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3.225 1
Electronic Materials
Silicon Age: Communications
Computation
Automation
Defense
..
Factors:
Reproducibility/Reliability
Miniaturization
Functionality
Cost
..
H.L. Tuller-2001
Pervasive technology
3.225 2
What Features Distinguish Different Conductors?
Magnitude: agnitude!
metal; semiconductor; insulator
Carrier type:
electrons vs ions;
negative vs positive
Mechanism:
wave-like
activated hopping
Field Dependence:
Linear vs non-linear
H.L. Tuller-2001
varies by over 25 orders of m
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3.225 3
How Do We Arrive at Properties That We Want?
Crystal Structure:
diamond vs graphite
Composition
silicon vs germanium
Doping
n-Si:P vs p-Si:B
Microstructure
single vs polycrystalline
Processing/Annealing Conditions
Ga1+xAs vs Ga1-xAs
H.L. Tuller-2001
3.225 4
Interconnect
Resistor
Insulator
Non-ohmic device
diode, transistor
Thermistor
Piezoresistor
Chemoresistor
Photoconductor
Magnetoresistor
What is the Application?
H.L. Tuller-2001
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3.225 5
Origin of Conduction Range of Resistivity
Why?
E.A. Fitzgerald-1999
3.225 6
Response of Material to Applied Potential
I
V
e-V
I
Linear,
OhmicRectification,
Non-linear, Non-Ohmic
V=IR
V=f(I)
Metals show Ohmic behavior microscopic origin?
E.A. Fitzgerald-1999
R
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3.225 7
Microscopic Origin: Can we Predict Conductivity of Metals?
Drude model: Sea of electrons
all electrons are bound to ion atom cores except valence electrons
ignore cores
electrongas
E.A. Fitzgerald-1999
Schematic model of a crystal of sodium
metal.
From: Kittel, Introduction to Solid State Physics, 3rd
Ed., Wiley (1967) p. 198.
C.
3.225 8
Does this Microscopic Picture of Metals Give us Ohms Law?
F=-eE
E
F=ma
m(dv/dt)=-eE
v =-(eE/m)t
v,J,,I
t
t
E
No, Ohms law can not be only from electric force on electron!
Constant E gives ever-increasingv
E.A. Fitzgerald-1999
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3.225 9
Equation of Motion - Impact of Collisions
Assume: probability of collision in time dt = dt/ time varying field F(t)
v(t+dt) = (1- dt/) {v(t) +dv} = (1- dt/) {v(t) + (F(t)dt)/m}
v(t) + (F(t)dt)/m - v(t) dt/ (for small dt)
dv(t)/dt + v(t)/ = F(t)/m
Note: erm proportional to velocity corresponds to
frictional damping term
H.L. Tuller-2001
T
3.225 10
Hydrodynamic Representation of e- Motion
dp t
dt
p t F t F t
( ) ( )( ) ( ) ...= + + +
1Response (ma)
p=momentum=mv
Drag Driving Force Restoring Force...
dp t
dt
p teE
( ) ( )
Add a drag term, i.e. the electrons have many collisions during drift
1/ represents a viscosity in mechanical terms
E.A. Fitzgerald-1999
2
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3.225 11
In steady state,dp t
dt
( )=0
p t p et
( ) ( )=
1
p E =
p
t
-eE
If the environment has a lot of collisions,
mvavg
=-eE vavg
=-eE/m
=em
E.A. Fitzgerald-1999
E=Define v
Mean-free Time Between Collisions, Electron Mobility
e
3.225 12
vd
E
j = I/A
Adx
What is the Current Density ?
n (#/vol)
H.L. Tuller-2001
# electrons crossing plane in time dt = n(dxA) = n(vddtA)
# charges crossing plane per unit time and area = j
Ohms Law:
Dimensional analysis: (A/cm2)/(V/cm)=A/(V-cm)= (ohm-cm)-1 = Siemens/cm-(S/cm)
( )( ) ( EmnevnedtAedtAvnjdd
2
===( EjmneEj === 2
)
)
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3.225 13
Energy Dissipation - Joule Heating
Frictional damping term leads to energy losses:
Power absorbed by particle from force F:
P = W/t = (Fd)/t = Fv
Electron gas: P/vol= n(-eE)(-eE/m)= ne2E2/m = E2
= jE = (I/A)(V/l) = IV/vol
Total power absorbed: 2/R = I2R
How much current does a 100 W bulb draw?
I = 100W/115V = 0.87A
H.L. Tuller-2001
P = IV = V
3.225 14
Predicting Conductivity using Drude
ntheory from the periodic table (# valence e- and the crystal structure)
ntheory=AVZm/A,where AV is 6.023x1023 atoms/mole
m is the densityZ is the number of electrons per atom
A is the atomic weight
For metals, ntheory~1022 cm-3
If we assume that this is correct, we can extract
E.A. Fitzgerald-1999
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3.225 15
~10-14 sec for metals inDrude model
Extracting Typical for Metals
E.A. Fitzgerald-1999
3.225 16
Thermal Velocity
So far we have discussed drift velocity vD and scattering time related to the applied electric field
Thermal velocity vth is much greater than vD
kTmvth2
3
2
1 2=
m
kTvth
3=
Thermal velocity is much greater than drift velocity
x
x
xL=vD
E.A. Fitzgerald-1999
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3.225 17
Resistivity/Conductivity-- Pessimist vs Optimist
L
WI
V
t
R = L/Wt = L/A (hm-cm)
= 1/ (hm-cm)-1 (Siemens/cm)
(Test your dimensions: =E/j=ne)
Ohms/square Note, if L=W, then R= /t independentof magnitude of L and W. Useful for working with films of
thickness, t.R R R
H.L. Tuller-2001
R=V/I;
3.225 18
How to Make Resistance Measurements
Rs
Rc1Rc2
I
V
V/I = Rc1 + Rs + Rc2
I.s
>> Rc1
+ Rc2
; no problem
II. For Rs Rc1 + Rc2 ; major problem 4 probes
H.L. Tuller-2001
For R
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3.225 19
How to Make Resistance Measurements
Rs
Rc4Rc1
I
V14
v23
Rc2 Rc3
4 probe method: Essential feature - use of high impedance
voltmeter to measure V23 no current flows through Rc2& Rc3 therefore no IR contribution to V23
Rs(2-3) = v23 /I = -1 (d23/A) = (d23/A)(Note: -resistivity is inverse ofconductivity)
H.L. Tuller-2001
3.225 20
How to Make Resistance Measurements - Wafers
IV
d d
R
R+dR
x
j = I/2R2 ; V = IR = Id/A = jd
V23 = 2d (I/2R2 ) dR = (- I/ 2R) 2d = I/4d
d d
= (2d/I) V23 ; = (/ln2) V/I for d >>x
Si
H.L. Tuller-2001
Id
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3.225 21
Example: Conductivity Engineering
Objective: increase strength of Cu but keep conductivity high
v
m
e
m
ne
===
l2
Scattering length
connects scattering time
to microstructure
Dislocation
(edge)
l decreases, decreases, decreases
e-
E.A. Fitzgerald-1999
3.225 22
Can increase strength with second phase particles
As long as distance between second phase< l, conductivity marginally effected
Example: Conductivity Engineering
L
S
L+S
Sn Cu
L
X Cu
+L +L
+
Smicrostructure
Material not strengthened, conductivity decreases
dislocation
LL>l
Dislocation motion inhibited by second phase;
material strengthened; conductivity about the same
E.A. Fitzgerald-1999
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- - - - - - - - -
3.225 23
Scaling of Si CMOS includes conductivity engineering
One example: as devices shrink
vertical field increases
decreases due to increased scattering at SiO2/Si interface increased doping in channel need for electrostatic integrity: ionized
impurity scattering
SiO2
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3.225 25
Experimental Hall Results on Metals
Valence=1 metals look like
free-electron Drude metals
Valence=2 and 3, magnitude
and sign suggest problems
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3.225 1
Response of Free e- to AC Electric Fields
Microscopic picture
e-ti
OZ eEE =B=0 in conductor,
and )()( BFEFrrrr
>>tieeEtpdt
tdp = 0
)()(
tieptp = 0)(0
00 eE
ppi =
try
1
0
0 =ieE
p
>>1/, p out of phase with E
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3.225 3
Momentum represented in the complex plane
Response of e- to AC Electric Fields
real
imaginary
p
p (1/)
Instead of a complex momentum, we can go back to macroscopic
and create a complex J and
ieJtJ = 0)( 02
00
)1
(
E
im
ne
m
nepnevJ
===
m
ne
i
200 ,1
==
E.A. Fitzgerald-1999
3.225 4
Low frequency (1/)
electron has nearly 1 collision or less when
direction is changed
J imaginary and 90 degrees out of phase with
E, is imaginary
Response of e- to AC Electric Fields
Qualitatively:
1, electrons out of phase, electrons too slow, less interaction,transmission =r0 r=1
Hzcmx
cmxc 14
8
1014 10
105000
sec/103,sec,10 ==
E-fields with frequencies greater than visible light frequency expected to be
beyond influence of free electrons
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3.225 5
Need Maxwells equations
from experiments: Gauss, Faraday, Amperes laws
second term in Amperes is from the unification
electromagnetic waves!
