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Home , Band diagram, Density of states, Energy gap, Fermi level

Calculating Band Structure

Nearly free • Assume plane wave solution for • Weak potential V(x) • edge

Tight binding method • Electrons in local atomic states (bound states) • Interatomic interactions >> lower potential • Unbound states for electrons • = difference between bound / unbound states

Crystal Field Splitting • Group theory to determine crystalline • Crystalline symmetry establishes relevant energy levels • Field splitting of energy levels

However all approaches assume a crystal structures. Bands and energy gaps still exist without the need for crystalline structure. For these systems, Molecular Orbital theory is used.

• Energy bands consist of a large number of closely spaced energy levels. • Free electron model assumes electrons are free to move within the but are confined to the metal by potential barriers. • This model is OK for , but does not work for since the effects of periodic potential have been ignored. Kronig-Penny Model

• This model takes into account the effect of periodic arrangement of electron energy levels as a function of lattice constant a • As the lattice constant is reduced, there is an overlap of electron wavefunctions that leads to splitting of energy levels consistent with Pauli exclusion principle. A further lowering of the lattice constant causes the energy bands to split again

Energy bands for versus lattice constant. Formation of Bands

Periodic potential Inter- interactions Many more states Conduction / valence bands

Free electron model

Conduction band states

Valence band states

Bound states Conduction / valence bands

Conduction band states

Lowest Unoccupied Molecular Level (LUMO)

Valence band states

Highest Occupied Molecular Orbital (HOMO) Electrons fill from bottom up = filled valence band Example band structures

Ge Si GaAs Find:

Valence bands?

Conduction bands?

Energy Gap?

Highest Occupied Molecular Level (HOMO)?

Lowest Unoccupied Molecular Level (LUMO)? Simple Energy Diagram

A simplified energy used to describe semiconductors. Shown are the valence and conduction band as indicated by the valence band edge,

Ev, and the conduction band edge, Ec. The vacuum level, Evacuum, and the , , are also indicated on the figure. Metals, Insulators and Semiconductors

Possible energy band diagrams of a crystal. Shown are: a) a half filled band, b) two overlapping bands, c) an almost full band separated by a small bandgap from an almost empty band and d) a full band and an empty band separated by a large bandgap. Semiconductors

• Filled valence band (valence = 4, 3+5, 2+6) • at zero temperature

Metals Free electrons Valence not 4

Semiconductors Si, Ge Binary system III-V: GaAs, InP, GaN, GaP Filled p shells Binary II-VI: CdTe, ZnS, 4 valence electrons Eg Temperature Dependence Eg Dependence

Doping, N, introduces impurity bands that lower the bandgap. Energy bands in Electric Field

Electrons travel down.

Holes travel up.

Energy band diagram in the presence of a uniform electric field. Shown are the upper almost-empty band and the lower almost-filled band. The tilt of the bands is caused by an externally applied electric field. The effective mass

The presence of the periodic potential, due to the in the crystal without the valence electrons, changes the properties of the electrons. Therefore, the mass of the electron differs from the free electron mass, m0. Because of the of the effective mass and the presence of multiple equivalent band minima, we define two types of effective mass: 1) the effective mass for calculations and 2) the effective mass for conductivity calculations. Motion of Electrons and Holes in Bands

Electron excited out of valence band Temperature Light Defect …

Electron in conduction band state

Empty state in valence band (Hole = empty state) Electrons - holes

Electron in conduction band NOT localized

Hole in valence band Usually less Mobile (higher effective mass), but not always

Electron – hole pairs in different bands large separation Region Near Gap

e(k) In the region near the gap,

Local maximum / minimum dE/dk = 0 Conduction band

effective mass m* = h2/(d2E/dk2)

Electrons kx Minimum energy Bottom of conduction band

Holes Opposite E(k) derivative “Opposite effective charge” Valence Top of valence band band General Carrier Concentration

Probability of hopping into state

n0 = (number of states / energy) * energy distribution Conduction band

Gap

gc (E) = density of states

f (E) = energy distribution Valence band Density of states

The density of states in a semiconductor equals the density per unit volume and energy of the number of solutions to Schrödinger's equation.

