SEMICONDUCTOR PHYSICS & DEVICES
Electron & Hole Density:
SEMICONDUCTOR PHYSICS & DEVICES
⏞ ( ) ( ) ( ) ( ) ( )) ( : Fermi-Dirac-Probability, : density of states in cond. band ( ) ( ) ( )
Summary of the lectures held by Prof. Dr. A. Schenk
(
CONSTANTS & MATERIAL PARAMETER S
Planck’s constant: Dielectric constant (für cm):
(
(
)
⏞
)
) (
√
)
(
(
)
)
(
⁄
)
(
(
√
)
)
⁄
)
(
)
for
)
(
)
( ⏟
(
)
⁄
(
( (
) (
)
(
MATERIAL-PARAMETER GERMA NIUM Relative permittivity: Intrinsic carrier concentration:
MATERIAL-PARAMETER ANDERE
(
)
(
)
(
)
)
4. SEMICONDUCT OR IN EQU ILI BRIUM direct semicond.: if the minimum of the conduction band is at the same place like the maximum of the valence band. indirect semicond.: if it is not a direct semiconductor. This means a phonon is also needed actually get an electron from valence to conduction band. Problematic in optical systems. intrinsic: no impurities, lattices defects, ... noted as , extrinsic: aritificially doped semiconductor thermal equilibrium: no applied voltage or current,noted as donor atoms: create n-type semiconductor ⁄ fermi level: with ( ) eff. mass approx.: so quantum effects can be neglected
CHARGE NEUTRALITY
In case of an applied voltage we get
CHARGE CARRIERS I N S EMICONDUCTORS
and :
for n-type at low injection (
) and
:
( ⏟
→
(
( ) ( )
( )
( ( )
. )
(
)
)
(
)
(
)
⁄ ( ) for abd ( )⁄ ⁄ ( ) ( ) { ⁄ ( ) Appilcation 4: finite number of electron-hole pairs is generated at , but for . n-type semicond. with constant applied E-field in direction. Solution:
Application 3:
)
(
√
(
)
)
Diffusion Length: →
and get
)→
)
⁄
)
)
we have
(
(
Derivation: (
( )
)
⁄
)→
)
( )
konst and
POSITION OF THE FERMI -ENERGY -LEVEL
(
)
) and
(
Einstein Relation:
In the special case
(
⏟
( ) ( ) ⁄ Application 2: homogeneous n-type semicond. zero applied electr. field, therm. eq. for , uniform gen. rate for
)
(
and
for p-type at low injection (
(
GRADED I MPURITY DISTRI BUTION
√(
)
)
Application 1: n-type semicond., uniform conc. of excess holes:
TOTAL CURRENT DE NSIT Y
Complete ionization, if above expression is Doping of Semiconductors: p-doping: e.g. Boron with 3 instead of 4 valene electr. n-doping: e.g. Phosphorus, 5 instead of 4 valence electrons near degenerate: conducts always metallic
]
(
⏟
( ⏟
⏞ ⏟
⏟
APPLICATIONS OF AMBI POLA R TRANSPORT
charge neutrality in thermal equilibr. at complete ionisation: [
)
CARRIER DIFFUSI ON
Relative number of electrons/holes on the doping energy levels relative to the overall number of charge carriers.
hole mobility: effective electron mass: effective hole mass:
⁄ ⁄
( ⏟
→
)
IONIZATION & FREEZE OUT
)
⏟
with )
( )
:
Ambipolar Transport for p-type (for n-type replace n with p)
5. CARRIER TRANSPORT PH ENOMENA
)
⏟(
( )
and
( )
AMBIPOLAR TRANSPORT
)
CARRIER DRIFT
for intrinsic case:
electron mobility:
) material at low-level injection ( )
Continuity Equations: p.195ff
nt ns c cas
Relative permittivity: Intrinsic electron concentration: Eff. state density in conduct. band: Eff. state density in valence band: band gap: electron affinity:
Rel. permittivity of Doping density
for p-type (
-PRODUCT
MATERIAL-PARAMETER SILIZIU M jeweils bei
⁄
(
(
∫ (
)
)
hole conc.
