Chapter 6Lecture Notes

Semiconductors in Nonequilibrium Conditions

Excess electrons in the conduction band and excess holes in the valence band may exist in addition to the thermal-equilibrium concentrations if an external excitation is applied to the semiconductor. The creation of excess electrons and holes means that the semiconductor is no longer in thermal equilibrium.

Carrier generation and recombination

Generation ⇒ process whereby electrons and holes are created

Recombination ⇒ process whereby electrons and holes are coincided and lost.

A sudden change in temperature or optical excitation can generate excess electrons and holes creating a nonequilibrium condition.

Semiconductors in Nonequilibrium Conditions

Injection ⇒ A process of introducing excess carriers insemiconductors.

Generation and recombination are two types:

(i) Direct band-to-band generation and recombination and

(ii) the recombination through allowed energy states within the bandgap, referred to as traps or recombination centers.

Direct band-to-band generation and recombination

The random generation-recombination of electrons-holes occur continuously due to the thermal excitation.

In direct band-to-band generation-recombination, the electrons and holes are created-annihilated in pairs:

00 pn GG =00 pn RR

Thermal equilibrium:

=

0000 pnpn RRGG ===

At thermal equilibrium, the concentrations of electrons and holes are independent of time; therefore, the generation and recombination rates are equal, so we have,

Excess carrier generation and recombinationSymbol Definition n0, p0

n, p

δn = n- n0 δp = p- p0

Gn0, Gp0 Rn0, Rp0

pn gg ′′ ,

pn RR ′′ ,

τn0,τp0

Thermal equilibrium electron and hole concentrations (independent of time)

Total electron and hole concentrations (may be functions of time and/or position).

Excess electron and hole concentrations (may be functions of time and/or position).

Thermal electron and hole generation rates (cm-3s-1) Thermal equilibrium electron and hole recombination rates (cm-3s-1)

Excess electron and hole generation rates (cm-3s-1)

Excess electron and hole recombination rates (cm-3s-1)

Excess minority carrier electron and hole lifetimes (s)

Nonequilibrium Excess Carriers

( ) ( ),0 tnntn δ+= ( ) ( )tpptp δ+= 02innp ≠and

The excess electrons and holes are created in pairs,

pn gg ′=′ ( ) ( )tptn δδ =and

The excess electrons and holes recombine in pairs,

When excess electrons and holes are created,

pn RR ′=′

( )

The excess carrier will decay over time and the decay rate depends on the concentration of excess carrier.

( ) ( )[ ]tptnndt

tdni −∝ 2 ( ) ( ) ( )[ ]tptnn

dttdn

ir −= 2α

αr is the constant of proportionality for recombination.

or,

Nonequilibrium Excess Carriers

where τn0 is the excess minority carrier lifetime for electron and τn0 = 1/(αrp0)

( ) ( ) ( ) ( )[ ]tnpntndt

tndr δδαδ

++−= 00

( ) ( ) ( )[ ]tptnndt

tdnir −= 2α

( ) ( ),0 tnntn δ+= ( ) ( )tpptp δ+= 0 ( ) ( )tptn δδ =and

For low-level injection condition: ( ) ( )tnnp δ>>+ 00

For p-type material, 00 np >>

( ) ( ) 0ptndt

tndrδαδ

−=Thus,

( ) ( ) ( ) 00 /00 nr ttp enentn τα δδδ −− ==The solution is,

( ) ( ) ( ) ( )[ ]tnpntndt

tndr δδαδ

++−= 00

For high-level injection condition: ( ) ( )tnnp δ⟨+ 00

( )nnpr δατ

++=

000

1

( ) ( ) ( ) ( )[ ]0

00 τδδδαδ ntnpntn

dttndR rn =++=−=′

Examples

The recombination rate is a positive quantity.

And, for n-type

The excess recombination rate is determined by the excess minority carrier lifetime.

The recombination rate for electrons in p-type material,

( ) ( ) ( )0

0n

rntnptn

dttndR

τδδαδ

==−=′

The excess electrons and holes recombine in pairs, (in p-type material)

( )0n

pntnRR

τδ

=′=′

( )0p

pntpRR

τδ

=′=′

where τp0 is the minority carrier lifetime of holes.

