materials science and engineering - electronic and mechanical properties of materials

8/8/2019 Materials Science and Engineering - Electronic and Mechanical Properties of Materials

1/106

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


2/106

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


3/106

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?


R


4/106

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


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



5/106

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


2


6/106

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=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

)

)


7/106

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



8/106

3.225 15

~10-14 sec for metals inDrude model

Extracting Typical for Metals


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



9/106

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


10/106

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


11/106

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-


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



12/106

- - - - - - - - -

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


13/106

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



14/106

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


15/106

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

==


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



16/106

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)


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

===

=



17/106

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


18/106

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


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



19/106

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!


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-

)


20/106

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!


21/106

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


22/106

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

) )

)


23/106

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

)

)


24/106

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 =

H.L. Tuller-2001


25/106

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


26/106

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


27/106

3.225 1


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


E(kx,ky)

kx

ky

m

kkE

yx

2

)( 222 +=h

E. Fitzgerald-1999


28/106

3.225 3


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


29/106

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

) ] )


30/106

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


31/106

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


32/106

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


33/106

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

))

))

)


34/106

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


35/106

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


36/106

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


37/106

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 =

H.L. Tuller-2001


38/106

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*

E. Fitzgerald-1999


39/106

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

E. Fitzgerald-1999


40/106

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

E. Fitzgerald-1999

+


41/106

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

)


42/106

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

)


43/106

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

-+

E. Fitzgerald-1999


44/106

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


45/106

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

E. Fitzgerald-1999


46/106

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

E. Fitzgerald-1999


47/106

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

E. Fitzgerald-1999


48/106

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)


49/106

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

H.L. Tuller, 2001


50/106

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

+

=

+

=

E. Fitzgerald-1999


51/106

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

H.L. Tuller, 2001


52/106

3.225 9

Junction Fabrication Processes

H.L. Tuller, 2001

3.225 10

CMOS Devices

H.L. Tuller, 2001


53/106

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

E. Fitzgerald-1999


54/106

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


55/106

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


56/106

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


57/106

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.

H.L. Tuller, 2001


58/106

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


59/106

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.

H.L. Tuller, 2001


60/106

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


61/106

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


62/106

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


63/106

- -

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)

E. Fitzgerald-1999


64/106

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?

E. Fitzgerald-1999


65/106

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

)


66/106

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

E. Fitzgerald-1999


67/106

3.225 9

For a material with many dipoles:


)( 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


+=

=+==

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

)) )


68/106

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

E. Fitzgerald-1999


69/106

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

==

E. Fitzgerald-1999


70/106

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


- +

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

)

)

)


71/106

3.225 17

Usually Clausius-Mosotti necessary due to high density of dipoles


(

++==+

+ 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


72/106

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

+=

+==


73/106

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


74/106

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


75/106

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


76/106

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 (


77/106

3.225 7


E. Fitzgerald-1999

3.225 8


E. Fitzgerald-1999


78/106

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


79/106

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


80/106

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


81/106

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


82/106

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


83/106

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


84/106

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


85/106

3.225 23

Sensors for Exhaust Gas Monitoring

Requirements clear dependence on pO2

short response times < 100 ms

700


86/106

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


87/106

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).)


88/106

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


89/106

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


90/106

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


91/106

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=


92/106

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


93/106

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


94/106

3.225 13 H.L. Tuller-2001

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


95/106

3.225 15 H.L. Tuller-2001

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


96/106

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:


97/106

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


98/106

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


99/106

3.225 23 H.L. Tuller-2001

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


100/106

3.225 25 H.L. Tuller-2001

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]


101/106

3.225 27 H.L. Tuller-2001

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


102/106

1

3.225 1 E. Fitzgerald-1999

Magnetic Materials

The inductor

(

Law)sFaraday'(explicit1

Theorem)s(Gree

materials science and engineering - electronic and mechanical properties of materials

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