinsic
semiconductors or semiconductors with a low density of impurities
• for highly doped semiconductors the Fermi energy is shifted close to the band edges or may
even positioned inside the band; in this case the approximation breaks down and the
semiconductor is called degenerate semiconductor
• for the subsequent considerations it is important to note that the term “Fermi energy” is used in
literature differently; often the chemical potential at T = 0 is called Fermi energy; we will use for
the value of the chemical potential µ(T) rather the term “Fermi level” EF or EF(T) in order to
avoid confusion with the term “mobility” µ (“Lieber konsequent inkonsequent; nur nicht immer
dieses ewige Hin-und-Her”, Max Steenbeck)
• substituting the Fermi-Dirac distribution by the Boltzmann distribution the charge carrier
concentrations can be calculated easily; for example, we get for the electrons
n = ∫ DC (E)f (E)de =
∞
• and with
∫
0
(2m )32
∗
n
2π 2 h 3
∞
e EFkBT ∫ E − EC e −E/kBT dE
EC
π
x e dx =
2 (R.G., p. 148, # 3.225,3)
−x
398
• we find for the integral an analytical solution
3
2
 m∗nk BT  −(EC −EF )/k BT
−(E C −EF ) /k B T

n = 2
e
N
e
=
C
2 
2π
h


3
2
and
 m∗p k BT  −(EV −EF )/k BT
− (E V −EF ) /k B T

p = 2
e
=
N
e
V
 2πh 2 


where the effective densities of state NC and NV introduced here, depend only weakly on
temperature compared to the exponential factor
• in this way we get an extensive simplification: in this approximation it looks like that we have not
wide bands but two energy levels EC and EV; we will take advantage of this interpretation
elsewhere
• from the result we can conclude that the density of charge carriers depends on the position if the
Fermi level, however the product n— p not; taking into account, that the band gap is given by
EG = EC- EV we get
3
n ⋅ p = NCN V e −EG /kBT
(
)
3
 kBT 
∗ ∗ 2 −EG /k B T
= 4
⋅
m
e

nm p
2
 2πh 
399
• the product of the carrier densities is completely characterized by the effective mass and the
energy gap and has a characteristic value for each non-degenerate semiconductor at a given
temperature
• following the notation of thermodynamics the relation n— p = const. is often called law of mass
action
• in the thermal equilibrium new charge carriers are permanently generated by thermal activation;
after a short time they disappear again by recombination processes; during the time of its
existence they diffuse along a certain distance, which is called diffusion length
• note: so far we did not restrict our discussion to the case of intrinsic semiconductors; we will use
the above relations and in particular the law of mass action also for the doped semiconductors
• we consider now the case of intrinsic semiconductors for which the conduction electrons
exclusively accrue from the valence band; in this case we get n i = pi, where the index “i” denoted
quantities of the intrinsic semiconductor:
ni = pi = NCN V e −EC /2kB T
• band gap and calculated intrinsic charge carriers densities for some major semiconductors at
300K
400
EG [eV]
ni [m-3]
germanium
0.66
2.4 — 1019
Silicon
1.12
1.1 — 1016
gallium arsenide
1.42
1.8 — 1012
semiconductor
after Hunklinger
• for Si and GaAs the charge carrier density at room temperature is very low; it is nearly impossible
to observe intrinsic conduction at ambient temperatures in such materials, as the carrier density
due to lattice impurities is higher than the intrinsic one
• the position of the Fermi level and its temperature dependence can be derived in general from the
requirement of charge neutrality (all charge in a volume adds to zero); with ni = p i; for the intrinsic
conduction we get
 m∗p 
 NV  E C + E V 3
EC + EV 1
+ k B T⋅ ln  =
EF =
+ k B T⋅ ln ∗ 
m 
2
2
2
4
 NC 
 n
• at T = 0 the Fermi level is in the middle of the forbidden energy gap; if the effective masses of
electrons and holes are equal (valence and conduction band have the same curvature) the
position of the Fermi level does not change with temperature
• different effective masses cause a shift of the Fermi level; however, the temperature variation is
small compared to the energy gap
401
energy E
energy E
EC
EF
EV
EC
EF
EV
after Hunklinger
(a) density of states D(E),
Fermi-Dirac distribution f(E)
(b) density of states D(E),
Fermi-Dirac distribution f(E)
density of states D(E) and Fermi-Dirac distribution for the valence and conduction band at T > 0;
the electrons are shown in light blue, the holes in white; (a) semiconductor with equal densities of
states in the valence as well as in the conduction band: NC=NV; (b) semiconductor with NV > NC,
i.e. with a larger number of charge carriers at the valence band edge
• initially the number of excited electrons (or holes correspondingly) is increasing at the band edges
due to the parabolic shape of the density of states
• at higher excitation energies however, the number of excited electrons drops down again
because of the decreasing probability of occupancy
• as for intrinsic semiconductors the numbers of electrons and holes are equal, the light blue area
in the conduction band and the while area in the valence band are equal also
402
• for different densities of states (see graph on the right) this is only possible, if the Fermi level is
moving towards the band with the lower density of states
8.2. Doped Semiconductors
Doping
• in growing real crystals the contamination with impurities cannot be avoided; normally such
impurities are electrically active, i.e. they cause additional charge carriers in the bands
• in very pure samples of silicon or germanium one can find about 1018 and in GaAs samples 1022
charge carriers per cubic metre instead of the intrinsic values given elsewhere intrinsic
properties can be observed in germanium only
• in any case, for technical purposes the electrical conductivity is too low because of the low
density of charge carriers; for example, for silicon one finds for the intrinsic conductivity σi≡ n ieµn
the small value σi≡10-4(Ωm)-1
• doped semiconductors: simple examples the tetravalent elements silicon and germanium; the
atoms of these crystals show a tetrahedral bonding configuration to the next neighbours
• inserting pentavalent atoms such as P, As or Sb into the Si crystal, the tetravalent neighbouring
silicon atoms enforce the fourfold covalent bonding; the impurity atom has one excess electron
not needed for the tetrahedral binding
403
• this electron cannot take part in crystal binding but is (at low temperatures) still bonded to the
positive atomic core due to the Coulomb interaction
impurity atoms in silicon; (a) electron
donor: metaphorically speaking, the
unbound electron orbits the
positively charged atomic core or
arsenic; (b) electron acceptor: the
negatively charged atomic core of
the boron is apparently surrounded
by a positive charge
after Hunklinger
• as we will show later, the excess electrons are bounded that weakly that they already
disassociate from the atomic core at room temperature
• pentavalent impurity atoms in crystals with a fourfold binding are called donors as they yield an
electron at room temperature
• also trivalent impurity atoms such as B, Al, Ga or In can be embedded into the crystal structure
at regular lattice sites; however, one electron is missing to complete the fourth bond
• the missing electron is alternately (randomly) taken from one of the neighbouring silicon atoms;
from outside it looks as if a positive charge is orbiting the positively charged atomic core of the
impurity atom
404
• at room temperature the positive charge is yielded to the crystal lattice, or in other words one
electron is permanently transferred to the impurity atom; therefore, the trivalent impurity atoms
are called acceptors
• Which are the physical consequences of the existence of donors in the crystal lattice?
the optical absorption of donors can be described by means of the Bohr theory as in a simple
approximation the motion of the electron around the positive atomic core of the impurity atom is
similar to the hydrogen atom
• the eigenvalues of the energy are given by
1 m * e4
1
Eν = −
⋅
2 (4πεr ε 0 )2 h 2 ν 2
where ν is the principal quantum number (ν is used here in order to avoid confusion with the
density of particles), m* is the effective mass (not the free-electron mass as for the hydrogen
model), and the dielectric constant εr takes into account the screening of the Coulomb potential
due to the neighbouring silicon atoms
• the latter quantities cause a 100 to 1,000 times lower energy eigenvalue compared to the
hydrogen atom; it is worth to note, that the effective Bohr radius
4πε r ε 0h 2
a0 =
m * e2
is about 50 times larger than the radius of the hydrogen atom; about 1,000 lattice atoms are
enclosed by the electron orbit!