Continuity equations:

The continuity equation describes the behavior of excess carriers with time and in space in the presence of electric fields and density gradients.

Fp(x) Fp(x + Δx)

x x + ΔxArea, A

∼

Δx

The net increase in hole concentration per unit time,

( ) ( )pp

ppx

xxx

Rgx

xxFxFtp

−+Δ

Δ+−=

∂∂ ++

→ΔΔ+→

0lim

F =F+ represents hole particle flux (# / cm2 –s)

ppp Rgx

Ftp

−+∂

∂−=

∂∂ +

The hole flux, eJF pp /=+The unit of hole flux is holes/cm2-s

ppp Rg

xJ

etp

−+∂

∂−=

∂∂ 1

nnn Rg

xJ

etn

−+∂∂

=∂∂ 1

Similarly for electrons,

Continuity equations

These are the continuity equations for holes and electrons respectively

Recall: In one dimension, the electron and hole current densities due to the drift and diffusion are given by:

xneDnEeJ nnn ∂∂

+= μxpeDpEeJ ppp ∂∂

−= μ

( )nnnn Rg

xnE

xnD

tn

−+∂

∂+

∂∂

=∂∂ μ2

2( )pppp Rg

xpE

xpD

tp

−+∂

∂−

∂∂

=∂∂ μ2

2

nnnn RgxEn

xnE

xnD

tn

−+⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

+∂∂

=∂∂ μ2

2

pppp RgxEp

xpE

xpD

tp

−+⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

−∂∂

=∂∂ μ2

2

Or,

Substitute these in the continuity equations:

The thermal-equilibrium concentrations, n0 and p0, are not functions of time. For the special case of homogeneous semiconductor, n0 and p0 are also independent of the space coordinates. So the continuity may then be written in the form of:

nnnn RgxEn

xnE

xnD

tn

−+⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

+∂∂

=∂∂ δμδδ

2

2

nppp RgxEp

xpE

xpD

tp

−+⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

−∂∂

=∂∂ δμδδ

2

2

pppandnnn δδ +=+= 00

Time –dependent diffusion equations

Ambipolar transport equation

Under applied external electric field, the excess electrons and holes created by any mean say, light illumination, tend to drift in the opposite directions.Separation of these charged particles induces an internal electric field opposite to the applied external electric field. This induced internal electric field attract the electrons and holes move toward each other, holding the pulses of excess electrons and holes together. So, the electric field appears in the time-dependent diffusion equations is composed of the external and the induced internal electric fields.Electrons and holes drift or diffuse together with a single effective mobility or diffusion coefficient. This phenomena is called ambipolar transport.

ggg pn == RRRand pn ==

apps

EEx

Enpe⟨⟨

∂∂

=−

intint ;)(

εδδ

The effect of internal field

As the electrons and holes are diffusing and drifting together:

RgxEn

xnE

xnD

tn

nn −+⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

+∂∂

=∂∂ δμδδ

2

2

RgxEp

xnE

xnD

tn

pp −+⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

−∂∂

=∂∂ δμδδ

2

2

EliminatingxE∂∂

RgxnE

xnD

tn

−+∂∂′+

∂∂′=

∂∂ δμδδ

2

2

( )pnnp

pn

pn

μμμμ

μ+

−=′

( )pDnDpnDD

Dpn

pn

+

+=′

D′ is the ambipolar diffusion coefficient

μ′ is called the ambipolar mobility

Assumption:Charge neutrality (quasi-neutrality) δn ≈δp

Ambipolar transport equation

02

2

nnn

ngxnE

xnD

tn

τδδμδδ

−+∂∂

+∂∂

=∂∂

02

2

ppp

pgxpE

xpD

tp

τδδμδδ

−+∂∂

−∂∂

=∂∂

For p-type semiconductors

And for n-type semiconductors

For p-type semiconductors

For n-type semiconductors D′ ≈ Dp and μ′ ≈ - μp

D′ ≈ Dn and μ′ ≈μn

( ) ( )

( ) ( )

( ) ( )

