SEMICONDUCTOR PHYSICS & DEVICES

Summary of the lectures held by Prof. Dr. A. Schenk

Lukas Cavigelli, July 2011

lukasc@ee.ethz.ch

CONSTANTS & MATERIAL PARAMETERS

Elementary charge:

Boltzmann’s constant:

Planck’s constant:

Dielectric constant (für cm):

MATERIAL-PARAMETER SILIZIUM

(Materialparameter jeweils bei )

Relative permittivity:

Intrinsic electron concentration:

Eff. state density in conduct. band:

Eff. state density in valence band:

band gap:

electron affinity:

electron mobility:

hole mobility:

effective electron mass:

effective hole mass:

MATERIAL-PARAMETER GERMANIUM

Relative permittivity:

Intrinsic carrier concentration:

MATERIAL-PARAMETER ANDERE

Rel. permittivity of

Doping density [ ]

4. SEMICONDUCTOR IN EQU ILIBRIUM

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 CARRIERS IN S EMICONDUCTORS

Electron & Hole Density:

( ) ( ) ( ) ( ) ( ( ))⏞

: Fermi-Dirac-Probability, : density of states in cond. band

( )

(

) (

( )

)

Electron & Hole Concentration:

electron conc. ∫ (

) ⁄

√ (

( )

)

(

) (

( )

) (

) ⁄

hole conc. ∫ (

) ⁄

√ (

( )

)

( ( )

) (

( )

) (

) ⁄

- -PRODUCT

(

) (

)⏟ nt ns c cas

⏞

for intrinsic case:

( )⏟

(

)

IONIZATION & FREEZE OUT

Relative number of electrons/holes on the doping energy levels relative to the overall number of charge carriers.

( ( )

)

( ( )

)

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

CHARGE NEUTRALITY

charge neutrality in thermal equilibr. at complete ionisation:

√(

)

POSITION OF THE FERMI-ENERGY-LEVEL

( ) →

( )

( )

In the special case we have and get .

( ) →

( )

( )

5. CARRIER TRANSPORT PHENOMENA

CARRIER DRIFT

( ) →

( )

CARRIER DIFFUSION

TOTAL CURRENT DENSIT Y

GRADED IMPURITY DISTRIBUTION

In case of an applied voltage we get konst and ( )

( )

⏟ ⁄

→ ( )

( )

( )

Einstein Relation:

Derivation:

- → ( )

( ) ( )

→

( )

( )

( )

( )

Hall Effect: left out, see p.177ff

6. NON-EQUILIBRIUM EXCESS CARRIERS

Thermal equilibrium: ⏟

⏟

: excess electron and hole generation rates

: excess electron concentration

: excess minority carrier electron lifetime

: excess electron and hole recombination rates

CARRIER GENERATION & RECOMBINATION

for p-type ( ) material at low-level injection ( )

( )

for and ( ) :

( )

Continuity Equations: p.195ff

AMBIPOLAR TRANSPORT

Ambipolar Transport for p-type (for n-type replace n with p)

( )

⏟

( )

⏟

⏟

⏟⏞

⁄

⁄

( )

⏟

with ( )

⏟

and ( )

⏟

and

for p-type at low injection ( ) and :

for n-type at low injection ( ) and :

APPLICATIONS OF AMBI POLAR TRANSPORT

Application 1: n-type semicond., uniform conc. of excess holes:

( )

( )

( )

( ) ( ) ⁄ Application 2: homogeneous n-type semicond. zero applied

electr. field, therm. eq. for , uniform gen. rate for

( )

( )

( )

( ) ( ⁄ )

Application 3: 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:

( ) ⁄

√ (

( )

)

Diffusion Length:

√ √

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

7. PN JUNCTION

BASIC STRUCTURE

p-region ( ) n-region ( ) In fact we reassign , in p-region, …

: 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.

BASIC PROPERTIES

Built-in potential (Diffusionsspannung):

(

)

Derivation:

∫ ∫ {

Space Charge Width:

√ ( )

(

)

Maximum Electric Field: (for reverse applied bias)

( )

Capacitance:

√

( )( )

One-Sided Junctions: for -junction:

For non-uniformly doped junctions: redo all from scratch

8. PN JUNCTION DIODE

: Acceptor conc. p-region, maj. carrier hole

density in p-region in thermal equilibrium

: Donor conc. in n-region

⁄ : 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

CARRIER DENSITIES WI TH FORWARD VOLTAGE

Boundary Conditions:

(

)

(

)

( ) (

) ( ) (

)

Minority Carrier Distribution: p.275ff Ideal pn Junction Current:

( )

( (

) ) (

)

( )

( (

) ) (

)

( ) ( ) ( (

) )

(

)

(

√

√

)

Remark: is the cutoff current “Sp st om” Small Signal Model: p. 286ff

GENERATION-RECOMBINATION CURRENTS

Reverse-Bias Generation Current: (p. 297)

( )

Forward-Bias Recombination Current: (p. 300)

(

) (

)

(

)

Diode Current-Voltage Relationship:

( (

) )

: ideality factor. large diffusion dominates if small recombination dominates

JUNCTION BREAKDOWN

Voltage for Avalanche Breakdown in one-sided junction: p.305

with : doping conc. in weak doped half of the junction and ( ) has to be looked up in a table.