Response of Light to Interaction with Material
SI Units (MKS)
MHB
PED
t
D
cJ
cHx
t
B
cEx
B
D
rrrrrr
rrr
rrrr
4
4
14
1
0
4
+=+=
+=
===
00
00
0
;
0
rr
HMHB
EPED
t
DJHx
t
BEx
B
D
===+=
=+=
+=
===
rrrrrrrr
rrr
rrrr
Gaussian Units (CGS)
E.A. Fitzgerald-1999
3.225 6
Waves in Materials
Non-magnetic material, =0 Polarization non-existent or swamped by free electrons, P=0
t
EJBx
t
BEx
+=
=r
rr
rr
000
t
BxExx
=r
r)(
2
2
000
2
000
2 ][
t
E
t
EE
t
EJ
tE
+
=
+
=
For a typical wave,
)()()(
)(
2
000
2
0
)(
0
rErEirE
erEeeEeEE titiriktrki
====
Wave Equation
0
2
22
1)(
)()()(
i
rEc
rE
+==
)(
)(
)(
2
22
0
c
kv
ck
eErE rik
===
=
E.A. Fitzgerald-1999
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3.225 7
Waves slow down in materials (depends on ())
Wavelength decreases (depends on ())
Frequency dependence in ()
Waves in Materials
)1(11)(
0
0
0
i
ii
+=+=
m
ne
i
i
p
p
0
22
2
2
1)(
=+=
Plasma Frequency
For>>>1, () goes to 1
For an excellent conductor (0 large), ignore 1, look at case for
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3.225 9
Waves in Materials
For a material with any 0, look at case for>>1
( ) 2
2
1
p= p, is positive, k=kr, wave propagates
R
p
E.A. Fitzgerald-1999
3.225 10
Success and Failure of Free e- Picture
Success
Metal conductivity
Hall effect valence=1
Skin Depth
Wiedmann-Franz law
Examples of Failure
Insulators, Semiconductors
Hall effect valence>1
Thermoelectric effect
Colors of metals
K/=thermal conduct./electrical conduct.~CT
23
1thermvvc=
m
Tkvnk
T
Ec bthermb
v
v
3;
2
3 2 ==
=
m
Tnk
m
Tknk bbb
2
2
33
2
3
3
1=
=
m
ne 2=
T
e
kb2
2
3
=
Therefore :
~C!Luck: cvreal=cvclass/100;
vreal2=vclass
2*10
0
E.A. Fitzgerald-1999
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3.225 11
Wiedmann-Franz Success
Exposed Failure when
cv and v2 are not both
in property
Thermoelectric Effect
TQE =e
nk
ne
nk
ne
cQ b
bv
23
2
3
3===Thermopower Q is
Thermopower is about 100 times too large!
E.A. Fitzgerald-1999
3.225 12
Waves in Vacuum
0, =J
00; ==
2
2
00
2
t
EE
= Wave EquationFor typical wave:
trikeEE =0 2;2 ==k
2
00
2 =k
(21
00
== k
For constant phase: t)
== ckvphase
( 2100
= c
Example:
Violet light ( = 7.5 x 1014 Hz) = c/ = 400 nmk=2/ = 1.57 x 107 m -1
= 2 = 4.71 x 1015 s-1
After Livingston
)(kx-
)
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3.225 13
Waves in Materials; Skin Depth
The skin depth is defined by
21
2
=
ik += 22Conductive materials( 21ik
(
ii +=+= 12
1 21
( ( xxtitkxi eeEeEE ==00
0 t
E
t
E
E
+
=
2
2
2
;
After Livingston
))
) )
3.225 14
Plasma Frequency
Remember: ik += 22
where ( 11
00 >>
=
ii
then
=2
2
2022 1
pk
where
212 m
nep Plasma Frequency
For > p; k is real number no attenuation!
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3.225 15
Compton, Planck, Einstein
light (xrays) can be particle-like
DeBroglie
matter can act like it has a wave-nature
Schrodinger, Born
Unification of wave-particle duality, Schrodinger
Equation
Wave-particle Duality: Electrons are notjustparticles
E. Fitzgerald-1999
3.225 16
Light is always quantized: Photoelectric effect (Einstein)
Photoelectric effect shows that E=h even outside the box
I,E,
e-
metal
block
Maximum
electron
energy,
Emax
c
Emax=h(-c)
!
For light with
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3.225 17
DeBroglie: Matter is Wave
His PhD thesis! =h/p also for matter
To verify, need very light matter (p small) so is large enough
Need small periodic structure on scale of to see if wave is there (diffraction)
Solution:electron diffraction from a crystal
N=2dsin
For small , ~/d, so must be on order ofd in order to measure easily
E. Fitzgerald-1999
3.225 18
must be able to represent everything from a particle to a wave (the twoextremes)
Unification: Wave-particle Duality
wave particle
( tkxiAe =
k and p known exactly
( txki
n
nnnea
=
=n to create a delta function in 2
generalized
( txki
n
nnnea =
E. Fitzgerald-1999
) )
)
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3.225 19
Quantum Mechanics - Wave Equation
Classical Hamiltonian
QM Operators
EzyxVm
p=+ ),,(
2
2
=i
ph
tiE
= h
1. and must be finite, continuous and single valued.2.
* real with dV* = probability of finding particle in volume dV.3. Average or expectation value of variable
=v
op dV *
tizyxV
m
=+ hh ),,(2
2
2
H.L. Tuller-2001
3.225 20
Time and Spatial Dependence of
Assume (x,y,z,t) separable into (x,y,z) and (t)
Applying separation of variables:
=
=+
tiV
m
1
2
22hh
= constant
Time-Dependent Equation:
( ) ( titi AeAet == h h=
Time-Independent Equation:
( 022
2 =+ Vmh
Solutions:n -eigenfunctions; n -eigenvalues
H.L. Tuller-2001
)
)
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3.225 21
Free Particle
One dimensional OV =
222
2 2k
m
dx
d==
hikxAe=
)(),( tkxiAetx = Momentum
kdxxi
px
hh =
= *
m
p
m
k
22
222
==hkp h= Crystal Momentum
H.L. Tuller-2001
3.225 22
Particle in Box
2
2 2;h
mkBeAe ikxikx =+=
Boundary Conditions:
0)()0( == d0)0( =+= BA BA =
0)()( == ikdikd eeAd 02 = ASinkd
d
n
k
=
...3,2,1=n
=
d
xnd
sin2 ...3,2,1=n
V
0=x dx =
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3.225 23
Particle in Box
zkykxkAzyxn 321 sinsinsin),,( =
2
3
2
2
2
1
2 kkkk ++= ;dnk ii = ...3,2,1=in
m
knnn
mdn
2)(
8
22
2
3
2
2
2
12
2hh
=++=n = Quantum numbers
Degeneracy
First excited state 112, 211, 121
Ground state E ; 1321 === nnn not zero!
H.L. Tuller-2001
3.225 24
Consequence of Electrons as Waves on Free Electron Model
Standing wave pictureTraveling wave picture
00
L
L
nk
e
ee
Lxx
ikx
Lxikikx
2
1
)()(
)(
===
+=+
Just having a boundary condition means that k and E are quasi-continuous,
i.e. for large L, they appear continuous but are discrete
E. Fitzgerald-1999
L
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3.225 25
Representation of E,k for 1-D Material
m
p
m
kE
22
222
==h
E
k
k=2/L
Quasi-continuousk
m
kE
m
k
dk
dE
=
=
2
2
h
h
states
electrons
EnEn-1
En+1m=+1/2,-1/2
All e- in box accounted for
EF
kF kF
Total number of electrons=N=2*2kF*L/2
E. Fitzgerald-1999
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3.225 1
Representation of E,k for 1-D Material
2
1
2
221)(
2
===
=
Em
k
m
LdE
dk
dk
dNEg
LkN F
hh
g(E)=density of states=number of electron states per energy per length
n, the electron density, the number of electrons per unitlength is determined by the crystal structure and valence
n determines the energy and velocity of the highestoccupied electron state at T=0
2or
222
nk
mEk
L
Nn F
FF ====h
m
k
dk
dE
mE
km
k
E2
22 2
;2
h
h
h
=
==
E. Fitzgerald-1999
3.225 2
Representation of E,k for 2-D Material
E(kx,ky)
kx
ky
m
kkE
yx
2
)( 222 +=h
E. Fitzgerald-1999
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3.225 3
Representation of E,k for 3-D Material
kx
ky
kz
(kx,ky,kz)
2/L
mkkk
E zyx2
)(2222 ++
=h
Fermi Surface or Fermi Sphere
kF
mk
v FF h=mk
E FF2
22h
=BF
FkE
T =
32
2)(
hmEm
Eg=( 3123 nkF =
E. Fitzgerald-1999
)
3.225 4
So how have material properties changed?