0,,)6(

0 :ionrecombinatcarrier excess No (5)

0' :generationcarrier excess No (4)

0 :Field Electric Zero(3)

0,0,0 :Carriers excess ofon distributi Uniform(2)

0 :stateSteady (1)

0000

00

2

2

2

2

===′∞==

===′

=

=∂

∂=

∂∂

=∂∂

=∂∂

=∂

∂=

∂∂

=∂

∂=

∂∂

pnnp

pn

np

pnRlifetimescarrierInfinite

pnR

gxpE

xnE

xp

xn

xnD

xnD

tp

tn

τδ

τδττ

τδ

τδ

δδ

δδδδ

δδ

Applications of ambipolar transport equation

Indirect recombination:

In real semiconductors, there are some crystal defects and these defects create discrete electronic energy states within the forbidden energy band.

Recombination through the defect (trap) states is called indirect recombination

The carrier lifetime due to the recombination through the defect energy state is determined by the Shockley-Read-Hall theory of recombination.

Shockley-Read-Hall recombination:

Shockley-Read-Hall theory of recombination assumes that a single trap center exists at an energy Et within the bandgap.

The trap center here is acceptor-like.

It is negatively charged when it contains an electron and is neutral when it does not contain an electron.

There are four basic processes

Capture rate ∝ (free carrier concentration )×(concentration of empty defects states)

Emission rate ∝ (trapped carrier concentration )×(concentration of empty conduction or valence band states)

Emission rate ∝ (concentration of trapped carriers)

The electron capture rate (process 1), ( )[ ] ttFncn NEfnCR −= 1

The electron emission rate (process 2), ( )tEtnen EfNER =

The relationship between the capture Cn and emission coefficients En can be determined by the principle of detailed balance.

Principle of Detailed Balance: Under equilibrium conditions each fundamental process and its inverse must self-balance independent of any other process that may be occurring inside the material.

Under thermal equilibrium, Rcn = Ren, which gives a relation between Cn and En.

In nonequilibrium, the net rate at which electrons are captured from the conduction band is given by,

encnn RRR −=′

Under steady state condition, there is no change of trap carrier concentration.

( )( ) ( )ppCnnC

nnpNCCRR

pn

itpnpn ′++′+

−=′=′

2

Thus

( ) ( )⎥⎦⎤

⎢⎣⎡ −−

=′⎥⎦⎤

⎢⎣⎡ −−

=′kT

EENp

kTEE

Nn vtv

tcc expandexpWhere,

The trap level energy is near the midgap so that n′ and p′ are not too different from the intrinsic carrier concentration ni.

inpn

At thermal equilibrium,

=′=′

200 inpnnp == so that 0=′=′ pn RR

( )( ) ( )

( )( ) ( )

( )( )( ) ( )inip

i

inip

i

pn

itpn

nnpnnnnnpnn

npnnnnp

ppCnnCnnpNCC

R

+++++−++

=

+++−

≈′++′+

−=′

δτδτδδ

ττ

0000

200

00

22

( ) ( )( ) ( )inip nnpnnn

nnpnR+++++

++=′

δτδτδδ

0000

200

For p-type semiconductors0n

nnR

τδ

=′

tnn NC

10 =τ

For n-type semiconductors0p

ppR

τδ

=′

Limits of extrinsic doping and low injection:

For high level injection,00 np

nppRRττ

δ+

=′=′

tpp NC

10 =τ

Example A step illumination is applied uniformly to an n-type semiconductor at time t = 0 and switched off at time t = toff (toff >> τp0). No applied bias. (1)Find the expression of the hole concentration and (2) Sketch hole concentration vs time for 0 < t < ∞ . (3) Also find the expression of the conductivity for 0 < t < ∞ .

Light

Illumination

0Time, t

po

toff

po+δp(∞)

t'

τp0g'

0

g and δp(t)

δp(t') = δp(0)exp(-t'/τp0)

02

2

ppp

pgxpE

xpD

tp

τδδμδδ

−′+∂∂

−∂∂

=∂∂

A semiconductor has the following properties: The semiconductor is a homogeneous, p-type material in thermal equilibrium for t ≤ 0. At t = 0, an external source is turned on which produces excess carriers uniformly at rate of . At, the external source is turned off. (1) Derive the expression for the excess-electron concentration as a function of time for 0 ≤ t ≤ ∞.

