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Band Theory of Solids ‐1 We concluded that, due to the periodic potential associated with the crystalline lattice, there are allowed and disallowed energy bands. Let us look at how carrier transport is affected if a band is filled with electrons or not. 1. If an allowed band is completely empty of electrons, obviously there are no electrons in the band to participate in electrical conduction. This can happen, for example, in a high‐energy band where the energies of the band are above the energies of the systems electrons. 2. Similarly, and surprisingly, if an allowed band is completely filled with electrons, those electrons can not contributive to electric conduction either. 3. Only electrons in a partially filled energy band can contribute to conduction.
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Band Theory of Solids ‐2 There is another way to view band structures that is particularly helpful in understanding how two systems interact when brought together. It turns out that if a quantum system has energy levels E1, E2, E3, then if two such identical systems (e.g. two atoms) are brought together, it can be shown that each energy level ill split into two levels.
E n E b , E n
±
Where En is an energy level slightly above or below the energy value E Where E is an energy level slightly above or below the energy value En of the isolated of the isolated system. E
System 1
E
System 2 E2
E2
E1
E1
System 1
E2+ E2‐ E1‐ E1+
System 2 E2 E1
~
~ x
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Band Theory of Solids ‐3 The splitting is due to the overlap of each system's wavefunctions (orbitals). For example, in the case of two atoms that can come together to form a molecule, the atomic orbitals associated with each atom begin to overlap as atoms are brought together. This can be seen by considering a simplified liner model of forming a lithium (Li) molecule. Lithium has the electronic configuration 1s2 2s1 and in forming the molecule Li2 the s shell atomic orbitals form antibonding and bonding molecular orbitals as depicted . In the ground configuration the bonding molecular state is filled with two 2s1 electrons (one from each atom) and the antibonding state is empty.
If N identical atoms are brought together, each energy level of an isolated atom, E1, E2, E3 will split into N levels forming quasi‐continous bands. © Nezih Pala
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Band Theory of Solids ‐4 When isolated atoms are brought together to form a solid, various interactions occur between neighboring atoms. The forces of attraction and repulsion between atoms will find a balance at the proper interatomic spacing for the crystal. In this process, important changes occur in the electron energy level configurations, and these changes result in the varied electrical properties of solids.
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Band Theory of Solids ‐5 •when two atoms are completely isolated from each other, they can have identical electronic structures •as the spacing between the two atoms becomes smaller, electron wave functions begin to overlap. The Pauli exclusion principle dictates that no two electrons in a given interacting system may have the same quantum state; thus there must be a splitting of the discrete energy levels of the isolated atoms into new levels belonging to the pair rather than to individual atoms.
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Band Theory of Solids ‐6
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Band Theory of Solids ‐7
Linear combinations of atomic orbitals (LCAO): The LCO when 2 atoms are brought together leads to 2 distinct “normal modes – a higher energy antibonding orbital and a lower energy bonding orbital . Note that the electron probability density is high in the region between the ion cores (covalent “bond”), leading to lowering of the bonding energy level and the cohesion of the crystal. If instead of 2 atoms, one brings together N atoms, there will be N distinct LCAO, and N closely spaced energy levels in a band. © Nezih Pala
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Band Theory of Solids ‐8
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Band Theory of Solids ‐9 Energy levels in Si as a function of atomic spacing. The core levels (n=1,2) in SI are completely filled with electrons. At the actual atomic spacing of the crystal, the 2N electrons in the 3s subshell and the 2N electrons in the 3p subshell undergo sp3 hybridization, and all end up in y , p the lower 4N states (valence band) while the higher lying 4N states (conduction band) are empty separated by a band gap.
1s2 ֲ 2N states 2s2 ֲ 2N states
2p6 ֲ 6N states 3s2 ֲ 2N states
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Band Theory of Solids ‐10 Two atoms Six atoms Solid of N atoms
Electrons must occupy different energies due to Pauli Exclusion principle.
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Intrinsic Materials –1 A perfect semiconductor crystal with no impurities or lattice defects is called an intrinsic semiconductor. In such material there are no free charge carriers at T= 0 K, since the valence band is filled with electrons and the conduction band is empty. At higher temperatures electron‐hole pairs are generated as valence band electrons are excited thermally across the band gap to the conduction band. These EHPs are the only charge carriers in intrinsic material. These EHPs are the only charge carriers in intrinsic material. If one of the Si valence electrons is broken away from its position in the bonding structure such that it becomes free to move about in the lattice, a conduction electron is created and a broken bond (hole) is left behind. The energy required to break the bond is the band gap energy Eg.
