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

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

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.

Formation of crystal grid

Basic types of primitives cells

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

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

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

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

Electron in a quantum well

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

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, 055.10-34 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

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.

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.

Energy bands

Energy bands

Energy bands

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.

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.

Conductors, insulators & semiconductors

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.

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

Indirect & direct semiconductors

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

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

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)

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

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).

Electrical conductivity in Semiconductors

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.

N-type Semiconductor

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.

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).

p-type Semiconductor

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

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.

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)

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

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

Carrier concentrations in intrinsic semiconductor

Carrier concentrations in exintrinsic n- type semiconductor

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

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

Calculations of Carrier concentrations For constant temperature the carrier concentrations is

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

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

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

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

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.

P-N junctions in equillibrium

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

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.

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

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

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.

Heterostructure

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

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

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.