MSE 141 Electronic MaterialsSemiconductors and DevicesDr. Benjamin O. ChanAssociate ProfessorFebruary 2012

Problem Set 3•Hummel 3rd edition•Chapter 8

▫1, 4, 6, 7, 9, 13, 15, 22

•Due Feb. 16•Final Exam

▫Thursday Feb. 23

Semiconductors•Active electrical component

▫Metals and their alloys are usually passive▫Non-linear I-V characteristics

Diodes, transistors, lasers Sensors, amplifiers, light source

•Can manipulate conductivity•Si the most common material

▫Ge: first material used▫Other semiconductors

GaAs, AlGaAs, GaN, InP, CdTe

Si Success•Ease of oxidation

▫Wet or dry oxidation Natural oxide forms after a few days when

exposed to air▫SiO2 layer provides good insulating and

masking functions Photolithography

Masking function Circuit Fabrication

Electrical insulation

Bond Modification•Hybrid states

▫2 s states and 6 p states combine to form 8 sp states

▫8 sp states split into 2 branches containing 1 s and 3 p states Lower s state: 1 electron per atom Lower p states: 3 electrons per atom

•Valid for covalently bonded materials

Overlapping sp Levels

Energy Bands•Valence band contains 4N electrons

▫Completely filled•Conduction band can accommodate 4N

electrons▫But of course it is empty!

•States are filled like water fills a vessel!

Band Structure of Si

Energy Gap at 0 KElement Eg (eV)

C (diamond) 5.48

Si 1.17

Ge 0.74

Sn (gray) 0.08

The Energy Gap•Decreases with atomic number

▫Artificial diamond: C-based IC▫Ge: good IR detector▫Sn easily conducts at room temperature

•Wavelength equivalent

)(24.1)(eVE

m

Temperature Dependence

▫where Eg(0) is the energy gap at 0 K▫ = 5 x 10-4 eV/K and▫D = Debye temperature (Table 19.2)

Temperature needed to reach 96% of the final value for heat capacity CV

T> D : classical region T< D : QM consideration

Dgg T

TETE

2

)0()(

Eg vs. T•Eg decreases with

increasing temperature

•For Si, ▫D=650K, ▫Eg reduction = -2.4

x 10-4 eV/K

Intrinsic Semiconductors•Intrinsic = pure•Electrical conduction

▫Electrons must be excited from the valence to the conduction band

▫Small Problem Thermal energy at room temperature (kBT) is

only 25.8 meV. How can electrons cross Eg (1.10 eV for Si at T=300K)?

Interband Transition•Electron goes from valence

to conduction band▫“hole” appears in valence band

•Intrinsic carrier concentration (298 K)▫Si: 1.5 x 1010 cm-3

▫GaAs: 1.1 x 106 cm-3

•Seatwork▫Determine the concentration of intrinsic

carriers relative to the number of atoms in Si and GaAs.

Ec

Ev

Fermi Energy•Probability of occupation= F(EF) = ½•T = 0K

▫E<Ev: F(E)=1▫E>Ec: F(E)=0

•EF located in the middle of the energy gap•At higher temperatures F(E) can become

less than 1 for E<Ev and F(E) can be bigger than 0 for E>Ec

Fermi Energy

Seatwork/Homework•Using the Fermi distribution, calculate

the probability of occupation for E > Ec for Si and GaAs at room temperature.

Why should EF be in the middle of the energy gap?

Density of Electrons and Holes

Electrons and Holes•Number of electrons in the conduction

band = number of holes in the valence band

•Expression for Nh similar to Ne

•Although me* ≠ mh*

Intrinsic Carrier Concentration

Mobility •Conductivity is not determined by carrier

concentration alone!

•Drift velocity v per unit electric field E

Ev

Conductivity• Recall

and

• Substituting m expression,eNe

EvN

Nvej

Ej

Temperature Dependence of Mobility, Carrier Density and Conductivity

Extrinsic Semiconductors•Extrinsic=Impure

▫No such thing as pure semiconductor▫Defective by nature

•Doping=Intentionally Impure▫Group III impurity

Missing electron=hole=p-type▫Group V impurity

Extra electron=n-type

Group V Impurity in Si

Intentional Impurities•Usually in the ppm range•Group V

▫Donates an electron to the material▫Impurity energy level just below conduction

band•Group III

▫Accepts an electron from the material▫Impurity energy level just above valence band

Donor and Acceptor Levels

Majority and Minority Carriers•Majority carrier

▫Usually from dopant (extrinsic carriers)•Minority carrier

▫Usually from interband transitions (intrinsic carriers)

•Valid for reasonably low T•The picture can change at high T

▫Intrinsic carriers become the majority!