405
• for ν = 1 one gets the ionisation energy denoted as Ed or Ea, respectively; for the donors in silicon
we have εr, Si = 11.7 and m*n ≡ 0.3m and get Ed ≡ 30 meV; as the vacuum for the hydrogen atom
corresponds to the conduction band in the semiconductor the ground state of the donor is 30meV
below the lower edge of the conduction band
energy E
n-type semiconductor
p-type semiconductor
Ed
ED
EC
acceptor bound
energy level
donor bound
energy level
Ea
EA
EV
(a)
x
(b)
x
after Hunklinger
energy levels of impurity atoms in semiconductors; (a) the ground state of the donor is just
below the lower band edge of the conduction band EC; the ionisation energy is Ed; (b) the
ground state of the acceptor is directly above the upper band edge of the valence band EV, Ea
is the ionisation energy of the holes
406
• for germanium we have εr, Ge ≡ 15.8 and m*n ≡ 0.15m, therefore the ionisation energy is as low
as Ed ≡ 9 meV
• donor and acceptor atoms themselves are not mobile; they do not contribute directly to the
conductivity; at low temperatures the impurity atoms are in the ground state and electrically
neutral; at room temperatures they are charged positively or negatively because of their low
ionisation energy
• in the frame of the simple hydrogen model the ionisation energy does not depend on the
particular donor or acceptor atom; for silicon and germanium one get the following figures
donor
P
As
Sb
Bi
silicon
45
49
39
69
germanium
12
12.7
9.6
-
acceptor
B
Al
GA
In
silicon
45
57
65
16
10.4
10.2
10.8
11.2
germanium
ionisation energies given in meV for
donors and acceptors in silicon and
germanium
• keeping in mind the very rough approximation given by the hydrogen model, we find a
surprisingly good agreement with the experimental values; apparently, the screening effect can
be described quite-well by the dielectric constant; the reason for that is the large radius of the
“hydrogen atom”
407
• the ionisation of the donors is small and corresponds approximately to the thermal energy at
room temperature; therefore, the donors are already ionised at that temperature; this is also true
for the acceptors
absorption coefficient α/cm-1
• from the hydrogen model one would further energy eigenvalues corresponding to the different
values of the principal quantum number ν; this is indeed the case as experiments on infrared
absorption show
25
Ge:Sb
20
infrared absorption of germanium
doped with 7 — 1026m-3 antimony
atoms at 9K
T = 9K
15
10
5
after Hunklinger
0
6
8
10
12
photon energy E/meV
14
408
• measurements where made at low temperatures that the non-binding electron sits in the ground
state; the spectrum shows a couple of absorption lines
• however, the spectrum is not that simple as could be expected from the simple hydrogen model;
the electrons of the donors are moving here in the crystal field with a cubical symmetry and not
in the Coulomb field with spherical symmetry
• above the ionisation energy of 9.6 meV (for germanium yields εr2m/m* ≡ 1,500) we can see the
transition into continuous states, namely the states of the conduction band
• similar spectra can be observed also for other impurity atoms as well as for other semiconductor
acting as the host crystal
• the theoretical description of the energy states of impurity atoms can be improved by taking into
account the solid state properties of the semiconductor, in particular the real behaviour of the
local electrical field (causing a splitting of the ground state and selection rules for the optical
transitions)
• comparing the influence of the different impurity atoms and the different host crystals we have to
take into account additionally the corresponding covalent bindings and the corresponding
effective masses as well as their directional dependence
409
germanium
energy levels of impurity atoms in germanium; the
number indicate the distance to the nearest band edge
in meV; the letters A and D denote acceptor or donor
levels, respectively (just in the case if the classification
of the level cannot be directly derived from its position)
after Hunklinger
• apparently the position of the impurity atoms which not belong to the third or fifth main group
cannot be explained by the simple model; in particular, chemical elements such as zinc,
chromium and copper can have several charge states with different energies; gold can form in
silicon four different levels and appear as donor as well as acceptor
410
Charge Carrier Density and Fermi Level
• in real semiconductor materials unwanted impurities cannot be completely eliminated; therefore,
also unintended donors and acceptors appear; the donors (placed at the upper edge of the
energy gap) will dispense the electrons to the acceptors (placed lower energies)
• this electron transfer does not create free charge carriers but the effect of the impurities
compensate each other
• the method to prepare highly resistive semiconductor materials where unavoidable impurities
are compensated by means of a systematic addition of depends is called compensation
• the energy of the quantum states forming band is practically independent of the occupation of
the states; the energy does not depend on whether the state is empty, occupied with one
electron or two electrons (with antiparallel spins)
• this is easy to understand as the electrons are delocalised (smeared out over the whole crystal);
they do not “see” each other; the single-electron approximation is valid
• in contrast to that the electrons at the impurities are specially localised; apart from very
sophisticated impurity atoms with different energy levels the impurity state can be empty, single
or double occupied (similar to the quantum states in the bands)
• in the case of a double occupation the energy of the impurity is increased significantly; this due
to the increased Coulomb repulsion for the localised electrons
• therefore, in terms of energy double occupied impurity states lie mostly inside the conduction
band; only the neutral or single charged impurity states of importance
411
• as the impurities are either neutral or ionised we can split their density nD and nA in a neutral and a
charged part:
nD = nD0 + nD+
nA = n0A + n−A
• the occupation of the ground state (the probability that the impurity atom is ionised) can be calculated
from the Fermi-Dirac distribution
−1
−1
0
0
nD  1 (ED −EF )/kBT 
nA  1 (EF −E A )/kB T 
=  ⋅e
+ 1
=  ⋅e
+ 1
nD  gD
nA  g A


the weightings gD and gA take into account the degeneration of the impurity states so that the usual
expression for the Fermi-Dirac statistics is a little modified
• for example, in a simple donor the electron can sit either spin-up or spin-down; these two possibilities
give the impurity state the two fold statistical weight gD= 2; the situation is more complicated for the
acceptor states where, in addition, the degeneration of the valence band has to be taken into account
• in order to simplify the further considerations we drop the problem of degeneration (we are only
interested in the general features)
• free charge carriers can be generated either by an excitation of electrons from the valence band or by
ionisation of impurity atoms
• the expressions for the excitation of the band states derived elsewhere for the intrinsic
semiconductors can be also used for the doped ones (note: we have not used assumption restricting
the results to intrinsic charge carriers)
412
• furthermore, we have to take into account the charge neutrality of the doped semiconductors:
n + n−A = p + nD−
• even for our simplified approach the situation is quite complex as we have to consider four
different charge carrier concentrations in a temperature dependent equilibrium
• we restrict ourselves to the simple bur realistic case where one type of impurity atoms dominates:
we consider a n-type semiconductor containing many donors but only a few acceptors nD>> nA
n = NC e − (E C −EF )/kBT (undegenerated semiconductor with EC-EF >> kBT assumed here)
nD = nD0 + nD+
nA = n0A + n−A
nD0
1
= (ED −EF )/k BT
nD e
+1
n + n−A = nD+ + p
• the doping is assumed to be that high that the contribution of the impurities to the free charge
carrier concentration dominates nD+ >> p, i.e. we have extrinsic conduction; the density of the
thermally excited electrons n i from the valence band can be neglected compared to the
contribution from the impurity atoms
413
• a further simplification results from the assumption of a n-type semiconductor nD>> nA; in this
case nearly all acceptors trap an electron from the donors and neutral acceptors practically not
appear; we can neglect n 0A and get n A ≅ n −A
• the charge neutrality can be written as n + n A = nD+
the occupation of states in a compensated ntype semiconductor; the states occupied by
electrons are shown in blue; as representatives
for the unwanted impurities (acting here as
acceptors), the density of states for two
acceptor levels is shown and labelled with ‘EA’
energy E
EC
EF
ED
‘EA’
EV
density of states D(E)
after Hunklinger
• the donors are partially ionised and have dispensed their electrons either to the conduction
band or to the acceptor levels
• the simplifications discussed above, allow us now to calculate the concentration of the
conduction electrons n and the temperature dependence
1


n = n − nA = (nD − n ) − n A = nD 1 − (ED −EF )/ kB T  − n A
+ 1
 e
+
D
0
D
414
• expressing the Fermi energy by means of the effective density of states NL one obtains
n(n + nA )
= NC e −Ed /kBT ,
nD − nA − n
where Ed= EC- ED is the energy difference between the conduction band and the donor level
electron density log n
• qualitative temperature dependence of the conduction electron density n
e
e − E d /2kB T
e − E d /kB T
α
energy E
− E g /2k BT
β
γ
δ
EC
ED
EF
reciprocal temperature T-1
electron density in the conduction band of a
n-type semiconductor (top) and the position
of the Fermi level (bottom) as a function of
the reciprocal temperature T-1; discussion of
the different temperature regions α to δ see
elsewhere
E C + EV
2
EV
after Hunklinger
415
• compensation region (region δ): at very low temperatures kBT << Ed n << nA<< nD is valid and
we get the approximate expressions
− E d /k B T
nD
n ≅ NC e
nA
nD
n
EF = EC + k BT ln ≅ EC − Ed + kBT ln
nA
NC
• for T 0 is the position of the Fermi level defined by the donors: EF≅ EL-Ed= ED; the donors are
