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Extrinsic Semiconductors Recombination Carrier Transport Minority Carrier Injection What are we learning today? What are we learning today?

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Page 1: Eng323 Lect02 My

✒ Extrinsic Semiconductors

✒ Recombination

✒ Carrier Transport

✒ Minority Carrier Injection

What are we learning today?What are we learning today?

Page 2: Eng323 Lect02 My

n-type Dopingn-type Doping• A pure Si lattice is entirely made up of Si atoms. Imagine if

some Group V elements like As or P are introduced in small quantities as impurities. This will ensure that the diamond structure of the host lattice is retained.

• As has 5 valence electrons. When As takes up its position in the lattice, 4 of these valence electrons bond with 4 neighbouring Si atoms, & the fifth electron is left unbonded.

• It is left orbiting around the As+ ion core. This being the outermost electron of As, the ionization energy required to free this electron is very low, typically ~ 0.05 eV.

• The average thermal energy of lattice vibration at a temperature T is ≈ 3kT = 0.07 eV at room temperature.

Page 3: Eng323 Lect02 My
Page 4: Eng323 Lect02 My

As+

e-

Arsenic doped Si crystal. The four valence electrons of As allow it tobond just like Si but the fifth electron is left orbiting the As site. Theenergy required to release to free fifth-electron into the CB is verysmall.

Page 5: Eng323 Lect02 My

• Thus at room temperature, this electron will be “free” in the semiconductor.

• In terms of the energy band diagram, this electron will be occupying a state in the CB.

• The addition of As introduces localized electronic states at the As site.

• The energy Ed of these states is ~0.05 eV below EC, meaning thereby that by gaining an energy 0.05 eV, an electron can free itself from the As atom and get released to occupy an empty state in the CB.

• The CB now has an electron, but the localized energy state Ed is now ionized (As+).

Page 6: Eng323 Lect02 My

x

As+ As+ As+ As+

EcEd

Ev

As atom sites every 106 Si atoms

Distanceintocrystal

~0.03 eV

CB

Electron Energy

Page 7: Eng323 Lect02 My

• As+ ions are immobile.

• As atoms are called donor atoms or donor impurity.

• At elevated temperatures (> 100K) all donor atoms are ionized.

• The process introduces free electrons in the CB. (Note: unlike in an intrinsic semiconductor, this process is NOT accompanied by creation of equal number of holes in VB)

• The Ed level created by the impurity is called donor level.

• If Nd is the donor atom concentration, there is a major contribution of electron concentration in CB due to Nd.

• Electrons (in CB) and equal number of holes (in VB) can also be produced by the intrinsic property of Si (ni).

• If Nd is >> ni, then it can be assumed that the total n ≈ Nd, & by law of mass action, p = ni

2/Nd.

Page 8: Eng323 Lect02 My

B-h+

(a) (b)

B-

Free

Boron doped Si crystal. B has only three valence electrons. When itsubstitutes for a Si atom one of its bonds has an electron missing andtherefore a hole as shown in (a). The hole orbits around the B- site bythe tunneling of electrons from neighboring bonds as shown in (b).Eventually, thermally vibrating Si atoms provides enough energy tofree the hole from the B- site into the VB as shown.

p-type Dopingp-type Doping• By adding an element of Group III as impurity, like B. Al, Ga,

etc, we can introduce holes in valence band.

Page 9: Eng323 Lect02 My

• The energy needed to free the hole from the influence of the B- ion is typically 0.05 eV.

• In terms of the energy band diagram, this hole will be occupying a state in the VB.

• The addition of B atoms introduces localized electronic states Ea (called acceptor states) at the B site.

• The energy Ea of these states is about 0.05 eV above Ev, meaning that by gaining an energy 0.05 eV, an electron can move from the top of VB to occupy an acceptor state Ea.

• The VB now has a hole, but the localized energy state Ea is now ionized (B-).

