NON-EQUILIBRIUM EXCESS CARRIERS IN SEMICONDUCTORS

NONEQUILIBRIUM EXCESS CARRIERS IN
SEMICONDUCTORS
10th June 2020
Manmohan Dash

We will discuss
1) generation and recombination of excess carriers in
a semiconductor
2) recombination rate and generation rate of excess
carriers, and excess carrier lifetime
3) Continuity equation
4) time dependent diffusion equation
5) ambipolar transport and corresponding equation

When voltage is applied or a current exists in
a semiconductor, its no more in equilibrium
1) Excess e- in the conduction band (CB) and excess
holes in the valence band (VB) may exist on top of
the thermal-equilibrium concentrations, due to
external excitation of the semiconductor.
2) We will discuss the behavior of nonequilibrium e- and
hole concentrations as functions of time and space
coordinates
INTRODUCTION

1) Excess e- and holes do not move independent of each
other. They diffuse, drift, and recombine with the same
effective diffusion coefficient, drift mobility, and
lifetime. This phenomenon is known as ambipolar
transport.
2) We will develop the ambipolar transport equation. It
describes the behavior of excess e- and holes. This is
central to the understanding of the electrical
properties and operation of the semiconductor
devices.
INTRODUCTION

generation and recombination of excess
carriers in a semiconductor
1) generation is the process in which e- and
holes are created
2) recombination is the process in which e-
and holes are annihilated
CARRIER GENERATION AND RECOMBINATION

Any deviation from thermal equilibrium
changes the e- and hole concentrations.
1) A sudden increase in temperature increases
the rate at which e- and holes are thermally
generated, their concentrations change with
time and new equilibrium values are reached.
2) An external excitation, such as light: can
generate e- and holes, creating a
nonequilibrium condition.

To understand the generation and
recombination processes:
A. we can consider direct band-to-band
generation and recombination
B. or the effect of allowed electronic energy states
within the bandgap, referred to as traps or
recombination centers.

Any deviation from thermal equilibrium changes
the e- and hole concentrations. In thermal
equilibrium:
1) Carrier concentrations are constant in time.
2) Carriers are generated: e- are continually thermally
excited from the VB into the CB by the thermal
process.
3) Carriers are recombined: e- move randomly through
the crystal in the CB, come close to holes and “fall”
into the empty states in the VB. This process annihilates
both the e- and the hole.
SEMICONDUCTORS IN EQUILIBRIUM

Net carrier
concentrations are
independent of time in
thermal equilibrium.
The rate of e- and
holes generation and
the rate of their
recombination must
be equal.

With Gn0: thermal-generation rate of e- s and Gp0: thermal-
generation rate of holes; (in units of #/cm3-s), for direct band-to-
band generation, e- s and holes, are created in pairs, so: Gn0 =
Gp0.
With Rn0: recombination rate of e- s and Rp0: recombination rate
of holes; (in units of #/cm3-s), in direct band-to-band
recombination, e- s and holes recombine in pairs, so: Rn0 = Rp0.
In thermal equilibrium, the concentrations of e- s and holes,
constant in time; generation and recombination rates are equal,
so Gn0 = Gp0= Rn0 = Rp0.

e- s in the VB may be excited into CB when, suitable
photons are incident on a semiconductor, an e- is
created in the CB, and a hole is created in the VB; an e-–
hole pair (EHP) is generated.
The additional e- s and holes created are called excess
electrons and excess holes.
We need to define certain parameters to continue our
discussion.
EXCESS CARRIER GENERATION AND RECOMBINATION

n0; thermal-equilibrium e- concentration -- independent of time
and also usually position.
n; total e- concentration -- may be function of time and/or
position.
n = n - n0; excess e- concentration -- may be function of time
and/or position.
g’n; excess e- generation rate, R’n; excess e- recombination rate
n0; excess minority carrier e- lifetime

p0; thermal-equilibrium hole concentration -- independent of time
and also usually position.
p; total hole concentration -- may be function of time and/or
position.
p = p - p0; excess hole concentration -- may be function of time
and/or position.
g’p; excess hole generation rate, R’p; excess hole recombination
rate
p0; excess minority carrier hole lifetime

Excess e- and holes are generated by external force at a
particular rate.
With g’n the generation rate of excess e- and g’p of excess
holes, for direct band-to band generation, excess e- and
holes are created in pairs, so: g’n = g’p.
When excess e- and holes are created, concentration of e- in
CB and holes in VB increase above their thermal equilibrium
values. So: n = n0 + n and p = p0 + p.

Since thermal equilibrium is no longer valid we can write: np ≠
n0p0 = ni
2 .

A steady-state generation of excess e- and holes does
not increase carrier concentrations. An e- in the CB may
“fall down” into the VB, leading to the process of excess
e-–hole recombination.
Recombination rate for excess e- (R’n) is equal to same
for excess hole (R’p) because excess e- and holes
recombine in pairs, so R’n = R’p.

