Chapter 2 Semiconductor Materials and Their Properties...Compound semiconductors: III-V and II-VI...
Transcript of Chapter 2 Semiconductor Materials and Their Properties...Compound semiconductors: III-V and II-VI...
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Chapter 2
Semiconductor Materials and Their Properties
In this chapter, we will cover the following topics
(1) Elemental and compound semiconductors (2) The valence bond model (3) The energy band theory (4) Concentration of electrons and holes including Fermi levels, energy distribution,
and temperature dependence
2.1 Elemental and Compound Semiconductors
Elemental semiconductors: two important ones: Si and Ge, both belong to group-IV
with 4 valence electrons in their outermost shell.
They crystallize in a diamond structure. Neighboring atoms are bound by covalent bonds.
Si is by far the widely used semiconductor for various device applications
Compound semiconductors: III-V and II-VI compounds.
III-Vs: GaN, GaP, GaAs, GaSb, InP, InAs, InSb
They crystallize in zinc blende structure. 8 valence electrons are shared between a pair of
nearest atoms. Therefore, the bonding has a covalent character. On the other hand, since
the group III elements are more electropositive and group V elements are more
electronegative. Hence, the bonding has a partial ionic character as well. But the covalent
nature is predominant.
II-VIs: ZnS, ZnSe, ZnTe, CdS, CdSe, CdTe
Crystal structures:
CdS and CdSe: wurtzite (two interpenetrating hexagonal close-packed lattices)
ZnTe and CdTe: zinc blende
ZnS and ZnSe: can be both (depeding on the substrate on which it is grown)
Bonding: mixture of covalent and ionic types. Stronger ionicity than III-Vs since group-
VI elements are considerably more electronegative than group II elements.
The ionic character has the effect of binding the valence electrons rather tightly to the
lattice atoms. Thus, the band gaps of these compounds are larger than those of the
covalent semiconductors of comparable atomic weights.
Ternary and quaternary compounds:
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Ga1-xAlxAs, ZnSxSe1-x, Zn1-xMgxSySe1-y, where 0 โค ๐ฅ โค 1, 0 โค ๐ฆ โค 1. The properties of the resulting compound vary gradually with the fraction yx, .
Advantages of compound semiconductors:
Wider choice of bandgap than elemental semiconductors: IR-visible-UV
Direct bandgap materials: optoelectronic applications, LEDs, lasers, sensors.
A major difficulty with compounds is that their preparation in single crystal form is more
difficult.
Two models can be used to study semiconductors
(1) Valence bond model which describes properties in domain of space and time. (2) Energy band model which describes properties in energy and momentum.
Energy band model is far more useful.
2.2 The Valence Bond Model
Use elemental semiconductors as example, Si and Se, they form diamond structure where
each atom bound to its four nearest neighbors by covalent bonds. These neighbors are all
equidistance from the central atom and lie at the four corners of the tetrahedron.
Fig.2.1 Illustration of covalent bond in tetrahedron.
Each bond has two electrons with opposite spin so that the central atom appears to have
eight electrons with opposite spins.
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Fig.2.2 An 2-D illustration of the diamond structure.
At ๐ = 0K, all electrons are tightly bound in the bonds โ a perfect insulator
At higher temperatures, lattice vibration occasionally shake loose some electrons so they
can move freely inside the crystal. The vacant side created by the broken bond has a net
positive charge known as hole (a particle of positive charge). Both electrons and holes are
responsible for semiconductor conductivity.
One way to visualize the movement of a hole is to consider the neighboring bond
electrons jumps over to the vacant site to create another vacant at the position from which
the electron came from. This is equivalent to hole moving from one site to another.
This picture has its limitation. It fails to explain the wave nature of the hole and the Hall
effect.
The above described generation of electrons and holes (both are called carriers) is due to
the thermal excitation. This is the case for intrinsic semiconductors where number of
electrons is equal to that of holes. The carrier concentration depends on temperature.
There is another way of introducing conduction carriers into the semiconductors โ
dopping of impurity atoms intentionally.
For Si and Ge, elementals from group-III and V are commonly used as impurities.
The doped semiconductors are called extrinsic semiconductors.
n-type semiconductors
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A small amount of group V elements (As, P, Sb) is added into Si. These impurities
occupy lattice sites that are normally occupied by Si atoms โ substitutional impurities
Fig.2.3 Illustration of group-V substitutional impurity in Si.
The 5th
electron is bound to the impurity atom only by weak electrostatic force.
Bohr theory of the hydrogen atom can be used to calculate the radius and the ionization
energy of its ground state
๐0 = 0.53 ๐๐ ๐0 ๐0๐๐โ ร
๐ธ๐ผ = 13.6 ๐0๐๐โ ๐๐ ๐0
2
eV
Two modifications to the hydrogen formula:
(1) effective mass: ๐๐โ < ๐0 (free electron mass)
(2) permittivity of the semiconductor: ๐๐ > ๐0 (permittivity of the free space)
The ionization energy is typically small, therefore, at room temperature all of them
should be ionized to become conduction electrons โ donors.
n-type semiconductor: majority carriers are electrons, minority carriers are holes.
p-type semiconductors
Group III elements occupy a substitutional position in the Si lattice (B, Al, In, Ga)
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Fig.2.4 Illustration of group-III substitutional impurity in Si.
Radius and ionization energy of the bound hole state can be calculated similarly.
These impurities cn contribute holes by accepting electrons โ acceptors.
p-type semiconductor: majority carriers are holes, minority carriers are electrons.
If a semiconductor is doped with both donors and acceptors, the extra electrons attached
the donors fall into the incomplete bonds of the acceptors so that neither electrons nor
holes will be produced โ compensation.
However, if ๐๐ > ๐๐ , we get n-type; if ๐๐ < ๐๐ , we get p-type.
Ambipolar semiconductors: can be doped to become both n-type and p-type. Elemental
and most of III-V compound semiconductors are ambipolar.
Unipolar semiconductors: can be doped either n-type or p-type, but not both. Many II-
VI compounds are such.
The unipolar behavior is due to the mechanism of self-compensation.
Consider ZnTe doped with iodine (I in group-VII):
Intention: I atoms replace Te atoms to make it n-type
Practice: For I atoms to replace Te atoms, temperature has to be raised, it will cause Zn
vacancies due to the evaporation of Zn atoms. Each Zn vacancy acts as a double acceptor.
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โ๐ธ Zn : energy required to create a Zn vacancy 2๐ธ๐ : energy released when two donor electrons drop into the Zn vacancy.
If 2๐ธ๐ > โ๐ธ Zn , the system energy is lowered, thus ZnTe will remain p-type.
2.3 Energy Band Model
The electronic states in a crystal is obviously different from that in an isolated atom. But
they are also related because a crystal is formed by binding atoms in a regular order.