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Intrinsic Materials –2 Since the electrons and holes are created in pairs, the conduction band electron concentration n (electrons per cm3) is equal to the concentration of holes in the valence band p (holes per cm3). Thus for intrinsic material
n= p= ni where ni is concentration of EHPs in intrinsic material or intrinsic concentration. ni depends on temperature (!) Obviously, if a steady state carrier concentration is maintained, there must be recombination of EHPs at the same rate at which they are generated. Recombination occurs when an electron in the conduction band makes a transition (direct or indirect) to an empty state (hole) in the valence band, thus annihilating the pair.
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Intrinsic Materials –3
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Extrinsic Materials –1 In addition to the intrinsic carriers generated thermally, it is possible to create carriers in semiconductors by purposely (controllably) introducing impurities into the crystal.
This process is called doping. By doping a crystal can be altered so that it has a predominance of either electrons or By doping, a crystal can be altered so that it has a predominance of either electrons or holes. There are two types of doped semiconductors:
•p‐type (mostly holes) •n‐type (mostly electrons)
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Extrinsic Materials –2 When a crystal is doped such that the equilibrium carrier concentrations n0 and p0 are different from the intrinsic carrier concentration ni, the material is said to be extrinsic. Donors
Acceptors
Dopants increasing electron concentration
Dopants increasing electron concentration
P (phosphorus)
B (Boron)
As (Arsenic)
Ga (Gallium)
Sb (Antimony)
In (Indium) Al (Aluminum)
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Extrinsic Materials –3
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Extrinsic Materials –4 When impurities or lattice defects are introduced into an otherwise perfect crystal, additional levels are created in the energy band structure, usually within the band gap.
n‐type semiconductors An impurity from V‐column of the periodic table (P, As, and Sb) introduces an energy level very near the conduction band in Si or Ge. This level is filled with electrons at T= 0 K, and very little thermal energy is required to excite these electrons to the conduction band. At T about 50–100 K virtually all of the electrons in the impurity level are "donated" to the conduction band. Such an impurity level is called a donor level. Thus semiconductors doped with a significant number of donor atoms will have n0>> ni or p0 at room temperature. This is n‐type material.
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Extrinsic Materials –5
n‐type
Semiconductor Si (Z= 14): 1s2 2s2 2p6 3s2 3p2 Dopant (donor) As (Z= 33): 1s2 2s2 2p6 3s2 3p6 3d10 4s2 4p3 © Nezih Pala
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Extrinsic Materials –6 The image cannot be display ed. Your computer may not hav e enough memory to open the image, or the image may hav e been corrupted. Restart y our computer, and then open the file again. If the red x still appears, y ou may hav e to delete the image and then insert it again.
Band diagram for n‐type semiconductor © Nezih Pala
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Extrinsic Materials –7 Atoms from III‐column (B, Al, Ga, and In) introduce impurity levels in Ge or Si near the valence band. These levels are empty of electrons at 0 K. At low temperatures, enough thermal energy is available to excite electrons from the valence band into the impurity level, leaving behind holes in the valence band. Since this type of impurity level "accepts" electrons from the valence band, it is called an acceptor level, and the column III impurities are acceptor impurities in Ge and Si. Doping with acceptor impurities can create a semiconductor with a hole concentration p0much greater than the conduction band electron concentration n0 or ni( this type is p‐ type material)
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Extrinsic Materials –8
p‐type
Semiconductor Si (Z= 14): 1s2 2s2 2p6 3s2 3p2 Dopant (acceptor) B (Z= 5): 1s2 2s2 2p1 © Nezih Pala
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Extrinsic Materials –9
Band diagram for p‐type semiconductor © Nezih Pala
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Extrinsic Materials –10 Example: Calculate the approximate energy required to excite the extra electron of As donor atom into the conduction band of Si (the donor binding energy).
Solution: Let’s assume for rough calculations that the As (1s2 2s2 2p2 3s2 3p6 3d10 4s2 4p3) atom has its four covalent bonding electrons rather tightly bound and the fifth “extra” electron g g y loosely bound to the atom. We can approximate this situation by using the Bohr model results, considering the loosely bound electron as ranging about the tightly bound "core" electrons in a hydrogen‐like orbit. We have to find energy necessary to remove that “extra” electron from As atom.
