Semiconductors

1
Section 5

• Semiconductors

2
Electrical Conductivity
3
Enormous Range

• Metals are good conductors because of their
  electron cloud
• Covalent bonded solids are good insulators
  because their electrons are tightly bound
• This explains the wide range of electrical
  conductivities
• What about materials that fall mid-range?

4
Metals - Interaction of Levels
5
Carbon - Interaction of Levels at Zero K
6
Metals Semiconductors at 0 K

• In metals-
• There is a partially filled highest occupied band
• In semiconductors-
• The highest occupied band is full
• This band is separated from the next vacant band
  by an energy gap

7
Quantum Solution - Carbon
8
Metals - Density of Occupied States
9
Metals - True Density of States
10
Metals - Interaction of Levels
11
S/cndtr - Density of Occupied States
12
S/cndtr - the Fermi Energy
13
Fermi-Dirac Distribution

• Probability that a state with energy E is
  occupied at temperature T - f(E)
• Concentration of electrons in the conduction band
  can be determined

14
The Fermi Level (FL)

• Metals - FL - probability of occupation 0.5
• S/cndtrs (at T 0K)
• Prob. in valence band (VB) 1
• Prob. in conduction band (CB) 0
• EF set at middle of band gap

15
Concentration of Electrons (CB)
16
Band Gaps in eV
17
S/cndtrs Metals n m-3

• n in metals (1028 m-3)

18
Temperature Dependence of s
19
Effect of Temperature Demo
20
Conduction of Holes
21
Generation of Holes - ni pi
22
Total Conductivity - s

• s e(nme pmh)
• Pure conductor n p
• e magnitude of electron charge
• n number of electrons
• p number of holes
• m mobility of charge carrier

23
Photon Excitation of Electrons

• Photon excitation is an alternative to random
  thermal excitation of electrons
• Valence electron absorbs incoming photon
• Photon energy must be greater than the band gap
• Photon energy-
• E hn hc/l

24
Photon Absorption
25
Photoconductivity

• A beam of light of right frequency produces a
  large number of conduction electrons
• This process is known as photoconductivity
• The reverse process is the basis for the LED or
  light emitting diode

26
LED 1

• Not all semiconductors are capable of producing
  light
• Energy momentum must be conserved in the
  process
• Electron hole must have same wavevector

27
LED 2

• Right photon frequency sorts energy conservation
• Photon has a negligible wavevector
• Wavevector k is often referred to as the crystal
  momentum
• De Broglie- l h/p but k 2p/l
• Hence, k is proportional to momentum p

28
Energy Bands in Solids

• QM - electrons in a solid have critical values
• a is the lattice dimension
• l 2a,a,a/3 etc are those critical values - like
  waves resonating on a string
• Wavevector depends on the energy band

29
Indirect Gap Semiconductor

• Difference in wavevector from valence to
  conduction band is k p/a (QM)
• Silicon - p/a 6x109 m-1
• Silicon - Egap 1.11eV
• Suitable photon to conserve energy - k 6x106
  m-1
• Cannot conserve momentum

30
Direct Gap Semiconductor

• Consider a hole at top of valence and an electron
  at bottom of conduction
• QM shows some materials have the same wavevector
  for these
• Examples are gallium arsenide gallium nitride
• These materials others are suitable as LEDs
• Gallium nitride works tremendously well even
  though riddled with crystal defects

31
Electron Effective Mass

• Electrons in a s/cndtr do not behave the same way
  as free electrons
• The presence of the band gap, other valence
  electrons lattice ions effect their behaviour
• Electrons behave as if they have a smaller mass
  than free electrons
• Difference measurably by response to applied EM
  fields

32
Effective Mass - Electrons Holes

• QM effect
• Happens because wavevector varies with energy
• Greater d2E/dk2 means smaller effective mass
• Band curvature w.r.t k
• Less important in metals

33
Effective Mass Calculation 1

• Electrons in a material consist of a wave packet
• Group velocity
• vg dw/dk
• From QM
• E hn hw/2p
• Hence-
• vg (2p/h)dE/dk

• a dvg/dt (2p/h)(d2E/dk2)(dk/dt)
• Electric field S
• Work done in dt is dE
• dE eSvgdt
• dE/dt eSvg eS(2p/h)dE/dk

34
Effective Mass Calculation 2

• But dE/dt (dE/dk)(dk/dt)
• Hence-
• dk/dt eS(2p/h)
• Substitute into the expression for the
  acceleration a

• a eS(4p2/h2)(d2E/dk2)
• But-
• a eS/me by Newtons 2nd Law
• Hence-
• me (h2/4p2)d2E/dk2-1

