© M.N.A. Halif & S.N. Sabki CHAPTER 3: CARRIER CONCENTRATION PHENOMENA CHAPTER 3: CARRIER...

© M.N.A. Halif & S.N. Sabki

CHAPTER 3: CARRIER CHAPTER 3: CARRIER CONCENTRATION PHENOMENACONCENTRATION PHENOMENA

Part IPart I


SUB-TOPICS IN CHAPTER 3:SUB-TOPICS IN CHAPTER 3:

Carrier Drift

Carrier Diffusion

Generation & Recombination Process

Continuity Equation

Thermionic Emission Process

Tunneling Process

High-Field Effect


Part IPart I

Carrier Drift

Carrier Diffusion

Generation & Recombination Process


CARRIER DRIFTCARRIER DRIFT

MobilityThe electron in s/c have 3 degree of freedom – they can move in a 3-D space. The K.E of electron is given by

kTvmK thne 2

3

2

1 2 (1)

From the theorem for equipartition of energy, ½ kT unit

energy per degree of freedom.

mn – effective mass of electron,

vth – average thermal velocity (~ 107cm/s at T=300K)


Electron in s/c moving rapidly in all direction, where thermal motion of an individual electron may be visualized as a succession of random scattering from collisions with lattice atoms, impurity atoms, and other scattering centers, as shown in Fig. 3.1(a).

Average distance between collisions – mean free path.

Average time between collisions – mean free time C.

For typical mean free path ~ 10-5cm, C = 10-15/vth~10-12s (or in 1ps).

Mobility (cont.)


When small electric field, E, is applied to s/c sample, each electron will experience a force –qE from the field and accelerated along the field (in opposite direction) during the time between collisions – additional thermal velocity component.

This additional component called drift velocity.

Combination displacement of an electron (due to random thermal motion) & drift component illustrated in Fig. 3.1(b).

Note that: net displacement of the electron is in the opposite direction of applied field.

Mobility (cont.)


Figure 3.1. Schematic path of an Figure 3.1. Schematic path of an electronelectron in a semiconductor. ( in a semiconductor. (aa) Random ) Random thermal motion. (thermal motion. (bb) Combined motion due to random thermal motion and ) Combined motion due to random thermal motion and an applied electric field.an applied electric field.

Without electric field

hole

Mobility (cont.)


The momentum applied to an electron is given by -qEC, and momentum gained is mnvn. Thus, using physics conservation of energy, electron drift velocity:

Ev nn Note that: vn is proportional to E

The proportionality factor may be written as

n

Cn m

q

(2)

(3)

• The proportionality factor also called electron mobility.• A similar expression may be written for holes in valence band may be written as: vp = p E

• Mobility is very important parameter for carrier transport – itdescribes how strongly the motion of an electron is influencedby an applied electric field.

Mobility (cont.)


From eq. (3), mobility is related directly to mean free time between collisions determined by the various scattering mechanism.Two MOST important mechanisms: lattice scattering and impurity scattering.Lattice scattering – results from thermal vibrations of the lattice atoms at any temperature, T>0K (it becomes dominant at high temp. – mobility decreases with increasing temp.) – theoretically mobility due to lattice scattering L decrease in proportion to T-3/2

Impurity scattering – results when charge carrier travels past am ionized dopant impurity (donor or acceptor). It depend on Coulomb force interaction.Impurity scattering depends on total concentration of ionization impurities (sum of +ve and –ve charge ions). It becomes less significant at higher temperatures.

Mobility (cont.)


The probability of a collision taking place in unit time, 1/C, - the sum of the probabilities of collision due to the various scattering mechanism:

IL

CCC

111

111

impurity ,lattice ,

(4)

(4a)

or

L – lattice scattering mobility

I – impurity scattering mobility

Mobility (cont.)


Electron mobility as a function of temp. for Si with 5 different donor concentration is given by Fig. 3.2.

For lightly doping (i.e 1014cm-3) – lattice scattering dominates and mobility decreases as the temp. increases.

For heavily doped (i.e 1019cm-3) – at low temp. impurity scattering is most pronounced. Mobility is increases as temp. increases.

For a given temp., mobility decreases with increasing impurity concentration (due to enhanced impurity scattering).

Mobility (cont.)


