Semiconductors

1
Semiconductors

• Chapter 8 of Solymar

2
Introduction

• Classification of solids by conductivity
• Conductors metals
• Insulators dielectrics
• Semiconductors
• Semiconductors led to the information age through
  transistors and integrated circuits
• Electrical properties of semiconductors can be
  controlled by the impurities we put in
• Purification a prerequisite
• Purity in ppb range
• Doping techniques required

3
Intrinsic Semiconductors

• Definition
• Intrinsic pure, undoped
• Extrinsic doped, with impurity
• Silicon as an example
• Silicon has 4 valence electrons
• They form 4 covalent bonds with 4 neighboring
  silicon atoms
• The 4 bonds are equally spaced, leading to a
  tetrahedral structure
• In band theory
• All 4 valence electrons are in the valence band
  at 0 K, so the valence band is full and the
  conduction band is empty
• A 1.1 eV energy gap between valence band and
  conduction band
• To excite electrons from valence band to
  conduction band, extra energy is needed thermal
  energy or optical energy

4
Questions

• What is the number of electrons in the conduction
  band at temperature T?
• What is the number of holes in the valence band
  at temperature T?
• How to do it?

Density of states Z(E)
Occupancy probability F(E)
Number of electrons at energy E Z(E)F(E)
Total number of electrons ?Z(E)F(E)dE
5
Density of States

• Assume electrons in the conduction band behave
  like free electrons, but with an effective mass
  m
• Density of states
• Reference at top of valence band (Fig. 8.1)
• For holes
• Total number of electrons
• The upper and lower limits

From bottom of the conduction band
to top of the conduction band
6
Simplifications
C.B.

• F(E) decays exponentially when E EF
• In conduction band, E EF
• We dont know the top of the conduction band
• Since F(E) is so small deep in conduction band,
  we can replace top of conduction band with ?
• Ce ( E Eg )1/2 only valid for the bottom of the
  conduction band
• Since F(E) is so small deep in conduction band,
  we can use the equation without too much error
• F(E) is difficult to integrate
• Since E EF in conduction band, E EF kT

EF
V.B.
7
Derivation

• Z(E), F(E), and Z(E)F(E) as functions of E (Fig.
  8.2)
• With all the simplifications
• With a new variable
• where
• This is the number of electrons in conduction band

8
Hole Concentration

• Integration from -? to 0
• Where
• For an intrinsic semiconductor

9
Intrinsic Semiconductor

• In an intrinsic semiconductor, Ne Nh
• From which
• kT small
• EF is roughly in the middle of the band gap in an
  intrinsic semiconductor

10
Extrinsic Semiconductors

• What happens if a small amount of impurity atoms
  is added to an intrinsic semiconductor
• If a small amount of Group V atoms is introduced
  into Si
• Antimony (Sb), arsenic (As), or phosphorous (P)
• Each Group V atom replaces a Si atom
• 4 of the valence electrons of Group V atoms are
  used for covalent bonding (Fig. 8.3)
• The 5th lone electron becomes loosely bound to
  the impurity atom, since it has 9 electrons in
  outer shell
• The lone electron can easily escape the impurity
  atom and becomes a free electron

11
Donor Atom

• The energy required to free the lone electron
  from the impurity atom into the conduction band
  is small
• The band gap represents the energy required to
  free a valence electron from a Si atom into
  conduction band
• Since Group V atoms donate electrons, they are
  called donors, and their energy level ED is
  called the donor level

12
The Hydrogen Model

• The energy of an electron in a hydrogen atom is
• A donor atom is similar to a hydrogen atom
• It loses and gains one electron
• When it loses the lone electron, it has 1 unit of
  positive charge
• When we apply the hydrogen model
• Donor level is at
• Table 8.1 Energy levels of donors and acceptors