How many carriers can you put in the material?•Limited by solid solubility curve for

impurity▫Hume Rothery rules!

•Crystal structure should be preserved▫Complete solubility not required!

Solid Solubilities of Impurities

Carrier Concentration•Intrinsic

•n-type

•p-type

Carrier Concentration vs. Temperature

Conductivity and Temperature

Fermi Energy Changing with Doping and Temperature

Determining Carrier Concentration•Indirect measurement

▫Resistivity is easier to measure compared to conductivity

▫Correlate resistivity with carrier concentration

•Direct Measurment▫Hall Effect

Resistivity Measurements

lRA

R IV

sample ammeter

voltmeter

Problems with Resistivity Measurements•Geometrical shape required for specimen

▫Uniform cross-sectional area•Contact resistance•Loading effects from meter

▫Voltmeter: should not draw current High impedance required

▫Ammeter: should not impede current flow Zero resistance desired

•Stray capacitance and inductance

Four Point Probe Technique(ASTM F84-84, F374-84)•Suitable for wafer geometry

▫Rectangular/circular/cylindrical/slab•Minimizes contact resistance and loading

effects•Four probes separated equally by distance

s▫Outer probes: constant current supply▫Inner probes: voltmeter

•Sheet resistance Rs)(CF

IVRs

Four Point Probe•Correction Factors

▫Geometrical in nature▫Something to do with s

and how big the sample dimensions are with respect to it

•Computing resistivity

▫w=sample thicknesswRs

Correction Factor Limit•Rs formula valid for w « a or d

•BONUS PROBLEM▫In the limit as d»s, show that

CF = (/ln 2) = 4.54

Why do you need a constant current source for the four point

probe set-up?(Why is a constant voltage source

not sufficient?)

Resistivity and Carrier Concentration

Empirically Generated Curves!•There are other ways to determine carrier

concentration!▫Non-destructive▫Destructive

Rutherford Backscattering (RBS) Secondary Ion Mass Spectrometry (SIMS) Neutron Activation Analysis (NAA)

Carrier Type Determination•Is the material n-type or p-type?•Seebeck Effect

▫Double thermocouple•Works for

▫Bulk material▫Films on high resistivity material▫Films on opposite type substrate

Hot Probe (ASTM F42-77(87))

Voltmeter/Ammeter

+

How it works…•Hot probe has greater carrier

concentration•Positive terminal of meter must be on the

hot probe▫If p-type, electrons flow from cold to hot

junction: meter records positive reading▫If n-type, electrons from flow from hot to cold:

meter records negative reading•If the tester leads are inverted, the

deflections will be inverted! (this can be confusing!?)

Determining Resistivity and Carrier Type Simultaneously

•Hall Effect Apparatus▫Can measure

concentrations down to 1012 electrons/cm3

▫Sensitivity is several orders of magnitude better than any chemical analysis

▫Applies to bulk material▫Pass a current in a material

immersed in a magnetic field

Hall Effect•Consider a rectangular block of n-type

semiconductor immersed in a magnetic field with a magnetic induction B in the z-direction

•Let a current density j flow through the material in the +x direction

•Flowing electrons experiencea Lorentz force deflecting them from their pathBvqF

Hall Effect• Lorentz force makes the electrons drift to one

side of the material• Accumulation of charge sets up an electric field

▫ Hall field

• Hall field builds up until it balances the Lorentz force

• The current density j in the material iseNvj x

zxy BvE

EeF

N and the Hall CoefficientWe can calculate the number of conduction electrons N (per unit volume)

•Define the Hall constant/coefficient

▫coefficient is inversely proportional to N▫sign indicates carrier type

NeRH

1

y

zx

eEBjN

Limitations of the Hall Technique•Heating Effects

▫An electromagnet is usually used▫Current required is in the AMPS range

typically•Do the measurements quickly•Demagnetization is necessary when

repeating the experiment

Capacitance-Voltage Technique•Junction or schottky barrier needed•The variation of capacitance with voltage

is indicative of carrier concentration•Depth profiling is possible

▫5 to 10 m▫Depends on carrier concentration

Affects reverse bias conditions

Metal-Semiconductor Contacts• Desired effect: Ohmic contact

▫ Obeys Ohm’s law (linear I-V characteristics)▫ Interconnection between devices▫ Interface to external circuit

• Another effect: rectifying contact▫ Schottky Barrier▫ Conducting in forward bias, non-conducting in reverse

bias▫ Undesirable for ohmic contacts▫ Can fabricate devices based on this effect

MOS (Metal-Oxide-Semiconductor) MIS (Metal-Insulator-Semiconductor)

What are the conditions that determine whether you will get an ohmic or

rectifying contact?