Figure 3.2.Figure 3.2. Electron mobility in silicon versus temperature for various Electron mobility in silicon versus temperature for various donor concentrations. Insert shows the theoretical donor concentrations. Insert shows the theoretical temperature dependence of electron mobility.temperature dependence of electron mobility.

Lightly doped

Heavily doped


Figure 3.3.Figure 3.3. Mobilities and diffusivities in Si and Mobilities and diffusivities in Si and

GaAs at 300 K as a function of impurity GaAs at 300 K as a function of impurity concentration.concentration.

Mobility reaches a maximum value at low impurity concentrations corresponds to the lattice scattering limitation.

Both electron & hole mobilities decrease with increasing impurity concentration.

Mobility of electrons is greater than holes due to the smaller effective mass of electrons.

Mobility (cont.)


EXAMPLE 1EXAMPLE 1

Calculate the mean free time of an electron having a mobility of 2000 cm2/ V-s at room temperature; also calculate the mean free path. Assume mn = 0.22mo in these calculation.


ResistivityResistivity

Refer to Fig. 3.4.3.4(a) – n-type s/c & its band diagram at thermal equilibrium. 3.4(b) – when biasing voltage is applied at right-hand-terminal.Assume that contact at both terminals are ohmic (there is negligible voltage drop at each of the contacts). Behavior of ohmic contact – Chapter 7.When E (electric field) is applied to s/c, each electron may experience a force of –qE. Thus, the force is equal to the negative gradient of the potential energy:

dx

dEqE C U)energy, potentialelectron of(gradient (5)

EC – conduction band energy


Figure 3.4.Figure 3.4. Conduction process in an Conduction process in an nn-type semiconductor (a) at thermal -type semiconductor (a) at thermal equilibrium and (b) under a biasing condition.equilibrium and (b) under a biasing condition.

Resistivity (cont.)Resistivity (cont.)


In the gradient of U, any part of the band diagram that is parallel to EC (e.g EF, Ei, and EV) may be used. But it’s convenient to use intrinsic Fermi level E i (when consider p-n junction in Chapter 4). From (5):

dx

dE i (6)

Where - electrostatic potential, and defined as

q

Ei (7)

Which represents the relationship between electrostaticpotential and potential energy, U.


For homogenous s/c (Fig. 3.4(b)) – U and Ei decrease linearly with distance, thus electric field constant –ve x-direction.

Electrons in cond. band move to the right – electron undergoes a collision, loses some or all of its K.E to the lattice & drops toward its thermal equilibrium position – this process will be repeated many times.

Hole behaves in the same manner but in the opposite direction.

Transport of carriers under applied electric field – drift current.


Figure 3.5.Figure 3.5. Current conduction in a uniformly doped semiconductor bar Current conduction in a uniformly doped semiconductor bar with length with length LL and cross-sectional area and cross-sectional area AA..

• From Fig. 3.5, with application of electric field, current density forboth electron and hole, Jk may be written as

EqkyqkvyqvA

IJ k

k

iki

kk

1

)(

(8)

Where for electron, k = n, y= -1; and hole, k=p, y=1


Total current flowing in s/c sample is sum of the electron and hole components, which is

Epnq

JJJ

pn

pn

)(

)(

1

pn pnq

(9)

From (9), conductivity = q(nn + pp). Thus, resistivity of semiconductor is given by

(10)

For extrinsic s/c, generally may be written as

kqk 1 (11)

For n-type (n>>p), k=n, and p-type (p>>n), k=p


Figure 3.6.Figure 3.6. Measurement of Measurement of resistivity resistivity using a four-point probe.using a four-point probe.

In practical, to measure resistivity – commonly used the four-point probe method (Fig. 3.6)

With thickness, W << d, thus

resistivity is govern by

cm )(

I

CFVW (12)

Where CF ~ ‘correction factor’ and it is depends on the ratio of d/s, s – probe spacing.


Figure 3.7.Figure 3.7. Resistivity versus impurity concentration Resistivity versus impurity concentration33 for Si and for Si and GaAs.GaAs.

Room temperature


EXAMPLE 2EXAMPLE 2

Find the resistivity of n-type Si doped with 1016 phosphorus atom/cm3 at T = 300K.


THE HALL EFFECTTHE HALL EFFECT

The “Hall effect” was discovered in 1879 by the American physicist, Edwin Hall (1855 – 1938). He discovered the "Hall effect" while working on his doctoral (PhD) thesis in Physics.