partially charged due to the compensation effect; therefore, the Fermi energy corresponds to the
donor energy at very low temperatures
• with increasing temperature the donors dispense also electrons to the conduction band, the
charge carrier density is increasing exponentially; the Fermi energy increases linearly
• impurity reserve (region γ): the electron density is increasing fast with temperature and
exceeds at least the density of acceptors n A, which is in our example much smaller than the
density of donors; consequently we have n A<< n << nD and
n ≅ nDNC e −Ed/2kB T
EF = EC + kB T ln
n
E 1
N
≅ E C − d − k BT ⋅ ln C
NC
2 2
nD
416
• the Fermi energy is about in the middle between the conduction band and the donor level; the
exponent responsible for the increase of the conduction electrons is reduced by a factor of two
compared to the case of very low temperatures
• to put it simply, the donors play here the role of an electron source (similar to the valence band in
intrinsic semiconductors); in this temperature range a significant fraction of the donors is still not
ionised; this situation is called impurity reserve
• impurity depletion (region β): at room temperature we have kBT ≅ Ed and in good approximation
exp (-Ed/kBT) ≅ 1, taking into account nA<< n we find n2 ≅ NC— (nD-n)
as n << NC we get nD-n ≅ 0: the density of the free charge carriers is given the number of
impurities and depends not on temperature
• as NA can be neglected it results
n ≅ nD = const,
NC
,
nD
i.e. the Fermi level is decreasing
EF ≅ E C − k BT ln
• as now all impurities are ionised (n ≅ nD) it is called impurity depletion
• intrinsic conduction (range α): with further increasing temperature charge carriers from the
valence band are excited and the assumption p << nD becomes invalid and intrinsic conduction
dominates the Fermi level moves to the middle of the band gap and the number of free
charge carriers increases rapidly with temperature, just as known from the intrinsic
417
semiconductors
n = NC e
EF ≅
− E g / k BT
EL + E V
2
• example: measured density of free electrons in n-type germanium as a function of reciprocal
temperature
charge carrier density n [m-3]
1024
Ge
1020
density of charge carriers in n-type germanium
as a function of the reciprocal temperature,
measured by means of the Hall effect; the
range of intrinsic conductivity is shown by a
dashed line; the concentration of the arsenic
dopant is increased stepwise by one order of
magnitude and covers the range from 1019 m-3
to 1024 m-3
1017
0.0
0.02 0.04 0.06 0.08 0.1
reciprocal temperature T-1 [K -1]
after Hunklinger
• the concentration of the donors can be directly read off the graphs at higher temperatures (range
β); more or less pronounced one can see the ranges of intrinsic conductivity, the impurity
depletion and of the impurity reserve
418
• an exception is the sample with the highest donor concentration (nD= 1023 m-3) where the charge
carrier density is nearly independent of the temperature; at such high impurity concentrations the
distance between the donor levels is that small that the neighbouring wave functions are
overlapping, consequently, the electrons of the donors are not longer localised
• this effect is known as metal-isolator transition: at increasing concentration of donors (or
acceptors) a transition from a semiconducting to a metallic phase appears
conductivity σ [Ω-1cm-1]
1000
Si:P
100
10
isolator
metal
electrical conductivity of a highly doped
silicon sample; the metal-isolator
transition can be observed at an electron
concentration of n = 3.74 — 1024 m-3
1
0
2
4
6
8
density of electrons n [1024 m-3]
after Hunklinger
419
• for the phosphorus-doped silicon the metal-isolator transition can be observed at
n = 3.74 — 1024 m-3; for the arsenic-doped germanium at n = 3.5 — 10 24 m-3; in such cases of
highly doped semiconductors the Fermi level is positioned in side of the conduction band and
the charge carrier concentration is independent of temperature; the sample shows metallic
behaviour
• most commonly the concentration of the free charge carrier is measured by means of Hall
experiments; for semiconductors both types of charge carriers contribute to; as already derived
elsewhere the Hall constant RH is given by
RH =
ρµρ2 − nµn2
e (ρµρ + nµn )
2
• the sign of the Hall constant is determined by the charge carriers contributing most to the
charge transport; if one type of charge carrier dominates (p << n or vice versa) their
concentration can be derived directly from the measured Hall voltage (in this case RH does not
depend on the mobility)
• compared to metals the Hall constant is quite large because of its relatively small concentration
of charge carriers
• example: Hall measurements at n-type and p-type indium antimonide (InSb) samples
420
Hall constant |RH| [cm3A-1s-1]
105
InSb
104
Hall constant of indium antimonide as a
function of reciprocal temperature; the graphs
for the n-type samples are shown in black, the
graphs for the p-type samples are blue
coloured
103
102
10
1
2
3
4
5
6
7
8
-1
-3
-1
reciprocal temperature T [10 K ]
after Hunklinger
• in the range of impurity depletion (right-hand side) one type of charge carriers is dominating; the
expression for the Hall constant can be simplified to
RH ≅ −1/en ≅ −1/enD
or
R H ≅ 1/eρ ≅ 1/enA
• in this case, the density of charge carriers is constant; the mobility does not affect the Hall
voltage
• at high temperatures (range of intrinsic conductivity) the Hall constant decreases exponentially
with the temperature as the density of charge carriers is increasing exponentially
421
• because of their extremely high mobility electrons contribute most to the conductivity and Hall
constant (see elsewhere); as a consequence the p-type samples a very remarkable temperature
behaviour: at the transition from the hole conduction to the intrinsic conduction the Hall constant
goes to zero and changes the sign afterwards
Mobility and Electrical Conductivity
• to get the complete understanding of the electrical conductivity the mobility of the charge carriers
µ = eτ/m* has to be discussed
• typical values of the mobility µn and µp for electrons and holes for some selected materials at
room temperature
mobility [cm²/Vs]
C
Si
Ge
GaAs
InSb
µn
1.800
1.900
3.800
9.200
80.000
µp
1.400
480
1.800
400
1.250
• effective masses as well as the mean free time between collision affect the mobility; in
semiconductors the electrons are mainly scattered by the phonons; the electron- electron
interaction can be neglected because of their low density compared to metals
• negatively and positively charged carriers act similar in terms of the mean times between
collisions as also the scattering of holes can be finally attributed to an electron scattering
422
• example: mobility of the electrons in aluminium- doped Mg2Ge
mobility µ/cm2V-1s-1
103
mobility of the electrons in aluminium- doped
Mg2Ge, measured by means of the Hall effect; the
samples contain 1.3 — 1022, 4.2 — 1022 and 8.2 — 1023
impurity atoms per m3; sample no.(3) shows metallic
behaviour
102
after Hunklinger
101
5
10
20
50 100 200
temperature / K
• no. (3) is the most highly doped sample and shows metal-like behaviour
• sample no. (1) was not knowingly doped but its impurity concentration is 1.3 — 1022m-3; for sample
no. (2) the aluminium concentration is 4.2 — 1022m-3
• apart from no. (3) the mobility's show a steep increase of mobility at low temperatures, a
maximum at about 50K too 100K followed by a steep decrease
• similar to metals at low temperatures the scattering processes at the impurity atoms are
dominating; as the impurity atoms are charged we have to take the Rutherford scattering in
order to describe the scattering mechanism correctly
423
• the integral cross section σsc is proportional to the forth power of the mean velocity νof the
scattering particles (see nuclear physics)
σ sc ∝
• note:
1
ϑ4
, with
3
m* 2
kBT =
ν → ν = 3kB T/m *
2
2
the kinetic theory of gases can be used here as the charge carriers in non degenerated
semiconductors obey the Boltzmann distribution
• for the mean free path l we get l −1 = n sc σ sc ∝
nsc
, where nsc is the density of charged impurity
4
ν atoms
• for simplicity we assume that the number of charged impurity atoms does not depend on
temperature:
1 ν nsc ν nsc
= ∝ 4 ∝ 3/2
τ l
ν
T
or
eτ T 3/2
µ=
∝
m * nsc
• in reality, the number charged scatter centres depends on temperature as well as on the detail of
the compensation mechanism in the sample, therefore the measured temperature dependence of
the mobility very often differs from the simple T3/2 law (see graphs elsewhere)
424
• nevertheless, we use the simple model in order to estimate the scattering cross- section of the
aluminium ions: for the mobility of sample (2) at 30K we find µ ≅ 500 cm²/Vs; with
1
1
e
σ sc =
=
=
nsc ν
l ⋅ hsc τ ⋅ νhsc m * µ
we get σsc ≈ 2.26 — 10-11 cm2 (for m* = me) and a scattering diameter of about 500Ǻ, which is
quite large
• note:
in metals the scattering capability of point defects is much less effective because of the
screening of the Coulomb potential by the free electron gas
• above the maximum, at higher temperatures (room temperatures), the collision of the charge
carriers with acoustic phonons are dominating; in order to find out the temperature dependence of
the mobility µ in this range of temperature we use again the relationship between mean free path
l and scattering cross- section rsc
• at room temperature the phonons with the Debye frequency ωD dominate; therefore the scattering
cross- section is independent of temperature; for T >> θ we get for the phonon density nph ∝ T
• taking into account
the reciprocal mean time between collisions is
1 ν
= = ν nphσ sc ∝ ν nph ∝ T 3 / 2
τ l
or
µ ∝ T −3 / 2
this temperature dependence is in good agreement with the experimental results
425
• the metal- like behaviour of sample no. (3) becomes apparent in the weak temperature
dependence as well as in the high values of the mobility at low temperatures
• in conclusions we consider the electrical conductivity; example: temperature dependence of ntype germanium
temperature T/K
electrical conductivity σ/Ω-1m-1
105
100 50
20
10
electrical conductivity of n- type germanium;
samples are identical with those used elsewhere
for the experimental determination of the charge
carrier density; the sample with the highest
electrical conductivity shows a metal- like
behaviour