Direct band-to-band recombination occurs spontaneously;
probability of an e- and hole recombining is constant with time.
The rate at which e- recombines must be proportional
to the e- concentration and also to the hole concentration.
(eg if there are no e- or holes, there can be no recombination)
So net rate of change in the e- concentration can be written as,
with n(t) = n0 + n(t) and p(t) = p0 + p(t);

The 1st term, rni
2 is thermal-equilibrium generation rate.
Excess e- and holes are created and recombined in pairs,
so n(t) = p(t).
(Excess e- and hole concentrations being equal we can say
excess carriers to mean either)
n0 and p0 (thermal-equilibrium parameters), are constant in time;
so

This can be solved if we impose the condition of low-
level injection (LLI).
LLI puts limits on magnitude of the excess carrier
concentration when compared with the thermal-
equilibrium carrier concentrations.
We know for Extrinsic n-type material, n0 >> p0, and for
extrinsic p-type material, p0 >> n0.

LLI means excess carrier concentration << thermal-
equilibrium majority carrier concentration.
High-level injection (HLI) means excess carrier
concentration becomes ~ to or > thermal-
equilibrium majority carrier concentrations.

Thus for p-type material (p0 >> n0) under LLI (n(t) << p0).
This has an exponential decay of “excess carrier concentration
n(t)” as the solution given by:
where n0=(rp0)-1 is a constant for LLI.
Thus this solution describes the decay of excess minority carrier e-,
and n0 is known as excess minority carrier lifetime.

The recombination rate of “excess minority carrier e-” (a
+ve quantity) can be written:
For direct band-to-band recombination, the “excess
majority carrier holes” recombine at the same rate, so
that for the p-type material:

For n-type material (n0 >> p0) under LLI (n(t) << n0),
minority carrier holes decay with a time constant
p0=(rn0)-1, where p0 is also known as “excess
minority carrier lifetime”.
The recombination rate of the majority carrier e- will
be the same as that of the minority carrier holes, so
we have

The generation rates of excess carriers are not
functions of e- or hole concentrations.
In general, the generation and
recombination rates may be functions of the
space coordinates and time.

The excess e- and holes do not move independent of
each other.
We saw the generation and recombination rates of excess
carriers. Now we need to know how the excess carriers behave in
time and space in presence of electric fields and density
gradients.
Excess e- and holes do not move independently: they diffuse and
drift with the same effective diffusion coefficient and effective
mobility. This phenomenon is known as ambipolar transport.
CHARACTERISTICS OF EXCESS CARRIERS

Extrinsic semiconductor under low injection: effective
diffusion coefficient (D) and mobility parameters () are
those of the minority carrier.
We are interested about D and , as they characterize the
behavior of excess carriers. For this we must develop the
continuity equations for the carriers and develop the ambipolar
transport equations.
Behavior of excess carriers has a profound impact on the
characteristics of semiconductor devices.
CHARACTERISTICS OF EXCESS CARRIERS

Fpx
+ is the hole-particle flux, with unit: # of
holes/cm2-s. For the x component of the particle
current density:
(by Taylor expansion of LHS, dx being small, only 1st
two terms are significant) Net increase in # of holes
per unit time in the differential volume element due
to the x-component of hole flux is:
CONTINUITY EQUATIONS

Carrier flux (of holes here)

Generation rate (g) and recombination rate (R) of holes
also contribute to the hole concentration.
Net increase in # of holes per unit time in the differential volume
element is thus:
p is density of holes. 1st term on RHS: increase due to hole flux,
2nd term: increase due to generation of holes, last term:
decrease due to recombination of holes.
pt includes both (i) thermal-equilibrium carrier lifetime and (ii)
excess carrier lifetime.

This leads to the continuity equation for the holes
(and by hindsight for the e-)
Hole continuity equation:
Similarly we have the
e- continuity equation:

For thermal equilibrium hole and e- current
densities, are given, in 1 dim, by
Hole current density:
e- current density:
Dividing by electronic charge we get corresponding particle flux.
TIME DEPENDENT DIFFUSION EQUATIONS

We spatially differentiate the results, substitute into
equation of continuity and use product rule:
Product rule
Hole time-dependent diffusion eqn:
e- time-dependent diffusion eqn :
They also describe the space and time behavior of the excess
carriers.

hole and e- concentrations are functions of both the
thermal equilibrium and the excess values.
thermal equilibrium concentrations, n0 and p0, are not functions of
time. In case of homogeneous semiconductor, n0 and p0 are also
independent of the space coordinates. Time dependent diffusion
equations take the form:
Hole time-dependent diffusion equation:
e- time-dependent diffusion equation :
Here the total concentrations and the excess concentrations appear
separately.