In 1880, full details of Hall's experimentation with this phenomenon formed his doctoral thesis and was published in the American Journal of Science and in the Philosophical Magazine.


Hall effect is used to measure the carrier concentration.

It is also one of the most convincing methods to show the existence of holes as charge carriers – measurement can give directly the carrier type.

Fig. 3.8 show the Hall effect set-up (consider a p-type sample). Using Lorentz force F = qv x B = qvxBz. (B: magnetic field)

There is no net current flow along y-direction (in steady-state), thus Ey exactly balances the Lorentz force:

qpR

BJRBqp

JE

H

zpHzp

y

1

(13)

Hall field

Hall coefficient (14)

THE HALL EFFECTTHE HALL EFFECT


Figure 3.8.Figure 3.8. Basic setup to measure carrier concentration using the Hall Basic setup to measure carrier concentration using the Hall effect.effect.


The measurement of the Hall voltage for a known current and magnetic field yields

AqV

WIB

qE

BJ

qRp

H

z

y

zp

H

1

All quantities in RHS can be measured, thus carrierconcentration and carrier type can be obtained directlyfrom Hall measurement.

(15)

RHS: right-hand-side


EXAMPLE 3EXAMPLE 3

A sample of Si is doped with 1016 phosphorus atom/cm3. Find a Hall voltage in a sample with W = 300m, A = 0.0025cm2, I = 1mA, and Bz = 10-4 Wb/cm2.


CARRIER DIFFUSIONCARRIER DIFFUSION

Carriers move from a high concentration region to low concentration region ~ called diffusion current.

From Fig. 3.9, current density may explain by mathematical formalism below:

2

)(

2

)(

2

1

th

th

vlnF

vlnF

Diffusion ProcessDiffusion Process

dx

dnqDJ

dx

dnD

dx

dnlv

dx

dnln

dx

dnlnFFF

nn

nth

)0()0( 2

121

F ~ average electron flow per unit area.l ~ mean free pathDn ~ diffusion coefficient

LHS:

RHS:

(17)

(16)


Figure 3.9.Figure 3.9. Electron concentration versus distance; Electron concentration versus distance; ll is the mean free is the mean free path. The directions of electron and current flows are indicated by path. The directions of electron and current flows are indicated by

arrows.arrows.


EINSTEIN RELATIONEINSTEIN RELATION

Rewrite Eq. (17) using theorem for equipartition of energy:

kTvm thn 2

1

2

1 2

q

kTlvD nthn

(18)

• Using (3), (16), & (18), Einstein relation may be written as

(19)

(relation of diffusivity & mobility)


DENSITY EQUATIONSDENSITY EQUATIONS

Total current density at any point is the sum of the drift & diffusion components:

dx

dkyqDkEqJ kkk

(20)

Where k = n, with y=1, and k=p, with y= -1.

• Total conduction current density is given by

Jcond = Jn + Jp


GENERATION & RECOMBINATION GENERATION & RECOMBINATION

For the direct-bandgap s/c in thermal equilibrium – the continuous thermal vibration of lattice atoms – cause bonds between neighboring atoms to be broken.

Bonds broken cause electron-hole pair.

Carrier generation – electron to make upward transition to cond. band & leaving a hole in valence band. It represented by the generation rate Gth (number of electron-hole pair generated/cm3/s) – Fig. 3.10(a).

Recombination – electron makes transition downward from cond. band. It represented by recombination rate Rth (Fig. 3.10(a)).

At thermal equilibrium cond. : Gth = Rth for pn = ni2 to be

maintained.

Direct RecombinationDirect Recombination


Figure 3.10.Figure 3.10. Direct generation and Direct generation and recombination of electron-hole pairs: recombination of electron-hole pairs: ((aa) at thermal equilibrium and () at thermal equilibrium and (bb) ) under illumination.under illumination.

nonothth pnRG

nonnon

thL

nono

pppnnn

GGG

ppnnR

;

))((

• The rate of generation & recombination in n-type is

(21)

• When we shine a light, it produced electron-hole pair at a rate GL, recombination and generation rate

(22)

(23)

nno & pno – electron and hole densities - proportionality constant


nop

p

non

thL

thLn

n

ppU

UGRG

RGGRGdt

dp

1

;

• The net change of hole concentration is given by

(24)

At steady-state, dpn/dt = 0;

(25)

And at low level injection, pno << nno, the net recombination is

U is net recombination, defined as

U = (nno + pno + ∆p)∆p

p – lifetime of the excess minority carriers.

