Understand and apply principles of operation and design of modern electronic devices: (i) (i) (ii) (iii (iii))
(iv) (v) (vi) (vii) (viii)
Equ Equation tions s desc descrribing ing dev deviice op operati ation; on; Appropriate de device mo models; Facto actors rs that that dete determ rmin ine e dev devic ice e per perfo form rman ance ce
Formulas Example problems or!ed out "he steps used used in the problem problem listed in in order #eminders #eminders of things to loo! out for in doing a problem Any rules used to solve problems
Electronic Properties of Silicon Quantum Theory The Waveform of a Small Particle $avefunction % hich describes particle distribution or amplitude as a function of space and time coords& (eiθ = cos θ + i sin θ to split.)
WAVE
FUNCTION
WAVE NUMBER
ANGULAR FREQUENCY
k =
2 π
λ
←
ω = 2 π
v ' frequency
←
SCHRODINGER ←
EQUATION
λ ' avelength
←
ℏ= h ∕ 2 π h is plan!s constant V(x is the potential energy of the particle m is its mass
"he *intensity+ of the ave and represents the probability that a particle exists at a certain point in space& CONSERVED
PARTICLE AKA: NORMALIZATION CONDITION , AS IN PROBABILITY TEORY!
!
∴
" ,E - . / 0 1 E " E $ A 2 E F U 3 - " 4 . 3 ←
For the in5nite square ell problem&
3ote: 0lane aves cannot be normali6ed in this manner because they have in5nite extent&
"x#ectation an$ Uncertainty 4f in the lowest energy level of well:
AVG POSITION OF
A PARTICLE
HEISENBERG
3ote that the expectation value ill alays be L/2 due to the mirror image symmetry about the center of the potential ell&
Uncertainty beteen position ( x) and momentum (#)& -an never !no the exact position and momentum of a particle at the same time&
UNCERTAINTY PRINCIPLE ENERGY – TIME
%t is the time interval required for an appreciable change to occur in the properties of the system under study
UNCERTAINTY RELATION
&tom' an$ the Perio$ic Tale "ner)y *e)eneracy
4n 7 and 8 dimensions is that several eigenfunctions have the same value of energy& "he value of the ground state energy level E9 (98& e2) is !non as the ioniation energy of hydrogen the energy required to completely remove the electron from the hydrogen atom&
IONIZATION ENERGY YDRO"EN
ao
WAVEFUNCTION HYDROGEN
BOR RADI#S •
SPIN
s =± ℏ / 2 •
-orresponds to the most prob radius for the electron
∀ energy
eigenfunction hich represents an electron there also exist to possible spin states& (spin up or spin don)& ∴ in addition to the avefunction e must also specify the spin&
Note+ Qualitatively, one can think of the electron wave possessing a circular polarization and thus a localized “current loop” leading to an intrinsic magnetic eld. Clockwise and counter clockwise rotation can be related to the two possible values of spin.
PAULI EXCLUSION PRINCIPLE
3o more than 9 electron can be in any given state (%(xt) and spin state) at the same time&
"lectron' in Cry'tal' The T,o &##roache' to -o$elin) "lectron' in Cry'tal'
.ne approach considers ho the energy levels of isolated atoms change hen they are brought together to form a crystal&
"he other approach examines the quantum mechanical properties of electrons in a crystal by attempting to solve the
/etals vs 4nsulators Each band has 73 (spin remember=) available electron states here 3 is the number of unit cells ma!ing up the crystal& .o, the an$' are /lle$ ,ith electron' determines hether a material is a metal or an insulator&
METALS
INSULATORS
Uppermost populated energy band is only partially 5lled (say half5lled) then there are plenty of higher energy states available for electrons to gain !inetic energy so that they can contribute to current >o and thus the crystal is able to conduct electricity& "he uppermost band is completely 5lled then there is no easy and continuous ay for charge carriers to gain energy because there is a forbidden energy band gap 0 ") before the next band of states becomes accessible& "he crystal is therefore insulating and cannot conduct electrical current&
1an$ "$)e Preliminary ?and Edge @iagram Energy band edge diagram for aemiconductor& Electronhole pairs are created hen carriers are excited from the valence band to the conduction band& ← ← ←
"v 0 2alence band edge "c 0 "he conduction band& ") 2 "he to band edges are separated by the band gap