The Fermi velocity is much higher than
kT even at T=0! Pauli Exclusion raises
the energy of the electrons since only 2
e- allowed in each level
Only electrons near Fermi surface can
interact, i.e. absorb energy and
contribute to properties
TF~104K (Troom~10
2K),
EF~100Eclass, vF2~100vclass
2
E. Fitzgerald-1999
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3.225 5
Effect of Temperature (T>0): Coupled electronic-thermal properties in conductors
Electrons at the Fermi surface are able to increase energy: responsible forproperties
Fermi-Dirac distribution
NOT Bolltzmann distribution, in which any number of particles can occupy
each energy state/level
Originates from:
...N possible configurations
T=0 T>0
EF
1
1)(
+
= Tk
EE
b
F
e
f
If E-EF/kbT is
large (i.e. far fromEF) than Tk
EEb
F
ef)(
=
E. Fitzgerald-1999
3.225 6
Fermi-Dirac Distribution: the Fermi Surface when T>0
~EF
f(E)
1T=0
T>00.5
kbTkbT
EAll these e- not
perturbed by T
fBoltz
Boltzmann-like tail, for
the larger E-EF values
v
vT
Uc
=
Heat capacity of metal (which is ~ heat capacity of free e- in a metal):
([ ( ( )2~~~ TkEgTkEgTkNEU bFbFb U=total energy ofelectrons in system
TkEgT
Uc bF
v
v
2)(2 =
= Right dependence, very close to exact derivation
E. Fitzgerald-1999
) ] )
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3.225 7
Electrons in a Periodic Potential
Rigorous path: H=E
We already know effect: DeBroglie and electron diffraction
Unit cells in crystal lattice are 10-8 cm in size
Electron waves in solid are =h/p~10-8 cm in size
Certain wavelengths of valence electrons will diffract!
E. Fitzgerald-1999
3.225 8
Diffraction Picture of the Origin of Band Gaps
Start with 1-D crystal again
~a
a1-D
sin2dn = d=a,sin=1
an
kk
an
==
=2
2
Take lowest order, n=1, and
consider an incident valence
electron moving to the right
xai
oo
xai
ii
eak
eak
====
;
;
Reflected wave to left:
Total wave for electrons with diffracted wavelengths:
xaix
aoia
ois
oi
sin2
cos2
===+=
=
akkk oi2
==
E. Fitzgerald-1999
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3.225 9
Diffraction Picture of the Origin of Band Gaps
Probability Density=probability/volume of finding electron=||2
xa
xa
s
a
22
22
cos4
sin4
==
a
a
Only two solutions for a diffracted wave
Electron density on atomsElectron density off atomsNo other solutions possible at this wavelength: no free traveling wave
E. Fitzgerald-1999
3.225 10
Assume electrons with wave vectors (ks) far from diffractioncondition are still free and look like traveling waves and see
ion potential, U, as a weak background potential
Electrons near diffraction condition have only two possiblesolutions
electron densities between ions, E=Efree-U
electron densities on ions, E= Efree+U
Exact solution using H=E shows that E near diffractionconditions is also parabolic in k, E~k2
Nearly-Free Electron Model
E. Fitzgerald-1999
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3.225 11
Nearly-Free Electron Model (still 1-D crystal)
m
p
m
kE
22
222
==h
E
k
k=2/L
Quasi-continuous
km
kE
m
k
dk
dE
=
=
2
2
h
h
states
/a-/a 0
Eg=2UDiffraction,
k=n/a
Away from k=n/a,free electron curve
k=2/a=G=reciprocal lattice vector
Near k=n/a,band gaps form, strong
interaction of e- with
U on ions
E. Fitzgerald-1999
3.225 12
Electron Wave Functions in Periodic Lattice
Often called Bloch Electrons or Bloch Wavefunctions
E
k/a0
Away from Bragg condition, ~free electron
m
kEe
mU
mH ikxo
2;;
22
222
22
2hhh
=
+
=
Near Bragg condition, ~standing wave electron
( ) ( ) ( )xUExuGxGxxUUm
H ooo ==+
= ;sinorcos;2
22
h
Since both are solutions to the S.E., general wave is
( )xueikxlatticefree ==
termed Bloch functions
E. Fitzgerald-1999
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3.225 13
Block Theorem
If the potential on the lattice is U(r) (and therefore
U(r+R)=U(r)), then the wave solutions to the S.E. are a
plane wave with a periodic part u(r) that has the periodicity
of the lattice
( ) ( )( ) ( )Rruru
ruer rik
+=
=
Note the probability density spatial info is in u(r):
( ) )(*2* ruruo =An equivalent way of writing the Bloch theorem in terms of:
((
(( ( )
( ( )reRre
r
eRrueRrRik
rik
RrikRrik
=+
=+=+
++
E. Fitzgerald-1999
3.225 14
Reduced-Zone Scheme
Only show k=+-/a since all solutions represented there
/a/a
E. Fitzgerald-1999
))
))
)
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3.225 15
Real Band Structures
GaAs: Very close to what we have derived in the nearly free electron model Conduction band minimum at k=0: Direct Band Gap
E. Fitzgerald-1999
3.225 16
Review of H atom
( ) ( ) ( )
EH
rR
=
=
Do separation of variables; each variable gives a separation constant
separation yields ml givesr gives n
l
After solving, the energy E is a function of n
( 222242
6.13
24 n
eV
n
eZE
o
=
=
h
ml and and give the shape(i.e. orbital shape)
l
The relationship between the separation constants (and therefore the quantum numbers are:)
n=1,2,3,
=0,1,2,,n-1ml=- , - +1,,0,, ,
(ms=+ or - 1/2)
l l l l
0
-13.6eV
U(r)
E. Fitzgerald-1999
)
in
-1
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3.225 17
Relationship between Quantum Numbers
s p d
Origin of the periodic table
s s p
E. Fitzgerald-1999
3.225 18
Bonding and Hybridization
Energy level spacing decreases as atoms are added
Energy is lowered as bonding distance decreases
All levels have E vs. R curves: as bonding distance decreases, ion core
repulsion eventually increases E
E
R
s
p
Debye-Huckel
hybridizationNFE picture,
semiconductors
E. Fitzgerald-1999
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3.225 1
Properties of non-free electrons
Electrons near the diffraction condition are not
approximated as free Their properties can still be viewed as free e- if an
effective mass m* is used
/a/a
2
2
2*
*
22
2
k
Em
m
kE
ec
ec
=
=
h
h
2
2
2*
*
22
2
k
Em
m
kE
ev
ev
=
=
h
h
Note: These
electrons have
negative mass!
m
kE
2
22h
=
E. Fitzgerald-1999
3.225 2
Band Gap Energy Trends
H.L. Tuller-2001
Note Trends: 1. As descend column, MP decreases as does Eg while ao increases.
2. As move from IV to III-V to II-VI compounds become more ionic,
MP and Eg increase while ao tends to decrease
II B III IV V VI
B N O
Al Si P S
Zn Ga Ge As Se
Cd In Sn Sb Te
MP (K) Eg (eV) aoA6 / 10 3.56 / 3.16
1685 / 1770 1.1 / 3 5.42 / 5.46
1231 / 1510 / ? 0.72 / 1.35/ ? 5.66 / 5.65 / ?
508 / 798 / ? 0.08 /0.18 / 1.45 6.45 / 6.09 / ?
IV / III-V / II-VI*
Fill in as many of the question marks as you can.
C
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3.225 3
Trends in III-V and II-VI Compounds
BandBand
GapGap
((eVeV))
Lattice Constant (A)Lattice Constant (A)
SiGeSiGe
AlloysAlloys
Larger atoms, weaker bonds, smaller U, smaller Eg, higher, more costly!