Surface States•Consider an n-type free surface

▫Surface states generate free electrons making it charged

▫Negative charge repels electrons near the surface, exposing ionized donors

▫Depletion layer results (layer with fewer electrons compared to the bulk material)

▫Energy bands curve upwards towards the surface indicating repulsion of electrons from the surface (potential barrier!)

Band Bending at the Semiconductor Surface

Band Bending for p-type Semiconductor•Consider p-type free surface

▫Holes migrate to free surface▫Band edges bent downward toward the

surface▫Potential barrier for holes is established

near the surface

The Work Function •Energy required to take an electron from

EF to ▫Appendix 4▫Difference between EF and vacuum (free

electron state) Metals: Ionization energy Semiconductors: depends on carrier type

▫Different from electron affinity

Metal/n-type Semiconductor Interface•Before contact

▫M> S

•Upon contact▫Electrons from conduction band of

semiconductor flow into conduction band of metal until EF equalizes

▫Metal is negatively charged▫Potential barrier for electrons forms▫EC in bulk semiconductor is lowered by M - S

Metal-Semiconductor Interface I

Metal-Semiconductor Interface II

Metal/p-type Semiconductor Interface•Before contact

▫M < S

•Upon contact▫Electrons diffuse from metal to semiconductor ▫Metal is positively charged▫Downward potential barrier for holes forms▫EC in bulk semiconductor is lowered by M - S

Contact Potential•The difference between work functions of

the metal and semiconductor (or any two materials in contact with each other)▫M- S

•Potential barrier height from metal side▫M- ▫ is the electron affinity

Currents•Diffusion Current

▫Current driven by concentration gradients▫Equilibrium state: Electrons flowing from the

semiconductor to the metal and vice versa must be equal

•Drift Current▫Current driven by electric fields (drift velocity)▫Bent bands imply an electric field in the

material: holes are swept in the direction of the field and electrons opposite to it

Current Across the Barrier•Metal/n-type semiconductor junction

▫DC bias: metal negative bias▫Metal becomes more negative▫Electrons in semiconductor repelled more▫Potential barrier increases making depletion

region wider▫Diffusion currents become negligible▫Voltage-independent drift current produces

small and constant electron current from metal to semiconductor (reverse bias)

Forward and Reverse Bias on Metal Semiconductor Interface

Forward Bias•Positive terminal on metal

▫Potential barrier reduces▫Electrons driven over the barrier▫Large current results from semiconductor to

metal▫Depletion layer reduces

•Rectifying property useful for converting AC to DC

I-V Characteristic

Comparison with pn Junction

Current Equations•Metal to semiconductor

▫A = contact area, C = constant•Semiconductor to metal

▫V = bias voltage•Saturation current

kTMS

MeACTI 2

kTeVSM

SMeACTI 2

kTS

SMeACTI 2

Net Current

• Forward bias: V>0▫ Inet increases exponentially▫ Watch out for the knee!

For Si, knee voltage = 0.7• Reverse bias: V<0

▫ Inet ~ -Is

▫ Graphs exagerrate Is so you can see it 3 orders of magnitude smaller than forward current (A – mA)

1 kTeVSMSSMnet eIIII

Advantages of Schottky Diode•Involves only one type of conduction

carrier (electrons)▫No mutual annihilation (recombination) of

electrons and holes can occur▫Can be switched more rapidly from forward to

reverse bias Suitable for microwave-frequency detectors

•Metal contact provides better heat removal ▫Helpful in high power devices

Metallization (Ohmic Contact)Metal/n-type interface▫ M < S

▫ Electrons flow from metal to semiconductor▫ Bands bend downward▫ No potential barrier forms impeding electron flow in

either direction▫ Current can be injected in and out of the semiconductor

without significant power loss▫ Ohmic contact is formed!

