Lecture 6 Introduction to Semiconductors
Kathy Aidala Applied Physics, G2 Harvard University 8 October, 2002
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Semiconductors • The electronics industry today is based on semiconductors, due to our well developed ability to affect the electronic properties of the solid. • Understanding semiconductors allows us to understand the functioning of circuit elements, as well grasp future possibilities and limitations.
• Reading: Sedra & Smith 3.3.1, Streetman Solid State Electronic Devices, Ch 3, appendix IV Wei
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Band Theory •
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Analogy to atoms – From chemistry, we are familiar with the idea of “electron clouds” orbiting the nucleus. – The energy of the different clouds, or levels, is discrete. Adding energy can cause an electron to “jump” into a higher level. In the same way, an electron can lose energy and emit a specific wavelength of light when falling to a lower energy level. (Atomic spectra) – Pauli Exclusion Principle: no two electrons can occupy the same exact state at the same time. This is why electrons fill the energy levels in the way they do. – Valence electrons are the electrons bound farthest from the nucleus
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Band Theory •
What is a crystalline solid? – A volume of atoms covalently bonded in a periodic structure with well defined symmetries. – Example: Silicon
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Where are the electrons? – Covalent bonds share electrons. The e- are delocalized, they can move around the crystal, orbit any atom, as long as there is an open state (cannot violate Pauli Exclusion) – This forms discrete energy bands. Solving Schroedinger’s Equation in the specific periodic structure reveals these bands.
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Specifics of Crystals Econduction EFermi
Egap Evalence
Atom • • • •
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Semiconductor
In an atom, electrons orbit in their shell, at a given energy. In a crystal, many electrons occupy a small energy band. There is a width to the energy band, which is why Pauli Exclusion is not violated. Within the band, electrons can move easily if there are available states, because the difference in energy is tiny. Between bands, electrons must get energy from another source, because the band gap can be significant. 5
Fermi Energy •
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The highest energy electron reached if you were to fill the solid with the intrinsic number of electrons at absolute zero. (No added thermal energy) Meaningful! There is a sea of electrons sitting beneath this energy. – If you bring two solids together with different Fermi energies, the electrons will move around to reach an equilibrium. (Foreshadowing: PN junction) – If you try to put a lower energy electron into a solid (at absolute zero) with a higher Fermi energy, it won’t fit. It cannot be done due to Pauli Exclusion. If the highest energy electron exactly fills a band, the Fermi Energy is near the center of the bands.
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Beyond 0 K: Fermi-Dirac Statistics •
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1 f (E) = E − EF ] 1 + Exp [ kT Wei
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Fermi Energy: The energy state whose probability of being occupied is exactly 1/2 . Electrons obey Fermi-Dirac statistics, which describe the probability of an electron being present in an allowed energy state. Note that if there are no states at a given energy (i.e. in the band gap) there will be no electrons, even if there is finite probability.
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Types of solids
Fermi level falls between Fermi level falls between bands, with a large band gap. bands, with a small band gap. SiO2: 9 eV. Si: 1.11 eV, Ge:0.67 eV, GaAs: 1.43 Wei
Fermi level falls inside the energy band. Easy for electrons to move around 8
Transport in Semiconductors Econduction Egap
• • •
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EFermi
Electrons that get excited into the conduction band carry current. The space left behind in the valence band is called a hole. Holes conduct current. In reality, it’s the movement of all the other electrons. The hole allows this motion. (Bubbles) - Holes can easily travel “up” in energy. - Holes have positive charge. - Current flows in the same direction as the holes move. - Holes have different masses and mobilities than electrons. 9
Intrinsic Semiconductor Summary Econduction Egap
EFermi
Evalence
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Fermi Level: All solids are characterized by an energy that describes the highest energy electron at 0K, the level which has 1/2 probability of being occupied at finite temperature. Semiconductors: A solid with its Fermi level exactly between bands, with a band gap small enough to be overcome at room temperature. Both electrons and holes carry current. 10
Controlling the properties of a Semiconductor Silicon: 4 valence electrons. Each Si atom bonds to four others. Doping • Replace some Si atoms with atoms that do not have four valence electrons. ee-
• These atoms will have an extra electron (group IV), or an extra hole (group III). • This increases the number of carriers and changes the Fermi level.