E. Fitzgerald-1999
3.225 4
Energy Gap and Mobility Trends
Material
GaN
AlAs
GaP
GaAs
InP
InAs
InSb
Eg(eV)K
3.39
2.3
2.4
1.53
1.41
0.43
0.23
n(cm2/Vs)
150
180
2,100
16,000
44,000
120,000
1,000,000
Remember that:*m
e= and 222* 11 kEhm =
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3.225 5
Metals and Insulators
EF in mid-band area: free e-, metallic
EF near band edge
EF in or near kT of band edge:semimetal
EF in gap:semiconductor
EF in very large gap, insulator
E. Fitzgerald-1999
3.225 6
Semiconductors
Intermediate magnitude band gap enables
free carrier generation by three mechanisms
photon absorption
thermal
impurity (i.e. doping)
Carriers that make it to the next band are
free carrier- like with mass, m*
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3.225 7
Semiconductors: Photon Absorption
When Elight=h>Eg, an electron can be promotedfrom the valence band to the conduction band
Ec near band gap
Ev near band gap
E
k
E=h
Creates a hole in the valence band
E. Fitzgerald-1999
3.225 8
Holes and Electrons
Instead of tracking electrons in valence band, more convenient to track missing
electrons, or holes
Also removes problem with negative electron mass: since hole energy increases as holes
sink, the mass of the hole is positive as long as it has a positive charge
Decreasing electron energy
Decreasing electron energyDecreasing hole energy
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3.225 9
Conductivity of Semiconductors
Need to include both electrons and holes in the conductivity expression
*
2
*
2
h
h
e
ehe
m
pe
m
nepene
=+=
p is analogous to n for holes, and so are and m*
Note that in both photon stimulated promotion as well as thermal
promotion, an equal number of holes and electrons are produced, i.e. n=p
E. Fitzgerald-1999
3.225 10
Thermal Promotion of Carriers
We have already developed how electrons are promoted in energy with T: Fermi-Dirac distribution
Just need to fold this into picture with a band-gap
EF
f(E)
1
E
g(E)
gc(E)~E1/2 in 3-D
gv(E)Despite gap, at non-zero
temperatures, there is some
possibility of carriers getting
into the conduction band (andcreating holes in the valence
band)
Eg
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+
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3.225 11
Density of Thermally Promoted of Carriers
=
cE
dEEgEfn )()(
Density of electron states per volume per dE
Fraction of states occupied at a particular temperature
Number of electrons per
volume in conductionband
( dEeEEemn TkEE
g
Tk
E
e b
g
b
F
= 21
2
3
2
*
2
2
2
1
h
Since2
0
2
1 =
dxex x , then TkETkEbe bgbF eeTkmn
= 23
2
*
22
hNC
((
TkEEe
e
Ef bFTk
EE
Tk
EEb
F
b
F>>
+= )(when
1
1)(
Tk
EE
Cb
gF
eNn
=
( 2123
2
*
2
2
2
1)( g
ec EE
mEg
=h
E. Fitzgerald-1999
)
))
)
3.225 12
A similar derivation can be done for holes, except the density of states
for holes is used
Even though we know that n=p, we will derive a separate expression
anyway since it will be useful in deriving other expressions
Density of Thermally Promoted of Carriers
( 2123
2
*
2
2
2
1)( E
mEg hv
=h )(1where,)()(
0
EffdEEgEfp hvh ==
Tk
E
bh b
F
eTkm
p
= 23
2
*
22
h
Tk
E
vb
F
eNp
=
E. Fitzgerald-1999
)
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3.225 13
Thermal Promotion
Because electron-hole pairs are generated, the Fermi level is
approximately in the middle of the band gap
The law of mass action describes the electron and hole
populations, since the total number of electron states is fixed in
the system
+==*
*
ln4
3
2gives
e
hb
g
Fm
mTk
EEpn
Since me* and mh* are close and in the ln term, the
Fermi level sits about in the center of the band gap
( TkE
veb
ib
g
emmTknnp 243
**2
3
222or
==
h
E. Fitzgerald-1999
)
3.225 14
Law of Mass Action for Carrier Promotion
( TkEhebi b gemmTknpn
== 23**3
2
2
24
h TkE
VCib
g
eNNn
=2;
Note that re-arranging the right equation leads to an expression similar to a chemicalreaction, where Eg is the barrier.
NCNV is the density of the reactants, and n and p are the products.
+ heNN gEVC
VC
iTk
E
VCNN
ne
NN
npb
g 2
==
Thus, a method of changing the electron or hole population without increasing the population
of the other carrier will lead to a dominant carrier type in the material.
Photon absorption and thermal excitation produce only pairs of carriers: intrinsicsemiconductor.
Increasing one carrier concentration without the other can only be achieved with impurities,also called doping: extrinsic semiconductors.
E. Fitzgerald-1999
)
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3.225 15
Intrinsic Semiconductors
Conductivity at any temperature is determined mostly by the size of the band gap
All intrinsic semiconductors are insulating at very low temperatures
*
2
*
2
h
h
e
ehe
m
pe
m
nepene
=+=Recall:
( TkEhei bgeen 2int
+= For Si, Eg=1.1eV, and let e and hbe approximately equal at 1000cm
2/V-sec (very good Si!).
~1010cm-3*1.602x10-19*1000cm2/V-sec=1.6x10-6 S/m, or a resistivity of about 106 ohm-m max.
One important note: No matter how pure Si is, the material will always be apoor insulator at room T.
As more analog wireless applications are brought on Si, this is a major issuefor system-on-chip applications.
This can be a
measurement
for Eg
E. Fitzgerald-1999
+
)
3.225 16
Extrinsic Semiconductors
Adding correct impurities can lead to controlled domination of one carrier type
n-type is dominated by electrons
p-type if dominated by holes
Adding other impurities can degrade electrical properties
Impurities with close electronic
structure to hostImpurities with very different
electronic structure to hostisoelectronic hydrogenic
xx
xx
Ge
Si
Px
xx
xx
xx
xAu
Si
deep level
Ec
Ev
Ec
Ev
Ec
Ev
ED EDEEP
-+
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3.225 17
Hydrogenic Model
For hydrogenic donors or acceptors, we can think of the electron or hole, respectively, as
an orbiting electron around a net fixed charge
We can estimate the energy to free the carrier into the conduction band or valence band
by using a modified expression for the energy of an electron in the H atom
2222
4 6.13
8 nnh
meE
o
n ==
2
*
22222
4*
222
4 16.131
88
22
mm
nnh
em
nh
meE
ro
ee
o
nr = = =
(in eV)
Thus, for the ground state n=1, we can see already that since is on the order of 10, thebinding energy of the carrier to the center is
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3.225 19
Expected Temperature Behavior of Doped Material (Example:n-type)
3 temperature regimes
ln(n)
1/T
Intrinsic ExtrinsicFreeze-out
Eg/2kb
Eb/kb
E. Fitzgerald-1999
3.225 20
Contrasting Semiconductor and Metal Conductivity
Semiconductors
changes in n(T) can dominate over
as T increases, conductivity increases
Metals
n fixed
as T increases, decreases, and conductivity decreases
=nem
2
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3.225 21
General Interpretation of
Metals and majority carriers in semiconductors
is the scattering length
Phonons (lattice vibrations), impurities, dislocations,
and grain boundaries can decrease
...11111
++++=gbdislimpurphonon
1
1
===
iii
iithth
ii
Nl
Nvv
l
where is the cross-section of the scatterer, N is the
number of scatterers per volume, and l is the average
distance before collisions
The mechanism that will tend to dominate the scattering will be the mechanism with the
shortest l (most numerous), unless there is a large difference in the cross-sections
Example: Si transistor, phonon dominates even though impurgets worse with scaling.
