1. Lecture 2nd – 3rd Solid state physics – Semiconductor Junctions
Optoelectronics
Lecture 2nd – 3rd
Solid state physics – Semiconductor Junctions

2. Contents Characteristics of solid state devices P – N Junctions
Solid and Crystal materials
Material properties – Energy states of the electrons – Schrodinger equation
Conductors – Insulators – Semiconductors
Intrinsic & extrinsic semiconductors
P – N Junctions
Junction in equilibrium
P – N Junction under bias
Energy bands in semiconductors
Alloys
Complex semiconductors
Heterojunctions

3. Solid and crystal Material properties
The properties of solid materials are determined by the behavior of the motion of a vast number of electrons, moving between positive ions.
The electron density is
Semiconductors atoms are organized in a crystal grid.
A crystal is characterized by having a well-structured periodic placement of atoms. The smallest assembly of atoms that can be repeated to form the entire crystal is called a primitive cell, with a dimension of lattice constant a. The primitive cell may have one atom or a group of atoms. The electrons (e) are moving inside this well-organized ion structure.

4. Formation of crystal grid

5. Basic types of primitives cells

6. Schrodinger equation
The motion of an electron inside a solid material is described from the Schrodinger equation
V: potential , Ψ wavefunction, h: reduced Planck constant, m: electron mass
The time independent Schrodinger equation ( dΨ/dt=0 ) takes the below form
E: electron energy

7. Schrodinger equation In a quantum well the potential ,
The Schrodinger equation takes the form below

8. Electron in a quantum well
(1.6)
The solution to Schrodinger equation has the below form
where
Making the necessary calculations
, and

9. Electron in a quantum well
The energy is quantiumed
and n is the quantum number of the electron energy
In the 3 dimension
, , and

10. Electron in a quantum well

11. Electron in a quantum well
The case of a quantum well of infinite potential barrier is an identical case, however is a good approximation for many cases as an electron inside a heterostructure

12. Fermions and bosons FERMIONS BOSONS
Particles following Fermi-Dirac distribution
The spin of the fermions is a half-integer ( 1/2, 3/2, ) of Planck ‘s reduced constant ħ ( ħ = h/2π =   1, Js )
The Fermions obey to the Pauli Exclusion Principle. Two electron or more electrons can NOT have the same quantum state.
Examples of Fermions particles are protons, electrons and neutrons
BOSONS
The spin of the Bosons is an integer multiple of Planck ‘s reduced constant ħ .
The Bosons do NOT obey to the Pauli Exclusion Principle.
The photons can occupy the same quantum state. The photons are Bosons

13. Energy bands
When the electrons come close to each other, their wavefunctions overlap and the electrons start to interact with one another.
According to the theory of the molecular orbits, when electrons come close to one another, they interact, the previous orbits are destroyed and new ones are created. The number of the new orbits is the same as the number of the destroyed ones.
The distribution of the energies of the new orbits forms regions where the density of states is very dense and other ones with no allowed state. The dense regions are called energy zones and the other ones are called forbidden zones.

14. Energy bands
As the energy increases the width of zones increases while the width of the forbidden region decreases.
The most crucial energy zones are the last occupied energy band named valance band, and the next energy band named conduction band.
The potential inside the crystal lattice is periodic U(r)=U(r+a)=U(r+2a)=…. .
So the free electron distribution is modeified due to the crystal lattice. The first zone band is called the Brillouin zone.

15. Energy bands

16. Energy bands

17. Energy bands

18. Conductors, insulators & semiconductors
In conductors, the valance band is half filled. So the free electrons, with specified energies, can move inside that zone.
By applying an external electric field, the free electrons of the half occupied valance band, can be accelerated by absorbing energy and jump to higher energy states inside the half-occupied valance zone.
In conductors, the conduction is half-filled or ovelpas with the valance band overlap
Insulators
In classical approach, the insulators do not have free electrons to accelerate and no electrical current is produced.
In quantum mechanical approach the higher energy zone is fully occupied. The electrons can NOT absorb enough energy because the next free energy state is in the next energy band, which is far away and requires a great amount of energy. Also due to the Pauli exclusion principle, two electrons can occupy the same energy state so the electrons can not change their moving state.
The conduction and valance band are far away from one another. The valance band is fully occupied while the conduction band is empty.

19. Conductors, insulators & semiconductors
In semiconductors, the energy gap between the valance and the conduction band is small and a few electrons can jump to the conduction band, by absorbing energy from an external electric field. In the conduction band there are many free electron states so the Pauli exclusion principle is not violated.
In conductors, the increase of temperature reduces their conductivity because the number of collisions between electrons-ions of the crystal lattice are increased. In conductors, ELECTRONS contribute to the material ‘s conductivity.
In Semiconductors, the increase of temperature increases their conductivity as the number of excited electrons is increased. In semiconductor, HOLES and ELECTRONS contribute to the material ‘s conductivity.
HOLE is the lack of an electron. More specifically is an unoccupied energy state, created by the excitation of an electron in the conduction band. The free energy states permit electrons in the valance band to move, making the free electron states to move in the opposite direction.

