Physical Electronics Slides of Chapter 6 (All Slides)blogs.cae.tntech.edu/bwabegaz42/files/2013/10/Physical-Electronics-Slides-of-Chapter-6...A semiconductor in thermal equilibrium,

PHYSICAL ELECTRONICS(ECE3540)

Wednesday, October 09, 2013Tennessee Technological University 1

CHAPTER 6 – CARRIER GENERATION AND RECOMBINATION

Chapter 6 – Carrier Generation and Recombination

Chapter 4: we considered the semiconductorin equilibrium and determined electron and holeconcentrations in the conduction and valencebands, respectively.

The net flow of the electrons and holes in asemiconductor will generate currents. The processby which these charged particles move is calledtransport.

Chapter 5: we considered the two basic transportmechanisms in a semiconductor crystal: drift themovement of charge due to electric fields, anddiffusion the flow of charge due to densitygradients.


Chapter 6 – Carrier Generation and Recombination

Chapter 6: we will discuss the behavior ofnon-equilibrium electron and holeconcentrations as functions of time and space.

We will develop the ambi-polar transportequation which describes the behavior of theexcess electrons and holes.

We can define two new parameters that apply tothe non-equilibrium semiconductor: the quasi-Fermi energy for electrons and the quasi-Fermienergy for holes.


Carrier Generation and Recombination Generation: is the process whereby electrons and holes

are created. Recombination: is the process whereby electrons and

holes are annihilated. Any deviation from thermal equilibrium will tend to

change the electron and hole concentrations in asemiconductor.

Examples: A sudden increase in temperature will increase the rate at which

electrons and holes are thermally generated so that theirconcentrations will change with time until new equilibrium valuesare reached.

An external excitation, such as light (a flux of photons), can alsogenerate electrons and holes, creating a non-equilibriumcondition.


The Semiconductor in Equilibrium Since the net carrier concentrations are

independent of time in thermal equilibrium, therate at which electrons and holes are generatedand the rate at which they recombine must beequal.

Let Gn0 and Gp0 be the thermal-generation rateof electrons and holes in #/cm3-s. For a directband-to-band generation, the electrons andholes are created in pairs, therefore: Gn0 = Gp0.


The Semiconductor in Equilibrium


Let Rn0 and Rp0 be the recombination rates of electronsand holes, respectively, for a semiconductor in thermalequilibrium, again given in units of #/cm3-s.

In direct band-to-band recombination, electrons andholes recombine in pairs, so that Rn0 = Rp0.

In thermal equilibrium, the concentrations of electronsand holes are independent of time; therefore, thegeneration and recombination rates are equal, so we haveGn0 = Gp0 = Rn0 = Rp0

Fig. 6.1: Electron-hole generation and recombination

-

+ +

-

Electron-Hole Generation

Electron-Hole Recombination

X

Excess Carrier Generation and Recombination Electrons in the valence band may be excited into the

conduction band when, for example, high-energyphotons are incident on a semiconductor.

When this happens, not only is an electron created inthe conduction band, but a hole is created in thevalence band; thus an electron-hole pair is generated.

The additional electrons and holes created are calledexcess electrons and excess holes.


Excess Carrier Generation and Recombination The excess electrons and holes are generated by an

external force at a particular rate. Let gn’ be the generation rate of excess electrons and gp’

be the generation rate of excess holes in units of #/cm3s For the direct band-to-band generation, the excess

electrons and holes are also created in pairs, so we musthave gn’ = gp’.

When excess electrons and holes are created, theconcentration of electrons in the conduction band andof holes in the valence band increase above their thermalequilibrium value.

We may write n = n0 + n and p = p0 + p. n0 and p0 arethe thermal-equilibrium concentrations, and n and pare the excess electron and hole concentrations.


Excess Carrier Generation and Recombination


Symbol Definition

no, po Thermal equilibrium electron and hole concentration (independent of time and position)

n, p Total electron and hole concentrations(may be functions of time and/or position)

n = n – n0 Excess electron concentration(may be function of time and/or position)

p = p – p0 Excess hole concentration(may be function of time and/or position)

gn’ , gp’ Excess electron and hole generation rates.

Rn’ , Rp’ Excess electron and hole recombination rates.

n0, p0 Excess minority carrier electron and hole lifetimes.Table. 6.1: Notations used in Carrier Generation and Recombination

Excess Carrier Generation and Recombination A steady-state generation of excess electrons and holes will not cause

a continual buildup of the carrier concentrations. As in the case ofthermal equilibrium, an electron in the conduction band may "falldown" into the valence band, leading to the process of excesselectron-hole recombination.

