ECE 4813 Dr. Alan Doolittle ECE 4813 Semiconductor Device and Material Characterization Dr. Alan Doolittle School of Electrical and Computer Engineering Georgia Institute of Technology As with all of these lecture slides, I am indebted to Dr. Dieter Schroder from Arizona State University for his generous contributions and freely given resources. Most of (>80%) the figures/slides in this lecture came from Dieter. Some of these figures are copyrighted and can be found within the class text, Semiconductor Device and Materials Characterization. Every serious microelectronics student should have a copy of this book!
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ECE 4813 Dr. Alan Doolittle
ECE 4813
Semiconductor Device and Material Characterization
Dr. Alan Doolittle School of Electrical and Computer Engineering
Georgia Institute of Technology
As with all of these lecture slides, I am indebted to Dr. Dieter Schroder from Arizona State University for his generous contributions and freely given resources. Most of (>80%) the figures/slides in this lecture came from Dieter. Some of these figures are copyrighted and can be found within the class text, Semiconductor Device and Materials Characterization. Every serious microelectronics student should have a copy of this book!
ECE 4813 Dr. Alan Doolittle
Doping Plasma Resonance
Free Carrier Absorption Infrared Spectroscopy
Photoluminescence Hall Measurements Magnetoresistance
Time of Flight
ECE 4813 Dr. Alan Doolittle
Plasma Resonance and Reflectivity Minimum
Electron/hole plasmas are ensembles of free carriers that can, at certain frequencies, oscillate in concert as a group.
Such a plasma resonance exists whose frequency / wavelength (ν=c/λ) is determined by the free carrier density.
Good for p (or n) > ~1018 cm-3 Since plasma resonances are hard to detect in practice,
most of the time, free carrier densities are determined by empirical reflectivity minimums…
… or free carrier absorption
nmK
qc Os
plasma
*2 επλ =
( )Cplasma BAn += λ
ECE 4813 Dr. Alan Doolittle
Free Carrier Absorption Free carrier absorption occurs within the conduction or
valance band (not between). For example a conduction electron is absorbs an IR photon and is promoted
into a higher energy state still inside the conduction band
In practice, empirical fitting is used
Generally measured using Fourier Transmission Infrared (FTIR)
lattice and ionized impurity scattering dominate the mobility
log T
log µ
µi
µlµ
IncreasingNi
iil N
TT5.1
5.1 ; == − µµ 10
100
1000
1014 1016 1018 1020 1022
Mob
ility
(cm
2 /V·
s)
Doping Density (cm-3)
µn
µp
Silicon
100
1000
1014 1015 1016 1017 1018 1019
µ n (c
m2 /V
·s)
ND (cm-3)
T=200 K250 K
300 K350 K
400 K450 K 500 K
n-Silicon
ECE 4813 Dr. Alan Doolittle
“Simple” Hall Effect Hall effect is commonly used during the development of new
semiconductor material
Resistivity, carrier concentration AND mobility can all be determined simultaneously
Lorentz Force – deflection of free carriers by an applied magnetic field
Temperature dependent Hall is very powerful and can elucidate scattering mechanisms (plotting mobility vs Ta), and determine dopant activation energies Compensated Dopant Freeze out regime – Arrhenius slope results in EA Uncompensated Dopant Freeze out regime – Arrhenius slope results in EA/2 At moderate temperatures, p~ (NA- ND) At elevated temperatures, p ~ ni
ECE 4813 Dr. Alan Doolittle
“Simple” Hall Effect
sVρ
VH
Id
w
B
Iε
θ
xy
z
)( BvqF
×+= ε
qwdpBIBv xy ==ε
∫ ∫∫ =−=−==0 0
0 w wyHV
qdpBIdy
qwdpBIdyVdVH ε
BIdVR H
H =
xx qwdpvqApvI ==
Hall coefficient [m3/C or cm3/C]
Lorentz Force
Hall voltage
Voltage induced by current I
In steady state, the magnetic force is balanced by the induced electric field
ECE 4813 Dr. Alan Doolittle
“Simple” Hall Effect Resistivity is simply found from the voltage drop
