Semiconductors : FermiDirac and Maxwell Boltzmann Distribution

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Semiconductors Fermi-Dirac and Maxwell
Boltzmann Distribution

• Marc Madou

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Quantization concept
In 1901, Max Planck showed that the energy
distribution of black body radiation can only be
explained by assuming that this radiation (i.e.
electromagnetic waves) is emitted and absorbed as
discrete energy quanta - photons.
The energy of each photon is related to the
wavelength of the radiation E h ? h c /
? where h Plancks constant (h 6.63 ??
10?34 Js) ? frequency (Hz s?1) c
speed of light (3 ? 108 m/s) ? wavelength (m)
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Example
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A new unit of energy
Since the energies related to atoms and photons
are very small, (EGREEN LIGHT 3.57 ? 10?19 J),
we have defined a new unit of energy called
electron Volt or eV One eV is the energy
acquired by an electron when accelerated by a 1.0
V potential difference.
1 eV 1.6 ?1019 J
Energy acquired by the electron is qV. Since q is
1.6 ? 10?19 C, the energy is 1.6 ? 10?19 J.
Define this as 1 eV. Therefore, EGREEN LIGHT
2.23eV
1 eV 1?? 1.6?1019 CV 1.6?1019 J
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Quantization concept (continued)
Niels Bohr in 1913 hypothesized that electrons in
hydrogen was restricted to certain discrete
levels. This comes about because the electron
waves can have only certain wavelengths, i. e. n?
2?r, where r is the orbit radius. ?
Quantization Based on this, one can show that
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Bohrs hydrogen atom model
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A numerical example
For the n 2 orbit, E2 ?3.4 eV and so on. The
number n is called the principal quantum number,
which determines the orbit of the electron.
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So, an important idea we got from Bohr model is
that the energy of electrons in atomic systems is
restricted to a limited set of values. The
energy level scheme in multi-electron atom like
Si is more complex, but intuitively similar.
Atomic configuration of Si

• Ten of the 14 Si-atom electrons occupy very deep
  lying energy levels and are tightly bound to the
  nucleus
• The remaining 4 electrons, called valence
  electrons are not very strongly bound and occupy
  4 of the 8 allowed slots.
• Configuration for Ge is identical to that of Si,
  except that the core has 28 electrons.

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Bond model

• Consider a semiconductor Ge, Si, or C
• Ge, Si, and C have four nearest neighbors, each
  has 4 electrons in outer shell
• Each atom shares its electrons with its nearest
  neighbor. This is called a covalent bonding
• No electrons are available for conduction in this
  covalent structure, so the material is and should
  be an insulator at 0 K

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2-dimensional (2D) semiconductor bonding model
No electrons are available for conduction.
Therefore, Si is an insulator at T 0 K.
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Simplified 2D representation of Si lattice

• How many atom-neighbors has each Si atom in a Si
  lattice?
• How many electrons are in the outer shell of an
  isolated Si atom?
• How many electrons are in the outer shell of a Si
  atom with 4 neighbors?

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Silicon Covalent Bond (2D Repr)

• Each Si atom has 4 nearest neighbors
• Si atom 4 valence elec and 4 ion core
• 8 bond sites / atom
• All bond sites filled
• Bonding electrons shared 50/50
• _ Bonding electron

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Si Bond ModelAbove Zero Kelvin

• Enough therm energy kT(k8.62E-5 eV/K) to break
  some bonds
• Free electron and broken bond separate
• One electron for every hole (absent electron of
  broken bond)

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(a) Point defect (b) Electron generation
At higher temperatures (e.g. 300 K), some bonds
get broken, releasing electrons for conduction. A
broken bond is a deficient electron of a hole. At
the same time, the broken bond can move about the
crystal by accepting electrons from other bonds
thereby creating a hole.
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Energy band model
An isolated atom has its own electronic structure
with n 1, 2, 3 ... shells. When atoms come
together, their shells overlap. Consider Silicon
Si has 4 electrons in its outermost shell, but
there are 8 possible states. When atoms come
together to form a crystal, these shells overlap
and form bands. We do not consider the inner
shell electrons since they are too tightly
coupled to the inner core atom, and do not
participate in anything.
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Development of the energy-band model
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Figure 2.19. Schematic energy band
representations of (a) a conductor with two
possibilities (either the partially filled
conduction band shown at the upper portion or the
overlapping bands shown at the lower portion),
(b) a semiconductor, and (c) an insulator.
Metal, Semiconductor, Insulator
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Energy band model
At T 0K No conduction can take place since
there are no carriers in the conduction
band. Valence band does not contribute to
currents since it is full. Both bond model and
band model shows us that semiconductors behave
like insulators at 0K.
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Visualization of carriers using energy bands
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Si Energy BandStructure at 0 K