104
103
102
101
after Hunklinger
100
0.00
0.02 0.04 0.06 0.08
reciprocal temperature T-1/K-1
0.10
• starting at low temperatures the electrical conductivity is increasing exponentially with the
increasing concentration of charge carriers; the temperature dependence of the mobility and of
the effective density of states is not very important
426
• in the range of the impurity depletion, where the concentration of the charge carriers is nearly
constant, the electrical conductivity is decreasing in temperature due to the temperature
dependence of the mobility
• at high temperatures the intrinsic conductivity appears which can be observed at the sample
with the smallest impurity concentration (lowermost graph)
• an exception is the sample with the highest doping (uppermost graph) of nD=1024m-3, where the
conductivity does nearly not vary with temperature; in this sample the concentration of the
charge carriers in practically independent of temperature as already discussed elsewhere
• note:
r
r
Ohm’s law j =r neµ E is only valid as long as the mobility µ does not depend on the
electrical field E; indeed, for semiconductors of technical importance the mobility
decreases for fields beyond 105 V/m
r
r
• in this case, the drift velocity ν D = µE reaches is threshold value of about 105 m/s
• reason for that is the electron- phonon coupling: at high fields the energy gain of the electrons in
the electrical field is sufficiently high to create optical phonons; because of its high density of
states this process is very efficient and limits the drift velocity of electrons and holes
• note:
this limitation of the drift velocity is important for the modern semiconductor devices as
there, because of their small size, fields up to 107 V/m may appear
427
8.3. Inhomogeneous Semiconductors
• we consider here the basic physical ideas which are important to understand the technical
applications of semiconductors
• semiconductors where the doping or the chemical composition varies spacially are called in this
regard inhomogeneous semiconductors
• example: p- n junction
p-n junction
• the spatial variation of the concentration of impurity atoms is a /conditio sine qua non/ for the
application of semiconductors in the solid state electronics
• to this end the impurity atoms are either implanted or diffused into well-defined region of the
host material
• for the lateral definition of the doped regions in the micrometer and nanometer rang several
methods are applied; by for the most widely used method is the photo lithography not discussed
in detail in the context of our basic considerations
• for the following considerations we assume abrupt transitions, where the concentration of
depends varies step-like; the technical implementation of such a transition is of course only an
approximate approach
428
• example: p-n junction
semiconductor
semiconductor
EPF
EA
EPV
(a)
EnC
ED
EnF
EPC
energy E
n- type
energy E
E
P
C
p- type
EnV
X
EnV
(b)
X
after Hunklinger
position of the energy levels of the impurity atoms, of the band edges and of the Fermi level at
the p- n junction; (a) position of the energy levels at separated p- type and n- type crystals; (b)
p- n junction in the (thermodynamic) equilibrium; the gap edges are shifted to each other by the
diffusion voltage VD; the dots denote the electrons, the open circles the holes
• EPC and EnC denote the edge of the conduction band, EPV and EnV the edge of the valence band of
the p- type or n-type semiconductor, respectively; EPFand EnFare the Fermi levels of the isolated
crystals or the respective Fermi levels in a sufficiently large distance from the interface between
the two differently doped crystal regions of the p- n junction
429
• depending on the doping the Fermi level is at room temperature either slightly below the donor
level or slightly above the acceptor level; if p-type and n-type semiconductors are in a close
contact, the Fermi levels adapt each other or more general, from the thermo dynamical point
of view: in the equilibrium the chemical potentials have to be the same all over the crystal
• this is archived by a diffusion of charge carriers from the respective regions of high concentration
into the regions of lower concentration: electrons are diffusing from the n- type into the p-type
semiconductor and the holes /vice versa/
• the small charge transfer by diffusion leaves behind on the p side and excess of (-) ionised
acceptors and on the n side an excess of (+) ionised donors; in this way a charge double layer is
forming which creates an electric field directed from n to p
• this electrical field inhibits diffusion and thereby main taints the separation of the two carrier
types; the electrical potential in the crystal takes a jump in passing the region of the junction
• the jump of the voltage causes a shift in the chemical potentials in such a way that an uniform
value is arising across the whole crystal; as a consequence the bending of the bands occurs
• the effect of the concentration gradient exactly cancels the electrostatic potential V and
effectively the net particle flow of each carrier type is zero (we neglect for the moment the
recombination processes)
430
(a)
(a) variation of the hole and electron concentration across
an unbiased (zero applied voltage) p- n junction; (b)
~
electrostatic potential V ( x) from acceptor (-) to donor (+)
ions near the junction interface; the potential gradient
inhibits the diffusion of holes from the p side to the n side,
and it inhibits diffusion of electrons from the n side to the
p side; the electrical field in the junction region is called
built- in electrical field
after Kittel
X
(b)
~
• the bending of the band can be described by means of the macroscopic potential V ( x)
• the diffusion voltage VD, the potential difference between the two differently doped regions of
the semiconductor characterises the p- n junction; its value is given by the Fermi levels of the
two doped crystals and here with essentially by the band gap of the semiconductor:
431
eVD = EFn − EPF
N 
N 
NN 
= EC − k B T ln C  − E V − ln V  = EG − k B T ln C V 
 nD 
 nA 
 nD ⋅ n A 
or
eVD = k B T ln
nD ⋅ nA
n2i
where the Fermi levels are calculated for the state of impurity depletion of the semiconductor
which is a good approximation at room temperatures; furthermore have used the law of mass
action as derived elsewhere
• as a first approximation we find eVD ≅ EG
• charge carriers which are moving in the region of the corresponding doping, i.e. electrons in the
n- region and holes in the p- region are called majority charge carriers (or minority carriers)
• notation:majority carriers: nn and pp
minority carriers: np and pn
• at a larger distances from the interface of the p- n junction we have
nn ≅ nD+ ≈ nD
and
p p = n−A ≈ n A
432
• in conformity with the law of mass action n(x) — p(x) = const. must be valid at any position of the
junction
pn — pp = nn — pn = nip i
taking into account the usual doping concentrations it follows that the density of majority carriers
is much higher than the density of the intrinsic charge carriers where as the density of minority
carriers is much lower than the density of the intrinsic carriers
charge carrier density log n, p
• modelling the p-n interface
p-type
n-type
semiconductor
semiconductor
−
A
n
pp
nD+
nn
ni ,pi
np
concentration of free charge carriers and
ionised impurity atoms at the p- n junction in
logarithmic scale; it is assumed that the
density of donors (tinted in grey) overbalances
the acceptor density (tinted in blue); the
depletion layers is shown schematically in
lighter colour tints
pn
x=0
after Hunklinger
433
• close to the interface of junction the density f charge carriers in changing rapidly; taking not
~
account the potential curve V ( x) , the spatial behaviour of the edge of the conduction band is
~
E C ( x) = EPC − eV( x );
(note: a similar expression can be found for the valence band)
• assuming that the semiconductor is in the state of impurity depletion in which the impurity
atoms are completely ionised, the density of the free charge carriers can be calculated (see
section 8.1)
h( x) = np e
~
eV ( x ) / k B T
and
p( x) = p p e
~
− V ( x ) / k BT
;
the corresponding graphs are shown in the figure above in logarithmic scale
• from the expression for the free charge carriers we can see the law of mass action n(x) — p(x)
is valid along the whole p- n junction; as a consequence, the concentration of the charge
carriers close to the p- n interface is significantly reduced; this situation is known as charge
carrier depletion region
• in this region the charge of the ionised donors or acceptors, respectively is not completely
compensated by the free charge carriers and a space charge appears
434
carrier concentration [log scale]
neutral region
space
charge
region
electrons
holes
p-doped
diffusion force on holes
E-field force on holes
neutral region
n-doped
E-field
x
diffusion force on electrons
E-field force on electrons
a p-n junction in thermal equilibrium with zero bias voltage applied; electrons and holes
concentration are reported respectively with blue and red lines; grey regions are charge
neutral; light red zone is positively charged; light blue zone is negatively charged; the
electric field is shown on the bottom, the electrostatic force on electrons and holes and the
direction in which the diffusion tends to move electrons and holes
435
• the space charge is given by
ρ(x) = e[nD+ − nn (x) + p n (x)]
for x > 0
n- type semiconductor
for x > 0
p- type semiconductor
ionised majority minority
donors carriers carriers
ρ(x) = − e[n+A + np (x) − pp (x)]
~
• the potential curve V ( x) and the space charge ρ(x) are interlinked by the Poission equation
~
∂ 2 V(x)
ρ(x)
=
−
∂x 2
ε0 εr
• from a mathematical point of view the solution of this equation is not that simple as the space
charge depends again on the potential; for the solution a self- consistency procedure must be
applied
436
• for simplicity we use here the Schottky model, which takes into account that in the region of
space charge close to the interface the concentration of free charge carriers is very low and
can be even neglected in a first approximation:
ρ(x) =
0
for
x < -dp
-enA
for
-dp < x < 0
enD
for
0 < x < dn
0
for
x > dn
where dn and dp denote the thickness f the corresponding space charge regions
• note:
as the regions of space charge must be in total electrically neutral the two areas
nA — dp and nDdn must be the same
• for an rectangular space charge the Poission equation can be easily solved region wise; for
0 < x < dn we get 2 ~
∂ V(x)
enD
−
≅
∂x 2
ε 0ε r
~
∂V
en
Ex = −
= − D (dn − x)
∂x
ε 0 εr
en
~
~
V(x) = Vn (∞ ) − D (dn − x) 2
2ε 0ε r
437
• similar equations can be derived for the n region; taking into account the neutrality of electrical
charge nD dn = nAdp and the continuity of the electrical potential at x = 0
en
en
~
~
Vp (-∞) + A dp2 = Vn (∞) − D dn2
2ε 0ε r
2ε 0ε r
or
~
~
Vn (∞) - Vp (- ∞) = VD =
e
(nA d2p + nDd2n )
2ε 0ε r
we find for the thickness of the space charges
2ε 0 ε r VD n A
1
dn =
⋅ ⋅
e
nD (n A + nD )
2ε 0 εr VD nD
1
and
dp =
⋅ ⋅
e
nA (nA + nD )
~
• schematic representation of ρ(x), Ex(x) and − V( x)
Schottky mode for the space charge
region; (a) space charge in a
rectangular approximation, a more
realistic graph for the charge
density is shown schematically by a
dotted line; (b) electrical field
strength Ex(x); (c) graph of the
~
electrical potential − V( x) , note the
negative sign of the potential curve
(a)
(b)
(c)
x
after Hunklinger
438
• in order to estimate the size of the space charge region we assume eVD ≅ EG ≅ 1eV and
nA ≅ nD ≅ (1020 … 1024)m-3; then the thickness of the space charge region is dn ≅ dp ≅ 1µm …
10nm; the electrical field strength is Ex ≅ (106 … 108)V/m
Electrical Currents in the p-n Junction
• even in thermal equilibrium there is a small flow of electrons from n to p side (and vice versa
holes flow from p to n) where the electrons and their life's by combination with holes at the other
side, this current is a consequence of the different concentration of charge carriers at both sides
of the junction and is called diffusion current (sometimes recombination current)
• the drift current is balanced by the field current (sometimes drift current) where electrons
which are thermally generated in the p region and which are pushed by the built- in field to the n
region (and vice versa the holes in the opposite direction)
• if jf is the field current and jd the diffusion current we have jf + jd = 0
both currents are composed of electrons and holes j f = j fn + jpf and j d = jdn + jdp
as neither the holes nor the electrons can accumulate somewhere in the crystal also the single
components compensate each other jnf + jdn = 0 and jpf + jpd = 0
439
• the field current is generated by the minority carriers which where pushed by the built- in
electrical field into the region of the majority carriers; assuming for simplicity that the region of
space charge is small (smaller than the mean free path) and that the rate of recombination is
also small there nearly all minority carriers reach the majority region
the field current does nearly not depend on the particular shape of the potential
• on the other hand, the majority carriers (for example electrons in the h region) must climb the
energy wall eVD from the low side of the barrier to the high side; only a few charge carriers can
successfully climb the wall, the Boltzmann factor gives us the fraction of the „successful“ charge
carriers
jd = α(T) e-eVD/kBT ,
where α is the amplitude depending weakly on temperature
• taking into account the thermal equilibrium the magnitude of the current densities are
| jt| = | jd| = α(T) e-eVD/kBT
Biased p-n Junction
• i the p-n junction is biased by an external voltage V a large current will flow if we apply a voltage
across the junction in one direction, but if the voltage is in the opposite direction only a very
small current will flow
• if an alternating voltage is applied the current will flow chiefly in one direction; the p- n junction
rectifies the current
440
• the applied external voltage affects the balance of field current and diffusion current or, in other
words the thermal equilibrium; if however, the p- n junction is in a steady state and not far away
from the thermal equilibrium, the following model works well:
the applied voltage V drops chiefly in the region of space charge (or depletion region)
where the concentration of free charge carriers is low and which is consequently shows
a high resistance
the remaining part of the semiconductor is nearly field- free:
(note: the sign of the bias voltage V is chosen such that a positive voltage is directed
opposite to the diffusion voltage VD and therefore reduces the potential difference)
• within this sign convention the rectification property of the p- n junction the polarity:
if the polarity of the p region is positive the p- n junction is operated in the forward
direction (or pass direction)
if the polarity of the p region is negative the p- n junction is operated in the reverse
direction
• influence of the bias voltage on the energy bands and the Fermi level
441
p- type
n- type
p- type
EρC
EρC
EnC
E nF
ρ
F
ρ
V
E
E
− e( VD + | V |)
energy E
− e( VD − | V |)
energy E
n- type
EρF
EnC
EρV
EnF
EnV
EnV
(a)
x
(b)
x
p-n junction biased with an external voltage V the quasi Fermi level of the electrons in
shown as a blue dotted line, the quasi Fermi level of the holes is shown as a black dotted
line; (a) forward direction: the voltage +V decreases the potential hill; (b) reverse direction:
the voltage –V increases the potential hill
after Hunklinger
442
• in the region of the space charge the charge carriers are not in the thermodynamic
equilibrium; there is no joint Fermi level; however, electrons and holes are in an equilibrium
state among themselves
• therefore, two separate quasi Fermi levels can be defined, which can be treated
independently; as we will see elsewhere this is of importance for semiconductor lasers
• How does the bias voltage affect the field current and the diffusion current?
• the field current is nearly unaffected; once in the sphere of influence the built- in pushes the
minority carriers through the region of space charge to the majority side; this is to a large
extend independent from the field strength; for the electrons for example
jnf ( V ) ≅ jnf (0)
• on the other hand the bias voltage modifies the height of potential hill; instead f eVD the
majority carriers have to overcome the potential wall VD-V
• therefore, the diffusion current will change
jdn (V) = α(T)e − e(VD − V)/kBT = jdn (0)e eV/kBT
and the two current components, field current and diffusion current, flow in opposite direction
we find the following net current for the electrons
jn (V) = jnd (V) - jnf = jnf e eV/kBT − jnf = jnf (e eV/kBT − 1)
where we have used | jn (0) |=| jn ( 0) |
d
f
443
• adding together the contributions of electrons and holes the total current is
j(V) = ( jnf + jpf )(eeV/kBT − 1) = j f (e eV/kBT − 1)
• current- voltage characteristic of the p- n junction (p- n diode)
500
current I/µA
400
p- n diode
300
current- voltage characteristic of a p- n
diode; notice the different scales in the
forward direction and in the reverse
direction
200
100
0.1
0.0
-0.1
after Hunklinger
-0.2
0.0
0.2
voltage V/volts
444
• the very non linear behaviour of the characteristic indicate clearly the rectifier properties of the pn diode
• foe the forward direction the potential wall is given by e(VD-V) which can be climbed more and
more easily with increasing voltage V
• for the polarity in the reverse direction the maximum current is given by the very small field
current; a further increase is impossible as the generation of minority carriers cannot be
controlled (it is given by temperature)
• note:
a further increase of the bias voltage at a p- n junction with a high doping in reverse
direction may cause a sudden high current just in reverse direction; for silicon diodes
with a concentration of 1025m-3 dopands or higher the critical voltage is about 2 volts
• at p-n junctions with high doping a high electrical field s can appear in the small depletion layer;
as a result the lower edge of the conduction band of the n-region is shifted below the upper edge
of the valence band; as a result electrons can tunnel from the valence band of the p- type
semiconductor into the conduction band of the n-type semiconductor
• the voltage at which the sudden large current flow in the reverse direction appears is known as
breakdown voltage or Zener voltage (after C. Zener); in electronic circuits Zener diodes are
used for the generation of constant reference voltages
445
I
reverse
breakdown
schematic diagram of the
current-voltage characteristic
of a p-n junction diode (not to
scale); Von is known as cut-in
voltage or on-voltage or diode
forward voltage drop, Vbr is
the breakdown voltage
forward
Vbr
Von
V
• capacitance of the depletion layer is given by the thickness of the zone of space charge and
the bias voltage; in the frame of the Schottky model we have simply to substitute the diffusion
voltage VD by (VD-V)
dn (V) = dn (0) 1 −
V
VD
and
d p (V) = d p (0) 1 −
V
VD
where dn(0) and dp(0) denote the thicknesses at zero bias voltage
446
• let be A the sectional area of the p- n junction, then we get from the Schottky model for the
stored charge Q
Q = -enD dn(V) — A and therefore for the capacitance of the space charge
C=
dQ
d dn(V) A 2e ε 0 ε r n A nD
= − enD A
=
⋅
dV
dV
2 VD − V nA + nD
• as the capacitance depends on the number of impurity atoms the voltage dependence of the
depletion layer is often used to experimentally determine the impurity concentration
• in electronic circuits p- n junctions are often used as tuneable capacitances, known as varicap
diodes or varactor diodes, in order to tune the resonance frequency in resonant circuits (RLC
circuits) for electronic filters or oscillators
Metal- Semiconductor Junction
• the electrical contact between a metal and a semiconductor is very important for electronic
circuitry; ideally electrons may enter or leave the semiconductor; indeed this is a simple
exceptional case
• instead of an ohmic contact very often a blocking contact is observed, which strongly
constrains the current flow; the reason for it are the different work functions Φ of both materials,
the value of which is given by the distance of the Fermi levels to the vacuum energy level
447
• in the case of a n- type semiconductor we have
Φsc > Φme :
ohmic junction,
Φsc < Φme :
blocking junction;
for p- type semiconductors it is just contrariwise; the indices “sc” and “me” denote here the
semiconductor or the metal, respectively