Excess carriers move together
due to attractive force
between them.
A pulse of excess e- and excess holes tend
to drift in opposite directions to applied
electric field Eapp. Any separation induces
an internal electric field Eint between two
sets of excess carriers.
The internal electric field will create a
force attracting the e- and holes back
toward each other as shown in Figure.
Thus E = Eapp + Eint.
AMBIPOLAR TRANSPORT PHENOMENA

So excess carriers drift or diffuse
together with a single effective
mobility or diffusion coefficient.
This is known as ambipolar
diffusion or ambipolar transport.
Relatively small internal electric
field is sufficient to keep excess e-
and holes drifting and diffusing
together. So we can assume that
|Eint|<<|Eapp|.
AMBIPOLAR TRANSPORT EQUATION
We need the Poisson’s
equation to relate the excess
carrier with the internal
electric field >>
Here s is the permittivity of the
semiconductor material

A very small difference in excess e- and hole density sets
up an Eint field sufficient to keep the carriers diffusing and
drifting together
1% difference in p and n, eg, results in non-negligible .E =.Eint
terms in diffusion equations.
Charge neutrality: excess e- and hole concentration are
balanced at any point in space and time. (This is an approximate
statement otherwise there would be no internal field to keep the
excess carriers together).

Diffusion equations can be used together to eliminate the
divergence terms.
We define the generation rates of e- and holes: gn = gp  g and
recombination rates:
Lifetimes include both thermal-equilibrium and excess carrier
values. We denote excess e- and hole concentrations by n. With
charge neutrality p ≈ n, diffusion equations takes the following
form. (skipping some algebraic details)

This is the ambipolar transport equation:
Here we have defined D’ as the Ambipolar diffusion coefficient and ’ as the
Ambipolar mobility.
By using Einstein’s relations we can cast (D’  Ambipolar diffusion coefficient )
into another form.
Due to time dependence of p and n (inside of n and p) ambipolar
transport equation is a nonlinear differential equation

The ambipolar transport equation can be simplified and
linearized for extrinsic semiconductor under low-level
injection.
We can write Ambipolar diffusion
coefficient as:
For p-type semiconductor, p0 >> n0, low-level injection means n <<
p0. Assuming Dn and Dp are on the same order of magnitude,
ambipolar diffusion coefficient becomes: D’ = Dn. Similarly
ambipolar mobility is simplified to ’ = n.
EXTRINSIC DOPING AND LOW INJECTION

Ambipolar transport equation reduces to a linear
differential equation with constant coefficients for
extrinsic semiconductor under low-level injection.
For an extrinsic p-type semiconductor under low injection, we
saw that ambipolar parameters reduce to minority carrier e-
parameter values, which are constants.
Similarly for extrinsic n-type semiconductor under low injection,
p0 << n0 and n << n0. The ambipolar diffusion coefficient
reduces to D’ = Dp and ambipolar mobility reduces to ’ = - p.
So the ambipolar parameters again reduce to minority-carrier
(hole) values, which are constants.

Generation and recombination rate
terms in the ambipolar transport
equation.
We already saw e- and hole recombination rates
are equal. With nt and pt being the mean e- and
hole lifetimes, respectively;

1/nt is the probability per unit time that an e- will encounter a
hole and recombine. 1/pt is the probability per unit time that a
hole will encounter an e- and recombine. For an extrinsic p-type
semiconductor under low injection, the concentration of majority
carrier holes will be essentially constant, even when excess
carriers are present.
Probability per unit time of a minority carrier e- encountering a
majority carrier hole will be essentially constant. So, minority
carrier e- lifetime, nt  n, will remain a constant for extrinsic p-
type semiconductor under low injection. Similarly for an extrinsic
n-type semiconductor under low injection, the minority carrier
hole lifetime, pt  p, will also remain constant.

For e- we can write g – R = gn -Rn = (Gn0 + gn’ ) - (Rn0 + Rn’ ).
With thermal equilibrium: Gn0 = Rn0, we have g – R = gn’ -
Rn’ = gn’ - n/n, where n is excess minority carrier e-
lifetime.
For holes this becomes: g – R = gp’ - Rp’ = gp’ - n/p where
p is excess minority carrier hole lifetime.
Generation rate for excess e- is equal to that for excess
holes. We define a generation rate for excess carriers as g’,
so gn’ = gp’ = g’.

Ambipolar transport equation takes the form
p-type semiconductor,
low injection
n-type semiconductor,
low injection
Since excess majority carriers, diffuse and drift with the
excess minority carriers; the behavior of the excess majority
carrier is determined by the minority carrier parameters as
we see in the equations above.

NON-EQUILIBRIUM EXCESS CARRIERS IN SEMICONDUCTORS

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