Elementary charge: Boltzmann’s constant:
(
∫
electron conc. (
CARRIER GE NERATION & RECOMBINATION
)
Electron & Hole Concentration:
Lukas Cavigelli, July 2011 [email protected]
(Materialparameter
: excess minority carrier electron lifetime : excess electron and hole recombination rates
-
( )
( )
( )
( )
( )
√ →
( )
( )
Hall Effect: left out, see p.177ff
6. NON-EQUILIBRIUM EXCESS CARRIERS Thermal equilibrium: ⏟
⏟
: excess electron and hole generation rates : excess electron concentration
√
Dielectric Relaxation Time Constant: time it takes to until neutrality is achieved after a burst of excess carriers: ⁄
OTHER SUBTOPICS Quasi-Fermi Energy Levels: see p.216ff Excess-Carrier Lifetime (Shockley-Read-Hall): p.219ff
⁄ : minority carriers conc. in p-region, th. equ. ⁄ : min. carriers (holes) c. in n-region, th. equ. : total minority carrier conc. in p-region, analogous ( ): min. carrier conc. in p-region at space charge edge : excess min. carrier conc. in p-region
7. PN JUNCTION BASIC STRUCTURE p-region ( n-region ( In fact we reassign
) ) ,
in p-region, …
CARRIER DE NSITIES WI TH FORWARD VOLTAGE Boundary Conditions:
Polarity: in case of n-type semicond.: reverse-biased is +-pole connected to semiconductor Ideal Junction Properties:
Zener Breakdown: Electrons start tunneling through the potential barrier.
)
(
)
(
(
)
)
)
(
(
( )
( (
: We have a reverse voltage, if n-region is connected to higher potential than the n region. Very little current flows. ; Forward voltage. p-region connected to higher potential. Large current flows.
( ( )
)
)
√
(
)
)
(
)
)
(
(
(
√
(
)
Remark: is the cutoff current “Sp Small Signal Model: p. 286ff
: distance from potential maximum to the junction : reduction of the Schottky barrier )
) √
)
st om”
GENERATION-RECOMBINATION CURREN TS
BASIC PROPERTIES
Reverse-Bias Generation Current: (p. 297)
Built-in potential (Diffusionsspannung): (
(
)
)
(
)
(
{
∫
Special cases with interfacial layer: p. 336 Current Voltage Relationship:
Forward-Bias Recombination Current: (p. 300)
Derivation: ∫
√
{
)
(
(
Space Charge Width:
)
(
)
( (
)
)
)
(
)
Diode Current-Voltage Relationship: √
)
)
Minority Carrier Distribution: p.275ff Ideal pn Junction Current: ( )
(
Schottky-Effect, “Bildkrafteffekt”: Maximum cutoff voltage declines with higher applied reverse voltage.
heterojunction: junction with different semicond. materials (
)
√
9. METAL-SEMICONDUCTOR JU NCTIONS
(
(
√
(
(
)
(
)
)
)
√
Schottky-Diode: ( ) Ohmic contact: with n-type, or with p-type This can be achieved through heavy doping. See page 345ff Heterojunctions: see p. 349
if p-semicond. band is bent down instead of up; electr. in semicond., holes in metal
Capacitance:
SCHOTTKY BARRIER DI O DE (
)(
)
One-Sided Junctions: for -junction: For non-uniformly doped junctions: redo all from scratch
8. PN JUNCTION DIODE
JUNCTION BREAKDOWN Voltage for Avalanche Breakdown in one-sided junction: p.305
: Donor conc. in n-region
: ideal Schottky barrier height, potential barrier seen by electrons moving from metal into the semiconductor : metal work function (table) : electron affinity of the semiconductor (table) : potential difference between conduction band and Fermienergy of the n-type semiconductor (
: Acceptor conc. p-region, maj. carrier hole density in p-region in thermal equilibrium with and
: doping conc. in weak doped half of the junction ( ) has to be looked up in a table.
)
METAL-SEMICONDUCTOR O HMIC CONTACTS
)
Maximum Electric Field: (for reverse applied bias) (
(
: Richardson constant Comparison Schottky pn Diode: p.341
: ideality factor. large diffusion dominates if small recombination dominates (
⏟
)
Remark: This does not really work out in terms of its units. Feel free to exchange eV and V (no calculation needed).