Consider we have many isolated atoms (well separated), there are no interactions
between the atoms. A system of these atoms will have discrete energy levels and each
level is degenerate.
If we bring these atoms close to each other, the interaction between atoms will become
stronger. As a result, the wave functions of electrons in the outermost shell will begin to
overlap. The degeneracy of each energy level will be removed. The initially discrete level
will now split into many energy levels. The separation of these split energy levels
depends on the distance between atoms. The stronger the wave functions overlap, the
larger the energy splitting.
If we have ๐ identical atoms bound to form a crystal, each energy level will split into ๐ energy levels. Since ๐ is usually very large, the density of these energy levels is high. It can be treated as they form a quasi-continuous energy band โ allowed bands separated by
forbidden band โ bandgap.
Fig.2.5 Illustration of energy band formation.
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The overlap of electron wave functions in inner shells is small, the interaction is weak,
therefore, the energy splitting can be neglected. The different properties of crystal and
constituent atoms are due to the different states of valence electrons. For example,
isolated neutral atoms (atom gas) do not conduct current. As they are bound together to
form a crystal, very different electronic properties can be determined (conductors,
semiconductors, insulators).
Another example, the optical spectrum associated with the transitions between different
levels in an isolated atom is discrete. However, the spectrum in a solid is continuous.
When ๐ atoms form a crystal, one degenerate level splits into ๐ energy levels which allow totally 2๐ electrons states according to Pauli principle that each energy level can be occupied by two electrons with opposite spin.
For atoms with one valence electron (Na, sodium and K, pottasium), the solids formed by
them have their energy band half filled, therefore, they are good conductors (metals). The
reason that only partially filled bands conduct current will be explained when we talk
about the energy band theory.
For atoms with two valence electrons (Mg), the outermost electrons are s2. Intuitively,
one would think that the 2๐ states will be completely filled therefore they are insulators. But the fact is these bands are overlapping each other with higher energy bands. As a
result both bands are partially filled, which leads to a good conductivity (metals).
For C(diamond), Si and Ge, the situation is more complicated. They all have 4 valence
electrons, s2 p
2. When they form a crystal, two energy bands should be produced. One
corresponds to s-state with 2๐ states. The other p-states with 6๐ states. It seems that the 6๐ band should be partially filled, therefore diamond, Si, and Ge are all good conductors. But they are not. Actually the orbit mixing between the s-state and p-state has led to a
new combination which result in two energy bands, each having 4๐ states. The lower one, called valence band, is then completely filled, leaving the upper one called
conduction band completely empty at low temperatures. Therefore, at low temperatures,
they act like insulators.
Electronic States in a Crystal
A detailed understanding of electronic states and the behavior of electrons in a crystal
requires calculations of quantum mechanics. The number of electrons involved in the
system is on the order of 1023
. This is a complex many body problem, an exact solution
of such a system is impossible.
The energy band theory is actually based on the single-eletron approximation. This
approximation takes into account the electron interaction by adding them into the
periodic potential field of the atoms. As a result, electrons can be treated independently,
while the interactions with other electrons are included in the potential field as a fixed
charge distribution.
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The Schrodinger equation becomes
โั2
2๐โโ2 + ๐ ๐ ๐ท ๐ = ๐ธ๐ท ๐
Where ๐โ is the effective mass which is different from the free electron mass due to the interaction with other electrons and ๐ ๐ is the periodic potential which includes the electron interaction with lattice atoms and other electrons.
Bloch Functions
F. Bloch proved that the solutions of the Schrodinger equation for a periodic potential
must be of a special form
๐ท๐ ๐ = ๐ข๐ ๐ ๐๐๐โ๐
where ๐ข๐ ๐ has the period of the crystal lattice with ๐ข๐ ๐+ ๐ป = ๐ข๐ ๐
This is a result of the Bloch theorem which states that the eigen functions of the wave
equation for a periodic potential are of the form of the product of a plane wave ๐๐๐โ๐ and a function ๐ข๐ ๐ with the periodicity of the crystal lattice. The proof of this theorem can be found in Solid State Physics by Kittel.
Compared to free electron wave function ๐๐๐โ๐, Bloch function indicates that the probability of finding an electron in a lattice space is different from point to point within
one primitive unit cell, but the same between the corresponding points of different unit
cells.
In fact, ๐ข๐ ๐ describes the behavior of an electron around a lattice atom. ๐๐๐โ๐
demonstrates that the electron is not localized, but can propagate throughout crystal.
For a crystal with infinitely large volume, the value of ๐ can vary continuously. Given a ๐, there exists a set of eigen energies ๐ธ๐(๐) and corresponding ๐ท๐ ๐, ๐ . The quantum number ๐ indicates different energy band, intuitively can be considered as they are originated from different atomic energy levels. ๐ธ๐(๐) is therefore a continuous function of ๐. Due to the periodicity of a crystal, an electronic does not have a unique value of ๐. In fact, ๐โฒ = ๐+๐ฒ can represent the same state as ๐ does where ๐ฒ is the so-called reciprocal lattice vector
๐ฒ = ๐1๐1 + ๐2๐2 + ๐3๐3 Where ๐1 ,๐2, ๐3 are integers and ๐1,๐2,๐3 are primitive vectors of the reciprocal lattice which can be constructed by the primitive vectors of the crystal lattice (๐1,๐2,๐3)
๐1 = 2๐๐2 ร ๐3
๐1 โ ๐2 ร ๐3 ,๐2 = 2๐
๐3 ร ๐1๐1 โ ๐2 ร ๐3
,๐3 = 2๐๐1 ร ๐2
๐1 โ ๐2 ร ๐3
It is easy to prove that
๐๐ โ ๐๐ = 2๐๐ฟ๐๐ Actually, the Bloch wave function can be written as
๐ท๐ ๐, ๐ = ๐ข๐ ,๐ ๐ ๐๐๐โ๐ = ๐ข๐ ,๐ ๐ ๐
โ๐๐ฒโ๐ ๐๐๐โฒ โ๐ = ๐ข๐ ,๐โฒ ๐ ๐
๐๐โฒ โ๐
where ๐ข๐ ,๐โฒ ๐ = ๐ข๐ ,๐ ๐ ๐โ๐๐ฒโ๐ has the same periodicity as the lattice since
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๐โ๐๐ฒโ ๐+๐ป = ๐โ๐๐ฒโ๐, ๐ฒ โ ๐ป = 2๐๐ Thus, ๐โฒ represents the same state as ๐ does.
Unlike the situation for free electrons, the nonuniqueness of ๐ suggests that strictly speaking, ั๐ will not carry the meaning of momentum. However, we will see later ั๐ can still be treated as if it is the momentum of an electron in a crystal.