E. Fitzgerald-1999
3.225 22
Estimate of T dependence of conductivity
~l for metals
~l/vth for semiconductors
First need to estimate l=1/N
2
1
x
Nl
ion
ionion
ph
=
x=0
++
=dx
dxxx
*
2*
2 Use for harmonic oscillator, get:
1
2
==
kTe
Exk
hh
Average energy of harmonic oscillator
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3.225 23
Estimate of T dependence of conductivity
1
1
2
==
==
T
kT
e
kE
k
e
Exk
h
hh
Therefore, is proportional to T if Tlarge compared to :
TNvxNvv
l
Txl
Tx
Te
ionFionFF
cond
T
11
111
1
2
2
2
==
+
For a metal:
For a semiconductor, remember that the carriers at the band edges are classical-like:
2
3
2
1
*
1
3
== T
T
T
m
kT
l
v
l
th
23*
= T
m
e
E. Fitzgerald-1999
3.225 24
Example: Electron Mobility in Ge
~T-3/2 if phonon dominated
(T-1/2 from vth, T-1 from x-section
At higher doping, the
ionized donors are the
dominate scattering
mechanism
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3.225 1
Minority Carrier Lifetimes,
Minority carriers (e.g. electrons (minority carrier) in p-type material with
majority holes
is the time to recombination: recombination time
means for system to return to equilibrium after perturbation, e.g. by
illumination
Ec
Ev
, l
Recombination
x
E
E. Fitzgerald-1999
Generation
Deep levels in semiconductors act as carrier traps and/or enhanced
recombination sites
Ec
Ev
Recombination through deep levelEdeep
3.225 2
Generation and Recombination
Generation
photon-induced or thermally induced, G=#carriers/vol.-sec
e.g. g = P/h
Go is the equilibrium generation rate
Recombination
R=# carriers/vol.-sec
Ro is the equilibrium recombination rate, balanced by Go
Net change in carrier density:
dn/dt = G - R = G - (n - n 0) / = G - n /
Under steady state illumination: dn/dt = 0
np(0) = np0 + G
fter turning off illumination:
np(0) = np0 + G e-t/
H.L. Tuller, 2001
g
t
np(t)
np0
np(0)
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3.225 3
Key Processes: Drift and Diffusion
Electric Field: Drift
Concentration Gradient: Diffusion
EenJAenvIEepJAepvIeedehhdh
====
;
;
neDJpeDJ
eehh
==
neDEenJ peDEepJ
eeeTOThhhTOT+= =
E. Fitzgerald-1999
3.225 4
Electrochemical Potential
qzjjj +=
jjj ckTln0 += =
xqzj j
jj
j
=
xc
qDzx jjjj
=
Note: Under equilibrium conditions0=
xj
Electrochemical Potential EF
Chemical Potential
Electrostatic Potential
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3.225 5
Continuity Equations
For a given volume, change in carrier concentration in time is related to J
GRTOT
GRTOT
GRdiffdrift
tp
tp
Jet
ptn
tn
Jet
ntn
tn
tn
tn
tn
+
=
+
=
+
+
=
1
1
1-D,
GRxp
DxE
ptp
GRxn
DxE
ntn
hh
ee
+
+
=
+
+
=
2
2
2
2
E. Fitzgerald-1999
3.225 6
Minority Carrier Diffusion Equations
In many devices, carrier action outside E-field controls properties--> minority
carrier devices
Only diffusion in these regions
e
h
h
e
nR
pR
GRxp
Dtp
GRxn
Dtn
=
=
+
=
+
=
type,-pin
type,-nin
2
2
2
2 Assuming low-level injection,
tn
tn
tn
tn o
+
=
therefore
materialtype-nin
materialtype-pinG
2
2
2
2
Gp
xp
Dtp
nxn
Dtn
hh
ee
+
=
+
=
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3.225 7
Use of Minority Carrier Diffusion Equations
Example: Light shining on a surface of a semiconductor
h
G at x=0 (assume infinite
absorption coefficient to simplify
example)
Gp
xp
Dtp
hh+
=
22
n-type
p(x)? 0
Steady state solution
=0 in bulk
x
hh
hh
axaxaxax
axaxhh
DaD
BeAeBeaAeaBeAep
Dp
xp
1
22
2
2
=
+=+
+=
=
try
Now use boundary conditions of the problem:
hhDx
BepA
px
==
==0
0,@ Units of length:minority carrier
diffusion length, Lh
hLx
h
hh
eGpGB Gpx
== ==
,0@
p
x
Gh
xp
eDJ hh
=
E. Fitzgerald-1999
3.225 8
Semiconductor Electronics
Single crystalline - largely Si
some III - V compounds
Dominated by many nearly identical, highly engineered junctions
DRAMS (today) 109 transistors
Microprocessors (2002) 108
transistors
Total 1018 yr 106/person/day
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3.225 9
Junction Fabrication Processes
H.L. Tuller, 2001
3.225 10
CMOS Devices
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3.225 11
The p-n Junction (The Diode)
Note that dopants move the fermi energy from mid-gap towards either thevalence band edge (p-type) or the conduction band edge (n-type).
p-type material in equilibrium n-type material in equilibrium
p~Nan~Nd
n~ni2/Na
p~ni2/Nd
+=Cd
bgFNN
TkEE ln
=Va
bFNN
TkE lnEc
Ec
EFEv
EF
Ev
What happens when you join these together?
E. Fitzgerald-1999
3.225 12
-
--
-+++
+
Holes diffuse
Electrons diffuse
+++
+-
--
-
-
--
-+++
+-
--
-+++
+
An electric field forms due to the fixed nuclei in the lattice from the dopants
Therefore, a steady-state balance is achieved where diffusive
flux of the carriers is balanced by the drift flux
E
Drift and Diffusion
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3.225 13
Joining p and n
EcEF
Ev
pn
Carriers flow under driving force of diffusion until EF is flat
-
--
-+++
+
Holes diffuse
Electrons diffuse
E. Fitzgerald-1999
3.225 14
-
--
-+++
+-
--
-+++
+
W: depletion or space charge width
Metallurgical junction
E
VVbi
dxx
E = )(
dxxEV = )(
pand xNxN =
xp xn
)(
2
ada
dbiorp
NNN
N
e
Vx
+=
)(
2
add
abiorn
NNN
N
e
Vx
+=
ad
dabior
NN
NN
e
VW
+=
2
Space Charge, Electric Field and Potential
E. Fitzgerald-1999
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What is the built-in voltage Vbi?
EcEF
Ev
p n
eVbi=EFn-EFp
=
=
dV
ib
V
nbFn
NN
nTk
N
pTkE
2
lnln
=
=
V
ab
V
bFpN
NTk
N
pTkE lnln
=
2ln
i
dabbi
n
NN
e
TkV
We can also re-write these to show that eVbi is the barrier to minority carrier injection:
Tk
eV
npb
bi
enn
=Tk
eV
pnb
bi
epp
=
nn
np
pn
pp
eVbi
eVbi
E. Fitzgerald-1999
Qualitative Effect of Bias
Applying a potential to the ends of a diode does NOT increase current through
drift
The applied voltage upsets the steady-state balance between drift and
diffusion, which can unleash the flow of diffusion current
Minority carrier device
EcEF
Ev
nn
np
pn
pp
eVbi
eVbi
Tk
VVe
npb
abi
enn
)(
= TkVVe
pnb
abi
epp
)(
=
+eVa
-eVa
E. Fitzgerald-1999
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Current Flow - Recombination, Generation
H.L. Tuller, 2001
Forward bias (+ to p, - to n) decreases depletion region, increases diffusion
current exponentially
Reverse bias (- to p, + to n) increases depletion region, and no current flows
ideally
EcEF
Ev
nn
np
pn
pp
eVbi-e|Va|
Qualitative Effect of Bias
Ec
EF
Ev
nn
np
pn
pp
eVbi+e|Va|
eVbi-e|Va|
eVbi+e|Va|
Forward Bias Reverse Bias
+ -Va
=
+= 11
22Tk
qV
o
Tk
qV
d
i
h
h
a
i
e
e b
a
b
a
eJeN
n
L
D
N
n
L
DqJ
TkD bi=
iii DL =
V
I
Linear,
OhmicRectification,
Non-linear, Non-Ohmic
V=IR
V=f(I)
Solve minority
carrier diffusion
equations on each
side and determine
J at depletion edge
E. Fitzgerald-1999
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Devices
Solar Cell/DetectorReverse Bias/Zero Bias
Jedrift
Ec
EF
Jhdrift Ev
LED/Laser
JediffEc Laser
EF population inversionEv
reflectors for cavityJhdiff
E. Fitzgerald-1999
Potential Wells - Heterojunction Lasers
Energy bands of a light-emitting diode under forward bias for a double
heterojunction AlGaAs-GaAs-AlGaAs structure.
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Transistors
Bipolar (npn)
EcEF
Ev
emitter
base
collectorJ
diffJ
drift
Barrier, controlled by VEB
VEB VBC
base
emitter
collector
E. Fitzgerald-1999
Field Effect
H.L. Tuller, 2001
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Transistors
FET sourcegate
drain
n p
x
n
x=metal is a MESFETx=metal/poly Si/oxide is a MOSFET
CMOS
E. Fitzgerald-1999
Polycrystalline Solar Cells
Local field enhances minority carrier capture reducedminority carrier lifetime
majority carriers experience potential barrier increasedresistivity; reduced effective mobility
boundaries intersecting p-n junction provide shorting paths increase Io, decrease Voc.