SCHOTTKY V. PN -DIODE
Forward Voltage Cutoff Current Max. Cutoff Voltage Switching Speed
Schottky
pn
to very good, just a few ps
multiple kV Diffusion Capacity, reverserecovery effect
Punch-Through Breakdown Voltage: p. 208 if B-C space charge region increases until it reaches the B-E space chage region. ( ) ( )
LOW-FREQUENCY CURRENT GA IN
10. BIPOLAR TRANSIST OR by region:
Currents:
: : :
√ (
with
)
Avalanche Breakdown:
(Alpha-)Cutoff Frequency: |
⁄
whereby is determined empirically.
where the BE diode is conducting and the CB diode is not. In act v mod , th BE d od s th “actual” pn junct on wh as the CB junction is just used to suck off the electrons.
SIMPLIFIED CURRENT RELATIONS ( ) ⏟
(
)
this formula is only valid if recombination in the collector can be neglected (which should usually be the case) : thermal equilibrium electron conc. in the base : base width; : electron diffusion coefficient
: common-base current gain; : common-emitter current gain ) Attention: the calculation of is extremely sensitive!! ( Modes of Operation: Cutoff: almost no cu nt, “sw tch d off” Forward active: usual mode, lin. amplification control volt.: , large curr.: Saturation: “sw tch d on”, no fu th ampl f. forward-active:
⁄ Beta-Cutoff frequency: : or respectively at low frequency Large Signal Switching, Schottky-Clamped Transistor, other bipolar Transistor Structures: see book Possible Implementation:
: diffusion of minority carrier electrons in the base at : diffusion of minority carrier electrons in the base at : recombination of excess minority carrier electrons with majority carrier holes in the base. is the flow of holes into the base to replace recombined holes. : diffusion of minority carrier hole in the emitter at : recomb. of cerriers in the forward-biased B-E junction : diffusion of min. carrier holes in the collector at : generation of carriers in reverse-biased B-C junction Currents are B-E junction currents only. Currents are B-C junction current only. Derivation of Current Gain:
: breakdown voltage between C and B, with E unconnected
all factors should be for strong amplification. emitter injection efficiency factor :
MINORITY CARRIER DIS TRIBUTI ONS
( (
⁄ ⁄
) )
→
Other Effects: high injection (p. 401), emitter bandgap narrowing (p. 403), current crowding (p. 405), non-uniform base doping (p. 406) Open Circuit Calculation: p. 411
EQUIVALENT CIRCUIT M ODE LS - Ebers-Moll Model: p. 412 - Gummel-Poon Model: p. 416 - Hybrid-Pi Model: p. 418
EXAMPLE
CALCULATION
( (
⁄ ⁄
√
→ (
)
(
⁄
( )
)
recombination factor : ( ( (
)
) )
Early-Effect / Base Width Modulation: p.397 shows effects of not constant, but ( )
(
⁄
)
( )
( )
( )
(
(
)
((
(
(
(
)
)(
)
)
)(
) (
) )
√
)
)
) √
(
: excess minority electron concentration in the base : minority electron conc. in the base in thermal equilibrium ( ): total minority electron conc. in the base
) √
( (
) )
FREQUENCY LIMITATION S : emitter-to-collector delay, : emitter-base junction capacity charge time, : base transit time, : collector depletion region transit time, : collector capacity charge time (
: Early voltage; saturation:
⁄ ⁄
: diffusion length; : real width of the base/emitter (i.e. and ) base transport factor :
NON-IDEAL EFFECTS
cutoff:
|→
)
(
)
Note: the npn transistor only works due to its geometry (small base)
: output conductance (
√
)
(
MOS-CAPACITOR
)
Exam: only n-channel = p-type MOSFET.
( (
)
)
( (
MODES OF OPERATION
)
Non-Saturation:
)
MOS-FET: NON-IDEALITIES
depends on
.
Miller-Effect:
𝐼𝐷𝑆
Everything below is for this case.