The recombination rate for excess electrons is denoted by Rn’, and forexcess holes by Rp’.

Both parameters have units of #/cm3-s. The excess electrons andholes recombine in pairs, so the recombination rates must be equal.We can then write Rn’ = Rp’.

In the direct band-to-band recombination that we are considering, therecombination occurs spontaneously: thus, the probability of anelectron and hole recombining is constant with time.

The rate at which electrons recombine must be proportional to theelectron concentration and must also be proportional to the holeconcentration.

If there are no electrons or holes, there can be no recombination.


Excess Carrier Generation and Recombination The net rate of change in the electron concentration can

be written as:(eq. 6.1)

where

rni2 (eq. 6.1) is thermal-equilibrium generation rate.

Since excess electrons and holes are created andrecombine in pairs, n(t) = p(t). (Excess electron andhole concentrations are equal and termed excess carriers.)

The thermal-equilibrium parameters, n0 and p0,independent of time, become: (eq. 6.2)


)]()([)( 2 tptnndt

tdnir

)()( 0 tnntn )()( 0 tpptp

)]())[(())]())((([))((0000

2 tnpntntpptnnndt

tndrir

Excess Carrier Generation and Recombination (Eq. 6.2) can easily be solved if we impose the condition of

Low Level Injection. In an extrinsic n-type material, we generally have no >> po

and, in an extrinsic p-type material, we generally have po>>no.

Low-level injection means that the excess carrierconcentration is much less than the thermal equilibriummajority carrier concentration.

Conversely, high-level injection occurs when the excesscarrier concentration becomes comparable to or greater thanthe thermal equilibrium majority carrier concentrations.

If we consider a p-type material (po >> no) under low-levelinjection (n(t) <<po), then (eq. (6.2)) is:

(eq. (6.2)Wednesday, October 09, 2013Tennessee Technological University 12

)())((0 tnp

dttnd

r

Excess Carrier Generation and Recombination The solution to the equation is an exponential decay

from the initial excess concentration, or(eq. 6.3)

where n0 = (rp0)-1 and is a constant for the low-levelinjection.

The above equation describes the decay of excessminority carrier electrons so that n0 is often referredto as the excess minority carrier lifetime.


00 /)0()0()( nr ttp enentn

Excess Carrier Generation and Recombination The recombination rate defined as a positive quantity--of excess minority carrier

electrons can be written as:

For the direct band-to-band recombination, the excess majority carrier holesrecombine at the same rate, so that for the p-type material

For an n-type material (n0 >> p0) under low-level injection (n(t) <<no), the decayof minority carrier holes occurs with a time constant p0= (rno)-1 where p0 is alsoreferred to as the excess minority carrier lifetime. The recombination rate of themajority carrier electrons is the same as that of the minority carrier holes, so we have

The generation rates of excess carriers are not functions of electron or holeconcentrations. In general, the generation and recombination rates may befunctions of the space coordinates and time.


00

' ))(()())((

nrn

tntnpdt

tndR

0

'' )(

npn

tnRR

0

'' )(

ppn

tpRR

Exercise1. Consider a semiconductor in which n0 = 1013

cm-3 and ni = 1010 cm-3. Assume that theexcess-carrier lifetime is 10-6 s. Determine theelectron-hole recombination rate if the excess-hole concentration is p = 5 x 1013 cm-3.


0

'' )(

ppn

tpRR

Solution1. n-type semiconductor, low-injection so that


13196

13

0

10510105' scmxxpR

p

Exercise2. A semiconductor in thermal equilibrium, has a

hole concentration of p0= 1016 cm-3 and anintrinsic concentration of ni = 1010 cm-3. Theminority carrier lifetime is 2 x 107 s.

(a) Determine the thermal-equilibrium recombinationrate of electrons.

(b) Determine the change in the recombination rate ofelectrons if an excess electron concentration of n =1012 cm-3 exists.


0

'' )(

npn

tnRR

Solutiona) thermal-equilibrium recombination rate of

electrons

and

Then


0

00

nn

nR

3416

210

0

2

0 1010

)10( cmpnn i

13107

4

0 105)102(

)10( scmx

xRn

Solutionb) the change in the recombination rate of

electrons if an excess electron concentration ofn = 1012 cm-3 exists

Therefore,


13187

12

00 105

10210

scmxx

nRn

n

10180 105105 xxRRR nnn

1318105 scmxRn

Continuity Equations The continuity equations for electrons and holes are

shown as a differential volume element in which aone-dimensional hole particle flux is entering thedifferential element at r and is leaving at x + dx.