along the length (no magnetic field),
Carrier density
Mobility
IV
sdw ρρ =
(r ~ 1 - 2, Hall scattering factor)
HH qRrn
qRrp −== ;
pHH
H
H
H
pp
H
rRRR
rpq
qRrp
µσµ
µµσµ
==
===
=
ECE 4813 Dr. Alan Doolittle
Detailed Hall Effect The Hall Coefficient for both electrons and holes
present in the same material is in general:
( ) ( )
( ) ( )
−+
+
−+
−
=22
2
2
2
npBnpq
npBnp
rR
np
n
np
n
H
µµµ
µµµ
ECE 4813 Dr. Alan Doolittle
Detailed Hall Effect The Hall Coefficient is in general:
At low fields (B<<1/µn)…
And at high fields (B>>1/µn)…
( ) ( )
( ) ( )
−+
+
−+
−
=22
2
2
2
npBnpq
npBnp
rR
np
n
np
n
H
µµµ
µµµ
qnror
qpr
npq
npr
R
p
n
p
n
H −≈
+
−
= 2
2
µµ
µµ
( )npqrRH −
=
collisionsbetween mean time theis τwhere2
2
τ
τ=r
ECE 4813 Dr. Alan Doolittle
Two Layer Hall Effect Sometimes, a semiconductor has two different
conduction layers (surface inversion, fermi-level pinning, substrate layers, n-p junctions or p+/n or n+/n layers)
The Hall coefficient is then a weighted sum of both layers and can be either positive or negative leading to confusion (shown for the low B field limit):
21
22
11
222
2
211
1
tttandtt
ttwhere
ttR
ttRR
total
totaltotal
totalH
totalHH
+=
+
=
+
=
σσσ
σσ
σσ
ECE 4813 Dr. Alan Doolittle
Two Layer Hall Effect (more detail) The Hall coefficient is then a weighted sum of
both layers and can be either positive or negative leading to confusion:
( ) ( )[ ]( ) ( )
+++
+++=
21221
22
21
22211
21221
222
2112
2221
211
BtRtRttBtRtRRRtRtRtR
HH
HHHHHHtotalH
σσσσσσσσ
Low B Field
High B Field
21
22
11
222
2
211
1
tttandtt
ttwhere
ttR
ttRR
total
totaltotal
totalH
totalHH
+=
+
=
+
=
σσσ
σσ
σσ
+
=1221
21
tRtRtRRR
HH
totalHHH
ECE 4813 Dr. Alan Doolittle
Two Layer Hall Effect (more detail) The Hall coefficient is then a weighted sum of
both layers and can be either positive or negative leading to confusion (generally):
( ) ( )[ ]( ) ( )
+++
+++=
21221
22
21
22211
21221
222
2112
2221
211
BtRtRttBtRtRRRtRtRtR
HH
HHHHHHtotalH
σσσσσσσσ
ECE 4813 Dr. Alan Doolittle
Hall Effect Measurements Two approaches:
Hall Bar (5 or 6 contacts)
Van der Pauw configuration Based on Conformal mapping theory Contacts assumed point sources Uniform thickness Cannot contain isolated (interior) holes
1 4 2 3
(6) 5
1
4
2
3
ECE 4813 Dr. Alan Doolittle
Hall Effect Measurements Van der Pauw configuration
Measure resistivity first by “perimeter measurements”… Example: determine R12,34 where current goes in 1 and leaves 2 and voltage is measured between
terminals 3 and 4. Next determine R23,14 where current goes in 2 and leaves 3 and voltage is measured between terminals 1 and 4.
Use:
…where F is a symmetry term derived from conformal mapping theory F is determined from:
1
4
2
3
Differs some from your text. For details see http://www.nist.gov/pml/semiconductor/hall_resistivity.cfm
FRRt
+
=
2)2ln(14,2334,12πρ
14,23
34,12
)2ln(
1
2cosh
)2ln(11
RR
Rwhere
eFRR
r
F
r
r
=
=
+−
−
ECE 4813 Dr. Alan Doolittle
Hall Effect Measurements Van der Pauw configuration
Now measure the Hall voltage using “Crossing configurations” Example: Apply the magnetic field and determine V13,24P where current goes in 1 and leaves 3 and
voltage is measured between terminals 2 and 4. Next reverse the field and determine V13,24N again. To find the sheet concentration (#/cm2) use:
…where we have intentionally left out the proportionality constant In reality, 8 resistivity and eight hall voltage measurements are made to reduce contact related offset voltage errors resulting in an equation that is of the form:
See text for important sample geometry considerations (if ignored, significant error can result)
1
4
2
3
( )
+∝
NP VVqIB
24,1324,13
ρ
( )( ) ( ) ( ) ( )[ ]
−+−+−+−=
−
NPNPNPNP VVVVVVVVqIBx
13,2431,2413,4213,4242,3142,3124,1324,13
8108ρ
Differs some from your text. For details see http://www.nist.gov/pml/semiconductor/hall_resistivity.cfm