√ : maximum width of the space charge region in inversion
MODES OF OPERATION BY MOS -CAP Accumulation: negative voltage at gate holes accumulate in channel region channel fills up with holes not conducting Depletion: pos. voltage at gate not yet enough electrons in channel region to be conducting Threshold: pos. voltage at gate ( ) #electr. in space charge region = #holes in substrate as if channel region not doped Inversion: gate voltage more electrons than holes in channel region like n-doped material, although actually p
DEPLETION LAYER THIC KNESS (
Saturation: does not depend on anymore. Linear ( amplification of . Usually the mode wanted.
)
𝑡 Miller-Plateau Other non-idealities: subthreshold conduction MOSFET already conducts for voltages channel-width modulation
accumulation: Meaning of the Threshold Voltage:
√
)
depletion: not accumulation, but not conducting (
)
MOS-FET: CURRENT -VOLTAGE CHARACT. Currents: MOS-FET conducts, if inversion: conducting,
√
( ) ( ( ) ( ) : channel length; : channel width; {
Threshold Inversion Point: point of maximum space charge with
. At this point:
) ( ( : drain current
√
. Then:
)
(√
(
)
√
(
))
oxide breakthrough avalanche breakdown in the blocking pn-diode (source-drain) punch-through breakdown the source-drain space charge region extends until it touches the other pn-junction short circuit
Work Function Differences: - metal gate -
gate
(
p-substrate: (
p-substrate:
)
Random Stuff: nmos (normally not conducting):
)
MOS-FET
-
gate p-substrate: : metal to semiconductor work function difference Flat-Band Voltage: ⁄
Exam: only n-channel = p-type MOSFET. Everything below is for this case.
⁄
: trapped oxide charge per unit area Threshold Voltage (Schwellspannung): when ( (
)
(
)
{
) : max. space charge density per unit area of depl. region (
)
advantage of MOSFET: no control power needed
Transconductance: influence of gate voltage on drain current:
√
ADMINISTRATIVE STUFF
)
MOS-FET: OTHER STUFF Cutoff-Frequency:
CAPACITANCE -VOLTAGE CHARACTERIST ICS
(
(
)
Exam: 4 KP, 2.5 h, with calculator, all writen material allowed In the calculator: use nnc instead of nc (variable used) to do: define additional units in the calculator
SEMICONDUCTOR PHYSICS & DEVICES
⏞ ( ) ( ) ( ) ( ) ( )) ( : Fermi-Dirac-Probability, : density of states in cond. band ( ) ( ) ( )
Summary of the lectures held by Prof. Dr. A. Schenk
(
CONSTANTS & MATERIAL PARAMETER S
Planck’s constant: Dielectric constant (für cm):
(
(
)
⏞
)
) (
√
)
(
(
)
)
(
⁄
)
(
(
√
)
)
⁄
)
(
)
for
)
(
)
( ⏟
(
)
⁄
(
( (
) (
)
(
MATERIAL-PARAMETER GERMA NIUM Relative permittivity: Intrinsic carrier concentration:
MATERIAL-PARAMETER ANDERE
(
)
(
)
(
)
)
4. SEMICONDUCT OR IN EQU ILI BRIUM direct semicond.: if the minimum of the conduction band is at the same place like the maximum of the valence band. indirect semicond.: if it is not a direct semiconductor. This means a phonon is also needed actually get an electron from valence to conduction band. Problematic in optical systems. intrinsic: no impurities, lattices defects, ... noted as , extrinsic: aritificially doped semiconductor thermal equilibrium: no applied voltage or current,noted as donor atoms: create n-type semiconductor ⁄ fermi level: with ( ) eff. mass approx.: so quantum effects can be neglected
CHARGE NEUTRALITY
In case of an applied voltage we get
CHARGE CARRIERS I N S EMICONDUCTORS
and :
for n-type at low injection (
) and
:
( ⏟
→
(
( ) ( )
( )
( ( )
. )
(
)
)
(
)
(
)
⁄ ( ) for abd ( )⁄ ⁄ ( ) ( ) { ⁄ ( ) Appilcation 4: finite number of electron-hole pairs is generated at , but for . n-type semicond. with constant applied E-field in direction. Solution:
Application 3:
)
(
√
(
)
)
Diffusion Length: →
and get
)→
)
⁄
)
)
we have
(
(
Derivation: (
( )
)
⁄
)→
)
( )
konst and
POSITION OF THE FERMI -ENERGY -LEVEL
(
)
) and
(
Einstein Relation:
In the special case
(
⏟
( ) ( ) ⁄ Application 2: homogeneous n-type semicond. zero applied electr. field, therm. eq. for , uniform gen. rate for
)
(
and
for p-type at low injection (
(
GRADED I MPURITY DISTRI BUTION
√(
)
)
Application 1: n-type semicond., uniform conc. of excess holes:
TOTAL CURRENT DE NSIT Y
Complete ionization, if above expression is Doping of Semiconductors: p-doping: e.g. Boron with 3 instead of 4 valene electr. n-doping: e.g. Phosphorus, 5 instead of 4 valence electrons near degenerate: conducts always metallic
]
(
⏟
( ⏟
⏞ ⏟
⏟
APPLICATIONS OF AMBI POLA R TRANSPORT
charge neutrality in thermal equilibr. at complete ionisation: [
)
CARRIER DIFFUSI ON
Relative number of electrons/holes on the doping energy levels relative to the overall number of charge carriers.
