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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l Melting point, latent heat, specific heat
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