(2) Determine the value of excess electron concentration at (i) t = 0, a. (ii) , (iii) and (iv) t = ∞.b.(3) Plot the excess electron concentration as a function of time.

132010' −−= scmg st 6102 −×=

st 6102 −×= st 6103 −×=

Example 6.2:

sscmD

sscmD

pp

nn7

02

60

2

10/10

10/25−

−

==

==

τ

τ

Example : A continuous illumination is applied to an n-type semiconductor at time x = 0 and no carrier generation for x > 0. That means g′ = 0 for x > 0. Determine the steady state excess concentrations for holes for x > 0 and the hole diffusion current.

n-type semiconductor

xx = 0

δp(x),

δp(0) Diffusionδn(0)

δn(x)

Excess concentration

x

Numerical: Consider p type Silicon at T = 300 K. Assume that τn0 = 5 × 10-7 s, Dn = 25 cm2/s and δn(0) = 1015 cm-3. Determine excess hole concentration profile.

02

2

ppp

pgxpE

xpD

tp

τδδμδδ

−′+∂∂

−∂∂

=∂∂

pp Lx

Lx

BeAexp−

+=)(δ

02

2

0p

pp

xpD

τδδ

−∂∂

=

pp Lx

Lx

BeAexp−

+=)(δ

Dielectric relaxation time constant:

( ) ( ) dtet τρρ /0 −= σετ =dwhere

If an amount of net charge is injected suddenly in a semiconductor, the free charge carriers of opposite sign try to balance the injected charge and establish charge neutrality. How fast the charge neutrality can be achieved is determined by the dielectric relaxation time constant, τd. This phenomenon is called dielectric relaxation. The excess net charge decays exponentially,

Example 6.6(a) Find the dielectric relaxation time constant if the resistivity of the sample is 1014 Ω-cm and relative permittivity is 6.3.

(b) Find the dielectric relaxation time constant of an n-type Silicon with Nd = 1016 cm-3.

tJEJE

∂∂

−=∇==∇ρσ

ερ .,,.

0)( =+ ρεσρ

dtd

Surface Effects

02

2

0p

ppg

xpD

τδδ

−′+∂∂

=

pp Lx

Lx

p BeAegxp−

++= τδ ')(pS

pB

S

B

pp

ττ

δδ

=

0,00;; =⇒≠∴→∞→∞→−

AbutBBeAexAs pp Lx

Lx

02

2

0p

ppg

xpD

τδδ

−′+∂∂

=

Bgxpgxp pp +=→=∞→ τδτδ ')0(;')(

Time and Spatial Dependence of Excess Carrier Concentration

Example : A finite number of carriers is generated instantaneously at time t = 0 and x = 0, but g′ = 0 for t > 0. Assume an n-type semiconductor with a constant applied electric field E0, which is applied in the +x direction. Determine the excess carrier concentration as a function of x and t.

The ambipolar equation for t > 0 and x > 0,

002

2

ppp

pxpE

xpD

tp

τδδμδδ

−∂∂

−∂

∂=

∂∂

The solution of the above equation,

( ) ( )( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−=

−

tDtEx

tDetxp

p

p

p

t p

4exp

4,

20

2/1

0 μ

πδ

τ

Quasi-Fermi level:If excess carriers are created in a semiconductor, the Fermi energy is strictly no longer defined. Under nonequilibrium condition, we can assign individual Fermi level for electrons and holes. These are called quasi-Fermi levels and the total electron or hole concentration can be determined using these quasi-Fermi levels. We can write,

⎟⎠⎞

⎜⎝⎛ −

=+kT

EEnnn FiFn

i exp0 δ

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=+

kTEE

npp FpFii exp0 δ

Example Given: n-type semiconductor at 300K. n0 = 1015 cm-3, p0 = 105 cm-3, ni = 1010 cm-3, δn = δp=1013 cm-3. Calculate (a) the Fermi energy level and (b) the quasi Fermi energy levels.