• Metal/p-type interface▫ M > S

Ohmic Contacts

Aluminum on Silicon• Common material for metallization on Si or SiO2

▫ Relatively good conductor▫ Low melting/annealing temperatureAl < Si▫ Ohmic contact formed with p-type Si

Improve contact by diffusing Al into Si generating a p+ (heavily doped) region at the contact point

▫ Rectifying contact on n-type Si Avoid rectification by heavily doping (n+) contact region

Depletion layer becomes thinner Makes electron tunneling possible! Ohmic contact formed!

p-n Junction•Diffuse n-type impurity into p-type

material (or vice versa)•EF equalizes

▫Electrons flow from n to p▫Holes flow from p to n▫Depletion layer is formed▫Electric field across junction builds up until no

net charge transfer occurs across the junction

p-n Junction

The Biased Junction• Forward bias

▫ p-side positive▫ Electrons driven from p to n, holes from n to p▫ Depletion layer narrows and barrier height decreases▫ Large electron flow from n to p

• Reverse bias▫ n-side positive▫ Electrons driven from n to p, holes from p to n▫ Depletion layer widens and barrier height increases▫ Small drift current

Forward and Reverse Bias

Shockley Equation• Ideal diode law

▫ where the reverse saturation current is

▫ Cep ,Chn = equilibrium concentration of electrons in p, holes in n

▫ D’s and L’s = diffusion constants and diffusion lengths

1 kTeVS eII

hn

hnhn

ep

epepS L

DCLDC

AeI

Diffusion•D related to mobility via Einstein relation

•Minority carrier diffusion length L

▫ tep = lifetime of electrons in p (before recombination)

ekT

D epep

epepep DL t

Is•Keep Is small

▫Keep minority carrier concentrations (Chn and Cep) low compared to electrons and holes introduced by doping

▫Selecting semiconductors with with large energy gaps and high doping levels will help!

Deviation from Ideal Behavior•Surface effects

▫Surface channels/depletion region form▫Leakage current generated

•Generation-recombination of carriers in the depletion region

•Tunneling of carriers between states in the bandgap

•High injection condition (np~pp or pn~nn)•Series resistance effect

Ideality Factor

where n is the ideality factorn=2 recombination

current dominatesn=1 diffusion current

dominates1<n<2 both currents

contribute

1 nkTeVS eII

Junction Breakdown• Usually under reverse bias• Junction conducts a large current (>Is)• Irreversible when current becomes too large (mA

range)• Reverse bias bigger than critical value can result

in impact ionization▫ Can excite other electrons into the conduction band▫ Results in breakdown (reverse current increases

rapidly)▫ Avalanche process▫ Depends on degree of doping

Higher doping = lower breakdown voltage

Zener Diode•Zener breakdown

▫Tunneling effect at low reverse voltages (<4V for Si diodes)

▫Occurs under heavy doping Barrier width very thin Some valence electrons on the p-side are

opposite empty conduction band states on the n-side

Zener Diode• Breakdown voltage

is quite consistent• Zener diode is not

destroyed by breakdown▫ Unless excessive

heat dissipation causes it to melt

• Useful in circuit protection devices, e.g., voltage regulators (power supplies)

Photodiodes•Bare p-n junction

▫Exposed to light of energy > Eg

▫Electron-hole pairs generated▫Electrons in the depletion layer are swept

to the n-side▫Holes are swept to the p-side▫External circuit can detect the presence of

these charge carriers

Solar Cell (Photodiode)

Carrier Collection

Basic Features•Open circuit voltage

▫Connect voltmeter across the solar cell▫Typically 500-700 mV

•Short circuit current▫Connect ammeter across solar cell (short

circuit!)▫More indicative of the carriers generated by

light•I-V Characteristics

▫Diode curve shifted down by light generated current: Ilight=Idark-IL

I-V Characteristics•Fourth quadrant curve

▫Negative power! (P=IV)▫Power generated!

•Efficiency▫Ratio of output power over input power

•Fill Factor▫How well diode curve fills box bounded by

I-, V- axes and Voc, Isc

I-V Curve•Desired solar cell operation at the

maximum power point▫Determined by the resistive load in the circuit

and the intensity of light impinging on the cell•Rated efficiency

▫Pin=1000 W/m2 on earth’s surface (high noon)▫Temperature dependent (std. temp. = 25C)

inPVI maxmax

Avalanche Photodiode•p-n photodiode operated at high reverse

bias (near breakdown)•Generated electron-hole pairs are

accelerated and ionize lattice ions▫Secondary electron-hole pairs generated▫10 to 1000 photocurrent gain