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Phosphorus doping Econduction
EFermi
Evalence
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Phosphorus has 5 valence electrons. P atoms will sit in the location of a Si atom in the lattice, to avoid breaking symmetry, but each will have an extra electron that does not bond in the same way. These electrons form their own band. Exactly where depends on the amounts of the two materials. The Fermi energy is at this band. This new band is located closer to the conduction band, because these extra electrons are easier to excite. 12
Boron Doping Econduction
EFermi Evalence
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Boron has 3 valence electrons. B will sit at a lattice site, but the adjacent Si atoms lack an electron to fill its shell. This creates a hole. These holes form their own energy band. The Fermi energy is at this band, This band is located closer to the valence band, because these extra holes are easy to “excite down” into the valence band.
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Doping
Econduction
Econduction EFermi
Evalence
Evalence
EFermi
• N-type materials: Doping Si with a Group V element, providing extra electrons (n for negative) and moving the Fermi level up. • P-type materials: Doping Si with a Group III element, providing extra holes (p for positive) and moving the Fermi level down.
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Equilibrium Concentrations: electrons N(E)
∞
n0 =
∫
f ( E ) N ( E )dE = N c f ( Ec ) =
Ec
Nc ≈ N c Exp[− ( Ec − EF ) / kT ] 1 + Exp[( Ec − EF ) / kT ]
n0 = equilibrium carrier concentration N ( E ) = density of states Wei
Carrier concentration
f(E)
3 2
2πm*n kT N c = 2 =effective density 2 h of states 15
Equilibrium Concentrations: holes N(E)
f(E)
Carrier concentration
Ev
Nv p0 = ∫ (1− f (E))N(E)dE = Nv (1− f (Ev )) = ≈ Nv Exp−(E [ F − Ev )/kT] 1+ Exp(E [ F − Ev )/kT] −∞ p0 = equilibrium carrier concentration N ( E ) = density of states
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2πm*p kT N v = 2 =effective density 2 of states h 16
Relationship to Intrinsic Semiconductors •The intrinsic electron and hole concentrations are equal because the carriers are created in pairs. − ( Ec − Ei ) / kT
− ( Ei − Ev ) / kT
ni = N c e = pi = N v e − ( Ec − E F ) / kT − ( E F − Ev ) / kT n0 p0 = N c e Nve
(
= Nc Nve
− ( Ec − Ev ) / kT
)(
= Nc Nve
− E gap / kT
)
= ni pi = ni2
This allows us to rewrite
n 0 = n ie
−(E F −E i )/ kT
p0 = n ie
−(E i −E F )/ kT
•As the Fermi energy moves closer to the conduction [valence] bands,n0 [p0] increases exponentially. Wei
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Temperature Dependence of Carrier Concentrations 3 3 2 2 π kT * * 4 −E g / 2kT (E F e n 0 = 2 2 (m n m p ) e h
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−E i )/ kT
The intrinsic concentration depends exponentially on temperature. The T3 dependence is negligible. Ionization: only a few donors [acceptors] are ionized. Extrinsic: All donors [acceptors] are ionized Intrinisic: As the temperaure increases past the point where it is high enough to excite carriers across the full band gap, extrinsic carriers eventually contribute more. 18
Drude Model of Conductivity • • •
Electrons are assumed to move in a direct path, free of interactions with the lattice or other electrons, until it collides. This collision abruptly alters its velocity and momentum. The probabilty of a collision occuring in time dt is simply dt/τ, where τ is the mean free time. τ is the average amount of time it takes for an electron to collide.
j = −nqv avg
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The current is the number of electrons*charge*area*velocity in a unit of time. For j = current density, divide by the area. This involve the drift velocity. At equilibrium, there is no net motion of charge, vavg = 0. With an applied electric field, there is a net drift of electrons [holes] against [with] the electric field resulting in an average velocity. This model allows us to apply Newton’s equations, but with an effective mass. The effective mass takes the interactions with the rest of the solid into account. 19
Drude Model v = v 0 + at
F = qE = ma
a=
qE m
Consider an electron just after a collision. The velocity it acquires before the next collision will be acceleration*time.
v avg
qEτ =− * m
nq 2τ j = σE = * E m
nq 2τ σ= * m
We want the average velocity of all the electrons, which can be obtained by simply averaging the time, which we already know is τ.
v avg qt µ= * =− We can also write this in terms of the mobility: σ = qnµ E Taking both holes and electrons into account, we end up with the formula m
j = q(nµn + pµ p )E
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Hall Effect •
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Moving electrons experience a force due to a perpendicular B field F =q E +v×B An electric field develops in response to this force. The sign of this field perpendicular to the flow of current determines the carrier type. Density and mobility can also be calculated.
(
)
J x Bz 1 Ey = µ= qnρ qn ρ = resistivity Wei
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