hole mobility: effective electron mass: effective hole mass:
⁄ ⁄
( ⏟
→
)
IONIZATION & FREEZE OUT
)
⏟
with )
( )
:
Ambipolar Transport for p-type (for n-type replace n with p)
5. CARRIER TRANSPORT PH ENOMENA
)
⏟(
( )
and
( )
AMBIPOLAR TRANSPORT
)
CARRIER DRIFT
for intrinsic case:
electron mobility:
) material at low-level injection ( )
Continuity Equations: p.195ff
nt ns c cas
Relative permittivity: Intrinsic electron concentration: Eff. state density in conduct. band: Eff. state density in valence band: band gap: electron affinity:
Rel. permittivity of Doping density
for p-type (
-PRODUCT
MATERIAL-PARAMETER SILIZIU M jeweils bei
⁄
(
(
∫ (
)
)
hole conc.
Elementary charge: Boltzmann’s constant:
(
∫
electron conc. (
CARRIER GE NERATION & RECOMBINATION
)
Electron & Hole Concentration:
Lukas Cavigelli, July 2011 [email protected]
(Materialparameter
: excess minority carrier electron lifetime : excess electron and hole recombination rates
-
( )
( )
( )
( )
( )
√ →
( )
( )
Hall Effect: left out, see p.177ff
6. NON-EQUILIBRIUM EXCESS CARRIERS Thermal equilibrium: ⏟
⏟
: excess electron and hole generation rates : excess electron concentration
√
Dielectric Relaxation Time Constant: time it takes to until neutrality is achieved after a burst of excess carriers: ⁄
OTHER SUBTOPICS Quasi-Fermi Energy Levels: see p.216ff Excess-Carrier Lifetime (Shockley-Read-Hall): p.219ff
⁄ : minority carriers conc. in p-region, th. equ. ⁄ : min. carriers (holes) c. in n-region, th. equ. : total minority carrier conc. in p-region, analogous ( ): min. carrier conc. in p-region at space charge edge : excess min. carrier conc. in p-region
7. PN JUNCTION BASIC STRUCTURE p-region ( n-region ( In fact we reassign
) ) ,
in p-region, …
CARRIER DE NSITIES WI TH FORWARD VOLTAGE Boundary Conditions:
Polarity: in case of n-type semicond.: reverse-biased is +-pole connected to semiconductor Ideal Junction Properties:
Zener Breakdown: Electrons start tunneling through the potential barrier.
)
(
)
(
(
)
)
)
(
(
( )
( (
: We have a reverse voltage, if n-region is connected to higher potential than the n region. Very little current flows. ; Forward voltage. p-region connected to higher potential. Large current flows.