• example: n- type semiconductor with Φsc > Φme
Φme
energy E
Φsc > Φme
Φsc
me
(a)
EC
ED
EF
energy E
Evac
n- type sc
Φme
me
x
Φsc
n- type sc
x
(b)
EF
me
(c)
Φsc
Φme
EC
ED
EF
energy E
Φsc < Φme
energy E
Evac
n- type sc
x
Φme
me
(d)
Φb Φsc
metal n- type semiconductor
junction before (left) and
after (right) bringing the two
materials in a close contact;
top: ohmic contact, the free
electrons at the boundary
layer are tinted in black;
bottom: Schottky junction,
the positive space charge in
the depletion region is
marked by the plus sign
n- type sc
x
after Hunklinger
448
• the situation of a n- type semiconductor with a larger work function than the metal is shown in the
top left figure (a), where the position of the vacuum potential, the gap edges, the donor states,
the Fermi levels as the work functions are plotted
• as soon as both materials are put in close contact electrons start to flow from the metal into the
semiconductor; as a result the potentials start to change in both materials to balance the two
Fermi levels
• similar to the p- n junction also here the bands of the semiconductor are bended close to the
metal- semiconductor interface as shown in the top right figure
• in the semiconductor close to the interface an enrichment of electrons can be observed as there
the Fermi level is just inside of the conduction band; under influence of an bias voltage electrons
can flow through the interface into the semiconductor (bottom left) ohmic behaviour is
observed
• if the work function of the metal is larger than that one of the semiconductor then, contrariwise
the electrons from the semiconductor to the metal to balance the Fermi levels
• as a consequence, at the boundary surface of the semiconductor a highly resistive depletion
layer appears which is blocking the current flow; this junction is known as Schottky junction
• the potential barrier between the semiconductor and the metal Φb is called Schottky barrier;
typical values are p-GaAs Φb ≅ 0.95eV, n-GaAs Φb ≅ 0.47eV
449
• Schottky barriers can be consider as one half of a p- n junction where the metal plays the role of
the p- type semiconductor and the formulas of the p- n junction can be used
• for a n- type semiconductor and a metal we have to take into account nD<<nA as the number of
states in the metal exceeds the number of states of the p- type semiconductor; for a metal- ptype semiconductor junction the argument is just the other way around
• the rectifying properties of Schottky junctions were already applied in the very early days of
radio engineering at the beg in of the 20th century in a device known as crystal detector
Semiconductor Heterostructures and Superlattices
• by means of thin film processes such as Molecular Beam Epitaxy (MBE) or Metal Organic
Chemical Vapor Phase Epitaxy (MOCVPE) it is possible to deposit layers of different
semiconductors with nearly perfect overall crystalline structure, which are known as hetero
junctions
• here it is important, that the lattice parameters of the systems GaP/Si, GaAs/Ge or InAs/GaSb
as well as for ternary and quaternary alloy as AlxGa1-xAs or GaxIn1-xAsyP1-y
• by changing the mixing ration the band gap can be adjusted in a certain range to the particular
application case; for the AlxGa1-xAs system the energy gap can be adjusted continuously
between 1.4eV (GaAs) and 2.2eV (AlAs)
• bringing together two semiconductors with different band gap at the interface a band bending
or band discontinuity can be observed
450
before contact
energy E
A
B
∆EL
EF
EL
EF
∆ EV
EV
equilibrium established
EL
energy E
EF
EV
hetero junction of two n- type semiconductors
with different band gaps; semiconductor A has a
stronger impurity doping than semiconductor B;
top: bands and Fermi levels of the original
materials before contact; bottom: after the
equilibrium is established the Fermi levels are
balanced; at the interface the band
discontinuities ∆EC and ∆EV as well as the band
bending can be observed; the degenerated
electron gas close to the interface in the B
semiconductor is finished in black
after Hunklinger
x
• as the transition from one band gap to the other one occurs within one atomic distance the
band discontinuity is very sharp; therefore, the associated electric fields can be in the order
of magnitude of the atomic fields of about 1010V/m
• the band discontinuity appears at both valence band and conduction band; for GaAs/Ge typical
values are ∆EV= 0.49eV and ∆EC= 0.28eV
• similar to the p- n junction also for the hetero junctions the Fermi levels are balanced in the
thermal equilibrium; consequently also the band bending can be observed
451
• depending on the charge carrier density the band bending extends over a length of a few
hundreds of Angstroms; the resulting electric fields are in the order of magnitude of 107 V/m
• isotype hetero junctions are structures with two different semi conducting materials but the
same type of doping; example for n- type material see figure above
• as in the thermal equilibrium the Fermi levels are balanced free electrons accumulate in the
semiconductor with the smaller band gap (semiconductor B) close to the interface
• it is even possible that close to the interface the Fermi level is inside of the conduction band and
we have a degenerated semiconductor there; on the opposite side of the interface (in the
semiconductor A) we have a depletion layer as the electrons from the stronger doped material
pass over to the weakly doped semiconductor B with the energetically lower quantum well
• this is also true if the material B is an intrinsic (e.g. undoped) semiconductor; then we have the
unusual situation of an undoped semiconductor where nearly all free electrons remain and where
this high concentration of electrons is specially separated from the donors
• note that in common semiconductors a high concentration of charge carriers always goes
together with a high density of impurity atoms
• in classical semiconductors the mobility of electrons is limited at low temperatures by the strong
impurity scattering where as in the hetero structures discussed here even at high charge carrier
densities high values of mobility can be expected as the electrons are moving in a region with a
low density of impurity atoms
• with the increasing progress in MBE technology indeed enormous mobilities at low temperatures
could be demonstrated
452
mobility µ/cm2V-1s-1
progress in increasing the electron mobility of
AlGaAs/GaAs hetero structure by improvement of MBE
technology from the mid 80’s to the mid 90’s; numbers
at the graphs denote the year of sample fabrication; the
largest values of mobility shown here are from a
Al0.35Ga0.65As/GaAs system
temperature T/K
after Hunklinger
• the sample with the largest value of electron mobility shown here had a complicated layer
structure in order to minimise the scattering processes at the interface; the dopands were
Silicon atoms in a Al0.35Ga0.65As layer which has a larger band gap than the GaAs material
• the electrons of the donor atoms are delivered to the energetically lower conduction band of the
adjacent intrinsic GaAs material
• as the epitaxial GaAs layer is to a large extent free of any impurities one can observe an
increase of the mobility by about four orders of magnitude compared with the conventionally
doped GaAs crystals
• systems, where different materials are put together periodically are called superlattices; a
sequence of the hetero structures discussed above is called Composition- modulated
Superlattice (modulation of doping)
453
energy E
Composition- Modulated Superlattice
E AC
EF
E BC
EBV
E AV
energy E
A
B
A
B
A
B
E AC
EF
E BC
E AV
EBV
spatial variation of the band gap in a
composition- modulated superlattice; top: band
edges and Fermi levels of the primary materials;
bottom: superlattice with quantum well structure
where the 2D electron gas (tinted in grey) is
located
after Hunklinger
z
• alternating sequence of layers of the semiconductor A with strong n- doping and layers of the
nearly intrinsic semiconductor B; because of the new band structure the electrons concentrate
in the quantum wells of the undoped semiconductor B where as in the respective highly doped
layers of the semiconductor A depletion zones occur
• the electrons capture in the quantum wells show different physical properties along and
perpendicular to the interfaces: in parallel to the interfaces the wave function of the electrons
can be described in terms of travelling Bloch waves
454
• perpendicular to the interfaces (z direction) the quantum well limits the propagation of the
electrons significantly; for the shake of discussion the eigenvalues of the electrons we
approximate the quantum well by a potential box in the z direction
• this corresponds exactly to the situation of the 2D electron gas discussed elsewhere; the
eigenvalues of the energy are given by
E j (k x ,k y ) =
h 2 (k 2x + k 2y )
2m * xy
+ Ej
where m*x,y denotes the effective mass for the electron propagation in the xy plane and Ej is the
transversal energy
• as the electrons can only move freely in the xy plane, such hetero structures provide a 2D
electron gas; note: a similar consideration is also true for the holes in the maxima of the
valence band
• for the 2D electron gas the density of states is constant (see elsewhere); for each sub band j
we have
D j (E) =
m * xy
πh
2
;
the existence of the step- like density of states can be demonstrated by means of optical
absorption experiment or photoluminescence
455
• note:
the transverse energy levels Ej are only sharp if the single quantum wells forming the
superlattice are sufficiently far from each other; if the distance between two quantum
wells is smaller than about 100Ǻ the overlap of the wave functions gets noticeable and
leads to a splitting of the energy level
• this effect is completely similar to the level splitting described by the “tight binding model”
discussed elsewhere and leads to the formation of so- called superlattice miniband structure
• superlattices offer the opportunity to systematically vary the distance between the quantum
wells or the overlap of the wave functions
Doping Superlattice
• doping superlattices are semiconductors which are alternately doped as n- type and p- type
material; the period of spatial repetition can be varied over a wide range, however a typical
value is some 100Ǻ