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Extrinsic Materials –11 We can approximate As dopant atom in Si lattice by using the Bohr model: the loosely bound “fifth” electron is ranging about the tightly bound "core" electrons in a hydrogen‐like orbit. The magnitude of the ground‐state energy (n= 1) of such an electron in the Bohr model is
E1
mq 4 2 K 2 2
Constant K in this case is K=4πεrε0 where εr is relative dielectric constant of Si
Approximation of As dopant atom in Si lattice.
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Extrinsic Materials –12 Besides relative dielectric constant of Si, we have to use conductivity effective mass of electron mn* in Si in the formula for energy E:
E
E
mn* q 4
2(4 0 r ) 2
2
mn* q 4 8( 0 r h) 2
1.18(9.1110 31 )(1.6 10 19 ) 4 8(8.85 10 12 11.8 6.63 10 34 ) 2
1.18 9.11 (1.6) 4 (10 107 ) 8 (8.85) 2 (11.8) 2 (6.63) 2 10 92
70.45 10 107 1.837 10 20 (J) 3.835 10 6 10 92
1eV 1.6 10 19 J E 0.1 eV © Nezih Pala
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Extrinsic Materials –13 Generally, the column‐V donor levels lie approximately 0.01 eV below the conduction band in Ge, and the column‐III acceptor levels lie about 0.01 eV above the valence band. In Si the usual donor and acceptor levels lie about 0.03‐0.06 eV from a band edge. When a semiconductor is doped n‐type or p‐type, one type of carrier dominates. For example, when we introduce donors, the number of electrons in conduction band is much higher than number of the holes in the valence band.
In n‐type material: holes –minority carriers electrons –majority carriers
In p‐type material: holes –majority carriers electrons –minority carriers
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Charge Carrier Statistics ‐1 In previous quantum well problems the allowed energy states were obtained, but here was no way to say which states would actually be “filled” by electrons. Here we will examine how many allowed states are near an energy of interest, and the probability that those states will actually be filled with electrons. Density of states and particle statistics concepts are indispensible in study of bulk materials as well as small material systems. The density of states is required as the first step in determining the carrier concentrations and energy distributions of carriers within a semiconductor. Integrating the density of states function g(E) between two energies E1 and E function g(E) between two energies E and E2 tells us the number of allowed states available tells us the number of allowed states available to electrons in the cited energy range per unit volume of the crystal. In principle, the density of states could be determined from band theory calculations for a given material. Such calculations however, would be rather involved and impractical. Fortunately, an excellent approximation for the density of states near the band edges can be obtained a simple and familiar approach of “particle in a box” problem.
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Charge Carrier Statistics ‐2 Remember the solution of Schrodinger’s equation for particle in a 3D box
r
8 sin k x x sin k y y sin k z z Lx Ly Lz
where
kx
nx Lx
ky
n y
kz
Ly
n z Lz
n x , n x , n x 1,2,3...
Each solution can be uniquely associated with a k‐space vector k = (nxπ/Lx)ax+(nyπ/Ly) ay+(nzπ/z) az where ax ,ay ,az are unit vectors directed along k‐space coordinate axes. In the figure, each point represents one solution of Schrodinger’s equation. © Nezih Pala
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Charge Carrier Statistics ‐3 Taking note of lattice arrangements of the solution dots, we can deduce that a k‐space “nit cell” of volume (π/Lx)(π/Ly) (π/Lz) contain one allowed solution. And therefore:
Lx Ly Lz Solutions 3 Unit volume of k - space C id i th t th Considering that there is no physical difference i h i l diff between wavefunction solutions which differ only in sign, the total number should be divided to 8. On the other hand, for electrons, two allowed spin states (spin up and spin down ) must be associated with each independent solution. We therefore obtain
Lx Ly Lz Solutions 4 3 Unit volume of k - space © Nezih Pala
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Charge Carrier Statistics ‐4 The next step is to determine the number of states with a k‐value between arbitrarily chosen k and k+dk. This is equivalent to adding up the states lying between the two k‐space spheres shown in the figure. Considering that the large dimensions of the system and close‐packed density of k‐space state s, the desired result is simply obtained by multiplying the k‐ space volume between the two spheres, 4πk2dk, times the last equation for the allowed states per unit k‐space volume.