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Effect of Traps (Defects) on Bands
Trapping (Fermi level in defect) creates depleted regions around defect
+=
C
dbgF
N
NTkEE ln
EFposition in semiconductor away from
traps in n-type material
EFpulled to mid-gap in defect/trap area
Ec
EF
Ev
EFpulled to trap level in defect
Etrap
Depleted regions; internal electric field
Edonor
E. Fitzgerald-1999
Other Means to Create Internal Potentials:
Different semiconductor materials have different band gaps and electron
affinity/work functions
Internal fields from doping p-n must be superimposed on these effects:
Poisson Solver (dE/dx=V=/)
EF
Vacuum level
12
Eg1Eg2
Thin films
Substrate
Potential barriers for holes
and electrons can be created
inside the material
Heterojunctions
E. Fitzgerald-1999
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Artificially Modulated Structures
H.L. Tuller, 2001
Quantum Wells
EC
hn=3
n=2n=1
EVL
If we approximate well as having infinite potential boundaries:
k=n for standing waves in the potential wellL
h
2k2 h2n2 We can modify electronicE=
2m* =8m*L2 transitions through quantum wells
E. Fitzgerald-1999
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Photodetectors/Solar Cells
E-h pairs generated by photons with energy
h Egare separated by the built-in potential gradient at the p-n junction.
The current voltage characteristics are given by
I=Io[exp(qV kT)1]Ip
where Ip is the photo-induced reverse current.
Junctions/Functions
H.L. Tuller, 2001
Junction Function ApplicationP/n
Metal/semiconductor
Injection/diffusion/collection
Blocking (reverse bias)
p-n rectifier, switch
p-n-p transistor
Acceleration/breakdown
Tunneling
Avalanche and tunnel diodes
Injection/confinement/recombination LED, injection laser
Generation/separation Solar cell, photodiode
Separation/confinement High electron mobiity devices
Quantum devices
M/I/S Inversion/depletion/accumulation MOSFET
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- -
3.225 1
The Capacitor
dA
CAQd
VCQ
AQd
EdxV
AQt
dxE
E
oo
o
d
d
oo
d
td o
o
===
==
===
=
2
2
2
2
+V
++
++++
--
- -I=0 always in
capacitor
E
V
td/2 d/2
E. Fitzgerald-1999
3.225 2
The Capacitor
The air-gap can store energy!
If we can move charge temporarily without current flow, can store even more
Bound charge around ion cores in a material can lead to dielectric properties
Two kinds of charge can create plate
charge:
surface charge
dipole polarization in the volume
Gauss law can not tell the difference
(only depends on charge per unit area)
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3.225 3
Material Polarization
+=+==
=+=
11
EP
EPED
or
oro
P is the Polarization
D is the Electric flux density or the Dielectric
displacement
is the dielectric or electric susceptibility
++
+ +++
+
------+ +
+++
- ---
-
---
EP
dA
C or=
All detail of material response is in rand therefore P
E. Fitzgerald-1999
3.225 4
Origin of Polarization
We are interested in the true dipoles creating polarization in materials (not
surface effect)
As with the free electrons, what is the response of these various dipole
mechanisms to various E-field frequencies?
When do we have to worry about controlling
molecular polarization (molecule may have non-uniform electron density)
ionic polarization (E-field may distort ion positions and temporarily create dipoles)
electronic polarization (bound electrons around ion cores could distort and lead to
polarization)
Except for the electronic polarization, we might expect the other mechanisms
to operate at lower frequencies, since the units are much more massive What are the applications that use waves in materials for frequencies below the
visible?
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3.225 5
Application for Different E-M Frequencies
Methods of detecting
these frequencies
Cell phones
=14-33cmDBS (TV)
=2.5cm
Other satellite, 10-50GHz
=3cm-6mm (mm wave)Fiber optics
=1.3-1.55m
MMIC, pronounced mimic
mm wave ICs
In communications, many E-M waves travel in insulating materials:
What is the response of the material (r) to these waves?
E. Fitzgerald-1999
3.225 6
Wave Eqn. with Insulating Material and Polarization
()( 0 EPED
tE
BxtPE
JBxtD
JHxtB
Exinsulatingononmag
rrrrr
rrrrrrrr
rr
=+=
=
++=
+=
=
2
2
22
2
00
2
tE
ctE
E rr
=
=
knc
kck
cc
rErE
erEeeEeEE
opticalr
r
r
titiriktrki
==
====
22
2
2
22
0
)(
0
)()(
)(
So polarization slows down the
velocity of the wave in the
material
E. Fitzgerald-1999
)
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3.225 7
Compare Optical (index of refraction) and Electrical Measurements of
Material Optical, n2
Electrical, diamond 5.66 5.68
NaCl 2.25 5.9
H2O 1.77 80.4
Only electronic polarization
Electronic and ionic polarisation
Electronic, ionic, and
molecular polarisation
Polarization that is active depends on material and frequency
E. Fitzgerald-1999
3.225 8
Microscopic Frequency Response of Materials
Bound charge can create dipole through charge displacement.
Hydrodynamic equation (Newtonian representation) will now have a
restoring force.
Review of dipole physics:
- +dr
dqp rr=Dipole moment:+q-q
prApplied E-field rotates dipole to align with field:
Exp rrr=TorquecosEpEpU rrrr ==Potential Energy
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3.225 9
For a material with many dipoles:
Microscopic Frequency Response of Materials
)( EpENpNP rrrrr ==(polarization=(#/vol)*dipole polarization)
=polarizability
0
so,
NEPo
== rr
Ep rr = Actually works well only for low density of dipoles, i.e. gases: little screeningFor solids where there can be a high density: local field
Eext
Eloc
For a spherical volume inside (theory of local field),
oextloc PEE 3
rrr
+=
E. Fitzgerald-1999
=
3.225 10
We now need to derive a new relationship between the dielectric constant and
the polarizability
Microscopic Frequency Response of Materials
+=
=+==
3
2 rextloc
extoextorextoextor
EEEEP
PEED
Plugging into P=NEloc:
(( ( 2
3
1
3
2
+=+
=ror
extrextoextorN
ENEE
Clausius-Mosotti Relation:oor
r N
332
1==
+
Where v is the volume per dipole (1/N)
Macro Micro
E. Fitzgerald-1999
)) )
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3.225 11
Different Types of Polarizability
Atomic or electronic,e
Displacement or ionic, i
Orientational or dipolar, o
Highest natural frequency
Lowest natural frequency
Lightest mass
Heaviest mass
oie +=ti
oeEE =As with free e-, we want to look at the time dependence of the E-field:KxeE
txm
tx
m
=
22
Response Drag Driving Force
Restoring Force
(mK
meE
mKmeE
xKxeExm
exxKxeExm
o
ooo
o
ooo
tio
=
=
==
==
222
2 )(
&&
So lighter mass will
have a higher critical
frequency
E. Fitzgerald-1999
+
)
3.225 12
Classical Model for Electronic Polarizability
Electron shell around atom is attached to nucleus via springs
+
Er
+
Er
prK K
r
tiolocii erreEZKrrmZ == assume,&&
Zi electrons,
mass Zim
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3.225 13
Electronic Polarizability
=i
oo
mZKmeE
r2
2
2
,oe
ieoe
meZ
=>
( 222
oei
em
eZ
=
( 222
oeoo
io EE
meZ
p
=
=
( 22 ;ioeoe
oo
mZK
meEr
=
=
;ti
oi epperZqdp === If no Clausius-Mosotti,( 222
2
11 nm
eNZNoeo
ioe
r =+=+=
r
oe
1
( 22
1oeo
im
eNZ+
E. Fitzgerald-1999
)
)
)
)
)
3.225 14
QM Electronic Polarizability
At the atomic electron level, QM expected: electron waves
QM gives same answer qualitatively
QM exact answer very difficult: many-bodied problem
( ) h 01102210
10
2
;EEf
me
e
==
E1
E0
f10 is the oscillator strength of the transition (1 couples to oby E-field)
For an atom with multiple electrons in multiple levels:
( ) h 000 2210
02
;EE
jf
me j
jjj
e
==
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3.225 15
Ionic Polarizability
Problem reduces to one similar to the electronic polarizability Critical frequency will be less than electronic since ions are more massive
The restoring force between ion positions is the interatomic potential
E(R)
R
Nuclei repulsion
Electron bonding in between ions
Parabolic at bottom near Ro
)(
2
)( 2
o
o
RRkR
EF
RRkE
=
=
=
Ro - +
klijklij CkxF ==
E. Fitzgerald-1999
3.225 16
Ionic Polarizability
- +
Eloc
+-
pu+u-
2 coupled differential eqns1 for + ions1 for - ions
(
( 222
222,
,
2
11
1
,
=
==
=
=
==
+=
+=
==
+
++
oi
i
oioo
oi
oi
oo
ti
o
ti
oloc
loc
M
e
Eewp
MK
MeEw
ewweEE
eEKwwM
MM
M
uuwuuw
&&
&&&&&&
Ionic materials always have ionic and
electronic polarization, so:
( 222
++=+= +
oi
eitotM
e
E. Fitzgerald-1999
)
)
)
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3.225 17
Usually Clausius-Mosotti necessary due to high density of dipoles
Ionic Polarizability
(
++==+
+ 222
3
1
32
1
oiootot
rr
Me
vN
By convention, things are abbreviated by using s and :
(
++=+
vnn
ooi
3
1
2
1
2
1,
2
2
++
=
+=
2
2,
1
22
2
2
soiT
T
sr
r
T
n2=
s
E. Fitzgerald-1999
)
)
]
3.225 18
Orientational Polarizability
No restoring force: analogous to conductivity
H
H
O
p +-
C
O O
p=0
+q
-q
For a group of many molecules at some temperature:
TkpE
TkU
bb eef cos== After averaging over the polarization of the
ensemble molecules (valid for low E-fields):
Tkpb
DC3
~2
Analogous to conductivity, the
molecules collide after a certain
time t, giving:
iDCo =1
E. Fitzgerald-1999
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3.225 19
Dielectric Loss
E. Fitzgerald-1999
For convenience, imagine a low density of molecules in the gas phase
C-M can be ignored for simplicity
There will be only electronic and orientational polarizability
i
nn
Nn
i
Nn
sor
o
DCsor
o
DCoer
+=
+==
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3.225 21
Dispersion
Dispersion can be defined a couple of ways (same, just different way)
when the group velocity ceases to be equal to the phase velocity
when the dielectric constant has a frequency dependence (i.e. when d/d not 0)
k
Dispersion-free
Dispersion
kc
r =
g
r
p vk
c
kv =
===
g
r
p vk
c
kv =
==
)(
E. Fitzgerald-1999
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3.225 1
Spontaneous Polarization
Remember form of orientational polarization:
kT
C
kT
por
==3
2
With C Curie constant
Define a critical temperature Tcby
k
NCT
c
03
=
Noting further
Thus
H.L. Tuller, 2001
or
03
orcN
T
T=
Fig. 1. The Curie-Weiss law illustrated for (Ba,Sr)TiO3From L.L. Hench and J.K. West, Principles of ElectronicCeramics, Wiley, 1990, p. 243.