Page 10: Eng323 Lect02 My

x Distanceinto crystal

Ev

Ea

B atom sites every 106 Si atoms

VB

Ec

Electron energy

~ 0.05eV

B - B - B - B -

h+

Energy band diagram for a p-type Si doped with 1 ppm B. There areacceptor energy levels just above Ev around B- sites. These acceptorlevels accept electrons from the VB and therefore create holes in the VB.

Page 11: Eng323 Lect02 My

• B- ions are immobile.

• B atoms are called acceptor atoms or acceptor impurity• At elevated temperatures (> 100K) all the acceptor atoms are

ionized.• The process introduces holes in the VB. (Note: unlike in an

intrinsic semiconductor, this process is NOT accompanied by creation of equal number of electrons in CB)

• The Ea level created by the impurity is called acceptor level.

• If Na is the acceptor concentration, then at room temperature, there is a major contribution to hole concentration in VB due to Na. Electrons (in CB) and equal number of holes (in VB) can also be produced by the intrinsic property of Si (ni).

• If Na is >> ni, then it can be assumed that the total p ≈ Na, and by law of mass action, n = ni

2/Na.

Page 12: Eng323 Lect02 My

Compensation DopingCompensation Doping• It is possible to compensate for excessive doping of donor

impurities by adding some acceptor impurities and vice versa and thus help achieve the required semiconductor properties.

• If Nd > Na, and (Nd – Na) >> ni, the effective electron concentration in the compensated material is n ≈ (Nd – Na) and the corresponding p = ni

2/n

Page 13: Eng323 Lect02 My

• If Na > Nd, and (Na – Nd) >> ni, the effective hole concentration in the compensated material is p ≈ (Na – Nd) and the corresponding n = ni

2/p

• The above arguments assume that the temperature is high and that all donors and acceptors are ionized. This is always the case at room temperatures (300K) and above.

• It is wrong to assume that if both dopings are high, both n and p will be high. This is because, the probability of both electrons and holes recombining will be very high and a sizeable number of electron – hole pairs will be lost through recombination.

Page 14: Eng323 Lect02 My

Conductivity in a SemiconductorConductivity in a Semiconductor• In the presence of an E-field, electrons in conduction band

gain energy from the field and move in a direction opposite to that of the field.

• They also collide due to thermal vibrations of lattice and lose energy. This results in electrons drifting, with a drift velocity vde.

• Similar process results for holes in valence band, in an opposite direction, with a drift velocity vdh.

• Total current density

where

n = electron conc. in CB and p = hole conc. in VB,e = electronic charge

Page 15: Eng323 Lect02 My

• The drift mobilities are defined as

• τ is the mean free time between scattering events, m* is the effective mass of electrons and holes.

• Considering the fact that J = σ. εx , where σ is the conductivity, and εx is the electric field,

Page 16: Eng323 Lect02 My

• The total conductivity of this extrinsic semiconductor is

• At low temperatures, not all donors are ionized, and the equation is not valid.

n-type Semiconductor:n-type Semiconductor:

• The total conductivity of this extrinsic semiconductor is

• At low temperatures, not all acceptors are ionized.

p-type Semiconductor:p-type Semiconductor:

Page 17: Eng323 Lect02 My

VB

CB

As As AsAs+Eg

T < Ts Ts < T < Ti

(b)T=T2(a)T=T1

T > Ti

(c)T=T3

As+EFAs+ As+ As+ As+

EF

As+ As+ As+

(a) Below Ts, the electron concentration is controlled by the ionization ofthe donors. (b) Between Ts and Ti, the electron concentration is equal tothe concentration of donors since they would all have ionized. (c) At hightemperatures, thermally generated electrons from the VB exceed thenumber of electrons from ionized donors and the semiconductor behavesas if intrinsic.

EF

Temperature Dependence of ConductivityTemperature Dependence of Conductivity

Page 18: Eng323 Lect02 My

• For an intrinsic semiconductor, we know that

• n or p is produced by thermal excitation across a bandgap energy Eg.