Obviously, ๐ธ๐(๐) various with ๐ periodically and for different n, ๐ธ๐(๐) varies within different energy intervals separated by energy gaps where electron states are forbidden.
Fig. 2.6 ๐ธ๐(๐) as a function of ๐.
If we take the complex conjugate of the Schrodinger equation, H remains unchanged,
therefore ๐ธ๐(๐) should have even symmetry with respect to ๐, i.e. ๐ธ๐ ๐ = ๐ธ๐(โ๐) since
๐ท๐โ ๐ = ๐ข๐
โ ๐ ๐โ๐๐โ๐ is the eigen function of ๐ธ๐(๐) also.
Brillouin Zone
Since ๐โฒ = ๐+๐ฒ represents the same electron state as ๐ does with ๐ฒ being the reciprocal lattice vector. We can actually limit ๐ within a primitive unit cell in the reciprocal lattice, because ๐ and ๐โฒ point to the equivalent points within different primitive unit cell in the reciprocal lattice. Now for an fixed electron state, there exists a
unique corresponding ๐.
For example in 1-D, we can limit โ ๐/2 < ๐ < ๐/๐.
A Brillouin zone is a special kind of primitive unit cell in a reciprocal lattice. It is defined
as a Wigner-Seitz cell in the reciprocal lattice, which is constructed by drawing
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perpendicular bisector planes in the reciprocal lattice from the chosen center to the
nearest equivalent reciprocal lattice sites.
The Wigner-Seitz cell is a primitive cell which maximally demonstrates the symmetry of
the crystal. For discussions on how to construct the Wigner-Seitz cell, one can refer to
Solid State Physics by Kittel.
State Density in k -Space
Consider a crystal with a dimension of ๐ฟ1 ร ๐ฟ2 ร ๐ฟ3 with the dimension of a primitive cell of ๐1 ร ๐2 ร ๐3. For any state ๐ = ๐1๐ + ๐2๐ + ๐3๐ , the periodic condition requires that
๐ท๐๐ 0,๐ฆ, ๐ง = ๐ท๐๐ ๐ฟ1,๐ฆ, ๐ง ,๐ท๐๐ ๐ฅ, 0, ๐ง = ๐ท๐๐ ๐ฅ, ๐ฟ2, ๐ง ,๐ท๐๐ ๐ฅ,๐ฆ, 0 = ๐ท๐๐ ๐ฅ,๐ฆ, ๐ฟ3 Thus
๐1๐ฟ1 = ๐1 2๐ ,๐2๐ฟ2 = ๐2 2๐ , ๐3๐ฟ3 = ๐3 2๐ .
Since, ๐ฟ1 = ๐1๐1, ๐ฟ2 = ๐2๐2, ๐ฟ3 = ๐3๐3, then
๐๐ =๐๐๐๐
2๐
๐๐, ๐ = 1,2,3
with ๐๐ โค๐๐
2 since โ ๐/2 < ๐๐ < ๐/๐. Therefore, โ๐๐ =
2๐
๐ฟ๐, then one state in ๐ -space
takes a volume of
โ๐1โ๐2โ๐3 =2๐
๐ฟ1
2๐
๐ฟ2
2๐
๐ฟ3= 2๐ 3
๐
where ๐ is the volume of crystal.
Taking into account that each state can accommodate two electrons with opposite spin,
the density of states in ๐-space is then
๐๐ =2๐
2๐ 3,
and within the Brillouin zone there are totally ๐ = ๐1๐2๐3 allowed ๐ states, which equals the number of primitive cells.
It can be shown that if the volume of a primitive cell in lattice is ๐บ, the volume of a reciprocal primitive cell is 2๐ 3/๐บ.
Motion of Electrons in a Crystal
The time-dependent solution to the Schrodinger equation is
๐ท๐ ๐, ๐ก = ๐ข๐ ๐ ๐๐ ๐โ๐โ๐๐ก
where ๐ = ๐ธ/ั. Due to the linearity of the time-dependent Schrodinger equation, the superposition of the above wave function with different ๐
๐ท ๐, ๐ก = ๐ถ๐๐
๐ข๐๐ ๐ ๐๐ ๐๐ โ๐โ๐๐ ๐ก
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will still be a solution to the Schrodinger equation. Since ๐๐ โs are closely packed in ๐-space and can be treated as continuous, the superposition becomes an integral
๐ท ๐, ๐ก = ๐ถ ๐ ๐/2
โ๐/2
๐ข๐ ๐ ๐๐ ๐โ๐โ๐ ๐ ๐ก ๐3๐
Now at some instant of time, one tries to look at a wave packet that consists of Bloch
functions in the neighborhood of ๐, then the above integral can be rewritten as
๐ท ๐, ๐ก = ๐ถ ๐+ ๐ผ โ๐
โโ๐
๐ข๐+๐ผ ๐ ๐๐ ๐+๐ผ โ๐โ๐ ๐+๐ผ ๐ก ๐3๐ผ
Since ๐ผ < โ๐ are small, then ๐ข๐+๐ผ ๐ โ ๐ข๐ ๐ and
๐ ๐+ ๐ผ = ๐ ๐ + โ๐๐ ๐ โ ๐ผ +โฏ which is related to energy by ๐ = ๐ธ/ั. Therefore
๐ท ๐, ๐ก = ๐ข๐ ๐ ๐๐ ๐โ๐โ๐ ๐ ๐ก ๐ถ ๐+ ๐ผ
โ๐
โโ๐
๐๐ ๐ผโ ๐โโ๐๐๐ก ๐3๐ผ
The factor in front of the integral is a Bloch function. The integral represents a wave
packet with a center position ๐ = โ๐๐๐ก which moves with the velocity
๐ =๐๐
๐๐ก= โ๐๐ ๐ =
1
ัโ๐๐ธ ๐
which is the group velocity of electron in a crystal.
Now letโs consider a 1-D situation, with
๐ถ ๐ + ๐ = ๐ถ, ๐ < โ๐
0, ๐ > โ๐
Then
๐ท ๐ฅ, ๐ก = ๐ถ๐ข๐ ๐ฅ ๐๐ ๐โ๐ฅโ๐ ๐ ๐ก ๐๐ ๐ โ ๐ฅโโ๐๐๐ก ๐๐
โ๐
โโ๐
= ๐ถ๐ข๐ ๐ฅ ๐๐ ๐โ๐ฅโ๐ ๐ ๐ก
sin โ๐ ๐ฅ โ ๐๐/๐๐ ๐ก
โ๐ ๐ฅ โ ๐๐/๐๐ ๐ก 2โ๐
Hence, the wave packet probability
๐ท ๐ฅ, ๐ก 2 โ ๐ข๐ ๐ฅ 2
sin โ๐ ๐ฅ โ ๐๐/๐๐ ๐ก
โ๐ ๐ฅ โ ๐๐/๐๐ ๐ก
2
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Fig.2.7 1-D electron distribution described by a wave packet with a wave vector
distribution within ๐ + โ๐.