133
00
=
=
ckT
CNN
c
c
TT
T
=
3
3.225 2
Each unit cell a dipole!
Large PR(remnant polarization, P(E=0)
Coercive Field EC, electric field required to bring P back to zero.
Ferroelectrics
E
RRo
E
Two equivalent-energy atom positions Can flip cell polarization by applying
large enough reverse E-field to get over
barrier
E
P
normal dielectric
Ps
Ec
PR
E. Fitzgerald-1999
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3.225 3
Ferroelectrics
Confused atom structure creates metastable relative positions ofpositive and negative ions
E. Fitzgerald-1999
3.225 4
Ferroelectrics
Applications
Capacitors
Non-volatile memories
Photorefractive materials
H.L. Tuller, 2001
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3.225 5
Characteristics of Optical Fiber
Snells Law
n1
n2
1
2Refraction
Boundary conditions for E-M wave gives
Snells Law:
2211 sinsin nn =
n2
n11
2
Internal Reflection: 1=90
2
11
2 sinn
nc
==
Glass/air, c=42
E. Fitzgerald-1999
3.225 6
Attenuation
Absorption
OH- dominant, SiO2 tetrahedral mode
Scattering
Raleigh scattering (density fluctuations) R~const./4 (
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3.225 7
Characteristics of Optical Fiber
E. Fitzgerald-1999
3.225 8
Characteristics of Optical Fiber
E. Fitzgerald-1999
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3.225 9
Colors Produced by Chromium
Above: alexandrite, emerald, and ruby.
Center: carbonate, chloride, oxide.
Below: potassium chromate and ammonium dichromate.
H.L. Tuller, 2001
3.225 10
Electron distribution in the ground state of a chromium atom (A) and a trivalent chromium ion (B).
Chromium Electronic Structure
H.L. Tuller, 2001
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3.225 11
Interaction of the d orbitals of a central ion with six ligands in
an octahedral arrangement.
Octahedral Environment of Transition Metal Ion
H.L. Tuller, 2001
3.225 12
The splitting of the five 3d orbitals in a tetrahedral and an octahedral ligand field.
Note: hen the element is a mid-gap dopant, transitions within this element lead to
absorption and/or emission via luminescence
Crystal Field Splitting
H.L. Tuller, 2001
W
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3.225 13
Optical Transitions in Ruby
Optical absorption spectrum tied to Cr transitions in ruby.
H.L. Tuller, 2001
3.225 14
Optical Transitions in Emerald
Optical absorption spectrum tied to Cr transitions in emerald.
H.L. Tuller, 2001
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3.225 15
Designer Wavelengths
Variation of band-gap energy with composition x of In1-xGaxAs.
H.L. Tuller, 2001
3.225 16
Band-Gap Colors
Mixed crystals of yellow cadmium sulfide CdS and black
cadmium selenide CdSe.
H.L. Tuller, 2001
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3.225 17
Light Sources
Photoluminescence
Cathodoluminescence
Electroluminescence
H.L. Tuller, 2001
3.225 18
Energy onversion torage onservation
Emissions Smoke stack Automotive
Challenges for New Millenium
Needed: dvances in sensors,actuators and energy conversiondevices.
CSC
a
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3.225 19
Harsh Environments Ceramic Sensors
AutomotiveEmissions
FactoryEmissionsProcess Control
AerospacePerformance
3.225 20
Electroceramics
Ceramics:
Traditionally admired for their stability Mechanical Chemical Thermal
Exhibit other key functional properties
Electrical, Electrochemical, Electromechanical Optical Magnetic
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3.225 21
Electroceramics Versatility
Atmosphere dependent conductivity (I.Kosacki and H.L. Tuller, Sensors &Actuators B 24-25, 370 (1995).)
High Strain (Pb0.98La0.02(Zr0.7Hf0.3)1-xTixO3 AFEFE System (C. Heremans and H.L. Tuller, J. Euro.Ceram. Soc., 19, 1139 (1999).)
Semiconducting; Electrochemical; Piezoelectric;Electro-optic; Magnetic, ...
3.225 22
Feedback Control System
System
Actuator
Sensor
Chemical
Signal
SignalElectrical
Power
Chemical
Species
Micro-Processor
Other Input
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3.225 23
Sensors for Exhaust Gas Monitoring
Requirements clear dependence on pO2
short response times < 100 ms
700
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3.225 25
3-Way Catalyst Conversion Efficiency
3.225 26
Potentiometric Gas Sensor
PO2(Ref)PO2(Exhaust)
E
E = (kT/4q) ln {PO2/ PO2(Ref)}
Nernst Potential
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3.225 27
Auto Exhaust Sensor
Requirements
Sensitivity
Reproducibility
Robust
Low cost
3.225 28
Miniaturization (e.g. biological implants)
Integration - logic, amplification, telemetry
Portability - low power dissipation
Rapid response
Cost
Sensor Trends and Challenges
Neural recording/stimulation microprobe. Probes15m thick 2-4mm long. (Najafi and Hetke, IEEETrans. Biomed. Eng. 37, 474 (1990).)
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3.225 1 H.L. Tuller-2001
Measurement of Gas Sensor Performance
Si wafer
ZnO film
H2
H2H2H2
Pt electrode
SiO2 layer
ElectricalMeasurement
Gas sensing materials:1. Sputtered ZnO film (150 nm(Massachusetts Institute of Technology)
2. Sputtered SnO2 film (60 nm)(Fraunhofer Institute of Physical Measurement Techniques)
Target gases:H2, CO, NH3, NO2 , CH4
Operating temperature:320 - 460 C
3.225 2 H.L. Tuller-2001
Mechanisms in Semiconducting Gas Sensor
Bulk:
OO= 2e+ VO
..+ 1/2 O2(g)
Induce shallow donors: density related to PO2
n2 [VO..] PO2
1/2 = KR(T) n = (2 KR(T))1/3PO2
-1/6
Bulk electronic conduction
modulate
Change in stoichiometry
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3.225 3 H.L. Tuller-2001
Resistive Oxygen Sensors Based on SrTiO3
m
1
2kT
E
pOeA
semiconducting oxide
Electrode
U I
2OpExhaust
3.225 4 H.L. Tuller-2001
Influence of Dopants on Electrical Conductivity of SrTiO3
Sr2+ Ti4+ O2-3
Acceptors: Al, Ni, Fe
Donors: Nb, Ta, Sb, Y, La, Ce, Pr, Nd, Pm, Sm, Gd
log(/(cm)-1
)T = 800 C
log(pO2 / bar)
0.995 1,005
-16 -4 0-5
-4
-3
-2
-1
0
1
donoracceptor
-20 -12 -8
donordopedacceptordoped
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3.225 5 H.L. Tuller-2001
Temperature Independence: High Acceptor Concentration in SrTiO3
10-20 10-15 10-10 10-5 100
0,1
1
750C
800C
900C
850C
950C
electricalconductivity/(cm)-1
pO2 / bar
m = 0,2
Sr(Ti0,65Fe0,35)O3
Response times
T / C t90 / ms
900 6.5
800 26
750 83
700
[1] Menesklou et al, MRS fall
meeting, Vol. 604, p. 305-10 (1999).