13
Group III Impurity

• If a small amount of Group V atoms is introduced
  into Si
• Boron (B), aluminum (Al), indium (In)
• Each Group III atom replaces a Si atom
• All 3 valence electrons of Group III atoms are
  used for covalent bonding (Fig. 8.4)
• There is a missing electron, since it has 7
  electrons in outer shell
• It is called a hole
• An electron in a neighboring bond may jump over,
  so the hole moves to the next bond
• The hole is weakly bound to the impurity atom

14
Acceptor Atom

• In the band theory, this hole level is slightly
  higher than the energy level at the Si atom, but
• It is still low enough for other electrons to
  easily gain excess energy to jump into it
• Since Group III atoms accept electrons, they are
  called acceptors, and their energy level EA is
  called the acceptor level

15
Terminology

• A semiconductor can have either donors and
  acceptors
• When electrons are the main charge carriers, it
  is called n-type (negative type)
• When holes are the main charge carriers, it is
  called p-type (positive type)

16
Compensation

• By accident or purposely, a semiconductor can
  have both donors and acceptors
• Electrons from donors will first populate the
  acceptors and then go into the conduction band
• If it has more donor atoms than acceptor atoms,
  there will be electrons going into the conduction
  band its n-type
• If it has more acceptor atoms than donor atoms,
  its p-type
• Donor atoms fill up acceptor atoms
• Its called compensation

17
Charge Neutrality

• How to determine EF?
• Charge neutrality
• The semiconductor is electrically neutral, so the
  total amount of positive charge is equal to the
  total amount of negative charge
• NA- number of ionized acceptors
• ND number of ionized donors
• How to find NA- and ND
• NA- number of occupied acceptor states
• ND number of unoccupied donor states
• Ne and Nh are given by
• Replace them in charge neutrality equation and
  solve

18
Simplification

• Usually in an n-type semiconductor, Ne Nh, ND
  NA-
• EF as a function of ND and T
• Not valid for intrinsic semiconductor, Ne Nh
• Not valid for lightly doped semiconductor, ND
  NA
• Not valid when E EF kT
• If (EF ED)/kT is a large negative number
• EF increases with ln(ND)

19
Example

• For Si, Eg 1.15 eV, Eg ED 0.049 eV, ND
  1022 m-3
• Introducing
• Equation
• becomes
• Solve for EF
• EF 0.97 eV
• Considerably above the middle of Eg, so E EF
  kT

20
Variation of EF w/ Temperature

• Eg and ED are both temperature dependent, but
  lets for now ignore their effects
• If ND 1021 m-3, density of lattice atoms 1028
  m-3, Eg 1 eV, Eg ED 0.05 eV
• At low temperatures, all the conduction electrons
  come from donors and little contribution from
  lattice atoms (Fig. 8.5)
• Carrier concentration
• The semiconductor seems to have a band gap of Eg
  ED
• At high temperatures, all donors are ionized and
  many electrons jump from valence band to
  conduction band, Ne ND, and the semiconductor
  behaves like an intrinsic one
• Carrier concentration

21
p-Type Semiconductor

• Fig. 8.6
• Question
• Why the band gap decreases when temperature
  increases?

22
Scattering

• The conductivity due to one type of charge
  carriers is
• m needs to be effective mass me
• Ne can be calculated
• t is the mean free time
• What determine t?

23
Scattering Mechanisms

• Lattice atoms
• The higher the temperature, the larger the
  amplitude of atomic vibration, the higher the
  scattering probability, the smaller the mean free
  time
• Thermal mean free time
• Ionized impurities
• Scattering occurs when the electrostatic energy
  is comparable to the thermal energy
• Scattering cross section
• Impurity mean free time

24
Overall Mean Free Time

• Overall mean free time
• High t is desirable for high mobility, which
  means pure material and low temperature
• Effect of doping (Fig. 8.7)
• Mobility decreases with doping
• Smaller effective mass m means higher mobility
• The conductivity with both electrons and holes