The center peak is bound by ๐ฅ โ ๐๐/๐๐ ๐ก = ยฑ๐/โ๐, i.e. the half width โ๐ฅ is related to โ๐ by โ๐ฅโ๐ = ๐ - exactly what is required by the uncertainty principle. For a wave packet with a space expansion of lattice constant ๐, we have โ๐ = ๐/๐ covers the entire Brillouin zone, i.e. ๐ is completely uncertain, and vice versa. However, when we are dealing with the ranges of ๐ and ๐ that are much greater than the uncertainty โ๐ and โ๐, we can consider both ๐ and ๐ have precise values โ quasi classic.
Effective Mass
Suppose that an external field is applied, and the electron moves a distance ๐๐, the work done on the electron is
๐๐ค = ๐ญ โ ๐๐ = ๐ญ โ ๐๐๐ก = ๐ญ โ โ๐๐๐๐ก = ๐ญ โ โ๐๐ธ๐๐ก/ั The work done on the electron causes its energy to change
๐๐ธ ๐ = โ๐๐ธ โ ๐๐ = โ๐๐ธ โ ๐๐/๐๐ก ๐๐ก Since ๐๐ค = ๐๐ธ, therefore ๐ญ = ั ๐๐/๐๐ก = ๐๐/๐๐ก which is analogue to the Newtonโs law with ๐ = ั๐ representing the momentum of the electron, similar to that of free electron.
The acceleration of an electron in a crystal is ๐๐
๐๐ก=
1
ั
๐
๐๐ก โ๐๐ธ =
1
ั ๐๐
๐๐กโ โ๐ โ๐๐ธ =
1
ั2 ๐ญ โ โ๐ โ๐๐ธ
In tensor form, we can write ๐๐
๐๐ก=
1
๐โ ๐ญ
where the inverse effective mass tensor
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1
๐โ =
1
ั2
๐2๐ธ/๐๐๐ฅ2 ๐2๐ธ/๐๐๐ฅ๐๐๐ฆ ๐
2๐ธ/๐๐๐ฅ๐๐๐ง
๐2๐ธ/๐๐๐ฆ๐๐๐ฅ ๐2๐ธ/๐๐๐ฆ
2 ๐2๐ธ/๐๐๐ฆ๐๐๐ง
๐2๐ธ/๐๐๐ง๐๐๐ฅ ๐2๐ธ/๐๐๐ง๐๐๐ฆ ๐
2๐ธ/๐๐๐ง2
is a function of ๐. The tensor is symmetric, we can choose a proper coordinate system so that the tensor is diagonalized. Therefore, we can have for each direction
๐๐ผโ๐๐ฃ๐ผ๐๐ก
= ๐น๐ผ , ๐ผ = ๐ฅ,๐ฆ, ๐ง
which is analogue of Newtonโs 2nd
law.
Usually in semiconductors, we only deal with those states that are close to band edges (๐ธ๐ and ๐ธ๐ฃ) because electrons are distributed a few ๐๐ around the band edges.
For conduction band near ๐ธ๐ with ๐ = 0, we can have
๐ธ๐ ๐ = ๐ธ๐ 0 +1
2 ๐2๐ธ๐๐๐๐ฅ2
๐๐ฅ2 +
๐2๐ธ๐๐๐๐ฆ2
๐๐ฆ2 +
๐2๐ธ๐๐๐๐ง2
๐๐ง2 +โฏ
= ๐ธ๐ 0 +ั2
2 ๐๐ฅ
2
๐๐ ,๐ฅโ+
๐๐ฆ2
๐๐,๐ฆโ+๐๐ง
2
๐๐ ,๐งโ +โฏ
Since ๐ธ๐ 0 is a minimum, ๐๐ธ๐
๐๐๐ผ= 0, and
๐2๐ธ๐
๐๐๐ฅ2 > 0, therefore, ๐๐ ,๐ผ
โ > 0. Near the band
edge, ๐ธ๐ ๐ has a parabolic relation with ๐.
For crystals with cubic symmetry, we have ๐๐ ,๐ฅโ = ๐๐,๐ฆ
โ = ๐๐ ,๐งโ = ๐๐
โ at ๐ = 0. Then
๐ธ๐ ๐ = ๐ธ๐ 0 +ั2๐2
2๐๐โ
and ๐๐โ๐๐
๐๐ก= ๐ญ, the same expressions as free electrons except that we have to use the
effective mass ๐๐โ to replace the free electron mass.
For valence band near the top ๐ธ๐ฃ with ๐ = 0, we can have
๐ธ๐ฃ ๐ = ๐ธ๐ฃ 0 +1
2 ๐2๐ธ๐ฃ๐๐๐ฅ2
๐๐ฅ2 +
๐2๐ธ๐ฃ๐๐๐ฆ2
๐๐ฆ2 +
๐2๐ธ๐ฃ๐๐๐ง2
๐๐ง2 +โฏ
= ๐ธ๐ฃ 0 +ั2
2 ๐๐ฅ
2
๐๐ฃ,๐ฅโ+
๐๐ฆ2
๐๐ฃ,๐ฆโ+๐๐ง
2
๐๐ฃ,๐งโ +โฏ
= ๐ธ๐ฃ 0 +ั2๐2
2๐๐ฃโ +โฏ
for cubic crystal. Since ๐ธ๐ฃ 0 is a maximum, ๐2๐ธ๐ฃ
๐๐๐ฅ2 < 0 which will result in negative
electron effective mass (๐๐ฃโ < 0) near the top of the valence band. However, when we
look at the Newtonโs law expression of these electrons under an electric field ๐ฌ,
๐๐ฃโ๐๐
๐๐ก= ๐ญ = โ๐๐ฌ
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14
the holes are introduced as having positive charge, therefore, the above Newtonโs law
should be modified
โ๐๐ฃโ๐๐
๐๐ก= ๐
โ๐๐
๐๐ก= ๐๐ฌ
where the hole effective mass ๐โ = โ๐๐ฃ
โ is actually positive at the top of the valence
band.
The effective mass is directly related with the curvature of the energy band, and therefore
related with the energy band width.
Thus, near the top of the valence band we can write
๐ธ๐ฃ ๐ = ๐ธ๐ฃ 0 โั2๐2
2๐โ
Fig.2.8 Illustration of the band curvature with the effective mass.
Energy Band Structures
The energy band structure is normally described as the relationship between energy ๐ธ
and ๐ = ๐๐ฅ ,๐๐ฆ ,๐๐ง which is difficult to plot in 3-D. Typically, the relationship is
plotted along the principle directions of the crystal.