• Every valence site is occupied by an electron
• No electrons allowed in band gap
• No electrons with enough energy to populate the
  conduction band

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Insulators, semiconductors, and metals
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Band Model forthermal carriers

• Thermal energy kT generates electron-hole pairs
• At 300K
• Eg(Si) 1.124 eV
• gtgt kT 25.86 meV,

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Donor cond. electr.due to phosphorous

• P atom 5 valence elec and 5 ion core
• 5th valence electr has no avail bond
• Each extra free el, -q, has one q ion
• P atoms free elect, so neutral

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Band Model fordonor electrons

• Ionization energy of donor Ei Ec-Ed 40 meV
• Since Ec-Ed kT, all donors are ionized, so ND
  n
• Electron freeze-out when kT is too small

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Acceptor Holedue to boron

• B atom 3 valence elec and 3 ion core
• 4th bond site has no avail el (gt hole)
• Each hole, adds --q, has one -q ion
• B atoms holes, so neutral

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Hole orbits andacceptor states

• Similar to free electrons and donor sites, there
  are hole orbits at acceptor sites
• The ionization energy of these states is EA - EV
  40 meV, so NA p and there is a hole
  freeze-out at low temperatures

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Impurity Levels in Si EG 1,124 meV

• Phosphorous, P EC - ED 44 meV
• Arsenic, As EC - ED 49 meV
• Boron, B EA - EV 45 meV
• Aluminum, Al EA - EV 57 meV
• Gallium, Ga EA - EV 65meV

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Fermi-Diracdistribution fctn

• The probability of an electron having an energy,
  E, is given by the F-D distr fF(E)
  1exp(E-EF)/kT-1
• Note fF (EF) 1/2
• EF is the equilibrium energy of the system
• The sum of the hole probability and the electron
  probability is 1

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Fermi-Dirac probability function

• At T0 all states below EF are occupied, above EF
  are free
• When T increases some electrons get enough energy
  to get above EF
• Fermi function smoothened step

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Fermi-DiracDF (continued)

• So the probability of a hole having energy E is 1
  - fF(E)
• At T 0 K, fF (E) becomes a step function and 0
  probability of E gt EF
• At T gtgt 0 K, there is a finite probability of E
  gtgt EF

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Maxwell-BoltzmanApproximation

• fF(E) 1exp(E-EF)/kT-1
• For E - EF gt 3 kT, the exp gt 20, so within a 5
  error, fF(E) exp-(E-EF)/kT
• This is the MB distribution function
• MB used when E-EFgt75 meV (T300K)
• For electrons when Ec - EF gt 75 meV and for holes
  when EF - Ev gt 75 meV

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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

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Electron Density of States Free Electrons
g(E) density of available states
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Electron Conc. inthe MB approx.

• Assuming the MB approx., the equilibrium electron
  concentration is

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Electron and HoleConc in MB approx

• Similarly, the equilibrium hole concentration
  is po Nv exp-(EF-Ev)/kT
• So that nopo NcNv exp-Eg/kT
• Nc 2.8E19/cm3, Nv 1.04E19/cm3

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Position of theFermi Level

• Efi is the Fermi level when no po
• Ef shown is a Fermi level for no gt po
• Ef lt Efi when no lt po
• Efi lt (Ec Ev)/2, which is the mid-band

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EF relative to Ec and Ev

• Inverting no Nc exp-(Ec-EF)/kT gives Ec - EF
  kT ln(Nc/no) For n-type material
  Ec - EF kTln(Nc/Nd)
• Inverting po Nv exp-(EF-Ev)/kT gives EF - Ev
  kT ln(Nv/po) For p-type material EF -
  Ev kT ln(Nv/Na)

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Equilibrium electronconc. and energies
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Equilibrium hole conc. and energies
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References

• 1Device Electronics for Integrated Circuits, 2
  ed., by Muller and Kamins, Wiley, New York, 1986.
• 2Physics of Semiconductor Devices, by S. M. Sze,
  Wiley, New York, 1981.