20. Conductors, insulators & semiconductors

21. Indirect & direct semiconductors
The valance and conduction band are strongly dependent on the electrons momentum.
Different materials have different crystal lattices.
In some materials, a moving electron has different effective masses when moving in different directions, while in other materials the effective mass is independent of the electron ‘s moving direction.
When an electron jumps from conduction to valance band the momentum is conserved.
Indirect Bandgap Semiconductors
The minimum of the conduction band and the maximum of the valance band takes place for different directions and consequently for different wavenumbers and momentum.
In order to change the momentum of the electron, an extra particle must engage in the transition. This particle is the PHONON.

22. Indirect & direct semiconductors
The phonon is the molecules vibrations.
The engagement of an extra particle reduces dramatically the probability of a radiative transition
Silicon (Si) is an indirect semiconductor material
Direct Semiconductors
The minimum of the conduction band and the maximum of the valance band takes place in the same directions and consequently for same wavenumbers and momentum.
In this case, a transition from conduction to valance band requires no change in the momentum. So no PHONON are produced
The only produced particle is the PHOTON whose energy is given by
Gallium Arsenide (GaAs) is a direct semiconductor material

23. Indirect & direct semiconductors

24. EFFECTIVE ELECTRON MASS
When an electron moves in a crystal grid, the electrons are affected by the external Electrical Field, the internal electrical field induced by the other electrons and the periodic potential of the crystal. Consequently, the mass of a free electron is different from that of a moving electron inside a semiconductor.
The same principle applies for the holes.
Effective mass of electron inside a solid (near the band edge

25. Electrical conductivity
In solid materials, without external Electric Field, the carriers are in random stable thermal motion. So no total electrical displacement is observed and no electrical current is produced. J=0
With the application of external Electric Field, the carriers are moving, producing a current
The electron ‘s velocity is

26. Electrical conductivity in metals
σ=neμ is the electron special conductivity
n : electron density
τ : electron relaxation time (time between two collisions)
μ : electron mobility (is affected by the scattering of the crystal grid and the crystal inserted atoms)

27. Electrical conductivity in Semiconductors
The carriers in semiconductors are electrons and holes.
The conductance is given by
Intrinsic semiconductors: are perfect crystals with no impurities and
Intrinsic semiconductors are identical and DON NOT exist
In silica

28. Electrical conductivity in Semiconductors
In a standard temperature T the semiconductor is in equilibrium. In equilibrium the recombination rate is equal to the excitation rate.
If the heat excitation rate is and the recombination rate is
The recombination rate is proportional to electrons and holes
B : proportional constant depending on the recombination rate
Extrinsic semiconductors : The crystal grid has random impurities or impurities inserted on purpose (doping).

29. Electrical conductivity in Semiconductors

30. N-type Semiconductor
The Silica (Si) and Germanium (Ge) has four electrons on its external orbit. The electrons of the external orbit occupy energy states in the valance band while the conduction band is almost empty.
When atoms of Arsenide (As) or Phosphor (P) are inserted on purpose in the crystal grid, the semiconductor is doped. The Phosphor has five electrons on its external orbit. The four electrons of the Phosphor make four covalent bonds with the four electrons of the Silica, living one electron weakly bonded. The weakly bonded electron can easily be excited in the conduction band, contributing to the conductance of the material.
The Phosphor or Arsenide inserts new energy states in the forbidden zone beneath the conduction band. The electron ‘s occupying these states can easily be excited in the conduction absorbing a small amount of energy.
This is a n-type semiconductor because it has an excess of negative charges (electron). The impurities are called donors. Example of elements used as donors are Arsenide and Phosphor.

31. N-type Semiconductor

32. p-type Semiconductor
When atoms of Boron (B) are inserted on purpose in the crystal grid, we can form a p-type semiconductor. The Boron has three electrons on its external orbit. The three electrons of Boron make three covalent bonds with the three electrons of the Silica. However the fourth electron of the silica can not form a covalent bond, due to the lack of the electron. This lack of electron leaves an unoccupied energy state in the valance band. The surrounding electrons can occupy this state, leaving an unoccupied energy state somewhere else. So the unoccupied energy state can move.
The computation of the motion of electrons is a very complicated calculation because of the many electrons which interact with one another (almost full valance band). However, their motion can be described by a new particle with positive charge, moving in an empty band. This particle is named HOLE
The Boron inserts new energy states in the forbidden zone just above the valance band. The electrons of the valance band can be excited to these states leaving a hole in the valance band.

33. p-type Semiconductor
Due to the lack of positive charges (Holes) this is a p-type semiconductor. The impurities used to produce the p-type semiconductor are called acceptors. Example of acceptors are Boron (B) and Aluminium (Al).