• as in- between the n- region and the p- region a small intrinsic transition zone appears (or even
knowingly deposited), such superlattices are also called n-i-p-i structures
456
energy
ECA
EBC
EF
EF
EAV
EBV
energy
A
B
A
B
A
ECA
EF
E AV
B
EBC
Eeff
g
spatial variation of the band gap in a doping
superlattice; top: band edges and Fermi levels
of the primary materials; bottom: in the
superlattice a wavelike potential develops; the
effective band gap as well as the free charge
carriers (labelled by plus sign and minus sign)
are also shown
EF
EBV
after Hunklinger
• the doping superlattice shows a wavelike band structure; this is caused by the chemical potential,
which must have the same value all over the semiconductor
• because of the edge of the valence band are closer to the Fermi level; as a consequence excited
free electrons can be found in the minima of the conduction band and holes in the maxima of the
valence band
• the two species of charge carriers are spatially separated what hinders recombination processes
and increases significantly the life time of electrons and holes
• furthermore, the modulation of the band edges causes a decrease of the effective optical band
gap
457
8.4. Semiconductor Circuit Elements and Devices
• now a days data processing in date transfer is mainly based on integrated circuits fabricated in
silicon technology; optoelectronic devices such as optical detectors, light-emitting diodes, or
semiconductor lasers mostly make use of III-V semiconductors
• semiconductor electronics devices can be subdivided into two-terminal devices (diodes) and
three-terminal devices (transistors); in diodes the properties of the current flow between two
electrodes is used, while in transistors the current or voltage between two electrodes is controlled
by the external voltage at a third electrode
• most of the optoelectronic devices are diodes while in the field of data processing or power
electronics transistors are widely used
• depending on whether only one or two types of charge carriers are involved the devices are said
to be unipolar or bipolar
Devices based on p-n Junctions
Solar Cell
• in a p-n junction the ionising radiation produces free electrons and holes; the number of electrons
is proportional to the energy transmitted by the radiation to the semiconductor
458
• if the absorption of a photon takes place in the depletion region (space charge region), both
charge carriers are separated there by the existing electrical field
• thus an additional electrical current IL is generated, which adds on the field current flowing
without illumination, see p- n junction
I = IS ( e
diffusion current
eV
− 1) − IL
kBT
field current
current I [mA]
• current- voltage characteristic of α silicon solar cell of 4cm2 under illumination
rectangle of
maximum
power
current- voltage characteristic of a silicon solar
cell under illumination; the optimum working
point is where the area of the blue tinted
rectangle and therefore the supplied electrical
power is a maximum; on top left the schematic
of the electric circuit as well as the load
resistance is shown
after Hunklinger
voltage V[v]
459
• for the calculation of the current-voltage characteristic typical values as IL= 100mA and IS= 1nA
were used; the circuit schematic includes the load resistance the vale of which has to be
optimised
• the electric circuit is characterised by the off-load voltage (I=0) and the short-circuit current (U=0)
for I = 0 we get for the voltage at the p- n junction
 k B T IL
k B T  IL

V=
ln + 1 ≅
ln ≅ 0.5V (typical value)
e  IS 
e IS
and for V = 0
I = IL;
in this case the current is solely determined by the light-induced component and is directly
proportional to the illuminace (transmitted radiation energy per unit area)
• the solar energy exploitation is optional if the yield of electric power P = V—I reaches its
maximum, or in other words, the area of the blue rectangle must be as large as possible
• usually this is the case if the operating voltage is about 80% of the off-load voltage; they get a
high efficiency the load resister has to be matched to the parameters of the solar cell
460
• Why is the achievable efficiency of solar cells relatively small?
on the one hand photons with energies smaller than the band gap cannot contribute to the
photocurrent, on the other hand the energy excess of the energy-rich photons gets lost as they
are only able to generate one electron-hole pair
• the efficiency ŋ of solar cells depends strongly on the band gap of the semiconductor used and
on the local solar spectrum
• ideal efficiency of solar cells taking into account the relative positions of the solar spectrum and
the band gap of the corresponding semiconductor devices
efficiency ŋ [%]
ideal efficiency of solar cells as a function of the
band gap; the efficiency can be derived from the
intersection of the vertical lines for the band gap of
the semiconductor with the blue tinted graph; the
weak oscillations are caused by the absorption of
the Earth’s atmosphere; here the Reference Solar
Spectral Irradiance AM 1.5 was supposed
band gap EG [eV]
after Hunklinger
461
• for the solar spectrum the “Reference Solar Spectral Irradiance at Hir Mass 1.5” (AM 1.5) was used
what means that the sun is at an elevation of 41.8° above the horizon (zenith: 90° AM 1.0)
• without any further provisions (such as focussing of light or a more complicated set- up of
semiconductors with different band gaps in a row) the maximum efficiency that can be achieved is
31%
• this efficiency can be achieved theoretically only if the band gap is matched to the maximum of the
solar spectrum; in reality, the achieved efficiencies are significantly smaller
• solar cells from amorphous Silicon (a-Si) currently achieve an efficiency of 10%, from polycrystalline
Silicon 15%, and from monocyrstalline Silicon 20%
• high efficiencies of 25% one can get from solar cells with GaAs which consist of three layers
Photodiode
• also photo diodes are very often based on the p-n junction; as already discussed the irradiation of
light generates an additional photocurrent IL in the p-n junction
• as the absorption of one photon generates in each case one electron- hole pair at a given
wavelength the current IL in the p-n junction is proportional to the intensity of the incident light
462
current I [mA]
current-voltage characteristic of a photodiode at two
different intensities of the incident light; with increasing
intensity the characteristic is shifted downwards; the
dots indicate the typical working points; the circuit
schematic with the load resistance is also shown
after Hunklinger
voltage V [v]
• under increasing illumination the current- voltage characteristic is shifted downwards resulting in a
change of the voltage drop at the load resistor
• in order to get a signal which is largely independent of the bias voltage the photodiode is
operated in reverse biasing
Light-Emitting Diode (LED)
• LED are widely used in everyday life
• LEDs benefit from the effect that for diodes in forward-biasing the majority carriers recombine
after leaving the depletion region within the diffusion length; for semiconductors with direct band
gap the transition of an electron from the conduction band into p hole of the valence band is
463
very often attended bay a photon emission
• this process in just the reversal of the optical absorption process discussed elsewhere
• for semiconductors with indirect band gap (phonon-assisted transition processes) the recombination
is mostly radiation less; the recombination energy is completely transferred to the phonons (or to
the lattice)
• in order to use also such semiconductor materials so-called recombination centres are embedded
into the crystal lattice; such recombination centres are lattice defects with an energy level close to
the valance band which enable transitions under photon emission
epoxy lens/case bond wire
anode (long leg)
flat = cathode
flat spot (at cathode)
reflective cavity with
semiconductor die
photograph and schematic representation of a light-emitting diode
• LEDs are available practically for the whole visible spectrum; the wavelength of the emitted light
is given by the band gap; therefore, semiconductors with different band gaps are used
464
colour
wavelength λ [nm]
semiconductor material
infrared
λ > 760
red
610 < λ < 760
aluminium gallium arsenide (AlGaAs)
gallium arsenide phosphide (GaAsP)
aluminium gallium indium phosphide (AlGaInP)
gallium (III) phosphide (GaP)
green
500 < λ < 570
indium gallium nitride (InGaN)
gallium nitride (GaN)
gallium (III) phosphide (GaP)
aluminium gallium indium phosphide (AlGaInP)
gallium arsenide phosphide (GaAsP)
blue
450 < λ < 500
zinc selenide (ZnSe)
indium gallium nitride (InGaN)
silicon carbide (SiC) as substrate
Silicon (Si) as substrate under development
ultraviolet
230 < λ < 400
diamant (C)
Aluminium nitride (AlN)
Aluminium gallium nitride (AlGaN)
Aluminium gallium indium nitride (AlGaInN) down to 230nm
white
broad spectrum
gallium arsenide (GaAs)
aluminium gallium arsenide (AlGaAs)
Blue/ UV diode with yellow phosphor
semiconductors, commonly used for LED
465
• GaAs with a band gap of EG= 1.43eV is emitting in the infrared range; where as GaN with
EG=3.37eV is emitting in the blue spectral range
• note that the mixing system InxGa1-xN has a direct gap which can be varied from 0.7eV to
3.37eV depending on composition
• white light LEDs are based either on individual LEDs that emit three primary colours and then
mixing them or to convert monochromatic light from a blue or UV leas to broad spectrum white
light similar to fluorescent light bulbs
• currently, phosphor based LEDs are most advanced; here a LED of (mostly) blue colour is
coated with phosphor of different colours to produce white light
• it should be noted that LEDs can be also produced on the basis of heterostructures
Transistor
• transistors are widely used to amplify currents or voltages; there are basically two different types:
unipolar transistors and bipolar transistors; the latter one is in principle a combination of two p- n
junctions
• the bipolar transistor (or bipolar junction transistor - BJT) was invented in 1947 by J. Bardeen,
W. Brattain and W.B. Shockley, Nobel prize in physics 1956
• bipolar transistors are so named because of their operation involves both types of charge