(26)


Figure 3.11.Figure 3.11. Decay of photo excited carriers. Decay of photo excited carriers.

aa) ) nn-type sample under constant illumination. -type sample under constant illumination. ((bb) Decay of minority carriers (holes) with time. ) Decay of minority carriers (holes) with time. ((cc) Schematic setup to measure minority carrier ) Schematic setup to measure minority carrier

lifetime.lifetime.

Lpno

p

nonL

Gpp

ppUG

and,

• From (25) & (26) (in steady-state), generation rate is given by

pLpnon

tGptp

exp)(

• When the light is turn off, t = 0, theboundary cond. pn(0)Eq. (28), andpn() pno, thus

(27)

(28)

(29)


The derivation of the recombination rate is more complicated.

Et – called the intermediate-level states.There are 4 basic transitions takes place.Example of the indirect-bandgap s/c – Si.After indirect recombination process:

(i) Electron capture (ii) Electron emission (iii) Hole capture (iv) Hole emission

Figure 3.12.Figure 3.12. Indirect generation-Indirect generation-

recombination processes at recombination processes at thermal equilibrium.thermal equilibrium.

Indirect Indirect RecombinationRecombination

GENERATION & RECOMBINATION GENERATION & RECOMBINATION


The recombination rate is given by (from derivation in Appendix I, Sze, pg. 541):

)exp(exp

2

CnnCnp

nnpNvU

inninp

inntpnth

p

non

no

i

nontoth

pp

Cnn

ppNvU

)cosh(2

1

kT

EEC ti

Under low-injection condition in a n-type, so nn >> pn , then (30) can be written as

(30)

(31)

Where, vth – thermal velocity, Nt – concentration of the

recombination centre, - capture cross section (effectiveness of the centre to capture an electron or hole), and


Surface RecombinationSurface Recombination

A large number of localize energy states (generation-recombination centers) may introduced at the surface region. (Fig. 3.13).

It may enhance the recombination rate at the surface region by an energy called surface-state.

The kinetics of the surface recombination are similar to those in bulk centers.

Total number of carrier recombining at the surface per unit area and unit time:

)( nosstpths ppNvU

stpthlr NvS

(32)

And, the low-injection surface recombination velocity is defined as:

Where, ps – concentration at surface, Nst – recombination center density per unit area in the surface region.

(33)


Figure 3.13. Schematic diagram of bonds at a clean semiconductor surface.

The bonds are anisotropic and differ from those in the bulk.


Occurs by the transfer of the energy & momentum released by the recombination of electron-hole pair to a 3rd particle (either electron or hole).

Example shown in Fig. 3.14, the 2nd electron absorb the energy released by direct recombination – becomes an energetic electron.

It’s very important – carrier concentration is very high (results from high doping or high injection level). The rate of this recombination can be expressed as

Figure 3.14. Auger recombination.

Auger Recombination

22 or BnpRpBnR AugAug (34)

B – proportionality constant (strong temperature depending)


Summary of Part 1Summary of Part 1

In part 1 of carrier transport phenomena, various temperature process include drift, diffusion, generation, and recombination.Carrier drift – under influence of an electric field. At low field, drift velocity is proportional to electric field called Mobility.Carrier diffusion – under influence of carrier concentration gradient.Total current = (drift + diffusion) components.Four types of recombination process:

(i) Direct (ii) Indirect (iii) Surface (iv) Auger


"Science is a powerful instrument. How it is used, whether it is a blessing or a curse to mankind, depends on mankind and not on the instrument. A knife is useful, but it can also kill."

Albert Einstein


““Do what you can, with what Do what you can, with what you have, where you are” you have, where you are”

Theodore Roosevelt


Next Lecture:Next Lecture:

CHAPTER 3 PART 2:Continuity Equation Thermionic Emission Process Tunneling Process High-Field Effect

Date post:	17-Jan-2016
Category:	Documents
Upload:	david-gardner
View:	272 times
Download:	1 times

© M.N.A. Halif & S.N. Sabki CHAPTER 3: CARRIER CONCENTRATION PHENOMENA CHAPTER 3: CARRIER...

Documents