25
Mass Action Law

• The product of electron concentration and hole
  concentration
• It is a constant for a given semiconductor at a
  given temperature
• For an intrinsic semiconductor, Ne Nh Ni
• For extrinsic semiconductors, when Ne increases,
  Nh must decrease
• When Nh increases, Ne must decrease
• When Ne increases, the recombination rate for
  holes increases, so Nh decreases
• Law of mass action

26
III-V II-VI Compounds

• So far we focus on Si
• and Ge
• They are both Group IVA
• Compound semiconductor
• Compound of Group III
• V
• Compound of Group II
• VI
• The bonding in II-VI is more ionic than in III-V,
  which is more ionic than IV
• Most important III-V compound semiconductor is
  GaAs
• Eg for GaAs 1.40 eV, meaning more difficult to
  break a bond in GaAs than in Si (1.12 eV)
• Mobility in GaAs (Fig. 8.8)

27
Trend in III-V Semiconductors

• Iif P replaces As in GaAs, the band gap increases
• If Sb replaces As in GaAs, the band gap decreases
• If Al replaces Ga, band gap increases
• If In replaces Ga, band gap decreases
• The rule is
• The lower in the Periodic Table, the smaller the
  band gap
• The lower in the Periodic Table, the larger the
  atom and the weaker the binding force between the
  nucleus and the electrons, and the easier to
  excite an electron into the conduction band

28
Compound Semiconductors

• A compound of Group III and Group V is a binary
  semiconductor
• There are ternary and quaternary compound
  semiconductors
• Ternary AlxGa1-xAs GaAsyP1-y
• Quaternary AlxGa1-xAsyP1-y
• Doping in GaAs by Si
• If Si replaces Ga atoms, its n-type
• If Si replaces As atoms, its p-type
• Residual doping in GaAs

Residual carbon forced into Ga or As sites by
AsH3/TMG ratio
29
II-VI Compound Semiconductors

• The bonding is more ionic, so the band gap is
  even bigger (Table 8.2)
• ZnSe, 2.6 eV
• ZnS, 3.6 eV
• ZnTe, 2.35 eV

30
Non-Equilibrium Processes

• So far all the discussions assume that the
  semiconductor is in thermal equilibrium
• A simple way to disturb the equilibrium is light
  (Fig. 8.9)
• Three processes to produce excess carriers for
  enhanced conduction - photoconduction
• When the light is switched off, the number of
  carriers will fall gradually to the equilibrium
  value
• The time excess carriers reduce to e-1 of the
  original value is called the lifetime of carriers
  t
• If the semiconductor is locally illuminated, the
  illuminated region has more carriers than the
  surrounding regions and a diffusion current
  results

31
Real Semiconductors

• Our band theory was based on simple rectangular
  crystals
• Real crystals are more complicated
• Si and Ge have diamond structure
• E k curve for Si looks quite different from the
  simple theory (Fig. 8.10)
• Minimum is not at kx 0 as predicted in our
  model
• Indirect band gap needs a phonon to assist in
  photon absorption and emission
• E ky curve looks different from E kx curve
• There are three valence bands with maximum at kx
  0 heavy, light, and split-off
• There are three types of holes heavy, light,
  split-off
• There are three effective masses for holes
• In most device applications, its the average
  effective mass which matters

32
Amorphous Semiconductors

• In amorphous semiconductors, there are
  crystallites (small crystals)
• Crystallites lead to an energy gap
• Small arrays of atoms give rise to spread band
  edges
• Covalent bonds break off when orientation
  changes, so there are dangling bonds and
  distorted bonds
• They act as traps for both electrons and holes,
  and mobility is reduced
• Doping is difficult, since carriers from dopant
  atoms compensate the dangling bonds
• By introducing hydrogen into amorphous Si,
  hydrogen atoms can saturate the dangling bonds
• Doping amorphous Si thus becomes possible
• Amorphous Si is for solar cells, xerox drums, and
  optoelectronics

