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Solids and semiconductors. Physics 123. Bonding in solids. Atoms in solids organize themselves in crystal structures Positions of atoms are determined by a balance of electrostatic attraction and repulsion

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solids and semiconductors

Solids and semiconductors

Physics 123

Lecture XX

bonding in solids
Bonding in solids
  • Atoms in solids organize themselves in crystal structures
  • Positions of atoms are determined by a balance of electrostatic attraction and repulsion
  • Minimum of potential energy U0is called ionic cohesive energy and is equivalent to binding energy in nucleus

Lecture XX

metals
Metals
  • Metal have 1-2 e on the outer shell, they are loosely bound to the rest of the atom and can be considered “free” to move within the boundaries of metal  electron gas
  • Electrons in potential well – boundaries on metal surface  L is very large
  • Distance between energy levels inversely proportional to L2
  • Energy levels become energy bands

Lecture XX

metals4
Metals
  • Electrons are fermions, according to Pauli principle not more than one electron can exit for each quantum state
  • How much space does a free electron need to itself?
  • dxdp>h
  • In 3-D Phase space (3 spatial coordinates +3 momentum coordinates)
  • dxdydzdpxdpydpz =dVdP>h3
  • electrons = balls in phase space each occupying h3 of space
  • Actually two electrons can coexist in h3– spin up and spin down

Lecture XX

density of states
Density of states
  • Let’s calculate the number of states in unit volume between energy E to E+dE: g(E)dE
  • In momentum space think of a spherical layer of radius p and thickness dp
  • Total phase space volume of this layer V4pp2dp
  • Number of electrons that can live in this volume (number of available apartments)

2(spin)x(total volume)/(volume occupied by one electron)

  • Number of states per unit volume

Lecture XX

fermi energy
Fermi energy
  • Consider T=0K
  • All electrons must fall into the lowest possible quantum state, but respect each other’s privacy – Pauli principle
  • Suppose you have n electron per unit volume, what is the highest energy that they can have at T=0K - Fermi energy?

Lecture XX

fermi dirac probability function
Fermi-Dirac probability function
  • At T=0 all states below EFare occupied, above EF are free
  • When T increases some electrons get enough energy to get above EF
  • Fermi function – smoothened step

Lecture XX

density of occupied states
Density of occupied states
  • g(E) – density of available states
  • f(E)- probability to find electron with a certain value of E
  • Number of occupied states per unit volume

Lecture XX

energy bands
Energy bands
  • In conductors the highest energy band is partially filled allowing electrons to move freely – conduction band
  • In insulators the highest energy band is completely filled – valence band, there is an energy gap between valence and conduction band – Eg
  • Semiconductors are similar to insulators, but the energy gap is smaller

insulator

semiconductor

conductor

Conduction band

Conduction band

Eg

Eg

Valence band

Valence band

Lecture XX

intrinsic semiconductors
Intrinsic semiconductors
  • Since in semiconductors the energy gap is small, thermal energy can be enough for some electrons to jump to conduction band
  • Resistivity of semiconductor decreases (unlike metals) with temperature – more electrons in conduction band
  • Electrons leave vacancies behind – holes, which act as effective positive charge and also carry electric current

Conduction band

Eg

EF

Valence band

Lecture XX

semiconductors
Semiconductors

Si

Si

Si

Si

Si

  • Most commonly used semiconductors Si (Z=14), Ge (Z=32)
  • Si electron structure: 1s22s22p63s23p2
  • Ge electron structure: 1s22s22p63s23p63d104s24p2
  • Semiconductors have 4 electrons on outer shell
  • In crystal structure each atom bonds with 4 neighbors to share electrons

Lecture XX

semiconductors and doping
Semiconductors and doping

Si

Si

Si

Si

As

Ga

Si

Si

Si

Si

  • Doping – introduction of impurities with valence 3 (Ga) or 5(As)
  • As incorporates itself into the existing crystal structure sharing 4 of its e with Si- neighbors, one e is free to move around – n-type doping
  • Ga does the same, but instead of extra e it creates a vacancy – hole – p-type doping
  • Resistivity of doped semiconductor is much higher than that of intrinsic material

Lecture XX

p and n type semiconductors
P and n-type semiconductors
  • Impurities create extra levels in the band structure

Conduction band

Conduction band

Acceptor level

Allows e to jump there

Donor level

Gives e to conduction band

Valence band

Valence band

p-type

n-type

Lecture XX

p n junction
p-n junction

Si

Si

Si

Si

As

Ga

Si

Si

Si

Si

  • Suppose that you bring p-type and n-type semiconductor in contact
  • Electrons from n-type will readily fill the vacancies provided in p-type, thus creating the space charge. Mind that before materials were brought together they were electrically neutral

Q=-1e

Q=+1e

Lecture XX

p n diode
p-n diode
  • The current flows through p-n junction if electrons have vacancies to jump to, it does not flow in the opposite direction
    • Not entirely true, there still is so called “dark” current, because of thermal excitation to conduction band, this current grows with T

P-type

vacancies

+++

+++

+++

P-type

+++

+++

+++

current

No

current

+

-

-

---

---

---

Electron

flow

+

---

---

---

n-type

electrons

n-type

LED: e+hole=light

Reverse bias

Forward bias

Lecture XX

transistors
Transistors
  • npn or pnp junction – no current is flowing – logical zero
  • Small current (supply of electrons) on base (p in npn or n in pnp) opens the transistor – larger current is flowing – logical one

current

P-type

+++

+++

---

n-type

+++

+++

P-type

Lecture XX