Energt states with k Lx Ly Lz 3 between k and k dk 4 2
2
k 2m 2 kdk dE m E
or or
4k 2 dk
2mE 2 1 m dE dk 2 E
k2
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Charge Carrier Statistics ‐5 Therefore
Energt states with E Lx Ly Lz 2 3 between E and E dE
m 2mE dE
Then by definition
E t states t t with ith E Energt / VdE N ( E ) between E and E dE where V is the volume of the crystal and g(E) is the density of states. Thus, finally
N (E)
m 2mE 2 3
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Charge Carrier Statistics ‐6 To obtain the conduction and valence band densities of states near the band edges in real materials , the mass m of the particle in the forgoing derivation is replaced by the appropriate carrier effective mass. Also if EC is taken to be the minimum electron energy in the conduction band and EV the maximum hole energy in the valence band the E in the last equation must be replaced by E‐EC in treating conduction band states and by EV‐E in treating valence band states. Introducing the subscripts c and v to identify the conduction and valance band densities of states, respectively, we can then write in general
NC (E ) NV ( E )
mn* 2mn* ( E EC ) 2 3 m*p 2m*p ( EV E )
2 3
E EC E EV
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Charge Carrier Statistics ‐6 To gain appreciation of the last equations assuming m*=m we have
N ( E ) 6.8 10 21 ( EeV V0 ) 1/eVcm3 For example if E=0.1 eV and V0=0 then N(E)=2.15x1021 1/eVcm3 Thinking physically, if there are enough electrons to fill the various states, then the density of states N(E) is the density of electrons having energy E. However, this is not case in reality and therefore we should find the probability for electrons to have the energy E.
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Statistical Distributions –1 Given: System of N particles (air molecules, for example) in thermal equilibrium at temperature T. Question: How is the total energy E distributed over the particles? How is the total energy E distributed over the particles? or: How many particles have the energy E1, E2, etc.? How can we answer these questions?
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Statistical Distributions – 2 Maxwell‐Boltzmann statistics •identical particles •“far” apart (no overlap of ψ)
Bose‐Einstein statistics •identical particles •integral spin (bosons) •close together (overlapping ψ))
Distinguishable particles (e.g. molecules in a gas)
Indistinguishable particles ‐ Bosons (e.g. photons)
f ( E ) Ae
E kT
1
f (E) Ae
E kT
Fermi‐Dirac statistics •identical particles •odd half‐integral spin (fermions) •close together (overlapping ψ)
Indistinguishable particles ‐ Fermions (e.g. electrons)
1
f (E)
1
Ae
E kT
1
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Statistical Distributions – 3 Maxwell‐Boltzmann statistics
f ( E ) Ae
E kT
Bose‐Einstein statistics
f (E)
1 E
Ae kT 1
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f (E)
1 E
Ae kT 1
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The Fermi Level –2 f (E)
1 1 e
The function f(E),‐the Fermi‐Dirac distribution function,‐ reflects probability that an available energy state at energy level E will be occupied by an electron at temperature T.
E EF kT
The quantity EF is called the Fermi level Important property of the Fermi level: f for an energy E equal to the Fermi level energy EF ,the occupation probability is
1
( EF ) Ae
EF EF kT
1
1 1 11 2
Fermi level is an energy state which has always a probability of ½ of being occupied by an electron. © Nezih Pala
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The Fermi Level –3 Let’s take a look on what happens with Fermi‐Dirac distribution function when temperature changes.
1. T = 0 For any energy value E< EF:
f ( EF )
1 1 e 1
For any energy value E> EF:
f ( EF )
E EF 0
1 1 e
E EF 0
1 1 1 1 e 1 0
1 1 0 1 e 1
At T = 0 K the Fermi‐Dirac distribution function f(E) takes the simple rectangular form what means: At T= 0 K, every energy state up to the Fermi level EF is filled with electrons and all states above EF are empty. © Nezih Pala
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The Fermi Level –4 2. T > 0 There is some probability for states above the Fermi level to be filled. •At T >0,for E >EF probability that energy states above EF are filled f (E) ≠0 (> 0). •At T >0,there is a corresponding probability [1-f (E)]≠0 that states below EF (E > EF) are empty. Fermi function f (E) is symmetrical about the Fermi level EF for all temperatures:
the probability f (EF+ΔE) that a state ΔE above EF is filled is the same as the probability [1-f(EF+ΔE)] that a state ΔE below EF is empty.