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15
Fig. 2.9 Band structure along the principle directions of the crystal (Si).
Direct band gap: the minimum of the conduction band is located at the same ๐ as the maximum of the valence band.
Indirect band gap: the minimum of the conduction band and the maximum of the
valence band are located at the different ๐โs in ๐-space.
For direct band gap semiconductor, a photon with energy ั๐ = ๐ธ๐ = ๐ธ๐ โ ๐ธ๐ฃ can excite
an electron from the top of the valence band to the bottom of the conduction band. But
for indirect band gap semiconductor, such a transition requires that a photon with an
energy ั๐ > ๐ธ๐ = ๐ธ๐ โ ๐ธ๐ฃ, because photons have very small momentum and the
absorption of a photon needs to be a vertical transition in ๐-space.
Electrons and holes are populated at the bottom of the conduction band and top of the
valence band, respectively. For direct band gap semiconductors, the probability of
electrons and holes recombine to emit photons is much higher than that of indirect
semiconductors. Therefore, direct band gap semiconductors have important applications
in optical devices.
Indirect: Si and Ge
Direct: most III-V and II-VI compounds
Constant energy surface is plotted as an surface area in ๐-space where each point on the surface has the same energy.
For a semiconductor with its conduction band minimum at ๐ = 0 and isotropic effective mass, obviously the constant energy surface is spherical,
http://upload.wikimedia.org/wikipedia/commons/5/53/Fcc_brillouin.png
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16
๐ธ = ๐ธ๐ +ั2๐2
2๐๐โ
Fig. 2.10 Constant energy surface for conduction band minimum at ๐ = 0 and isotropic effective mass.
If the tensor of the effective mass is anisotropic, the constant energy surface is an
ellipsoid.
For Ge and Si, the minimum of conduction band is not at ๐ = 0, the center of the constant energy surface is not located at ๐ = 0. Due to the crystal symmetry, there should be more than one constant energy surfaces, e.g. for Si there are 6 constant energy surfaces
along 6 equivalent principle axis 100 , the centers of the 6 ellipsoids are located at about ยพ of the distance from the Brillouin zone center.
๐ธ๐ ๐ = ๐ธ๐ +ั2
2 ๐๐ฅ โ ๐๐ฅ0
2
๐๐โ +
๐๐ฆ2 + ๐๐ง
2
๐๐กโ
where ๐๐โ and ๐๐ก
โ are the longitudinal and transverse effective mass.
Fig.2.11 Constant energy surfaces in Si and Ge.
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17
For Ge, the centers of constant energy ellipsoids are along each of the 8 100 directions. The Brillouin zone boundary goes through the center of each ellipsoid. There are 8 half
ellipsoids within the 1st Brillouin zone, making 4 full ellipsoids within the 1
st Brillouin
zone.
The above description of energy band structure is detailed and emphasize on the ๐ธ โ ๐ relationship. There are times when the detail description is not necessary when we are
only interested in the band gap ๐ธ๐ and band alignment in real space.
At 0K the valence band is completely filled, the conduction band has no electrons. Under
this condition, the semiconductor behaves like an insulator.
This behavior is due to the fact that completely filled energy band as well as completely
empty band do not conduct electric current.
Fig. 2.12 Simplified band diagram.
Every moving electron contributes to electric current. But current is the total effect of all
electrons in a crystal. Within an energy band, if there is a state with momentum ๐ and energy ๐ธ ๐ , there must be another state with momentum โ๐ and ๐ธ โ๐ = ๐ธ ๐ due to the symmetry of the crystal. Obviously, โ๐๐ธ ๐ = โโ๐๐ธ โ๐ which implies that electron at ๐ has same magnitude but opposite velocity as the electron at โ๐. If the band is completely filled, then both states are occupied with electrons. The contribution to
current from the two electrons with ๐ and โ๐ will be cancelled. Hence, the total current is zero.
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18
Fig. 2.13 Illustration of the symmetry of ๐ธ โ ๐ relationship and the corresponding electron velocity.
Under the condition of applied external field, the distribution of electrons within a
Brillouin zone is unchanged, even all electrons in k -space move according to
ั๐๐
๐๐ก= โ๐๐ฌ
This is because some electrons will flow out of the Brillouin zone on the right side, more
electrons will flow into the Brillouin zone from the left side to fill up the empty states.
For partially filled energy band, it is easy to see why current is not zero under an electric
field.
Fig.2.14 Occupied ๐ธ โ ๐ states in partially filled band with zero and nonzero field.
Holes as empty states in valence band
The above analysis also provides an explanation for how an empty state in a valence band
acts as a hole โ a conductive carrier.
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19
For completely filled band, the current density
๐ฝ = โ๐
๐ ๐๐
4๐
๐=1
= 0
where ๐ is the volume since each band has 4๐ states. Now assume that some state ๐ in the valence band is empty, we then can write the current density
๐ฝ๐ฃ == โ๐
๐ ๐๐
4๐
๐=1,๐โ ๐
= โ๐
๐ ๐๐
4๐
๐=1
โ ๐๐ =๐๐๐
๐
One empty state in the valence band acts as if it carries a positive charge ๐ which conducts current.
Energy levels of impurities
Usually impurity levels lie in forbidden energy region. For intentionally doped donors
and acceptors, they are normally shallow levels in the range of ๐ธ๐ผ~0.001~0.01๐๐. At room temperature, they are all ionized.
Fig.2.15 Impurity levels of donors and acceptors in forbidden band.
But some impurities tend to give deeper levels in the forbidden band, e.g. Au in Si; O in
Ge. They serve as recombination centers.
Compensation
Fig.2.16 Semiconductors doped with both donors and acceptors.
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20
Electrons from the level ๐ธ๐ instead of going to the conduction band will drop into the acceptor states at ๐ธ๐ . Each such transition eliminates one electron-hole pair that would have been there if the transition could be prevented. Obviously, if ๐๐ > ๐๐ , p-type; and if ๐๐ < ๐๐ , n-type.
2.4 Equilibrium Carrier Concentration
In order to calculate the electron and hole concentrations in the conduction and valence
bands, we need to know the density of states and probability of occupancy of these states.
Density of States ๐ ๐ธ
As we have learned the density of states in ๐-space is ๐๐ =2๐
2๐ 3, we need to convert this
into the density of states in energy ๐ ๐ธ since the probability of occupancy described by the Fermi function is given in energy.