The parameter Fpx+ is the hole-particle flux, or flow,

and has units of number of holes/cm2-s. For the x component of the particle current density:


dxx

FxFdxxF px

pxpx .)()(

Fig. 6.2: Differential volume showing x component of the hole-particle flux.

Fpx+(x) Fpx

+(x+dx)

x x+dxdy

dz

Continuity Equations This equation is a Taylor expansion of Fpx

+(x + dx).where the differential length dx is small, so that onlythe first two terms in the expansion are significant.The net increase in the number of holes per unit timewithin the differential volume element due to the x-component of hole flux is given by:

If Fpx+(x) > Fpx

+ (x + dx), for example, there will bea net increase in the number of holes in thedifferential volume element with time.


dxdydzx

FdydzdxxFxFdxdydz

tp px

pxpx

)]()([

Continuity Equations The generation rate and recombination rate of holes will also affect

the hole concentration in the differential volume. The net increase inthe number of holes per unit time in the differential volume elementis then given by

where p is the density of holes. The first term on the right side of the equation is the increase in the

number of holes per unit time due to the hole flux, the second termis the increase in the number of holes per unit time due to thegeneration of holes, and the last term is the decrease in the numberof holes per unit time due to the recombination of holes.

The recombination rate for holes is given by p/pt where ptincludes the thermal equilibrium carrier lifetime and the excess carrierlifetime.


dxdydzPdxdydzgdxdydzx

Fdxdydz

tp

ptp

px

Continuity Equations If we divide both sides of the equation by the differential

volume dx.dy.dz, the net increase in the holeconcentration per unit time is :

The above equation is known as the continuity equationfor holes.

Similarly, the one-dimensional continuity equation forelectrons is given by:

where Fn- is the electron-particle flow, or flux, also given in units

of number of electrons/cm2-s.


plp

p pgx

Ftp

nln

n ngx

Ftn

Time-Dependent Diffusion Equations The hole and electron current densities are given as:

Dividing the hole current density by (+e) and the electron current density by (-e) to find the flux of each particle:


xpeDpEeJ ppp

xneDnEeJ nnn

xpDpEF

eJ

pppp

)(

xnDnEF

eJ

nnnn

)(

Time-Dependent Diffusion Equations Taking the divergence of the equations, and

substituting back into the continuity equations:

Keeping in mind that we are limiting ourselves to aone-dimensional analysis, we can expand thederivative of the product as


ptppp

pgxpD

xpE

tp

2

2)(

ntnnn

ngxnD

xnE

tn

2

2)(

xEp

xpE

xpE

)(

Time-Dependent Diffusion Equations In a more generalized three-dimensional analysis, the

above equation would have to be replaced by a vector identity.

The above equations are the time-dependent diffusion equations for holes and electrons, respectively. Since both the hole concentration p and the electron concentration n contain the excess concentrations, the equations describe the space and time behavior of the excess carriers.


tppg

xEp

xpE

xpD

ptppp

))()((2

2

tnng

xEn

xnE

xnD

ptnnn

))()((2

2

Time-Dependent Diffusion Equations The hole and electron concentrations are functions of

both the thermal equilibrium. The thermal equilibriumconcentrations, no and po. are not functions of time.

For the special case of a homogeneous semiconductor,no and po are also independent of the space coordinates,therefore:

Note that the Equations (6.29) and (6.30) contain termsinvolving the total concentrations, p and n, and termsinvolving only the excess concentrations, p and n.


tppg

xEp

xpE

xpD

ptppp

)())()(()(

2

2

tnng

xEn

xnE

xnD

ptnnn

)())()(()(

2

2

Ambipolar Transport If a pulse of excess electrons and a pulse of excess holes are

created at a particular point in a semiconductor with anapplied electric field, the excess holes and electrons will tendto drift in opposite directions.

Electrons and holes are charged particles, any separation willinduce an internal electric field between the two sets ofparticles. This internal electric field will create a forceattracting the electrons and holes back toward each other.

The electric field is then composed of the externally appliedfield plus the induced internal field.


intEEE app

Eapp

appn

x

---

+ + +

Fig. 6.3: The creation of an internal electric fieldas excess electrons and holes tend to separate.

Ambipolar Transport Since the internal E-field creates a force attracting the

electrons and holes, this E-field will hold the pulsesof excess electrons and excess holes together.

The negatively charged electrons and positivelycharged holes then will drift or diffuse together witha single effective mobility or diffusion coefficient.