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The Fermi Level –5
The Fermi‐Dirac distribution function The symmetry of the distribution of empty and filled states about EF makes the Fermi level a natural reference point in calculations of electron and hole concentrations in semiconductors. © Nezih Pala
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The Fermi Level –6 Fermi level in intrinsic material The symmetry of the distribution of empty and filled states about EF makes the Fermi level a natural reference point in calculations of electron and hole concentrations in semiconductors. For intrinsic material we know: the concentration of holes in the valence band is equal to the concentration of electrons i f l i h in the conduction band. d i b d
֝ the Fermi level EF must lie at the middle of the band gap in material.
intrinsic
Since f (E) is symmetrical about EF, the electron probability "tail" of f (E) extending into the conduction band is symmetrical with the hole probability tail [1 –f (E)] in the valence band. The Fermi level in intrinsic material is located at energy Ei
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The Fermi Level –7 For intrinsic material at T > 0 K
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The Fermi Level –8 Fermi level in n‐type material In n‐type material there is a high concentration of electrons in the conduction band compared with the hole concentration in the valence band.
֝ The distribution function f (E) lies above its intrinsic position on the energy scale. n‐type material has larger concentration of electrons at E t t i lh l t ti f l t t Ec and correspondingly smaller d di l ll hole concentration at Ev, than intrinsic material. Energy difference (Ec-EF) gives a measure of concentration of electrons in the conduction band.
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The Fermi Level –9 For n‐type material at T > 0 K
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The Fermi Level –10 Fermi level in p‐type material In n‐type material there is a high concentration of holes in the valence band as compared to the electron concentration in the conduction band.
֝ The distribution function f (E) lies below its intrinsic position on the energy scale. The [1 –f (E)] tail below Ev is larger than the f (E) tail above Ec in p‐type material. The value of (EF - Ev) indicates how strongly p‐type the material is.
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The Fermi Level –11 For p‐type material at T > 0 K
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Charge Carrier Concentrations at Equilibrium Our goal is to find carrier concentration n0 and p0 in semiconductor. We can find concentration of carriers if we know: 1. the distribution function f (E) (probability of carriers to occupy energy state) 2. the densities of states N (E) in the valence and conduction bands.
Then concentration of electrons in the conduction band at equilibrium:
n0
f ( E ) N ( E )dE
The subscript “0”used with the electron and hole concentration symbols (n0, p0) will indicate equilibrium conditions.
EC
where f (E) is Fermi distribution function; N (E) is the density of states; N(E)dE is the density of states (cm‐3) in the energy range dE. © Nezih Pala
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Concentrations at Equilibrium –2 Using quantum mechanics approach, it can be shown that density of states N(E) in the conduction band is proportional to √E:
N (E)
(mn* ) 3 / 2 2 E 2 3
Then concentration of carriers in the conduction band at equilibrium can be calculated as:
n0
EC
1 1 e
E EF kT
(we take this as given !)
n0
f ( E ) N ( E )dE
EC
(mn* ) 3 / 2 2 E ( m* ) 3 / 2 2 dE n0 n 2 3 2 3
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E 1 e
E EF kT
dE 90
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Concentrations at Equilibrium –3 At room temperature kT is 0.026 eV → (Ec-EF) >>kT → Fermi distribution function f (E) can be simplified:
1 e ( EC EF ) / kT 1 e ( EC EF ) / kT n0
(mn* ) 3 / 2 2 2 3
EC
E
E 1 e
E EF kT
x
Using standard integral
dE
1 / 2 ax
e
(mn* ) 3 / 2 2 kTF e 2 3
dx
0
Ee
E kT
dE
EC
2a 3 / 2
we can obtain concentration of carriers in the conduction band at equilibrium:
2mn* kT n0 2 2 h
3/ 2
e
E F EC kT
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Concentrations at Equilibrium –4 2mn* kT n0 2 2 h
3/ 2
e
E F EC kT
‐concentration of carriers in the conduction band at equilibrium.
or
n0 N C e
E F EC kT
2mn* kT N C 2 2 h
where
3/ 2
NC is the effective density of states located at the bottom of the conduction band Ec
We have found concentration of electrons in the conduction band at equilibrium.
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Concentrations at Equilibrium –5 The concentration of holes in the valence band at equilibrium is
p0 NV [1 f ( EV )] where constant NV is the effective density of states in the valence band:
2m*p kT NV 2 h2
3/ 2
Thus, the concentration of holes in the valence band is
p0 N V e
EV E F kT
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Concentrations at Equilibrium –6
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