• In the same manner for an n-type semiconductor at low temperature (when thermal excitation is not possible and the only way n can be generated is through donor states)

• The factor ½ arises because the donor occupation statistics is different from the usual F-D distribution function.

Page 19: Eng323 Lect02 My

• There are three temperature regions of interest:

1. Low temperature region (T<Ts). n reaches a maximum of Nd in this range, called ionization range

2. Medium temperature region (Ts<T<Ti). All donors are ionized and the temperature is still low enough, so no significant thermal generation of carriers, and n continues to remain at Nd in this extrinsic range.

3. High temperature region (T>Ti). In this range, the thermally generated carriers exceed Nd and the semiconductor behaves as if it is intrinsic. This range is called intrinsic range.

Page 20: Eng323 Lect02 My

ln(n)

1/T

Ti

Ts

INTRINSIC

EXTRINSICIONIZATION

ni(T)

ln(Nd)slope = − ∆ E/2k

slope = -Eg/2k

The temperature dependence of the electron concentration in an n-typesemiconductor.

Page 21: Eng323 Lect02 My

• Based on the expressions for Nc and Nv , the temperature dependence of n or p for an intrinsic semiconductor, or for a semiconductor in the intrinsic range, is given by

• ni at room temperature (300K) for Si is reported as 1.5 x 1010 cm-3 though of late, authors of many text books use a value of 1x1010 cm-3 or even 9.69x109 cm-3.

Si

Ge

GaAs

0oC200oC400oC600oC 27oCLLLL

1018

1 1.5 2 2.5 3 3.5 41000/T (1/K)

1015

1012

103

106

109

2.4× 1013 cm-3

2.1× 106 cm-3

1.45× 1010 cm-3

Intri

nsic

Con

cent

ratio

n(c

m-3

)The temperature dependence of the intrinsic concentration.

Page 22: Eng323 Lect02 My

Drift Mobility: Temperature & Impurity DependenceDrift Mobility: Temperature & Impurity Dependence

• At low temperatures, scattering by ionized impurities is predominant and mobility is scattering dependent.

• It can be shown that the ionized impurity limited scattering is given by

As+

rc

KE > |PE|e­ KE = 1/2mev2

KE ≈ |PE|

KE < |PE|

Scattering of electrons by an ionized impurity

where NI is the concentration of all ionized impurities. Mobility decreases with increasing scattering centres (NI). Increased T reduces scattering radius and increases mobility.

Page 23: Eng323 Lect02 My

• At high temperature, mobility is limited by lattice scattering.

• The lattice scattering limited mobility can be shown to be given by

• Mobility decreases with increasing temperature and increasing lattice vibration.

• Combined effect of two scattering processes:

• The scattering process having the lowest mobility determines the overall drift mobility

Page 24: Eng323 Lect02 My

10

100

1000

10000

50000

70 100 800

µ Ι ∝ T1.5

Ge

Nd =1013

Si

µ L∝ T -1.5

Nd =1014

Nd =1016

Nd =1017

Nd =1018

Nd =1019

Temperature (K)

Ele

ctro

nD

rift

Mob

ilit

y(cm

2V

-1s-

1 )

Log-log plot of drift mobility vs temperature for n-type Ge and n-typeSi samples. Various donor concentrations for Si are shown. Nd are incm-3. The upper right inset is the simple theory for lattice limitedmobility whereas the lower left inset is the simple theory for impurityscattering limited mobility.

Page 25: Eng323 Lect02 My

50

100

1000

2000

1015 1016 1017 1018 1019 1020

Dopant Concentration, cm-3

ElectronsHoles

The variation of the drift mobility with dopant concentration in Si forelectrons and holes

Dri

ftM

obil

ity(

cm2

V-1

s-1 )

Page 26: Eng323 Lect02 My

Drift Mobility: Temperature DependenceDrift Mobility: Temperature Dependence

log(n)

INTRINSIC

EXTRINSIC

IONIZATION

log( )

log( )µ ∝ T-3/2 µ ∝ T3/2

Latticescattering

Impurityscattering

1/TLow TemperatureHigh Temperature

TMetal

Semiconductor

LOG

AR

ITH

MIC

SCA

LE

Res

isti

vity

Temperature dependence of electrical conductivity for a doped (n-type) semiconductor.

