Question

How would you make a metal using a one-dimensional lattice of atoms? How about an insulator? Why do metals conduct but insulators do not? Suppose light is shining on a semiconductor. List the three excitation mechanisms from most probable to least probable.

Answer 1

Metals are good conductors (poor insulators). Electrons in the outer layers of metal atoms are free to move from atom to atom. So if one end of a piece of metal is made positive, the electrons will be attracted towards it and because they are free, they can move towards it.

Static charge only builds up on insulators. These are materials that will not allow the flow of charged particles (nearly always electrons) through them. Insulators are materials made from atoms that hold onto their electrons very strongly. The voltage across an insulator has to be extremely high before an electron is given enough energy to free itself and move through the material.

Static charge won't build up on conductors unless they are isolated because as soon as you put too many electrons in one place, they repel each other and spread out, reducing or eliminating the effect. On insulators, the charge can't spread out - so you get a noticeable effect.

You can give metal objects static charge as long as the whole object is insulated from the rest of the world so that charge cannot escape from it (even though the charge is spread evenly throughout the whole metal object).

Good examples are the metal dome on a Van de Graaff generator and the annoying shocks that you get from cars.

Semi-Conductors

Semi-conductors have far fewer free electrons than metals so do not conduct as well. However, if they are given energy electrons are able to free themselves from their atom and flow, which increases their ability to conduct. Some semi-conductors are light sensitive, as the light energy is able to free the electrons. There are about 5 naturally occurring semi-conductors.

Solids, Liquids and Gases

Although in circuits we deal with electrons carrying charge, in liquids and gases other particles are also able to carry charge, such as ions in the process of electrolysis.

Insulators Most solid materials are classified as insulators because they offer very large resistance to the flow of electric current. Metals are classified as conductors because their outer electrons are not tightly bound, but in most materials even the outermost electrons are so tightly bound that there is essentially zero electron flow through them with ordinary voltages. Some materials are particularly good insulators and can be characterized by their high resistivities: Resistivity (ohm m) Glass 1012 Mica 9 x 1013 Quartz (fused) 5 x 1016 This is compared to the resistivity of copper: Resistivity (ohm m) Copper 1.7 x 10-8 Energy band model Dielectric insulators

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Circuit Elements Electric circuits are considered to be made up of localized circuit elements connected by wires which have essentially negligible resistance. The three basic circuit elements are resistors, capacitors, and inductors. Only these passive elements will be considered here; active circuit elements are the subject of electronics. Resistor Capacitor Inductor Resistivity Superconductors
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Conductors and Insulators An external influence repels a nearby electron mobile - _electrons Cross- section of copper wire The electron's neighbors find it repulsive. If it moves toward them, they move away, creating a chain of interactions that propagates through the material at the speed of light. 4s Copper's valence electrons move freely throughout the solid copper metal. Ar Atoms of insulating materials hold on tightly to their outer electrons, like good parents watching all their children. Copper and other metals tend to be "poor parents" of their outer or "valence" electrons, and they are just out wandering the neighborhood. (Ar)3d104s In a conductor, electric current can flow freely, in an insulator it cannot. Metals such as copper typify conductors, while most non-metallic solids are said to be good insulators, having extremely high resistance to the flow of charge through them. "Conductor" implies that the outer electrons of the atoms are loosely bound and free to move through the material. Most atoms hold on to their electrons tightly and are insulators. In copper, the valence electrons are essentially free and strongly repel each other. Any external influence which moves one of them will cause a repulsion of other electrons which propagates, "domino fashion" through the conductor. Simply stated, most metals are good electrical conductors, most nonmetals are not. Metals are also generally good heat conductors while nonmetals are not. Resistivity Superconductors

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Optical properties of
semiconductors

band structure of crystalline solids by solution of Schroedinger equation (one e- approximation)

Solution leads to energy bands separated by an energy band

m*- effective mass (determined by curvature of E-k)

Band structure E(k) k- wave vector

Band structure of solids k(E) – is a function of the three dimensional wave vector (k) within the Brillouin zone.