con$uction an$ of a semiconductor $hen electrons are excited from the valence an$ • "he electron' promoted into the conduction band can no participate in electronic tran'#ort& • 4n addition the electrons in the valence band no have some empty states available for them to also participate in current >o& "he vacant states that are left in the otherise full valence band can be treated as if they ere particles called hole' &
3ear the top and bottom of bands they can be approximated by parabolas similar to a free particle&
EFFECTIVE MASS
HOLES
mp ' mn
ELECTRONS
mn
ELECTRON ENERGIES
At the top of the valence band At bottom of the conduction band 4n -onduction ?and 4n 2alence ?and
4f !B ' !BC then the band gap is direct
Intrin'ic 3 "xtrin'ic (N or P Ty#e intrin'ic material the number of electrons in the conduction band is equal to the number of holes in the valence band& • Fermi level lies very close to the middle of the band gap
*onor Im#uritie'+ eective mass and the dielectric constant of the semiconductor (or relative permittivity r) of the atom &cce#tor Im#uritie'+ #eplace the electron eective mass ith the hole eective mass&
BINDING ENERGY
ENERGY LEVEL VACUUM
WORK FUNCTION
• • •
•
qG
extrin'ic material one type of carrier has a greater concentration (maDority carriers) than the other (minority carriers)&
EB
the energy of an electron that has been Dust freed from a material&
"he dierence beteen the Fermi level and EB (similar to the binding energy of an atom)&
N TYPE /aDority -arrier: Electrons /inority -arrier: ,oles doping of the crystal ith impurity atoms that $onate electron' to the con$uction an$ 0 As
• • •
•
ELECTRON AFFINITY
q H
@i beteen the conduction band edge E c and the vacuum level&
P TYPE /aDority -arrier: ,oles /inority -arrier: Electrons @oping of the crystal ith impurity atoms that acce#t electron' from the valence an$ ? Al Ia and 4n& etc
•
Fermi level of a semiconductor moves clo'er to the con$uction an$ e$)e (more electrons)
Fermi level is lo,ere$ to,ar$ the valence an$ e$)e for ptype doping (less electrons)
$,4-, ". U
∴ 4n
Fermi 4evel+
other ords the semiconductor becomes *intrinsic+ at very high temperatures&
Carrier Concentration' FERMI –DIRAC
•
DISTRIBUTION
•
HOLE PROB DISTRIB
9 f(E)
INTRINSIC
•
n ' p ' ni
CONC.
J /ass Action 1a
Note+ The intrin'ic carrier concentration increa'e' ,ith increa'in) tem#erature an$ $ecrea'e' ,ith increa'in) an$ )a# Note that the #ro$uct of electron an$ hole concentration' in a 'emicon$uctor i' con'tant for a )iven tem#erature (ma''0action la, Thermal e6uilirium concentration of electron' in the conduction band
DENSITY OF STATES
4f " 2 "F 77 5 1T
(.nly by li!e 7 8x)
E$$ECTI%E D O$ S IN
CARRIER
TE COND AND %AL
CONCENTRATIONS
BANDS
4n terms of intrinsic concentration and Fermi level •
INTRINSIC
FERMI LEVEL POSITION
•
intrinsic Fermi level lies very close to the center of the band gap ith only a slight oset due to the dierence in electron and hole eective mass& 4n thermal equilibrium there is no net current >o and thus the Fermi level must be constant throughout the material or device&
3a and 3d K represent the concentration of ioni6ed dopant atoms&
SPACE-CHARGE NEUTRALITY
∴
4f e start ith an electrically neutral semiconductor in thermal equilibrium it must remain neutral regardless of the number of holes electrons or ioni6ed impurities present&
S P A C E - C H A R G E N E U T R A L I T Y A N D M A S S A C T I O N L AW M U S T H O L D AT E Q U I L I B R I U M
HOLE CONCENTRATION IN A SEMICONDUCTOR
ELECTRON CONCENTRATION IN A SEMICONDUCTOR
H OWEVER: LOOK AT N
OR P
AGE – H AS S HORTCUTS T YPE P
Char)e Tran'#ort AVG THERMAL ENERGY
"his causes random motion of the charge carriers resulting in collisions or scattering ith the lattice impurities and beteen carriers etc& 4n thermal equilibrium e have seen that this type of random motion does not produce any net current >o&
THERMAL VELOCIT Y
←
DRIFT VELOCITY
←
τ 0 mean free time+ time interval beteen scattering events "x 0 con'tant electric /el$ in x direction
@escribes ho easily an electron can move in response to an applied electric 5eld& "he electron and hole mobilities (L n and Lp respectively) each depend on the total $o#ant concentration
MOBILITY
DRIFT CURRENT DENSITY
CONDUCTIVITY
σ =¿
R ESISTIVITY
p=1 / σ