Page 27: Eng323 Lect02 My

EF

EC,V

Fermi “filling” function

Energy band to be “filled”

Moderate T

Metal

EF

EC

EV

Conduction band(Empty)

Valence band(Filled)

Eg

T > 0

Insulator

EF

EC

EV

Conduction band(Partially Filled)

Valence band(Partially Empty)

T > 0

Semiconductor

• At T = 0, valence band is filled with electrons and conduction band is empty, leading to zero conductivity.

• At T > 0, electrons thermally “excited” from valence to conduction band, leading to partially empty valence and partially filled conduction bands.

• At T = 0, lower valence band is filled with electrons and upper conduction band is empty, leading to zero conductivity.

– Fermi energy EF is at midpoint of large energy gap (2-10 eV) between conduction and valence bands.

• At T > 0, electrons are NOT thermally “excited” from valence to conduction band, leading to zero conductivity.

• Overlapping Ev, Ec

• At T = 0, energy levels below EF are filled with electrons, while all levels above EF are empty.

• Electrons are free to move into “empty” states of conduction band with only a small electric field E, leading to high electrical conductivity!

• At T > 0, electrons have a probability to be thermally “excited” from below the Fermi energy to above it.

Page 28: Eng323 Lect02 My

Degenerate & Non-degenerate semiconductorsDegenerate & Non-degenerate semiconductors

• The general expression for electron concentration in CB is

• Nc is a measure of the density of states in CB.

• If n << Nc, then there are fewer electrons than the available states and Pauli’s Exclusion Principle can be neglected. The statistics is then described by Boltzmann’s statistics.

• All equations formulated so far here are with this condition, and such semiconductors for which n << Nc or p << Nv are called non-degenerate semiconductors.

Page 29: Eng323 Lect02 My

CB

g(E)

E

Impuritiesforming

a band

(a) (b)

EFp

Ev

EcEFn

Ev

Ec

CB

VB

• In a degenerate semiconductor, n>> Nc or p >> Nv brought about by heavy doping.

• Such large concentration of impurity states, result in their energy states forming band of energy that overlap with the conduction band (for n-type) or the valence band (for the p-type).

Page 30: Eng323 Lect02 My

Recombination & Minority Carrier InjectionRecombination & Minority Carrier Injection• Direct & Indirect Recombination• Recombination is a process wherein an electron in the CB meets a hole in

the VB and fills it. The electrons and holes that take part in the process are consequently annihilated.

• On the energy band diagram, recombination is represented by returning an electron in CB (higher energy state) into a hole in the VB (lower energy state).

• In order for the conservation of energy to be valid for this recombination process, the excess energy is lost in the form of photon of energy E = hν. Conservation of linear momentum must also be met in this recombination. The wavefunction of the electron in the CB will have a certain momentum hkcb . Similarly, the electron wavefunction in the VB will have a certain momentum hkvb.

• Momentum conservation requires kcb = kvb.

Page 31: Eng323 Lect02 My

Ev

Ec

CB

VB

ψ cb(kcb)

ψ vb(kvb)hυ = Eg

Distance

Energy

Direct recombination in GaAs. kcb = kvb so that momentum conservationis satisfied

Page 32: Eng323 Lect02 My

• In elemental semiconductors like Si and Ge, the electronic states with kcb = kvb occur deep in the middle of VB, which are fully occupied, so direct recombination is impossible.

• In compound semiconductors like GaAs and InSb, the electronic states with kcb = kvb occur right at the top of VB, which are essentially empty (contain holes), so direct recombination is possible.

• Consequently, a free electron in the CB edge Ec of GaAs can drop down to an empty electronic state at the top of valence band and maintain kcb = kvb.