Brillouin zone – depends on crystal structure and corresponds to unit cell of the reciprocal lattice

Forbidden energy region (gap) – no energy states
Energy bands are only permitted above (conduction band) and below (valence band) the gap

ECB and EVB contain several bands
each band has different effective mass (m*)

Light absorption in semiconductor

⦁   Energy conservation
⦁   Momentum conservation

Dispersion relation for quasi free electrons and photons for one dimensional case

photons – linear dispersion

electrons- quadratic dispersion p= hk

Semiconductor band gap (Eg ) –
the distance between valence band maximum and conduction band minimum.

Direct band gap – ECB minimum and EVB maximum at the same k value

Indirect band gap – ECB minimum and EVB maximum at the different k value

direct vs. indirect semiconductor

Direct (perpendicular) transitions: dipole-allowed interband transitions
Indirect transitions: „inclined“ transitions within the 1st Brillouin zone: the k- conservation can not be realized by a reciprocal lattice vector . Phonon supplies the missing momentum to the electron.

f
k = ki = kf   E
- Ei
= h + Eph
(phonon absorption)

Ef - Ei = h - Eph (phonon emission) kf = ki + kph

Character of optical excitation process

Direct - (perpendicular), dipole-allowed interband transitions

Energy conservation Momentum conservation

(provided by reciprocal lattice vector)

Indirect - phonon assisted with small probability and weak resulting absorption

(phonon absorption) (phonon emission)

ABSORPTION COEFFICIENT A AS FUNCTION OF THE ENERGY OF THE IMPINGING LIGHT

Photon energy
E – photon energy
h – Planck’s constant (4.135667516(91)×10−15 eV s) c – speed of light (299.79 m s-1)
λ - wavelenght

The absorption coefficient α, is a property of a material which defines the amount of light absorbed by it.
The inverse of the absorption coefficient, α–1, is the average distance traveled by a photon before it gets absorbed.

direct semiconductors
- square root dependence on photon energy

indirect semiconductors
- quadratic dependence on the photon energy

Optical properties

Relation of absorption coefficient () and light intensity (I) (Lambert-Baer´s Law)

- absorption coefficient
I0- intensity of incoming light x- distance to the surface
Exponential decay of intensity profile of absorbed light

Penetration depth and absorption coefficient
The wavelength-dependent value of “” determines how far the light enters the semiconductor. the light intensity vs. distance for a few typical examples of absorption behavior.

Penetration depth (x) – the inverse of the absorption coefficient (α-1) – average distance at which traveled by a photon before it gets absorbed

    low α carrier generation
through the material

10-6   10-5   10-4   10-3   10-2   10-1 cm

I0 – the intensity of incoming light

The absorption coefficient of a semiconductor material at a given wavelength determines the spatial region in which most of the light is absorbed.

For high absorptivity, most of the light is absorbed close to the semiconductor surface.

The   low   absorption   coefficient   of   indirect   semiconductors   leads   to   carrier   generation throughout the material for the curve where α = 10 cm-1.

Semiconductors with direct energy gap are generally characterized by:
⦁   a high absorption coefficient in the relevant energy range for photovoltaics;
⦁   most of the sunlight is absorbed within a small range beneath the surface
⦁   possibility to fabricate thin film solar cells;

Indirect semiconductors
⦁   need more material to absorb most of the sunlight; (Si, Ge, GaP)
⦁   thicker layers are needed;
⦁   higher material costs and increased demands on purity increase prize

The plot of the absorption coefficient for a series of semiconductors allows identification of thin film solar cell absorber material:
weak absorption of crystalline Si (x-Si) in the IR to visible range prohibits the use in thin film solar cells.
III-V compound sc, the steep increase of the absorption coefficient with the photon energy,
reaching values of α > 104 cm-1 within about 0.2eV beyond the fundamental absorption edge, makes these materials candidates for thin film applications.
ternary chalcopyrites CuInS2 and its selenide -even steeper increase of α.
amorphous hydrogenated silicon (a-Si:H) has a considerably increased absorption compared to x-Si and an optical gap shifted by about 0.6eV compared to the crystalline material which allows application in thin film devices with in principle higher photovoltages.

