Silicon thermal properties

Thermal properties are dominated by phonons

Silicon phonon dispersion, DOS

Different speeds of sound for different directions and polarizations causes dispersion of pulses.

http://lamp.tu-graz.ac.at/~hadley/ss1/phonons/table/dos2cv.html

http://lamp.tu-graz.ac.at/~hadley/ss1/phonons/table/dos2h.html

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Thermal conductivity Uj K T= − ∇

Imperfections in the crystal or grain boundaries decrease the mean free path and the thermal conductivity.

At high temperatures, the mean free path is limited by Umklapp processes. At low temperatures the Umklapp processes freeze out and the mean free path is limited by imperfections.

Student project

Help make the materials property page for Si.Calculate the phonon dispersion relation for the diamond crystal structure.Make table of the phonon density of states of silicon.Plot the thermal properties as a function of temperature.

Institute of Solid State PhysicsTechnische Universität Graz

Silicon electronic structure

Conduction band

Valence band

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Density of electrons in the conduction band

XΓL

3/2

exp300

F cc

B

E ETn Nk T

− =

Nc = effective density of states in conduction band at 300 K

Ec

Ev

Electrons in the conduction band

T = 300 K

×1012

electrons in the conduction band

T = 300 K

3

exp300

F cc

B

E ETn Nk T

− =

Density of electrons in the conduction bandDensity of holes in the valence band

3/2

exp300

v Fv

B

E ETp Nk T

− =

3/2

exp300

F cc

B

E ETn Nk T

− =

Conduction bands

Valence bands

Density of states

Ev Ec

Eg

Intrinsic carrier concentration

3

exp300 2

gi v c

B

ETn N Nk T

= −

~ 5 x 1022 atoms/cm31/T

log 1

0(n i

) cm

-3

GaAs

Si

Ge

300 K

in p n= =

Extrinsic semiconductors

The introduction of impurity atoms that can and electrons or holes is called doping.

n-type : donor atoms contribute electrons to the conduction band. Examples: P, As in Si.

p-type : acceptor atoms contribute holes to the valence band. Examples: B, Ga, Al in Si.

Ionization of dopants

Easier to ionize a P atom in Si than a free P atom

4

2 2 208nmeE

h nε= −

2*0

0r

mm

εε ε

Ionization energy is smaller by a factor:

Ionization energy ~ 25 meV

Bandgap narrowing

http://www.pvlighthouse.com.au/calculators/band%20gap%20calculator/band%20gap%20calculator.aspx

for Si: µn = 1500 cm2/Vsµp = 450 cm2/Vs

Drift

For E = 1000 V/cm vd = 106 cm/s

, , d n n d p pv E v Eµ µ= − =drift velocity:

* *sce e

m m vτµ − −

= =

( ), , d n d p n pj nev pev ne pe E Eµ µ σ= − + = + =

Drift

, , d n n d p pv E v Eµ µ= − = ( ), , d n d p n pj nev pev ne pe E Eµ µ σ= − + = + =

Solid state electronic devices, Streetman and B

anerjee

Mobility calculator

http://www.pvlighthouse.com.au/calculators/mobility%20calculator/mobility%20calculator.aspx

,d e ev Eµ=

,d h hv Eµ=

High fields

Emission of optical phonons causes the saturation of electron velocity.There are no semiconductors without optical phonons.

http://lamp.tu-graz.ac.at/~hadley/ss2/linearresponse/dielectric.php

Optical properties

Index of refractionExtinction coefficient

Absorption coefficient

pn junction

semiconductors in contact

Abrupt junction: the doping changes abruptly from p to n

Built-in voltage

2ln D Abi B

i

N NeV k Tn

=

Depletion width

W

Vbi ~ 1V

W ~ 1µm

Emax ~ 104 V/cm

vsat ~ 107 cm/sec

The electric field pushes the electrons towards the n-region and the holes towards the p-region.

Diffusion sends electrons towards the p-region and holes towards the n-region.

Diodes

exp 1sB

eVI Ik T

= −

solar cellsPhotodectectorsLEDslaser diodes (CD, AFM, bar code) signal diodessurge protectionZener diodesthermometersvariable capacitors