carriers: electrons and holes where as in unipolar transistors only one carrier type is involved
in the charge flow
466
• the PNP (where the letters “P” and “N” refer to the majority charge carriers inside the different
regions of the transistor) consist of emitter, base and collector
emitter
base
collector
holes
IE p++
electrons
IB
VEB
(a) emitter circuit
recombination
VBC
RL
collector circuit
emitter
collector
base
(b)
p++ IC
energy E
~
VE
n
bipolar transistor with forwardbiased E- B junction and
reverse-biased B- C junction,
(a) schematic representation of
the set- up and the circuit of the
PNP- type of transistor; the
regions of depletion are tinted
in grey; the relevant voltages
and branch currents are shown;
the plus sign indicates a high
doping; (b) band scheme of the
transistor; the position of the
Fermi level in the different
regions is shown by a dashdotted line
x
after Hunklinger
467
• for the function of the transistor it is important that the thickness of the base is sufficiently small
(d < 1µm) so that the recombination does not play an important role
• this holds similarly also for the NPN type transistor where in contrast to the PNP not the holes but
the electrons carry the main current
• the emitter- base junction is operated in forward biasing and is feeding holes into the base
region; the holes diffuse towards the base- collector transition where as the base- collector
transition is operated in reverse biasing
• as they are minority charge carriers in the n- type base they can pass unhindered the basecollector transition; the holes are sucked of at the collector and flow through the load resistance
RL into the collector circuit
• it should be noted again: the generation of the holes is controlled by the emitter- base voltages
VEB, but they are not allowed to recombine to a large extent with the numerous electrons in the
base region
• this can be achieved if the width of the base region is smaller than the diffusion length of the
holes; in this case the base current is very small and most of the emitter current flows to the
collector
• emitter current and collector current are about the same and therefore independent of the basecollector voltage VBC and consequently also independent of the value of the load resistance RL
• the voltage at the load resistance VL= ICRL can be much greater than the input signal V~E in the
emitter circuit; in other words, the signal is amplified
468
• the amplification can be roughly estimated as follows:
(a)
~
if the emitter voltage VE = UE + UEb and the emitter current IE are given, then we get for the
base- emitter junction (see p- n junction)
(
)
IE = IS,E e eVE / kBT − 1 ≅ IS,E e eVE / k BT
where IS,E is the saturation current of the emitter- base junction at reverse- biasing
(b)
if α is the fraction of holes which diffuse through the base region without recombination and
reach the collector, then the collector current is given by
IC = IS,C + α IE ≅ α IE
where the IS,C is the saturation current of the base- collector junction, which is small as the
base- collector diode is operated in reverse biasing
(c)
the voltage at the load resistance is given by
VL = ICR L = α R L IE
(d)
the amplification factor results from
dUL eα
~ = k T R L IE
dUE
B
k T
• with typical values α ≅ 1, IE = 10mA, B ≅ 0.025V (at T = 300K), and R L = 1kΩ the
e
amplification factor is 400!
469
• here, we have considered the so- called common- base circuit (or grounded- base circuit) of the
BJT amplifier where the base is grounded the emitter serves as the input and the collector is the
output; this circuit is used for the amplification of voltage or power
• the common- emitter circuit of the BJT amplifier is typically used to amplify currents; here the
base serves as the input, the collector is the output, and the emitter is common to both
• MOSFET stands for Metal- Oxide- Semiconductor Field- Effect Transistor
• metal- oxide- semiconductor boundary for a weakly doped p- type semiconductor
metal
semiconductor
metal
EF
EF
VB
(b)
(a)
oxide layer
VB
(c)
semiconductor
oxide layer
EF
EF
schematic representation of the
mode of operation of a metaloxide- semiconductor boundary;
the electrons in the conduction
band are marked by dots, the
holes in the valence band by
open circles; a positive voltage
VB is applied, the counter
electrode is not shown here;
VB
(d)
after Hunklinger
(a) position of gap edges and the Fermi level in absence of the field; (b) depletion region caused by
the positive bias voltage; (c) n- type inversion layer at higher positive bias voltage; (d) inversion
470
layer with a degenerated electron gas indicated by the black- tinted area
• if a positive biased voltage is applied to the metal electrode the holes in the semiconductor are
rejected where as the electrons are attracted; because of the existing negatively charged
acceptors a negative space charge is developed similar to the p- n transition
• this causes a band bending and to a depletion of holes close to the boundary; with increasing
voltage VB the bands drop further down and the electron density at the boundary layer
increases
• as here over a very small distance the conduction mechanism alters from p- type conduction to
n- type conduction the layer is revered to as inversion layer
• the inversion layer is separated from the p- type substrate by the insulating effect of the
depletion region
• at a further increase of the bias voltage the valence band edge drops below the Fermi level and
a degenerated Fermi gas with metallic character appears
• as this well- conducting channel is quiete small, the inversion layer can be considered as an
experimental realisation of a 2D electron gas (see quantum Hall effect else where)
471
after Hunklinger
MOSFET; the metal electrodes (tinted black) are direct contact to the n+- type regions of source
and drain; the depletion region (white) separated the channel and the n+-type region from the
p- type substrate; (a) gate voltage VG= 0; (b) the positive gate voltage causes the conductive
channel (tinted in dark blue)
• two highly doped n- type regions covered with a thin oxide layer are embedded into a p- type
material; the oxide layer opened at the n- type source and the n- type drain in order to get a
good electrical contact to source and drain
• the third metal contact to the gate is separated by a 100nm thick oxide layer from the p- type
substrate; this contact forms the metal- oxide- semiconductor (MOS) boundary (see above)
• without a bias voltage at the gate is one of the p- n junction is reverse- biased and causes a high
resistance
472
• at a sufficiently high positive gate voltage the low-resistive, metallic conductive channel appears;
the channel as well as source and drain are separated from the p- type substrate by a depletion
region; the width of the depletion region varies with the local electrical potential
• by means of the gate voltage the transistor can by switched from the current-less state to the
current- carrying state and back
• the performance of the MOSFET largely on the quality of the SiO2 layer and the quality of the
boundary; at variations of the gate voltage possible defects are also recharged so that only the
voltage can only partially affect the inversion layer
• the required high quality can be currently achieved only with Silicon; GaAs shows that many
defects that working MOSFET’s cannot be produced
Semiconductor Laser (Solid-State Laser)
• another application of heterostructures with outstanding technical importance is the solidstate laser
• we know already from the discussion of the photo diode: if the diode is forward- biased, then the
electrons and holes recombine under the emission of photons after crossing the space charge
region
• in order to get laser action a population inversion is needed at the p- n junction; as the
electrons and holes are preferably concentrated at the band edges the occupation probability
at the band edges EC and EV must be investigated
473
• the required population inversion is given by
f (E = EC ) > f (E = EV )
where
−1
E V −EF 
EC −EnF 


f (EL ) = 1 + e kBT 
, f (E V ) = 1 + e kBT 






therefore, we get population inversion for
p
−1
EnF − EpF > EL − E V = EG
EF
EC
EV
(a)
EF
EL
energy E
energy E
• this means, the quasi- Fermi levels EnF and EpF have in- band positions
EC
(b)
x
x
after Hunklinger
band scheme of a p++- n++
junction; (a) position of
Fermi levels in the thermal
equilibrium without an
external bias voltage; (b)
p++- n- - junction in forward
bias with a voltage V; there
appears a population in
version resulting in a laser
action
474
• in- band position of the quasi- Fermi levels can be achieved at very high dopting; indeed, GaAs
diodes show laser action if the injection current is sufficiently high
• however, the injection current can be significantly reduced, if a double hetero structure is used
EF
energy E
energy E
electron
EC
EFn
|eV|
EV
(a)
(b)
x
hole
x
double hetero structure
injection laser from p-AlGaAs
/ i-GaAs / n-AlGaAs (i-GaAs is
semi-insulating GaAs); (a)
band scheme of the hetero
junctions in the thermal
equilibrium; (b) band scheme
at a bias voltage V
after Hunklinger
• the active layer in the middle is weakly doped GaAs and is a semi-insulating material, which is
embedded in between AlGaAs which has a wider band gap
• the AlGaAs layers are of the p- type at the left side and of the n- type at the right side
• the unbiased hetero structure shows the band scheme sketched above; in forward-bias at a
sufficiently high voltage appear “band valleys”
• the quasi- Fermi levels of electrons and holes are an in- band positions for the active layer in the
middle so that a population in version accures
475
• holes flow from the p- type region and electrons flow from the n- type region into the active
layer; the band discontinuities there prevent the flow off of charge carriers and cause an
increased recombination under emission of photons
• this effect is known as electrical confinement; in addition to that there is also an optical
confinement; this is due to the higher refraction index of AlGaAs compared to GaAs
• in this way, the generated light is totally reflected and confined in the active layer; without any
further provisions the optical resonator is completed by the strongly reflecting boundary
surfaces between the semiconductor and the air
• commercial solid- state lasers are based on such devices
476