Assume that the constant energy surface is spherical so that the effective mass is a scalar
๐ธ ๐ = ๐ธ๐ +ั2๐2
2๐๐โ
A constant energy surface of ๐ธ โ ๐ธ๐ , the radius of the spherical surface is ๐ =
2๐๐โ ๐ธ โ ๐ธ๐ /ั. The volume encircled by this energy surface
4๐
3๐๐3 =
4๐
3๐ 2๐๐
โ ๐ธ โ ๐ธ๐ 3/2
ั3
The total number of states
2
2๐ 34๐
3๐๐3 =
2
2๐ 34๐
3๐ 2๐๐
โ ๐ธ โ ๐ธ๐ 3/2
ั3
=8๐
3
2๐๐โ 3/2
3 ๐ธ โ ๐ธ๐
3/2
= ๐๐ ๐ธ ๐ธโ๐ธ๐
0
๐๐ธ
The density of state in energy
๐๐ ๐ธ =4๐ 2๐๐
โ 3/2
3 ๐ธ โ ๐ธ๐
1/2
for conduction band ๐ธ > ๐ธ๐ . Similarly
๐๐ฃ ๐ธ =4๐ 2๐
โ 3/2
3 ๐ธ๐ฃ โ ๐ธ
1/2
for valence band ๐ธ < ๐ธ๐ฃ .
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21
Fermi-Dirac Distribution
The probability of occupancy of a state with energy ๐ธ by an electron
๐ ๐ธ =1
1 + exp ๐ธ โ ๐ธ๐น /๐๐
where ๐ธ๐น is the Fermi energy
Fig.2.17 Fermi-Dirac distribution
At ๐ = 0๐พ,
๐ ๐ธ = 0, ๐ธ > ๐ธ๐น1, ๐ธ < ๐ธ๐น
,
in general,
๐ ๐ธ =
>
1
2, ๐ธ < ๐ธ๐น
=1
2, ๐ธ = ๐ธ๐น
<1
2, ๐ธ > ๐ธ๐น
In extreme cases when ๐ธโ๐ธ๐น
๐๐โซ 1, ๐ ๐ธ = exp โ ๐ธ โ ๐ธ๐น /๐๐ - Maxwell-Boltzmann
Distribution, and when ๐ธโ๐ธ๐น
๐๐โช 1, ๐ ๐ธ = 1.
Electron concentration
The number of electrons in an energy interval dE within the conduction band
๐๐ = ๐ ๐ธ ๐ ๐ธ ๐๐ธ =4๐ 2๐๐
โ 3/2
3 ๐ธ โ ๐ธ๐
1/2 1 + exp ๐ธ โ ๐ธ๐น๐๐
โ1
๐๐ธ
Total electron concentration distributed within the conduction band
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22
๐ =4๐ 2๐๐
โ 3/2
3 ๐ธ โ ๐ธ๐
1/2๐ธtop
๐ธ๐
1 + exp ๐ธ โ ๐ธ๐น๐๐
โ1
๐๐ธ
This expression can be simplified if we assume that ๐ธ๐น lies below ๐ธ๐ by more than 3๐๐ so
๐ ๐ธ โ exp โ ๐ธ โ ๐ธ๐น /๐๐ , an analytical expression can be obtained
๐ = ๐๐ exp โ ๐ธ๐ โ ๐ธ๐น /๐๐ where
๐๐ = 2 2๐๐๐โ๐๐/2 3/2
representing the density of states required at the conduction band edge ๐ธ๐which gives the concentration in the conduction band after multiplying with the probability of occupancy
at band edge. In reality, the density of states at ๐ธ๐ is zero as given by ๐ ๐ธ โ ๐ธ โ ๐ธ๐ 1/2.
Hole concentration
The probability of a state not occupied by an electron
๐ ๐ธ = 1 โ ๐ ๐ธ =1
1 + exp ๐ธ๐น โ ๐ธ /๐๐
which can be reviewed as the probability of a state occupied by a hole. Similarly, the hole
concentration
๐ =4๐ 2๐
โ 3/2
3 ๐ธ๐ฃ โ ๐ธ
1/2๐ธ๐ฃ
๐ธbott om
1 + exp ๐ธ๐น โ ๐ธ
๐๐ โ1
๐๐ธ
Assuming that ๐ธ๐น lies above ๐ธ๐ฃ by more than 3๐๐ so ๐ ๐ธ โ exp โ ๐ธ๐น โ ๐ธ /๐๐ ,
an analytical expression can be obtained
๐ = ๐๐ฃ exp โ ๐ธ๐น โ ๐ธ๐ฃ /๐๐
where
๐๐ฃ = 2 2๐๐๐ฃโ๐๐/2 3/2
representing the density of states required at the valence band edge ๐ธ๐ฃ.
For semiconductors with anisotropic effective mass and multiple equivalent energy
minima in conduction band, all expressions concerning 2๐๐โ 3/2 should be modified as
๐ 8๐๐ฅโ๐๐ฆ
โ๐๐งโ
1/2 or ๐ 8๐๐
โ๐๐กโ2 1/2 where ๐ is the number of equivalent energy valleys
(minima). We thus define a density-of-states effective mass with
2๐๐โ 3/2 = ๐ 8๐๐
โ๐๐กโ2 1/2
such that ๐๐โ = ๐ 2๐๐
โ๐๐กโ2 1/3.
For intrinsic semiconductors (no doping, ๐๐ = 0 and ๐๐ = 0), we should have ๐ = ๐ =๐๐ , and we let ๐ธ๐ = ๐ธ๐น - the Fermi level of intrinsic semiconductor, then
๐๐ = ๐๐ exp โ ๐ธ๐ โ ๐ธ๐ /๐๐ = ๐๐ฃ exp โ ๐ธ๐ โ ๐ธ๐ฃ /๐๐ We can write in general
๐ = ๐๐exp ๐ธ๐น โ ๐ธ๐ /๐๐ , ๐ = ๐๐exp ๐ธ๐ โ ๐ธ๐น /๐๐
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23
The electron-hole concentration product
๐๐ = ๐๐2 = ๐๐ exp โ ๐ธ๐ โ ๐ธ๐ /๐๐ ๐๐ฃ exp โ ๐ธ๐ โ ๐ธ๐ฃ /๐๐
= ๐๐๐๐ฃexp โ ๐ธ๐ โ ๐ธ๐ฃ /๐๐
= 32 ๐2๐๐
โ๐โ
4
3/2
๐๐ 3exp โ๐ธ๐/๐๐
= ๐2 ๐๐ 3exp โ๐ธ๐/๐๐
both ๐๐โ and ๐
โ are density-of-states effective masses.
The above expressions for electron and hole concentrations are valid for nondegenerate
semiconductors because the carrier concentrations are low so that the Fermi-Dirac (FD)
distribution function can be reduced to Maxwell-Boltzmann (MB) distribution.
For degenerate semiconductors, FD cannot be reduced to MB, the integral for n and p are
to be carried out numerically.