?y considering a bar of material ith crosssectional: ← resistance 8 ← area & ← length 4
OHMS LAW
Scatterin) •
4attice 'catterin) increases ith temperature as the vibrations of the lattice become greater& "his type of cattering causes the mobility to scale as a poer la ith temperature T00n9 ,ith n ty#ically :;<=2>< Im#urity or $efect 'catterin) on the other hand decreases as temperature increases& "his is due to the increased thermal velocity of carriers hich ma!es them less susceptible to interaction ith impuritiesMdefects
*rift Velocity Saturation •
• •
•
At very high electric 5elds the drift velocity becomes comparable to the thermal velocity& NE O "hermal Energy ,ot -arriers "his is caused by increased scattering at high 5elds hich limits further increase in the drift velocity& 4n silicon "he saturation velocity for both electrons and holes is approximately 9BPQ cmMs
DIFFUSION CURRENT DENSITY
DIFFUSION CONSTANTS
*i?u'ion Current •
•
$hen there is a carrier concentration gradient& charge carriers >o from a region of high concentration to lo concentration resulting in a net current&
(!b)
-----------T OTAL E LECTRON AN D H OLE C URRENT D E N S I T I E S -------(INCLUDING BOTH DRIFT AND DIFFUSION)
@eneration an$ 8ecomination -
EXCESS E
nB and pB denote the thermal equilibrium carrier concentrations
P
CONCENTRATIONS
RECOMBINATION NET RATE
Rn 'Rp ' R
"xam#le' Particle "ner)ie' (Quantum Well Energy of a Free 0article 2
E=
2
ℏ k
=ℏω =hv
2m
p=ℏk =
h λ
de roglie relation
Energy of a 1oun$ Particle (4n5nite Suantum $ell) 4nside the ell the potential energy is 6ero hereas at the boundaries it rapidly increases to in5nity and the particle cannot escape or exist outside the ell&
?ecause particle cannot exist outside of this ell&
-anCt use cos it ouldnCt equal B at the origin&
n is a non6ero positive integer
Therefore9 y con/nin) the #article to a AoxB it' ener)y level' ecome 6uantie$+
•
ENERGY OF •
PARTICLE
Energy levels are inver'ely proportionally to the idth of the ell squared; i&e& more con5nement (smaller ell) leads to higher energy levels& 4n addition the energy levels become increasingly spaced out 7
IN I N$INITE
•
&ELL
as n increases due to the factor of n 4t also tells us the loest energy a particle can have hen con5ned to a region of space and that this must be a 5nite value !non as the )roun$ 'tate ener)y
"ner)y2Time Uncertainty Fixed
Excited state •
•
0article that has been excited to some higher energy level En and then decays&
And the energytime uncertainty:
4mposes the condition:
∴ After
the particle reaches its ground state no further decay can occur and e are bac! to a 5xed stationary state hich does not change in time as in the previous case& -an !no sharpness of the output spectrum of 1E@s can be& int& hv
"unneling : 0otential ?arriers -onsider a free particle: ith energy " impinging on a potential energy barrier of idth DE4 and height !o
0article must have plane ave solutions since R EGION "
R E"ION II R E"ION III
the potential energy is 6ero also a re>ected ave in the opp (neg) direction to the incoming particle& ,lassically forbidden rgn the NEJ0E: a classical particle could never exist in this region (3ot enough 0E to cross barrier&)
coeXcient t&
,oever ,e can have a plane ave in #I3 44 through S/& (-ave vector becomes comple comple epns instead become regular 0increasing and decaying1 eponentials.) "he amplitude of the transmitted ave is: 4f Y1 is fairly large (in other ords if the arrier i' tall an$ ,i$e ): 0robability of "ransmission: T GtGD can be approx& as: T : ex#(0H4
∴ 0article
has a 5nite 0r of transmission through the barrier that depends sensitively on the height and idth of the barrier& 4n particular if the arrier ecome' thin9 the tran'mi''ion can be quite signi5cant& "his inherently quantum mechanical phenomenon is !non as tunnelin)<
Carrier Concentration Calculation ?asic
?asic 7
-alculating @iusion -onst from @opants
0hotoconductor (Ienregen)
'(nctions )n* Dio*es PN Junction 4deal ?ehaviour 3on4deal ?ehaviour (@eviations from 4deal)
-etal0Semicon$uctor
*e'i)n Con'i$eration' $ith respect to device and integrated circuit performance
Prolem "xam#le'
1i#olar Junction Tran'itor (1JT
FieldEect @evices /.<
Con'tant'
Chart' *o#ant *en'ity v' 8e'i'tivity
-oility v' Carrier
