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Bohr quantized the atom…. An atom has a set of energy levels Some (but not all) occupied by electrons. Not really dealing with isolated atoms, but 3D solids. As atoms approach each other, each affects the other Energy levels are altered, splitting into bands

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Bohr quantized the atom…

An atom has a set of energy levels

Some (but not all) occupied by electrons

Not really dealing with isolated atoms, but 3D solids

As atoms approach each other, each affects the other

Energy levels are altered, splitting into bands

Each atom in the system produces another energy level in the band structure


Electronic structure of Solids

Overlapping levels

Outer levels begin to interact

gas

Broadening of energy levels as atoms approach

Degenerate:All electrons in an orbital have the same (lowest) energy

Electron energy


Solid

Gas

E=0

N=5

N=4

N=3

Electron energy

N=2

Internuclear distance

3.67Å

continuum

E=0

Overlapping bands

N=4

N=3

Electron energy

N=2

Energy bands for solid sodium at internuclear distance of 3.67Å


Solid

Gas

E=0

N=5

N=4

N=3

Electron energy

N=2

Internuclear distance

3.67Å

Immediate implication for X-Ray microanalysis…

Electron transitions from split levels (bands) will result in photon emission energies that do not reflect the discrete degenerate level…


Conduction band:

First empty band above the highest filled band

Valence band:

Outermost band containing electrons

Conduction band

Empty band

Bandgap

Electron energy

Outermost band containing elelectrons

Valence band

Partially full

Bandgap

Full

Bandgap

Full

nucleus


Transitions from the valence band involved in characteristic X-ray emission will be energy shifted depending on bond lengths, etc.

Resulting X-Rays will not be monochromatic

These will be Kα X-rays for ultra-light elements

Conduction band

Empty band

Bandgap

Electron energy

N=2 (L)

Valence band

Partially full

N=1 (K)

nucleus


Classification of solids: X-ray emission will be energy shifted depending on bond lengths, etc.

Conductors

Insulators

Semiconductors

Conductors:

Outermost band not completely filled

Essentially no band gap

overlap

lots of available energy states if field is applied

Metals and Alkali metals


Insulators: X-ray emission will be energy shifted depending on bond lengths, etc.

Valence band full or nearly full

Wide band gap with empty conduction band

Essentially no available energy states to which electron energies can be increased

Conduction band Empty

Eg

Wide bandgap

Valence band Full

Dielectric breakdown at high potential


Semiconductors: X-ray emission will be energy shifted depending on bond lengths, etc.

Similar to insulators but narrow band gap

At electrical temperatures some electrons can be promoted to the conduction band

Most are cubic Diamond FCC (single element)

Zinc blende (FCC ZnS)

Mark McClure, UNC-Pembroke

S

Zn

Conduction band Almost Empty

Conduction band Empty

Eg

bandgap

Valence band Almost Full

Valence band Full

T > 0K

T = 0K

Some common band gaps:

Element gap (ev)

Ge 0.6

Si 1.1

GaAs 1.4

SiO2 9.0


Semiconductors are either X-ray emission will be energy shifted depending on bond lengths, etc.intrinsic or extrinsic

Intrinsic Semiconductors: Pure state

Example: Covalently bonded, tetravalent Si lattice

Promotion of an electron to the conduction band leaves “hole” in the valence band = electron-hole pair

Apply an electric field and the electron will migrate to +

The hole will migrate to – (that is, the electron next to the hole will be attracted to the +, leaving a hole toward -)

+

-

Ec

Eg

Ev

Net propagation of hole


Extrinsic Semiconductors: X-ray emission will be energy shifted depending on bond lengths, etc.

Doped with impurity atoms

p-type

n-type

n-type

Dope Si with something like pentavalent antimony (5 valence electrons)

Narrows the band gap relative to Si

easy to promote Sb electron

Majority carriers are electrons in conduction band

Minority carriers are holes in valence band

Lattice doped with donor atoms

localized energy levels just below conduction band


E X-ray emission will be energy shifted depending on bond lengths, etc.c

Ed

Ev

Si

Si lattice with n-type dopant

Sb


p-type X-ray emission will be energy shifted depending on bond lengths, etc.