Temperature dependence of intrinsic carrier concentration
๐๐ = ๐ ๐๐ 3/2exp โ๐ธ๐/2๐๐
For most semiconductors, the band gap g
E decreases with the increase of temperature ๐,
๐ธ๐ ๐ = ๐ธ๐0 โ ๐๐ accurate for 100๐พ < ๐ < 400๐พ but inaccurate for low temperatures,
where ๐ธ๐0 is the extrapolated value of the band gap at ๐ = 0๐พ. The intrinsic concentration
๐๐ = ๐1 ๐๐ 3/2exp โ๐ธ๐0/2๐๐
where ๐1 = ๐exp ๐/2๐ .
Carrier concentrations in extrinsic semiconductors
Ionization of impurities
As donors in semiconductors can lose one electron to become positively charged e.g.
As0 โ ๐ โ As+ Let
๐๐ : total donor concentration ๐๐
0: neutral donor concentration
๐๐+ : ionized donors
We obviously should have ๐๐ = ๐๐0 + ๐๐
+. The occupation probability of impurity level
๐ธ๐ is different from the regular FD distribution because of the spin degeneracy of the donor levels. When a donor is ionized, there are two possible quantum states
corresponding to each of the two allowed spins. An electron can occupy any one of these
states with the condition that as soon as one of them is occupied, the occupancy of the
other is prohibited.
Taking this into consideration, the FD distribution needs to be modified for impusities
๐ ๐ธ๐ =๐๐
0
๐๐= 1 +
1
2exp
๐ธ๐ โ ๐ธ๐น๐๐
โ1
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24
The concentration of ionized donors
๐๐+ = ๐๐ โ ๐๐
0 = ๐๐ 1 +1
2exp
๐ธ๐ โ ๐ธ๐น๐๐
โ1
Similarly for acceptors with ionization process
Al0 + ๐ โ Alโ we should have ๐๐ = ๐๐
0 + ๐๐โ, occupation probability of the acceptor level ๐ธ๐
๐ ๐ธ๐ =๐๐โ
๐๐= 1 +
1
2exp
๐ธ๐ โ ๐ธ๐น๐๐
โ1
For n-type semiconductors, ๐ธ๐ โ ๐ธ๐ is on the order of ๐๐ at room temperature (๐ =300K), and for nondegenerate semiconductors, ๐ธ๐ โ ๐ธ๐น > 3๐๐. The ratio of electrons attached to the impurities to that in the conduction band
๐๐0
๐=๐๐ 1 +
12 exp
๐ธ๐ โ ๐ธ๐น๐๐
โ1
๐๐exp โ๐ธ๐ โ ๐ธ๐น๐๐
Under normal condition, ๐ธ๐ โ ๐ธ๐น > 3๐๐, then exp ๐ธ๐โ๐ธ๐น
๐๐ โซ 1 and
๐๐0
๐=
2๐๐๐๐
exp ๐ธ๐ โ ๐ธ๐น๐๐
~๐๐๐๐
- on the same order, since ๐ธ๐ โ ๐ธ๐~๐๐. For typical doping ๐๐ < 1017/๐๐3, most
impurities are ionized at room temperature with less than 1% remain unionized. We
should have ๐ = ๐๐ .
Consider a nondegenerate semiconductor to which impurities have been introduced. To
keep the discussion in general, we assume there are donors (๐๐ ) and acceptors (๐๐ ). Based on the condition of charge neutrality,
๐๐+ + ๐ = ๐๐
โ + ๐ which can be separated according to mobile and immobile charges as
๐ โ ๐ = ๐๐+ โ ๐๐
โ.
At ๐ > 100๐พ, we should have practically all impurities ionized, ๐๐+ = ๐๐ and ๐๐
โ = ๐๐ , thus
๐ โ ๐ = ๐๐ โ ๐๐ . Since ๐๐ = ๐๐
2, we arrive
๐ =1
2 ๐๐ โ ๐๐ + ๐๐ โ ๐๐ 2 + 4๐๐
2
๐ =1
2 ๐๐ โ๐๐ + ๐๐ โ ๐๐ 2 + 4๐๐
2
For intrinsic situation where ๐๐ = ๐๐ = 0 as well as for completely compensated doping where ๐๐ = ๐๐ , they reduces to ๐ = ๐ = ๐๐ .
For net n-type doping where ๐๐ โ ๐๐ โซ 2๐๐ , we obtain
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25
๐ โ ๐๐ โ ๐๐ โ ๐๐ if ๐๐ โซ ๐๐ and ๐ = ๐๐2/๐๐ (case of strongly extrinsic)
For net p-type doping where ๐๐ โ ๐๐ โซ 2๐๐ , we obtain ๐ โ ๐๐ โ ๐๐ โ ๐๐ if ๐๐ โซ ๐๐ and ๐ = ๐๐
2/๐๐ (case of strongly extrinsic)
2.5 The Fermi level and Energy Distribution of Carriers
Intrinsic semiconductors (๐ = ๐), we have
๐๐exp โ๐ธ๐ โ ๐ธ๐๐๐
= ๐๐ฃexp โ๐ธ๐ โ ๐ธ๐ฃ๐๐
Then
๐ธ๐ =๐ธ๐ + ๐ธ๐ฃ
2+
1
2๐๐ln
๐๐๐๐ฃ
=๐ธ๐ + ๐ธ๐ฃ
2+
3
4๐๐ln
๐โ
๐๐โ
where ๐ธ๐+๐ธ๐ฃ
2 is the center of the band gap. Obviously, for ๐
โ = ๐๐โ , ๐ธ๐ is at the midpoint
exactly; for ๐โ > ๐๐
โ , ๐ธ๐ is above the midpoint; for ๐โ < ๐๐
โ , ๐ธ๐ is below the midpoint.
As a good approximation, 3
4๐๐ln
๐โ
๐๐โ is small for most semiconductors, ๐ธ๐ is roughly at
the midgap for intrinsic case.