( ( )
)
)
√
(
)
)
(
)
)
(
(
(
√
(
)
Remark: is the cutoff current “Sp Small Signal Model: p. 286ff
: distance from potential maximum to the junction : reduction of the Schottky barrier )
) √
)
st om”
GENERATION-RECOMBINATION CURREN TS
BASIC PROPERTIES
Reverse-Bias Generation Current: (p. 297)
Built-in potential (Diffusionsspannung): (
(
)
)
(
)
(
{
∫
Special cases with interfacial layer: p. 336 Current Voltage Relationship:
Forward-Bias Recombination Current: (p. 300)
Derivation: ∫
√
{
)
(
(
Space Charge Width:
)
(
)
( (
)
)
)
(
)
Diode Current-Voltage Relationship: √
)
)
Minority Carrier Distribution: p.275ff Ideal pn Junction Current: ( )
(
Schottky-Effect, “Bildkrafteffekt”: Maximum cutoff voltage declines with higher applied reverse voltage.
heterojunction: junction with different semicond. materials (
)
√
9. METAL-SEMICONDUCTOR JU NCTIONS
(
(
√
(
(
)
(
)
)
)
√
Schottky-Diode: ( ) Ohmic contact: with n-type, or with p-type This can be achieved through heavy doping. See page 345ff Heterojunctions: see p. 349
if p-semicond. band is bent down instead of up; electr. in semicond., holes in metal
Capacitance:
SCHOTTKY BARRIER DI O DE (
)(
)
One-Sided Junctions: for -junction: For non-uniformly doped junctions: redo all from scratch
8. PN JUNCTION DIODE
JUNCTION BREAKDOWN Voltage for Avalanche Breakdown in one-sided junction: p.305
: Donor conc. in n-region
: ideal Schottky barrier height, potential barrier seen by electrons moving from metal into the semiconductor : metal work function (table) : electron affinity of the semiconductor (table) : potential difference between conduction band and Fermienergy of the n-type semiconductor (
: Acceptor conc. p-region, maj. carrier hole density in p-region in thermal equilibrium with and
: doping conc. in weak doped half of the junction ( ) has to be looked up in a table.
)
METAL-SEMICONDUCTOR O HMIC CONTACTS
)
Maximum Electric Field: (for reverse applied bias) (
(
: Richardson constant Comparison Schottky pn Diode: p.341
: ideality factor. large diffusion dominates if small recombination dominates (
⏟
)
Remark: This does not really work out in terms of its units. Feel free to exchange eV and V (no calculation needed).
SCHOTTKY V. PN -DIODE
Forward Voltage Cutoff Current Max. Cutoff Voltage Switching Speed
Schottky
pn
to very good, just a few ps
multiple kV Diffusion Capacity, reverserecovery effect
Punch-Through Breakdown Voltage: p. 208 if B-C space charge region increases until it reaches the B-E space chage region. ( ) ( )
LOW-FREQUENCY CURRENT GA IN
10. BIPOLAR TRANSIST OR by region:
Currents:
: : :
√ (
with
)
Avalanche Breakdown:
(Alpha-)Cutoff Frequency: |
⁄
whereby is determined empirically.
where the BE diode is conducting and the CB diode is not. In act v mod , th BE d od s th “actual” pn junct on wh as the CB junction is just used to suck off the electrons.
SIMPLIFIED CURRENT RELATIONS ( ) ⏟
(
)
this formula is only valid if recombination in the collector can be neglected (which should usually be the case) : thermal equilibrium electron conc. in the base : base width; : electron diffusion coefficient
: common-base current gain; : common-emitter current gain ) Attention: the calculation of is extremely sensitive!! ( Modes of Operation: Cutoff: almost no cu nt, “sw tch d off” Forward active: usual mode, lin. amplification control volt.: , large curr.: Saturation: “sw tch d on”, no fu th ampl f. forward-active:
⁄ Beta-Cutoff frequency: : or respectively at low frequency Large Signal Switching, Schottky-Clamped Transistor, other bipolar Transistor Structures: see book Possible Implementation:
: diffusion of minority carrier electrons in the base at : diffusion of minority carrier electrons in the base at : recombination of excess minority carrier electrons with majority carrier holes in the base. is the flow of holes into the base to replace recombined holes. : diffusion of minority carrier hole in the emitter at : recomb. of cerriers in the forward-biased B-E junction : diffusion of min. carrier holes in the collector at : generation of carriers in reverse-biased B-C junction Currents are B-E junction currents only. Currents are B-C junction current only. Derivation of Current Gain:
: breakdown voltage between C and B, with E unconnected
all factors should be for strong amplification. emitter injection efficiency factor :
MINORITY CARRIER DIS TRIBUTI ONS
( (
⁄ ⁄
) )
→
Other Effects: high injection (p. 401), emitter bandgap narrowing (p. 403), current crowding (p. 405), non-uniform base doping (p. 406) Open Circuit Calculation: p. 411
EQUIVALENT CIRCUIT M ODE LS - Ebers-Moll Model: p. 412 - Gummel-Poon Model: p. 416 - Hybrid-Pi Model: p. 418
EXAMPLE
CALCULATION
( (
⁄ ⁄
√
→ (
)
(
⁄
( )
)
recombination factor : ( ( (
)
) )
Early-Effect / Base Width Modulation: p.397 shows effects of not constant, but ( )
(
⁄
)
( )
( )
( )
(
(
)
((
(
(
(
)
)(
)
)
)(
) (
) )
√
)
)
) √
(
: excess minority electron concentration in the base : minority electron conc. in the base in thermal equilibrium ( ): total minority electron conc. in the base
) √
( (
) )
FREQUENCY LIMITATION S : emitter-to-collector delay, : emitter-base junction capacity charge time, : base transit time, : collector depletion region transit time, : collector capacity charge time (
: Early voltage; saturation:
⁄ ⁄
: diffusion length; : real width of the base/emitter (i.e. and ) base transport factor :
NON-IDEAL EFFECTS
cutoff:
|→
)
(
)
Note: the npn transistor only works due to its geometry (small base)
: output conductance (
√
)
(
MOS-CAPACITOR
)
Exam: only n-channel = p-type MOSFET.
( (
)
)
( (
MODES OF OPERATION
)
Non-Saturation:
)
MOS-FET: NON-IDEALITIES
depends on
.
Miller-Effect:
𝐼𝐷𝑆
Everything below is for this case.
√ : maximum width of the space charge region in inversion
MODES OF OPERATION BY MOS -CAP Accumulation: negative voltage at gate holes accumulate in channel region channel fills up with holes not conducting Depletion: pos. voltage at gate not yet enough electrons in channel region to be conducting Threshold: pos. voltage at gate ( ) #electr. in space charge region = #holes in substrate as if channel region not doped Inversion: gate voltage more electrons than holes in channel region like n-doped material, although actually p
DEPLETION LAYER THIC KNESS (
Saturation: does not depend on anymore. Linear ( amplification of . Usually the mode wanted.
)
𝑡 Miller-Plateau Other non-idealities: subthreshold conduction MOSFET already conducts for voltages channel-width modulation
accumulation: Meaning of the Threshold Voltage:
√
)
depletion: not accumulation, but not conducting (
)
MOS-FET: CURRENT -VOLTAGE CHARACT. Currents: MOS-FET conducts, if inversion: conducting,
√
( ) ( ( ) ( ) : channel length; : channel width; {
Threshold Inversion Point: point of maximum space charge with
. At this point:
) ( ( : drain current
√
. Then:
)
(√
(
)
√
(
))
oxide breakthrough avalanche breakdown in the blocking pn-diode (source-drain) punch-through breakdown the source-drain space charge region extends until it touches the other pn-junction short circuit
Work Function Differences: - metal gate -
gate
(
p-substrate: (
p-substrate:
)
Random Stuff: nmos (normally not conducting):
)
MOS-FET
-
gate p-substrate: : metal to semiconductor work function difference Flat-Band Voltage: ⁄
Exam: only n-channel = p-type MOSFET. Everything below is for this case.
⁄
: trapped oxide charge per unit area Threshold Voltage (Schwellspannung): when ( (
)
(
)
{
) : max. space charge density per unit area of depl. region (
)
advantage of MOSFET: no control power needed
Transconductance: influence of gate voltage on drain current:
√
ADMINISTRATIVE STUFF
)
MOS-FET: OTHER STUFF Cutoff-Frequency:
CAPACITANCE -VOLTAGE CHARACTERIST ICS
(
(
)
Exam: 4 KP, 2.5 h, with calculator, all writen material allowed In the calculator: use nnc instead of nc (variable used) to do: define additional units in the calculator