Dope Si with something like trivalent indium (3 valence electrons)

Incomplete bonding with Si

Nearby electron from Si can fill hole

Majority carriers are holes in the valence band

Minority carriers are electrons in the conduction band

Lattice doped with acceptor atoms

localized energy levels just above valence band


E X-ray emission will be energy shifted depending on bond lengths, etc.c

Ea

Ev

Si

Si lattice with p-type dopant

In


E X-ray emission will be energy shifted depending on bond lengths, etc.c

Eg

Ev

Valence band

Fermi Level:

That energy level for which there is a 50% probability of being occupied by an electron

Conduction band

Ec

Intrinsic

Eg

Ef

Ev

Valence band

Conduction band

Ef

n-type

Recombination

Electron-hole pairs not long lasting

Electron encountering hole can “fall” into it

Free time = microsecond or less


- X-ray emission will be energy shifted depending on bond lengths, etc.

-

-

-

-

-

-

Depletion width W

+

+

+

+

+

+

p

n

Direction of built-in field

Space-charge layers

The p-n junction

Single crystal of semiconductor

Make one end p-type (dope with acceptors)

Make the other end n-type (dope with doners)

The junction of the two leads to rectification

Current only passed in one direction (diode)

In the region of the junction

Recombination = depletion of region with few charge carriers

Results in “built-in” electric field


eV X-ray emission will be energy shifted depending on bond lengths, etc.0

-

-

-

-

-

-

-

Depletion width W

+

+

+

+

+

+

p

n

Direction of built-in field

Space-charge layers

Energy band diagram for p-n junction at equilibrium

Ecp

Ecn

Efp

Efn

Evp

Evn

Apply eV0 to get diffusion


Energy band diagram for p-n junction – applied forward bias

Ecp

eV

Ecn

Efp

Efn

Evp

Evn

Depletion width reduced

Built-in field reduced

Barrier height reduced

Diffusion current increased

Apply small Vto get diffusion

+

-

-

-

-

-

-

-

-

If Vforward = V0

No barrier

Pass large current in one direction

Depletion width W

+

+

+

+

+

+

p

n

Direction of built-in field

Space-charge layers


Energy band diagram for p-n junction – applied reverse bias

Ecp

eV

Evp

Ecn

Depletion width increased

Built-in field increased

Barrier height increased - Diffusion current decreased

Evn

-

+

-

-

-

-

-

-

-

Becomes Capacitor

No current passed

Depletion width W

+

+

+

+

+

+

p

n

Direction of built-in field

Space-charge layers


So: bias

Can use reversed bias p-n junction as voltage regulator

Zener diode

Voltage too high? Overcome gap energy and pass current

Can use forward bias p-n junction for rectification

AC → DC transformer

Analog-to-digital conversion

LED

Recombination – “tune” bandgap to achieve photon emission at the required wavelength

GaAs (IR) GaInN (blue) GaAsP (red) YAG:Ce (white)

Ternary and quaternary compounds allow precise bandgap engineering

PIN diode (p and n sections separated by high resistance material)

light detection

X-ray detection

electron detection

-Each of these serve to excite electron-hole pairs

-Bias properly and get amplification rather than simple propagation


Bipolar transistor = pair of merged diodes - NPN or PNP bias

base

base

N

P

N

P

N

P

collector emitter collector emitter

Three voltages (NPN)

Collector = + relative to base (collects electrons)

Emitter = - relative to base (emits electrons)

Small adjustments of the current on the base results in large changes in collector current.

= current amplifier

Amplify weak signals

Use small currents to switch large ones


Simple optical encoding: bias

Generate sine wave by LED passing ruled slide

Phototransistor sees varying light intensity

current output varies with base current

Diode rectifies

AC→DC

Square waves

Digital output to counter


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