185
3.225 6 H.L. Tuller-2001
Oxygen Sensor in Thick Film Technology
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3.225 7 H.L. Tuller-2001
0 5 10 15 20 25 30 351E-5
1E-4
1E-3
0,01
0,1
T = 850C
X=0.03
X=0.03
Sr1-x
LaxTiO
3porous ceramic
/S/cm
t / h
Transient Behavior of Porous Sr1-xLaxTiO3 for x=0.005 and x=0.03
T = 850 C
3.225 8 H.L. Tuller-2001
Mechanisms in Semiconducting Gas Sensor
Interface - Gas adsorption
2e+ O2(g) O(s)
Induce space charge barrier
1. Surface conduction
2. Grain boundary barrier
Grain boundary barrier
modulate
2=
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3.225 9 H.L. Tuller-2001
Sensor Configuration
A single 9 mm2 chip sensor array with: four sensing elements with interdigitated structure electrodes
heater
temperature sensor
3.225 10 H.L. Tuller-2001
Schematic Cross Section of Mounted Sensor
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3.225 11 H.L. Tuller-2001
Resistance onse to Gas Environment
ZnO film (150 nm)
Electrode: Pt(200 nm)/Ta(25 nm) film
Insulation layer: SiO2 layer (1 m)
Substrate: Si wafer
Si wafer
ZnO film
H2
H2 H2H2
Pt electrode
SiO2 layer
ElectricalMeasurement
0 20 0 0 0-100
102030405060708090
100110
-100
102030405060708090100110
MFC2 Temp NO2 NH3
Feuchte CO NO2kl H2
Pt-100resistance/G
asflow/sccm
time / h
0 20 0 0 0
100k
Temp:360C, H2, CO, NH
3(10, 50 and 100 ppm), NO
2(0.2, 0.4, and 2 ppm)
ZnO(Ar:O2=7:3) 1
[Pfad: \alphamis sy Mess ungenmess platz _1] M.J gle/27.02.2001
S1219aS1219b
S1219c
S1219d
resistance/Ohm
M9710746 20VDatum: 23.02.2001- 27.02.2001
Steuerdatei:allgas_h2.stg
Meprotokoll:273Schematic of Gas Sensor Structure
3.225 12 H.L. Tuller-2001
MicroElectroMechanical Systems - MEMS
Micromachining - Application of microfabrication tools, e.g. lithography, thinfilm deposition, etching (dry, wet), bonding
Bulk Micromachining Surface Micromachining
Resp
4 6 8
4 6 8
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Gas Sensors and MEMS
Miniaturization Reduced power consumption Improved sensitivity Decreased response time Reduced cost
Arrays Improved selectivity
Integration Smart sensors
3.225 14 H.L. Tuller-2001
Microhotplate
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Microhotplate Sensor Platform
NIST Microhotplate Design
3.225 16 H.L. Tuller-2001
Microhotplate Characteristics
Milli-second thermal rise and fall times
programmed thermal cycling
low duty cycle
Low thermal mass
low power dissipation
Arrays
enhanced selectivity
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3.225 17 H.L. Tuller-2001
Harsh Environment MEMS
High temperatures
Oxidation resistant
Chemically inert
Abrasion resistant
Wide band gap semiconductor/insulator
3.225 18 H.L. Tuller-2001
Photo Electro-chemical Etching - PEC
materials versatility e.g. Si, SiC, Ge, GaAs, GaN,etc.
precise dimensional control down to 0.1 mmthrough the use of highly selectivep-n junctionetch-stops
fabrication of structures with negligible internalstresses
fabrication of structures not constrained byspecific crystallographic orientations
Features:
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3.225 19 H.L. Tuller-2001
+
-
+
-
h+ h+h+h+semiconductor
Photo Electro-chemical Etching - PEC
Electro-chemicaletching
p-type
+
-
Light source
Photo electrochemical etching
+
-
h+h+semiconductor
electrolyte
Light source
n-type
3.225 20 H.L. Tuller-2001
Examples ...
Arrays of stress free4.2 m thick cantileverbeams.
Photoelectrochemicallymicromachined cantileversare not constrained tospecific crystal planes ordirections.
Similar structuressuccessfullymicromachined from SiCby Boston MicroSystemspersonnel
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3.225 21 H.L. Tuller-2001
Smart Gas Sensor
A Self Activated Microcantilever-based Gas Sensor
1. A device capable of sensing a change in environment and
responding without need for a microprocessor
2. A device has both gas sensing and actuating function by
integration of semiconducting oxide and piezoelectric thin films.
Micro-
Processor
Actuator
Sensor
Chemical
Environment
Microfluidic structure
3.225 22 H.L. Tuller-2001
Smart Gas Sensor
1. Semiconducting oxide thin films for high gas sensitivity
: Microstructure (Nano-Structure) and Composition
2. Piezoelectric thin films for providing actuating function
3. Thin film electroceramic deposition methods for integrating with
silicon microcantilever beam
: Compatibility with Si micromachining technology
4. Microcantilever structures for the self activated gas
: High performance in chemical environment
sensor
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Resonant Gas Sensor
Resonant Frequency: fR= 1/2l (o/o)1/2where l= resonator thickness, o= effective shear modulus and o=
density
Mass change causes shift in resonant frequency : (m0- m) / m o (f + f) / f
Gas Sensor elements :
(I) Active layerinteracts with environment
- stoichiometry change translates into mass change
(II) Resonatortransduces mass change into resonance frequency change
f
mElectrode
Electrode
Resonator
Active layer
3.225 24 H.L. Tuller-2001
Choice of Piezoelectric Materials
Temperature limitations of piezoelectric materials
Material Max OperatingTemperature (
oC)
Limitations
Quartz 450 High loss
LiNbO3 300 Decomposition
Li2B4O7 500 Phase transformation
GaPO4 933 ? Phase transformation
La2Ga5SiO4(Langasite)
1470 ? Melting point
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Design Considerations
Bulk conductivitydependent on temperature and PO2 contributes to resonator electrical losses
Modify bulk conductivity - how?
Stabilityto oxidation and reduction process
limited oxygen non-stoichiometry
slow oxygen diffusion kinetics
Defect chemistry and diffusion kinetics study
fR (T): Temperature dependence of resonant frequency
need to differentiate from mass dependence
Minimize @ intrinsic and device-levels
3.225 26 H.L. Tuller-2001
Langasite : Bulk Electrical Properties
Single activation energy in the temperature range 500 -900 C
Extrapolated room temperature conductivity: = 4.410 -18
S cm-1
8 9 10 11 12 1310
-7
10-6
10-5
10-4
Y-cut
0
= 2.1 S cm-1
EA = 105 kJ mol
-1
104/T [1/K]
[Scm-1
]
900 800 700 600 500
T [C]
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Langasite : fR (T)
Temperature dependence of the resonance frequency (fR)of a resonator device with difference mass loads.
0 100 200 300 400 500 600 700 800
1,71
1,72
1,73
1,74
1,75
1,76
1,77
Contact 1 (fCo1
)
Contact 2 (fCo2
)
Calculation fCo1
+ f(mCo2
)
T [C]
fA[MHz]
3.225 28 H.L. Tuller-2001
Ongoing Activities
Resonator (Langasite) -- H.Seh & H. Fritze
Defect chemistry
Oxygen diffusion/exchange studies
Bulk conductivity dependence on T and PO2
Active Layer (PCO) -- T. Stefanik
Transport-Defect chemistry correlations
Gas Sensor --
Add active layer (PCO) using PLD nanocrystalline vsmicrocrystalline
Sensor testing
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1
3.225 1 E. Fitzgerald-1999
Magnetic Materials
The inductor
(
Law)sFaraday'(explicit1
Theorem)s(Gree
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