This phenomenon is called ambipolar diffusion orambipolar transport:


tnRg

xnE

xnD

)()()( '

2

2'

Ambipolar Transport


tnRg

xnE

xnD

)()()( '

2

2'

pnnp

pn

pn

)('

pDnDpnDD

Dpn

pn

)('

The ambipolar diffusion coefficient D’ and the ambipolar mobilitycoefficient μ’ are defined from Einstein’s relation as follows:

Applications of Ambipolar Transport: Illustration of the behaviorof excess carriers in semiconductors, in PN junction diodes and otherdevices.

Exercise3. Consider an infinitely large, homogeneous n-

type semiconductor with a zero applied electricfield. Assume that, for t < 0, thesemiconductor is in thermal equilibrium andthat, for t≥0, a uniform generation rate existsin the crystal. Calculate the excess carrier concentration as a

function of time assuming the condition of lowinjection (where excess carrier concentration is muchlower than thermal equilibrium majority carrierconcentration n<<p0 (p type) or p<<n0 (n type) )

Wednesday, October 09, 2013Tennessee Technological University 31tnRg

xnE

xnD

)()()( '

2

2'

Solution The condition of a uniform generation rate and a

homogeneous semiconductor implies that:

Therefore, the ambipolar transport equationreduces to:

The solution of the differential is:


0)()(2

2

xp

xp

tppg

p

)(

0

'

)1()( 00

' p

t

p egtp

tpRg

xpE

xpD

)()()( '

2

2'

Exercise4. Consider an n-type Silicon at T = 300K doped

at Nd = 2 x 1016cm-3. Assume that p0 = 10-7sand g' = 5 x 1021 cm-3s-l. Determine the timedependence of excess carriers in reaching asteady-state condition. Does low injectionapply?


)1()( 00

' p

t

p egtp

Solution4. Using the equation:

Therefore,

Note 1: for t ∞, the steady state excess hole andelectron concentration of 5x1014cm-3 exists.

Note 2: p<<n0 therefore low injection is valid.


)1()( 00

' p

t

p egtp

3101410721 ]1[105)1)(10()105()( 77

cmexextp

tt

Dielectric Relaxation Time Constant


• Quasi-neutrality condition: the concentration of excess holes is balanced by an equal concentration of excess electrons.

• If a uniform concentration of holes p is suddenly injected into aportion of the surface of a semiconductor, we will instantly havea concentration of excess holes and a net positive charge densitythat is not balanced by a concentration of excess electrons.

How is charge neutrality achieved and how fast?• Dielectric Relaxation Time Constant

N typep holes

Fig. 6.4: The injection of a concentration of holes into a small region at the surface of an n-type semiconductor

Dielectric Relaxation Time Constant There are three defining equations to be considered. Poisson's equation is: (gradient of electric field equals net

charge density divided by permittivity)

From Ohm’s Law : (current density equals conductivitytimes electric field)

And from continuity equation, neglecting the effects ofgeneration and recombination:

The parameter is the net charge density and the initialvalue is given by e(p). We will assume that p isuniform over a short distance at the surface. Theparameter is the permittivity of the semiconductor.


E.

EJ

tJ

.

Dielectric Relaxation Time Constant Taking the divergence of Ohm's law and using

Poisson's equation: Substituting the above equation into the continuity

equation,which can be rearranged into the first order equation:

Whose solution is:

Dielectric Relaxation Time Constant is:


EJ ..

dtd

t

0

dtd

d

t

et

)0()(

d

Exercise5. Assume an n-type Silicon semiconductor with a

donor impurity concentration of Nd =1016cm-3

and mobility n = 1200cm2/V.sec . Calculate the dielectric relaxation time constant.


Solution5. The conductivity is found as

The permittivity of Silicon is calculated as:

The dielectric relaxation time constant is then:


11619 )(92.1)10)(1200)(10*6.1( cmNe dn

cmFr /)10*85.8)(7.11( 140

scm

cmFd

131

14

10*39.5)92.1(/)10*85.8)(7.11(

Picture Credits Semiconductor Physics and Devices, Donald Neaman, 4th

Edition, McGraw Hill Publications. B. Van Zeghbroeck, Principles of Semiconductor Devices,

Dept. of ECE, University of Colorado, Boulder, 2011http://ecee.colorado.edu/~bart/book/book/toc2.htm


Documents

Physical Electronics Slides of Chapter 6 (All Slides)blogs.cae.tntech.edu/bwabegaz42/files/2013/10/Physical-Electronics-Slides-of-Chapter-6...A semiconductor in thermal equilibrium,