1
Carriers Concentration and Current in
Semiconductors
Prof.P. Ravindran,
Department of Physics, Central University of Tamil
Nadu, India
http://folk.uio.no/ravi/semi2013
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
Carrier Transport



Two driving forces for carrier transport: electric field and spatial
variation of the carrier concentration.
Both driving forces lead to a directional motion of carriers superimposed
on the random thermal motion.
To calculate the directional carrier motion and the currents in a
semiconductor, classical & nonclassical models can be used.

The classical models assume that variation of E-field in time is
sufficiently slow so that the transport properties of carriers (mobility or
diffusivity) can follow the changes of the field immediately.

If carriers are exposed to a fast-varying field, they may not be able to
adjust their transport properties instantaneously to variations of the
field, and carrier mobility and diffusivity may be different from their
steady-state values  nonstationary
Nonstationary carrier transport can occur in electron devices under
both dc and ac bias conditions.

P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
2
Classical Description of Carrier Transport
Assume thermal equilibrium for a semiconductor having a spatially
homogeneous carrier concentration with no applied E-field. No driving
force for directional carrier motion. The carriers not in standstill condition
but in continuous motion due to kinetic energy. For electron in the
conduction band, E  32 k T  m2 v where vth is the thermal velocity, mn* is the
conductivity effective electron mass.
 The average time between two scattering events is the mean free time and
the average distance a carrier travels between collisions is the mean free
path.  Fig. 2.5 (a)
 Applying V, the E-fields adds a directional component to the random motion
of the electron.  Fig. 2.5 (b)
*
n
kin
B
2
th
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
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



The mean electron velocity: vn= -μnE
The directed unilateral motion of carriers caused by
E-field is drift velocity.
Similarly, vp = μpE
A change in E-field instantaneously results in a
change of the drift velocity.
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
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Fick’s First Law: relating diffusion current
to carrier concentration gradient.
dn
e  De
dx
e = electron flux, De = diffusion coefficient of electrons, dn/dx =
electron concentration gradient
Electron Diffusion Current Density
dn
J D,e  ee  eDe
dx
JD, e = electric current density due to electron diffusion,
Where:
e = electron flux, e = electronic charge,
De = diffusion coefficient of electrons,
dn/dx = electron concentration gradient
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
Hole Diffusion Current Density
dp
JD,h  eh  eDh
dx
JD, h = electric current density due to hole diffusion, e = electronic
charge, h = hole flux, Dh = diffusion coefficient of holes, dp/dx =
hole concentration gradient

Total Electron Current Due to Drift and Diffusion
dn
Je  en eEx  eDe
dx
Je = electron current due to drift and diffusion, n = electron
concentration, e = electron drift mobility, Ex = electric field in the x
direction, De = diffusion coefficient of electrons, dn/dx = electron
concentration gradient
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
Total Currents Due to Drift and Diffusion
dp
Jh  ephEx  eDh
dx
Jh = hole current due to drift and diffusion, p = hole concentration,
h = hole drift mobility, Ex = electric field in the x direction,
Dh = diffusion coefficient of holes,
dp/dx = hole concentration gradient
dn
Je  en eEx  eDe
dx
Je = electron current due to drift and diffusion, n = electron concentration
e = electron drift mobility, Ex = electric field in the x direction,
De = diffusion coefficient of electrons,
dn/dx = electron concentration gradient
Jtotal = Jh+Je
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
Einstein Relation:
diffusion coefficient and mobility are related!
De
kT

e
e
and
Dh
kT

h
e
De = diffusion coefficient of electrons,
e = electron drift mobility,
Dh = diffusion coefficient of the holes,
h = hole drift mobility
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
Carrier diffusion due to doping level gradient.
This is a common device fabrication step.
· represents electrons (majority carriers in this
case)
Exposed
As+ Donor
n2
Vo
Ex
n1
Diffusion occurs until
an electric field builds
up!
Note: the As+
are fixed,
non-mobile
charges!
Diffusion Flux
Drift
Net current = 0
We call this the built-in
potential.
Fig. 5.32: Non-uniform doping profile results in electron diffusion
towards the less concentrated regions. This exposes positively charged
donors and sets up a built-in field Ex . In the steady state, the diffusion of
electrons towards the right is balanced by their drift towards the left.
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
Built-In Potential and Concentration