34. p-type Semiconductor

35. Exitons
Energy states just beneth the conduction band. They exist in intrinsic and extrinsic materials. The exiton is a pair of hole and electron which come close due to Coulomb interaction. The pair of hole and electron rotates around their common center of gravity. The orbital radius of the pair is inversely proportional to their effective masses.
The effective mass of the exciton is

36. Carriers concentration
The properties of materials and the semiconductors are defined by the carriers concentration.
Metals : carriers ‘s concentration (n)
Semiconductor p or n type : carrier concentration is defined by the majority carriers concentration. It is proportional to the impurities concentrations.

37. Carriers concentration
In order to compute the carrier’s concentrations in each energy band we need to now
Distribution of energy states Z(E) (or N(E))
Occupation probability of an electron energy state F(E)

38. Fermi energy
The electrons are FERMIONS. The occupation of the energy states from the electrons are subjected to the Pauli exclusion principle.
The electrons occupy the lowest energy states first and continue occupying higher energy states until all the electrons have occupied a single energy state. The highest occupied energy state is called the Fermi Energy
The Fermi Energy is the energy of the highest occupied energy state of a Fermion system in absolute zero. (n = density of electorns per cubic meter)
The probability of an electron to occupy a state of an energy E is given

39. Fermi energy
The Fermi energy is also defined as the energy which has ½ occupation probability.
For if
All the states with energy lower than are full while the others are empty
For
The energy states with are mostly occupied.
For half of the states are occupied
For most of the energy states are unoccupied
cmf

40. Carrier concentrations in intrinsic semiconductor

41. Carrier concentrations in exintrinsic n- type semiconductor

42. Calculations of Carrier concentrations
Carrier density
For a n-type semiconductor is

43. Calculations of Carrier concentrations
Carrier density in p-type
In intrinsic semiconductor O

44. Calculations of Carrier concentrations
For constant temperature the carrier concentrations is

45. Majority carrier
The optoelectronic devices operate with carrier concentration’s higher than the concentration in thermal equilibrium
If an excess of carriers is created by excitation, the carrier concentration returns to original values when the excitation mechanism stops (thermal equilibrium).
The carrier recombination rate is greater than the thermal production rate (T constant)
jcnemwnoicj,we with

46. Majority carrier
The optoelectronic devices operate with carrier concentration’s higher than the concentration in thermal equilibrium
If an excess of carriers is created by excitation, the carrier concentration returns to original values when the excitation mechanism stops (thermal equilibrium).
The carrier recombination rate is greater than the thermal production rate (T constant)
jcnemwnoicj,we with

47. Carrier diffusion and drift
If there is an excess of holes in a n-type semiconductor, then the carriers diffuse in space due to carrier diffusion
Flow of holes
The diffusion of carriers causes a electrical current flow density
If there is an external field then

48. P-N junctions
Opticoelectrical set-ups consisted of different materials
Bring close a p-type semiconductor with a n-type semiconductor we form a P-N junction or a diode
Homojunctions
Heterojunctions

49. P-N junctions in equillibrium
Holes start diffusing form p-type to n-type and electrons from n-type to p- type
When carriers enter the opposite region they became minority carriers and they recombinate with majority carriers
For the above reason there is formed a space free of carriers (depletion region) where there only ions of Donors and Acceptors.
The ions produce an intrinsic potential and consequently an intrinsic Electrical Field.

50. P-N junctions in equillibrium

51. P-N junctions forward bias
When a junction is forward bias the potential barrier is reduced from to
Electrons and holes diffuse from n-type to p-type and from p-type to n-type respectively. Further, the carriers recombinate in the depletion region.
The lost carriers are replenished by battery.
Total carriers density
with saturation current

52. P-N junctions reversed bias
When a junction is reversed bias the potential barrier is increased from to
The diffusion current is reduced due to higher barrier.
The current is attributed to drift current.

53. P-N junction capacitance
The depletion area forms a capacitance
The capacitance restricts the junctions functionality especially at high frequencies

54. Deviations For high reverse bias voltages is P-N junctions collapses
The collapse is attributed to
Zener effect
Thin depletion zone
Tunneling effect
Avalanche effect
Wide depletion zone
Impurity ionization

55. Heterostructure
The Homojunctions have small efficiency and small output power.
For higher output power we use heterostructures. In these structures a thin layer is sandwiched between two layers of higher energy gap
Heterojunctions provide higher and more efficient carrier restriction and the carrier recombination is more effective than homojunction.
Also the different material layers provide greater change in the material’s refractive index. Heterostructures provide better and more efficient waveguide of the produced radiation due to their greater change in the refraction index.

56. Heterostructure

57. Quantum wells Semiconductors struct with a multilayer form
Very small layer’s width (10nm)
Example GaAs between GaAlAs
In quantum well the one direction of the struct is comparable to De Broglie thermalized electron wave
The energy levels in such a struct are given by
with

58. Quantum wells Semiconductors struct with a multilayer form
Very small layer’s width (10nm)
Example GaAs between GaAlAs
In quantum well the one direction of the struct is comparable to De Broglie thermalized electron wave
The energy levels in such a struct are given by
with

59. Quantum wells Increase of energy gap
Higher density of states at the lowest point of the conduction band
Increase two or three times of energy excitons bond with higher transition probability.