•Suitable for low-light-level and very high frequency applications

Tunnel Diode•Narrow depletion layer due to heavy

doping•Occupied valence band of p-side higher

than occupied conduction band of n-side▫Small reverse bias▫Electrons tunnel from p to n side

•Empty valence band lower than occupied conduction band of n-side▫Small forward bias▫Electrons tunnel from n to p side

Tunnel Diode

Negative Resistance•Moderate forward bias: non-conducting

▫Negative resistance region▫Current drops with voltage

•Normal forward bias: conducting•Applications

▫Positive resistance dissipates energy▫Negative resistance generates energy▫Oscillators with zero net resistance won’t lose

energy

Transistors•Bipolar junction

transistor▫npn or pnp▫Back-to-back pn

junctions▫Three terminal device

Base Emitter Collector

BJT’s•Biasing conditions for amplifier

applications▫Base-emitter forward biased▫Base-collector strongly reverse biased▫Electrons injected into emitter, amplified at

collector

Amplification•Forward biased emitter = small resistivity•Reverse biased base collector = much

larger resistivity•Power is larger in larger at the base

collector▫Current is identical in both junctions▫Power gain!

AC Amplification•Base voltage controls current from

emitter to collector▫Large forward bias decreases potential barrier

and electron injection into the p area is relatively high

▫Small forward voltage results in smaller electron injection

•Strong collector bias mimics and magnifies base voltage

Transistor Switch•The electron flow from emitter to

collector can be arrested via an appropriate base voltage

Desirable Characteristics (npn)•Large electron density in the emitter

requires it to be heavily doped (n-type)•Recombination with holes in the base area

should be kept to a minimum▫Base should be lightly doped (p-type)▫Base should be thin (~1m)

Beneficial side-effect: higher frequency response•Doping of collector usually not critical

▫Lightly doped for high gain and low capacitance

Collector I-V Characteristics

MOSFET•Metal-Oxide-Semiconductor Field Effect

Transistor•FET

▫Channel connecting a source S to a drain D▫Channel has the same type as source and drain▫Unipolar: only one type of charge carrier▫Gate voltage applied to channel controls the

flow of charge carriers through it▫Gate has MOS structure: gate voltage produces

field controlling channel width

Depletion-Type MOSFET

Types of MOSFET•Depletion type

▫Normally on: channel is conducting when VG = 0

▫As VG becomes more and more negative, electrons accumulating at the oxide interface repel electrons in the channel into the substrate until the channel is pinched off (turned off!)

•Enhancement type▫Normally off: no channel when VG=0▫Channel forms when VG>0: holes are driven

from oxide interface

Enhancement-Type MOSFET

More MOSFET’s•NMOSFET

▫n channel•PMOSFET

▫p channel•CMOSFET

▫p and n channels integrated into a single chip▫Complementary MOS▫Dominant technology for IT

Low voltage (0.1V), low power consumption, short channel length (high speed performance)

JFET•Junction FET

▫Channel consists of a pn junction▫Normally on: zero bias provides maximum

source to drain current▫Reverse bias increases depletion layer until

it pinches off the channel

JFET

Types of JFET• n-JFET

▫ n-type channel• p-JFET

▫ p-type channel• BIFET

▫ Combined bipolar transistors with JFET• MESFET

▫ Uses a metal semiconductor junction• MODFET

▫ Modulation doped (thin AlGaAs film on undoped GaAs substrate)

MESFET

Advantage of GaAs Over Si•Faster switching speeds

▫Electron mobility 6-fold compared to Si▫Hole mobilities lower

Stick to n-type material•More setbacks?

▫Si has larger thermal conductivity More power needed for faster switching rates

▫Electron mobilities are comparable at high fields

Quantum Structures•How small can you make transistor?•Quantum Dot

▫3-D confinement (0-D freedom)•Quantum Wire

▫2-D confinement (1-D freedom)•Quantum Well

▫1-D confinement (2-D freedom)

Quantum Dot

Q Dot Device•GaAs sandwiched in wider gap AlGaAs

▫AlGaAs conduction band higher than GaAs▫Potential barrier between GaAs regions

•At sufficiently large bias filled conduction band levels of n-GaAs levels out with empty conduction band levels in the middle n-GaAs▫Tunneling can occur

Q Dot Band Structure

Negative Differential Resistance•Tunneling can come to a near standstill at

slightly higher voltages▫Available empty energy levels not aligned

with free electrons

Q Dot I-V Characteristic