Extrinsic semiconductors
n-type: ๐ = ๐๐+, then
๐๐exp โ๐ธ๐ โ ๐ธ๐น๐๐
= ๐๐ 1 + 2exp ๐ธ๐น โ ๐ธ๐๐๐
โ1
Introduce ๐ = ๐๐
2๐๐
1/2
exp โ๐๐
2๐๐ where the ionization energy ๐๐ = ๐ธ๐ โ ๐ธ๐ . The
above equation can be written as 1
1 + 2exp ๐ธ๐น โ ๐ธ๐๐๐
= 2๐2exp
๐ธ๐น โ ๐ธ๐๐๐
Then
๐ธ๐น = ๐ธ๐ + ๐๐ln 4 + ๐2 โ ๐
4๐
And
๐ = ๐๐ ๐
2 4 + ๐2 โ ๐
Now let us discuss the above expressions under different temperatures corresponding to:
(1) Weak ionization ๐ โช 1 ๐ธ๐น = ๐ธ๐ + ๐๐๐๐ 1/2๐ (๐ธ๐น above ๐ธ๐ ) and ๐ = ๐๐๐ (few donors are ionized) (2) Complete ionization ๐ โซ 1
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26
๐ธ๐น = ๐ธ๐ + ๐๐๐๐ 1/2๐ = ๐ธ๐ + ๐๐๐๐ ๐๐/๐๐ (๐ธ๐น below ๐ธ๐ ) and ๐ = ๐๐ (all donors are ionized)
Since ๐ is a function of temperature, given ๐๐ and ๐๐ (type of donors), we can determine a temperature which divides the situations of weak ionization and complete ionization
(transition temperature from weak to complete ionizations: ๐ = 1). This temperature can be solved by
๐ = ๐๐
2๐๐
1/2
exp โ๐๐
2๐๐ = 1
And keep in mind that ๐๐ = ๐๐ ๐ .
It can be seen that as T increases, ๐ increases, and ๐ธ๐น moves from above ๐ธ๐ to below ๐ธ๐ . In fact, ๐ธ๐น decreases approximately linearly with the increase of temperature.
It is obvious that as the temperature continues to increase, ๐ธ๐น approaches ๐ธ๐ . At this point, the hole concentration approaches the electron concentration and cannot be
neglected any more, we must use
๐ = ๐ +๐๐ With ๐๐ = ๐๐
2, then
๐ =1
2๐๐ 1 + 1 +
4๐๐2
๐๐2
1/2
(1) ๐๐/๐๐ โช 1, ๐ = ๐๐ complete ionization (2) ๐๐/๐๐ โซ 1, ๐ = ๐๐ intrinsic
The transition temperature from complete ionization to intrinsic can be determined by
setting ๐ = ๐๐ .
Temperature dependence of carrier concentration in n-type semiconductors
(1) At low temperatures, ๐ธ๐น > ๐ธ๐ , ๐ < ๐๐ , ๐ โช ๐, weak ionization (2) At moderate temperatures, ๐ธ๐น < ๐ธ๐ , ๐ = ๐๐ , ๐ โช ๐, complete ionization (3) At high temperatures, ๐ธ๐น โ ๐ธ๐ , ๐ โ ๐๐ โซ ๐๐ , ๐ โ ๐, intrinsic
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27
Fig. 2.18 Temperature dependence of carrier concentration
Compensation
Consider semiconductors doped with both donors and acceptors, the temperature
dependence of the carrier concentration is different from that of doped with either donors
or accepters only at low temperatures.
Assume ๐๐ > ๐๐ , there will be ๐๐electrons make a transition from ๐ธ๐ to ๐ธ๐ . The remaining donors ๐๐ โ ๐๐ can be excited into the conduction band.
The neutrality condition is then ๐ + ๐๐ = ๐ + ๐๐
+
since all acceptors are ionized (occupied by electron) ๐๐โ = ๐๐ . Since donors are
partially ionized at least, ๐ธ๐น is in the neighborhood of ๐ธ๐ much closer to ๐ธ๐ than ๐ธ๐ฃ, thus, ๐ โช ๐, or
๐ + ๐๐ = ๐๐+ =
๐๐
1 + 2exp ๐ธ๐น โ ๐ธ๐๐๐
But exp ๐ธ๐นโ๐ธ๐
๐๐ =
๐
๐๐exp
๐๐
๐๐ where ๐๐ = ๐ธ๐ โ ๐ธ๐ is the ionization energy. Rearrange
the above equation
๐ ๐ + ๐๐
๐๐ โ ๐๐ โ ๐=๐๐2
exp โ๐๐๐๐
Letโs assume light compensation, i.e. ๐๐ โซ ๐๐ , we can examine two cases (1) ๐ โช ๐๐ (extremely low temperature)
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๐ =๐๐ ๐๐ โ ๐๐
2๐๐exp โ
๐๐๐๐
Since ๐ = ๐๐exp โ๐ธ๐โ๐ธ๐น
๐๐ , we have
๐ธ๐น = ๐ธ๐ + ๐๐ln๐๐ โ ๐๐
2๐๐
๐ธ๐น does not depend on the parameters of conduction band, only depends on ๐,๐๐ ,๐๐ , in this case. This is because the electrons are mainly distributed between the donor and
acceptor levels. The temperature dependence of ๐ is then
ln๐ = ๐๐๐๐ ๐๐ โ ๐๐
2๐๐โ๐๐๐
1
๐
(2) ๐ โซ ๐๐ (slightly higher temperature) We still have ๐ โช ๐๐ , then
๐ = ๐๐๐๐
2
1/2
exp โ๐๐๐๐
This is the same expression for electron concentration at low temperature (weak
ionization) when the semiconductor was doped with donors only. This is because the
electrons are mainly distributed between the donor level and the conduction band, the
existence of small quantity of acceptors does not alter the distribution
๐ธ๐น =๐ธ๐ + ๐ธ๐
2+
1
2๐๐๐๐
๐๐2๐๐
The temperature dependence of ๐ is then
ln๐ = ๐๐ ๐๐๐๐
2
1/2
โ๐๐2๐
1
๐
Apparently, the slopes for ln๐~1
๐ are different between the two cases: a factor of ยฝ. The
slope at low temperature is twice as much as that at slightly higher temperature. At the
transition temperature, we should have ๐ = ๐๐ . Based on this behavior, one can tell whether the semiconductor is compensated or not.
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Fig.2.19 Temperature dependence of n for compensated semiconductors.
As temperature increases beyond the weak ionization, we can have all donors ionized, but
not all electrons will be in the conduction band, actually, ๐ = ๐๐ โ ๐๐ (complete ionization). Further increase the temperature, the semiconductor reaches the intrinsic
region where ๐ = ๐ = ๐๐ โซ ๐๐ โ ๐๐ . The major difference between a compensated and uncompensated semiconductor is the temperature dependence at the low temperature
region.
Energy Distribution of Carriers
Both ๐ = ๐๐exp โ๐ธ๐โ๐ธ๐น
๐๐ and ๐ = ๐๐ฃexp โ
๐ธ๐นโ๐ธ๐ฃ
๐๐ do not tell how electrons and holes
are distributed within the conduction and valence bands. If we need to know the
concentration as a function, we need ๐ ๐ธ ๐ ๐ธ where ๐ ๐ธ is the density of states and ๐ ๐ธ is the FD function.
Fig.2.20 Illustration of ๐ ๐ธ , ๐ ๐ธ , and ๐ ๐ธ ๐ ๐ธ .