35
Electrons in a Metal

• Assume constant potential throughout the metal
• Solve Schrodinger equation
• Energy-
• E h2k2/8p2me

• d2E/dk2 h2/2p2me
• For metals, therefore-
• me me
• In metals, conduction electrons are located near
  the centre of a band

36
Electrons in Semiconductors
37
n - type Semiconductors

• Phosphorous (Z 15)
• Configuration -
• 1s22s22p63s23p3
• 5 electrons can take part in bonding
• Use modified Bohr model with screening

38
Donor Electron 1

• Binding energy - Ed
• Ed -13.6x(me/me)x(1/e2)x(Z2/n2)
• Donor electron is 3p state
• Experimentally Ed c.45 meV
• Hence Z 3
• Remember p electron orbital loops inside inner
  electrons

39
Donor Electron 2

• Donor electron is weakly attached to the
  phosphorus atom
• Donor electron occupies a state Ed (c. 40/50 meV)
  below the conduction band
• In a pure crystal there are no levels in the
  forbidden band
• Not so for doped crystals

40
Fermi Level at T 0K

• Probability of occupation of EF 0.5 in metals
• No electrons in conduction band at T 0K
• Place EF midway between donor state and
  conduction band

41
Fermi Level at High Temp

• At high temp, number of intrinsic carriers
  similar to donors
• EF should move down to middle of band gap
• Assume EF Eg - Ed/2 still applies at high T

42
Conduction Electrons 1

• Concentration n Cexp(-(Eg - EF)/kBT)
• At room temp Eg - EF is much less than Eg/2 (i.e.
  Eg/2 is value for intrinsic s/cndtrs)
• Actually at at T 0 K
• Eg - EF Ed/2

43
Conduction Electrons 2

• Hence concentration n Cexp(-Ed/2kBT)
• Ed 45 meV
• At T 300 K
• n 1025 exp(-0.94)
• n 4x1024 m-3
• Ed would have to be around 250 meV for agreement
  with next slide

44
Conduction Electrons 3

• At room temperature FD shows that most donor
  impurities are ionised
• Hence, number of conduction electrons n-
• n Ndonors nintrinsic
• At room temp ni 1016 (Slide 17)
• For 1 in a million donors, Nd 5x1022 m-3
  (Turton Q5.14)
• S/cndtr manufacture requires purity of 1 in 1011

45
p - type Semiconductors

• Aluminium (Z 13)
• Configuration -
• 1s22s22p63s23p1
• Only 3 electrons can take part in bonding
• Use modified Bohr model with screening

46
Acceptor State 1

• Use screened Bohr Theory with Z 3 as for
  phosphorous
• mh/me 0.54 for silicon
• See slide 34/35
• Ea -(0.54/0.43)x45 57meV

47
Acceptor State 2

• Acceptor state is Ea above valence band
• At room temperature a valence electron is easily
  excited up to this level
• A hole is left behind in the valence band
• This hole can be conducted through the material
  like a conduction band electron

48
Fermi Level at T 0K

• Probability of occupation of EF 0.5 in metals
• All electrons are in the valence band at T 0K
• Place EF midway between acceptor state and
  valence band

49
Hole Concentration p-type

• A similar analysis to slide 38/39 gives a value
  for p
• p Cexp(-Ea/2kBT) as EF Ea/2
• Ea 57 meV
• C 1025m-3 , T 300K, kB 1.38x10-23 JK-1
• p 3 x 1024 m-3
• Ea depends on the level of doping

50
Majority Minority Carriers

• n - type semiconductor - most current carried by
  electrons - majority carrier
• Thermally produced holes also take part in the
  conduction process - minority carrier
• Minority carriers very important in transistor
  operation

51
Law of Mass Action 1

• Irrespective of the level or type of doping, it
  can be shown that-
• np nipi constant

52
Law of Mass Action 2
53
The Classical Hall Effect 1

• Crossed Ex Bz fields applied to material
• Electrons majority carriers
• Negative charges accumulate on top
• Positive on bottom
• Hall field EH set up

54
The Classical Hall Effect 2

• At equilibrium-
• eEH Bzevd
• But Jx nevd
• Hence-
• EH RHJxBz
• Where-
• RH 1/ne is the Hall coefficient

55
Classical Hall Effect - p-type
56
Quantum Hall Effect(1980) 1

• Occurs in very thin conducting layers at very low
  temperatures high B fields
• Define Hall resistivity rH BzRH (units?)
• Klaus von Klitzing - Nobel prize 1985

57
Quantum Hall Effect 2

• rH as a function of Bz is stepped
• Conductivity-
• sH 1/ rH
• sH i(e2/h) where i is a small integer
• Fractional QHE discovered in 1982