Calculation of the number of states with wavenumber less than k Fermi-surface (3-D) ky Allowed state • K-space for k-vector – Set of allowed k vectors • Fermi surface – Electrons occupy 2 k all kf states less x than Ef*2m/h

– kF ~ wavelength 2p/L of electron wavefunction

Volume in lattice Area of sphere / k states in spheres 3  4pk  1  3  F    kF L3  N   3  6p 2  3  (2p / L)  Density of states http://ece-www.colorado.edu/~bart/book/book/chapter2/ch2_4.htm

3 Number of states: N  2 kF L3 6p 2

Density in energy:

Kinetic energy of electron:

Density of states / energy:

In conduction band, Nc:

Different m* in conduction and valence band Density of States in 1, 2 and 3D Probability density functions The distribution or probability density functions describe the probability that particles occupy the available energy levels in a given system. Of particular interest is the probability density function of electrons, called the Fermi function.

The Fermi-Dirac distribution function, also called Fermi function, provides the probability of occupancy of energy levels by . Fermions are half- integer spin particles, which obey the Pauli exclusion principle. Fermi-Dirac vs other distributions

Maxwell-Boltzmann: Noninteracting particles

Bose-Einstein:

Intrinsic: Ec – Ef = ½ Eg

High temperature: Fermi ~ Boltzmann Carrier Densities

The density of occupied states per unit volume and energy, n(E), ), is simply the product of the density of states in the conduction band, gc(E) and the Fermi-Dirac probability function, f(E).

Since holes correspond to empty states in the valence band, the probability of having a hole equals the probability that a particular state is not filled, so that the hole density per unit energy, p(E), equals: Carrier Densities

Product of density of states and distribution -- defines accessible bands -- within kT of Ef Carrier Densities Electrons

Holes Limiting Cases

0 K:

Non-degenerate semiconductors: semiconductors for which the is at least 3kT away from either band edge. Intrinsic Semiconductor Intrinsic semiconductors are usually non-degenerate Mass Action Law

The product of the electron and hole density equals the square of the intrinsic carrier density for any non-degenerate semiconductor.

The mass action law is a powerful relation which enables to quickly find the hole density if the electron density is known or vice versa Doped Semiconductor Add alternative element for electron/holes

Si valence = 4 P valence = 5 B valence=3

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Si = Si = Si = Si = Si = Si = Si = Si = Si = Si = Si = Si = Hole

=

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Si = Si = Si = Si = Si = Si = P = Si = Si = Si -- B = Si =

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Si = Si = Si = Si = Si = Si = Si = Si = e- Si = Si = Si = Si =

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Si = Si = Si = Si = Si = Si = Si = Si = Si = Si = Si = Si =

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Pure Si Phosphorous Boron n-doped p-doped All electron paired Electron added to Positive hole added to Insulator at T=0 conduction band conduction band Dopant Energy levels

P 0.046eV As 0.054eV

Easily ionized Energy required = easily donate electrons to donate electron Si Eg=1.2eV Au 0.54eV Cu 0.53eV Large energy bad. Add scattering Cu 0.40eV Donate no carriers Au 0. 35eV

Au 0. 29eV Cu 0.24eV

B 0.044eV

Energy required to donate hole Carrier concentration in thermal equilibrium

• Carrier concentration vs. inverse temperature

Region of Thermally activated Intrinsic carriers Functional device

ne N(carriers) = N(dopants) Activation of dopants

1/T(K) Dopants and

 k 3  2k 2 • Free electron metal: n   F ,e  F e  2  F  3p  2m

• Intrinsic semiconductor Ec – n(electrons) = n(holes) – Fermi energy = middle Ef Ev

Ec • n-doped material Ef – n(electrons) >> n(holes) – Fermi level near conduction band Ev

• p-doped materials Ec – n(electrons) >> n(holes) – Fermi level near conduction band Ef Ev Fermi Energy is not material specific but depends on doping level and type Mobility and Dopants

• Dopants destroy periodicity e – Scattering, lower mobility

10000 e GaAs Mobility e (cm2/V-s) 1000 h Si 100 1E14 1E15 1E16 1E17 1E18

Dopant Concentration (cm-3) Doping / Implantation

Implants: (1)NBL (isolation) (2) Deep n (Collector) (3) Base well (p) (4) Emitter (n) (5) Base contact

• Simple bipolar = 5 implants • Complicated CMOS circuit >12