Absorption coefficient vs absorption length for h ~ Eg + 0.2eV

semiconductor   CuInSe2   x-Si   InP   GaAs   a-Si:H
 /cm -1   2x105   103   5x104   1.5x104   104
x /µm penetration
depth   0.05   100   0.2   0.7   1

Excess carriers

We consider here absence of surface or bulk recombinations

Excess carrier concentration in EVB and ECB depends on:
⦁   Carrier life time
⦁   Absorption profile
⦁   Temperature

Excess carriers
Intrinsic carrier concentration similar to Si ni= pi = 1010 cm-3

For n-type doping with majority carriers concentration n = 1016 cm-3

 n2

Mass action law:


(1010 )2
p   1016
 104 cm3
p   i   n

Optical excitation perturbs this relation

Minority carriers concentration p=104 cm-3
Stationary excess carrier concentration

P- photon flux 1017cm-2s-1
for h=2eV (red light)

p  n  P
x

AM 1.5 at 84.4 mWcm-2
   1017
106
s1s

- carrier lifetime 1µs
p    103
[ cm3 ]

X- absorption of photons 10-3 cm3
within a volume of 1 cm-3 x 10 µm depth
p  1014 cm3

ni- intrinsic carriers SC   ni=1010cm-3
n- electrons in doped SC in the dark   n=1016cm-3
p- holes in doped SC in the dark   p=104cm-3

n- electrons in doped SC created by illumination   n=1014cm-3
p- holes in doped SC created by illumination   p=1014cm-3

n*- electrons in doped SC under illumination
n*=n + n=1016+1014   n*=1016+1014
p*- holes in doped SC under illumination
p*=p + p=104+1014   p*=104+1014
For majority carriers change by illumination is only 1%
For minority carriers change is illumination is drastical – ten orders of magnitude For n-type semiconductor:
⦁   concentration of electrons coming from doping and thermal excitation is much higher than concentration
of electrons coming from illumination
⦁   cocentration of holes coming from illumination is much higher than holes coming by thermal excitation

spatially dependent carrier concentration profiles in equilibrium (dark) and under illumination in comparison with the light absorption profile.

Whereas the excess majority carrier profile changes little (the change has been magnified in the figure), the excess minority carrier concentration p* deviates strongly from the constant dark concentration (p).

Quasi Fermi levels, definitions

For stationary illumination and sufficiently long carrier life time, excess minority and majority carriers exist stationary at the respective band edges. Their excess carrier concentration relation defines a new quasi equilibrium and attempts have been made to describe this situation in analogy to the dark equilibrium terminology. Therefore one describes the Fermi level for an illuminated semiconductor in the framework of the equations derived for the non illuminated semiconductor. For n-type and p-type semiconductors, EF was given by

which can be written, based on the approximations derived as

Carrier concentration for illumination:
n*(x) = n + n   p*(x) = p + p

n EF

*(x) 

ECB

⦁   kT ln
NCB

n *

p EF

*(x) 

EVB

⦁   kT ln
NVB

p *

knowing:
because      we can write:

   We can write:

Quasi Fermi level for e- is energetically located above above the dark Fermi level

F
p
E* ( x) 
EVB
⦁   kT ln
NVB

p*

EF 

E   
EVB E
⦁   kT ln

⦁   kT ln
NVB

p
NVB

VB   F

F
p
F
E* ( x)  E
p
⦁   kT ln
NVB
p

⦁   kT ln
NVB

p*

F
p
F
E* ( x)  E
⦁   kT (ln


NVB
p
⦁   ln
NVB )
p*

F
p
F
E* ( x)  E

*

⦁   kT ln


p*

p EF ( x)  EF
⦁   kT ln
p

p*  p  p

F
p
F
E* ( x)  E
⦁   kT ln
p  p p

F
n
F
E* ( x)  E
⦁   kT ln
p  p p   p
Quasi Fermi level for h+ is energetically located below the dark Fermi level

F
p
F
E* ( x)  E
⦁   kT
ln[1 
p ]
p