• The photon energy released in the process makes it a clear choice as material for light emission.

Page 33: Eng323 Lect02 My

• Elemental semiconductors like Si and Ge undergo indirect recombination. The Ec to Ev transition process usually involves a third body, usually an impurity atom or a crystal defect (like dislocations, vacancies) that create a localized energy state somewhere in the middle of the bandgap.

• These are called recombination centres, that capture an electron in the CB. The bound electron, waits for a hole in the VB with which it can recombine.

• In this type of two stage transitions, energy is lost in the form of lattice vibrations, called phonons.

• Sometimes, the impurity states of defect states act as mere trapping centres rather than recombination centres.

Page 34: Eng323 Lect02 My

Ec

Recombinationcenter

CB

VB

Phonons

(a) Recombination

Trappingcenter

Et

CB

VB

Ec

Ev

Et Et

Ev

Er Er Er

(b) Trapping

Recombination and trapping. (a) Recombination in Si via a recombination center which has a localized energy level at Er in the bandgap, usually near the middle. (b) Trapping and detrapping of electrons by trapping centers. A trapping center has a localized energy level in the band gap.

Page 35: Eng323 Lect02 My

Minority Carrier LifetimeMinority Carrier Lifetime• What happens when an n-type Si doped with 5x1016 cm-3

donors is uniformly illuminated with appropriate wavelength light to generate electron-hole pairs (EHPs) ?

Ec

Ev

Ed

CB

VB

Low level photoinjection into an n-type semiconductor in which∆ nn < nno

Page 36: Eng323 Lect02 My

• For this n-type semiconductor, electrons are majority carriers & holes are minority carriers. We use a subscript n to define n or p type semiconductor.

• nno = electron conc. in n-type at thermal equilibrium in the dark (majority carrier conc.)

• pno = hole conc. in n-type at thermal equilibrium in the dark (minority carrier conc.)

nno.pno = ni2

• When the semiconductor is illuminated, excess EHPs are generated. Electron & hole conc. dynamically changes.

• Suppose at any instant:nn = electron concentration in n-type = nno + Δnn

pn = hole concentration in n-type = pno + Δpn

where Δnn and Δpn are the excess majority & minority conc.

Page 37: Eng323 Lect02 My

• Since the excess carriers are created as EHPs,

Δnn = Δpn

• Obviously the law of mass action is not obeyed:

nn.pn ≠ ni

• Also

• A low level injection causing a 10% change in nno will have the following effects:

There is no significant change in nn but pn the minority carrier concentration is altered significantly.

Page 38: Eng323 Lect02 My

Low level injection in an n-type semiconductor does not affectnn but drastically affects the minority carrier concentration pn.

Page 39: Eng323 Lect02 My

• What happens when the illumination is removed?

--

-

--

-

--

--

- ---

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- ----

-- -

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--

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---+

+-

--

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n-type semiconductor inthe dark.pn = pno<< nno

A

Illumination with hυ >Egcreates excess holes:pn = pno+∆ pn = ∆ nn

B

Illumination

In dark afterillumination. Excessholes are disappearingby recombination.

C

Illumination of an n-type semiconductor results in excess electronand hole concentrations. After the illumination, the recombinationprocess restores equilibrium; the excess electrons and holes simplyrecombine.

Page 40: Eng323 Lect02 My

• Once the illumination is removed, the excess holes recombine with the available electrons & disappear, in a short period of time.

• The rate of recombination can be described by a minority carrier lifetime τh, which is the average time the hole exists in the VB from the time it is generated till it recombines with an electron in the CB.

• When both generation and recombination takes place simultaneously,

Rate of increase in excess hole conc. = Photogeneration rate - Recombination rate of excess holes

Page 41: Eng323 Lect02 My

• If the illumination is removed at time t = 0, Gph = 0 and the solution is:

Illumination

0 Time, tpno

Gph

toff

pno+∆ pn(∞ )

t'

τhGph

G and pn(t)

∆ pn(t') = ∆ pn(0)exp(-t'/τh)

0

Page 42: Eng323 Lect02 My

Diffusion & Conduction EquationsDiffusion & Conduction Equations• Electrons in CB & holes in VB can be considered as particles.