kT 
n
2 
V2  V1 
ln 
 
e n1 
Exposed
As+ Donor
n2
Vo
Ex
n1
V2 = potential at point 2, V1 = potential at point 1,
n2 = electron concentration at point 2,
n1 = electron concentration at point 1
Diffusion Flux
Drift
Net current = 0
Built-In Field in Nonuniform Doping
kT
Ex 
be
Fig. 5.32: Non-uniform doping profile results in electron diffusio
towards the less concentrated regions. This exposes positively ch
donors and sets up a built-in field Ex . In the steady state, the dif
electrons towards the right is balanced by their drift towards the
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)
http://Materials.Usask.Ca
Ex = electric field in the x direction,
b = characteristic of the exponential doping profile,
e = electronic charge .
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
Carrier creation: Photoinjected charge carriers
If we shine light on a
semiconductor, we will
generate new charge carriers
(in addition to those thermally
generated) if Ephoton>Egap.
If the light is always on and of
constant intensity, some steady
state concentration of
electrons and holes will result.
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
Carrier creation: Photoinjected charge carriers
Let’s consider the case of n-type material
Consider an n-type semiconductor with a doping concentration of 5 x 1016
cm-3.
What are the carrier concentrations?
Let’s define some terms;
nno majority carrier concentration in the n-type
in the dark (only thermally ionized carriers)
(i.e. the electron concentration in n-type)
pno minority carrier concentration in the n-type
in the dark (only thermally ionized carriers)
(i.e. the hole concentration in n-type)
semiconductor
semiconductor
Note: the no subscript implies that mass action law is valid!
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
When we have light:
With light of Ephoton>Egap hitting the semiconductor, we get
photogeneration of excess charge carriers.
nn  excess electron concentration such that::
nn = nn-nn0
&
pn  excess hole concentration such that::
pn = pn-pn0
Note that photogenerated carriers excited across the gap can only be
created in pairs
i.e. pn = nn and now
(in light) nnpn≠ni2 i.e. mass action not valid!
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors

Carrier density change under illumination
If the temperature is constant,
nn0 and pn0 are not time dependent, so
dnn dn n

dt
dt
and
dpn dpn

dt
dt
Consider the case of ‘weak’ illumination, which creates a 10% change in nn0
i.e. nn = 0.1nn0
Or if the doping level is nno=5 x 1016cm-3, then

nn = 0.1nn0= 0.5 x 1016cm-3
And pn =nn = 0.5 x 1016cm-3
Which change is more important? Majority or minority?
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
Recall the intrinsic carrier concentration
For Si
ni is roughly 1.5x1010cm-3
At room temperature
Since pno=ni2/nn0
= (1.5x1010)2/5x1016
pno =4500 cm-3
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
pn =nn = 0.5 x 1016cm-3
An extremely important
concept!
Minority carrier
concentration can be
controlled over many
orders of magnitude
with only a small change
in majority
concentration.
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
Carrier creation followed by recombination
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
Carrier creation followed by recombination
Mostly majority
carriers in the
dark
Almost equal
Carrier concentration
In light
The extra minority carriers
recombine once the
generation source is removed.
How quickly do
the carriers
recombine?
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
Minority carrier lifetime h for n-type
h = average time a hole exists in the valance band from its generation
until its recombination
And so 1/ h is the average probability (per unit time) that a hole will
recombine with an electron.
h depends on impurities, defects and temperature.
The recombination process in a real semiconductor usually involves a
carrier being localized at a recombination center.
 can be short (nanoseconds) allowing fast response (e.g. switch)
or slow (seconds) for a photoconductor or solar cell
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
Excess Minority Carrier Concentration
dpn
pn
 Gph 
dt
h
pn = excess hole (minority carrier) concentration,
t = time,
Gph = rate of photogeneration,
h = minority carrier lifetime (mean recombination time)
h = average time a hole exists in the valance band from its
generation until its recombination
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
Carrier concentration versus time with pulsed illumination
t’ is time after
illumination is
removed
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
Continuous illumination provides increased conductivity
Often used as a switch
or motion detector
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
Carrier diffusion away from high concentration
holes in this p-type
example
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors
Carrier motion: via diffusion (due to concentration gradient)
and drift (due to electric field)
Both diffusion and
drift occur in
semiconductors.
Note here that holes
(minority carriers)
drift and diffuse in
the same direction;
but electrons
(majority carriers)
do not!
With light we alter minority carrier concentration
P.Ravindran, PHY02E – Semiconductor Physics, February 2014: Carriers Concentration, Current & Hall Effect in Semiconductors