Zener Breakdown: Electrons start tunneling through the potential barrier.

9. METAL-SEMICONDUCTOR JUNCTIONS

heterojunction: junction with different semicond. materials

if p-semicond. band is bent down instead of up; electr. in semicond., holes in metal

SCHOTTKY BARRIER DIODE

: 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 Fermi- energy of the n-type semiconductor

( )

Remark: This does not really work out in terms of its units. Feel free to exchange eV and V (no calculation needed).

Polarity: in case of n-type semicond.: reverse-biased is +-pole connected to semiconductor Ideal Junction Properties:

√ ( )

√

( )

Schottky-Effect, “Bildkrafteffekt”: Maximum cutoff voltage declines with higher applied reverse voltage.

√

√

{

: distance from potential maximum to the junction : reduction of the Schottky barrier

Special cases with interfacial layer: p. 336 Current Voltage Relationship:

( (

) )

(

) (

) (

)

⏟

: Richardson constant Comparison Schottky pn Diode: p.341

METAL-SEMICONDUCTOR OHMIC CONTACTS

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

SCHOTTKY V. PN-DIODE

Schottky pn

Forward Voltage

Cutoff Current

Max. Cutoff Voltage to multiple kV

Switching Speed very good, just a few ps

Diffusion Capacity, reverse-recovery effect

10. BIPOLAR TRANSISTOR

by region: : : :

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:

cutoff: saturation:

LOW-FREQUENCY CURRENT GA IN

Currents:

: 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:

all factors should be for strong amplification. emitter injection efficiency factor :

( ⁄ )

( ⁄ )

→

⁄

⁄

: diffusion length; : real width of the base/emitter

(i.e.

and )

base transport factor :

→

(

)

( ⁄ )

recombination factor :

(

)

(

)

(

)

( ⁄ )

NON-IDEAL EFFECTS

Early-Effect / Base Width Modulation: p.397 shows effects of not constant, but ( )

: Early voltage; : output conductance

Punch-Through Breakdown Voltage: p. 208 if B-C space charge region increases until it reaches the B-E space chage region.

( )

( )

with √

( )

Avalanche Breakdown:

⁄

whereby is determined empirically. : breakdown voltage between C and B, with E unconnected

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 MODELS

- Ebers-Moll Model: p. 412 - Gummel-Poon Model: p. 416 - Hybrid-Pi Model: p. 418

MINORITY CARRIER DIS TRIBUTIONS

( ) ( )

(( (

) ) ( ) )

( ) (

)

( (

) ) (

)

( ) (

) (

) √

: excess minority electron concentration in the base

: minority electron conc. in the base in thermal equilibrium

( ): total minority electron conc. in the base

FREQUENCY LIMITATIONS

: emitter-to-collector delay, : emitter-base junction capacity charge time, : base transit time, : collector depletion region transit time, : collector capacity charge time

( )

( )

(Alpha-)Cutoff Frequency:

|

| →

Beta-Cutoff frequency: ⁄

: or respectively at low frequency Large Signal Switching, Schottky-Clamped Transistor, other bipolar Transistor Structures: see book Possible Implementation:

EXAMPLE CALCULATION

( ⁄ ) ( ⁄ )

√ √

(

)

(

)

√ (

)

√

( )

Note: the npn transistor only works due to its geometry (small base)

MOS-CAPACITOR

Exam: only n-channel = p-type MOSFET. Everything below is for this case.

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

( ) √

Threshold Inversion Point: point of maximum space charge

with . At this point: . Then: √

Work Function Differences:

- metal gate p-substrate: (

)

- gate p-substrate: (

)

- gate p-substrate:

: metal to semiconductor work function difference Flat-Band Voltage:

⁄ ⁄

: trapped oxide charge per unit area

Threshold Voltage (Schwellspannung): when

( )

( )

( ) : max. space charge density per unit area of depl. region

( )

√

CAPACITANCE-VOLTAGE CHARACTERIST ICS

( ) ( ) ( )

( ) ( )

√

: maximum width of the space charge region in inversion

accumulation:

depletion: not accumulation, but not conducting ( )

inversion: conducting,

MOS-FET

Exam: only n-channel = p-type MOSFET. Everything below is for this case.

MODES OF OPERATION

Non-Saturation: depends on .

Saturation: does not depend on anymore. Linear amplification of . Usually the mode wanted. ( )

Meaning of the Threshold Voltage:

MOS-FET: CURRENT-VOLTAGE CHARACT.

Currents: MOS-FET conducts, if

( )

{( ( )

)

( ) ( ( )

: channel length; : channel width; : drain current

Transconductance: influence of gate voltage on drain current:

{ ( )

MOS-FET: OTHER STUFF

Cutoff-Frequency:

( )

MOS-FET: NON-IDEALITIES

Miller-Effect:

Other non-idealities: subthreshold conduction

MOSFET already conducts for voltages channel-width modulation

√

(√ ( ) √ ( ))

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

Random Stuff: nmos (normally not conducting):

advantage of MOSFET: no control power needed

ADMINISTRATIVE STUFF

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

𝐼𝐷𝑆

𝑡 Miller-Plateau