• When such particles have some distribution in space, with a certain concentration gradient, then diffusion occurs.

• Particle flux is defined as the number of particles crossing unit area per unit time

• If the particle has a charge Q, current density J = Q.Γ

• The flux of electrons is governed by Fick’s First Law:

• -ve sign indicates that the direction of flux is opposite to the direction of increasing conc. Const. of proportionality is De.

Page 43: Eng323 Lect02 My

n1

n2

xo xo+ℓxo-ℓ

n(x,t)

n(x,t)

x

Net electron diffusion flux

Electric current

xo+ℓxo-ℓxo

(a) (b)

Page 44: Eng323 Lect02 My
Page 45: Eng323 Lect02 My

Ex

n-Type Semiconductor

x

Hole Drift

Hole Diffusion

Electron Drift

Electron Diffusion

Semitransparent electrode

Light

When there is an electric field and also a concentration gradient, chargecarriers move both by diffusion and drift. (Ex is the electric field.)

• In the figure, there are

Diffusion current due to electrons along -ve x

Drift current due to electrons along +ve x

Diffusion current due to holes along +ve x

Drift current due to holes along +ve x

Page 46: Eng323 Lect02 My

• Einstein’s Relation:

Page 47: Eng323 Lect02 My

Continuity EquationsContinuity Equations• Excess carrier concentration decays exponentially as a

function of time when optical illumination is removed. What happens when the optical illumination is constantly present?

• Consider an n-type semiconductor with an element of volume A.dx across which hole concentration is p(x,t).

x x+δ x

Jh

x

Jh + δ Jh

p(x,t)

Ay

z

Semiconductor

Page 48: Eng323 Lect02 My

• Without illumination, at thermal equilibrium, the value of hole concentration is pno .

• -ve sign indicates that some excess holes are lost along x due to recombination, as they move along x.

• Due to continuous illumination, there also exist continuous photogeneration at a constant rate, Gph and a recombination rate due to minority carrier lifetime

• Net rate of increase in hole concentration

This is the Continuity equation for holes.

Page 49: Eng323 Lect02 My

• Jh is the hole component of current density. It is made up of drift and diffusion components.

• There is a similar equation for net rate of excess electron concentration.

• If we consider uniform illumination everywhere, then Jh is uniform along x. This gives

Page 50: Eng323 Lect02 My

Steady State Continuity Equations:Steady State Continuity Equations:• Consider a continuous illumination at one end of a long n-

type slab. There is no bulk photogeneration, Gph = 0.

• In such a case, under steady state condition,

• Jh contains drift & diffusion components, Assuming that the drift component is negligible (say, electric field is negligible), then simplifying the expression for Jh & noting thatpn(x) = pno + Δpn(x)

• Lh is called diffusion length of holes = (Dhτh)1/2

Page 51: Eng323 Lect02 My

n-type semiconductor

Light

xo∆ pn(x)

∆ pn(0)

Diffusion

(a)

∆ nn(0)

∆ nn(x)

Excess concentration

x

• We assume a weak injection, one end is illuminated. The layer that absorbs the light is very thin. The excess holes created will have a steady state continuity equation. Suppose the illumination causes excess hole conc. at x = 0 to be Δpn(0).

• Solution is given by

Page 52: Eng323 Lect02 My

• Obviously at the other end of the long bar, there are no excess holes. The only hole conc present is the equilibrium value pno.

• The concentration gradient causes a diffusion hole current, which are prominent only near the illuminated end.

• There will be a steady state distribution of excess electrons for this example

where Le is the diffusion length of electrons = (Deτe)1/2

Page 53: Eng323 Lect02 My

(b)

Currents (mA)

0

− 420

x (µ m)

Diffusion

ID,e

Idrift,e

ID,h Drift4

40 60 800