33
Mobility Measurement

• Mobility definition
• Measurements of voltage and distance give the
  electric field
• Velocity measured by the time needed to move from
  point A to point C
• Haynes and Shockley method (Fig. 8.13)
• With S open, a current flows across R. With S
  closed, there is a sudden increase of current
  through R
• Holes are injected into the n-type semiconductor.
  They travel under the influence of B1
• When they arrive at C, the current through R
  rises again
• The time needed is recorded

34
Four-Point Probe

• A less direct method (Fig. 8.14)
• Ne measured by Hall measurement
• Conductivity s by 4-point probe
• A current I is passed between contacts 1 and 4
• The voltage drop V is measured between contacts 2
  and 3
• With equal probe spacing d
• At low fields, m is a constant
• At high fields, m is a function of the electrical
  field

35
Hall Measurement

• Its discussed in Chapter 1 of Solymar
• It measures charge density
• Four contacts are made on the sample
• Two opposite contacts pass a certain current
• The other two contacts measure the Hall voltage

36
Energy Gap Measurement

• Conductivity as a function of temperature
• For an intrinsic semiconductor
• Plot lns vs 1/T, the slope is Eg/2kT

37
Extrinsic Semiconductor

• At high temperatures, it is intrinsic in behavior
• At low temperatures, it is pseudo-intrinsic
• Apparent band gap Eg Eg ED for n-type
• At intermediate temperatures, all impurity atoms
  are ionized
• Conductivity s is less dependent on temperature
• In a ln s vs. 1/T plot, there are three regions
  with slopes of -Eg/2, 0, and -(Eg ED)/2 (Fig.
  8.17)

38
Optical Transmission

• Monochromatic light onto a semiconductor to
  measure transmission (Fig. 8.18)
• When wavelength is small, the incident photons
  have enough energy to excite electrons from
  valence band to conduction band, so transmission
  is low
• When wavelength is large, no absorption occurs
  and transmission is high
• The boundary is at l hc/Eg, at which there is a
  rise in transmission
• For GaAs
• Direct band gap maximum of V.B. and minimum of
  C.B. at the same k (Fig. 8.19)
• Indirect band gap maximum of V.B. and minimum of
  C.B. not at the same k

39
Phonon

• Si has a gradual transmission edge (Fig. 8.18)
• With an indirect band gap, momentum conservation
  comes in
• Phonon is the quantum mechanical equivalent
  (particles) of momentum (lattice vibration)
• Phonon with frequency w has the energy of hw
• The momentum operator is -ih?
• The momentum of an electron is hk
• For a free electron
• For an electron in a crystal, E k relation more
  complicated

40
Phonon and Photon

• For both phonons and photons
• Their momenta are hk
• v is the velocity of the wave
• For an optical wave with l 1 mm, f 3?1014 Hz,
  c 3?108 m/s
• k 6?1014 m-1
• For a sound wave (lattice vibration), zone
  boundary is at p/a. If a 3 nm
• k 1010 m-1
• This is much larger than photon momentum. Photon
  absorption is practically vertical in E k
  diagram
• In direct gap semiconductors, photon absorption
  is a two-particle process electron photon
• In indirect-gap semiconductors, it is a
  three-particle process electron, photon,
  phonon (less likely to happen)

41
Minority Carrier Lifetime

• Minority carrier lifetime
• If in Si, ND 1022 m-3, Ni 1016 m-3
• Nh 1010 m-3
• If 1015 m-3 electron-hole pairs are created by
  light
• Nh 1010 1015 1015 m-3, Ne 1022 1015
  1022
• Minority carriers are the ones which experience
  significant changes
• Photoconductivity
• Measurement of minority carrier lifetime (Fig.
  8.20)
• When light is switched off, the current DI decays
  exponentially
• tp is the lifetime of holes

42
HW Assignment

• 8.2, 8.6, 8.7, 8.9, 8.15, 8.17