Doped semiconductors: donor impurities
A silicon lattice with a single impurity atom (Phosphorus, P) added.
As compared to Si, the Phosphorus has one extra valence electron which, after all
bonds are made, has very weak bonding.
Very small energy is required to create a free electron from an impurity atom.
This type of impurity is called donor.
Note, that there is no hole created when a free electron comes from the impurity atom.
Free electron concentration in donor - doped semiconductors
When donor atoms are introduced into the semiconductor material, they are all ionized.
Each donor atom creates one free electron.
If the concentration of donor impurity (e.g. Phosphor) in Si is ND,
the concentration of free electrons,
n ≈ ND
For Si and other semiconductors, the typical doping levels are:
ND = 1015 cm-3 ….1018 cm-3
nD = 1015 cm-3 ….1018 cm-3 (compare to ni = 1.3×1010 cm-3 in intrinsic Si)
nD >> ni
Doping provides a flexible control over semiconductor conductivity.
The vast majority of microelectronic devices are based on doped semiconductors
Resistance of Donor-Doped Silicon sample
How much would be the resistance of the (1 cm×1cm× 1cm) Si sample
doped with donor impurities with concentration 2×1016 cm-3?
σ = qnµ ;
L 1 L
R=ρ = ×
A σ A
n = 2×1016 cm-3
µn = 1000 cm2/(V ×s)
q = 1.6 ×10-19 C
σ = 1.6 ×10-19 C × 2×1016 cm-3 × 1000 cm2/(V ×s)
σ = 3.2 (Ohm × cm)-1
ρ = 0.325 Ohm × cm
R = 0.325 (Ohm × cm) ×1 cm /(1cm ×1cm) = 0.325 Ohm
The resistance of a doped Si crystal can be significantly lower
than that of intrinsic Si
Doped semiconductors: acceptor impurities
A silicon lattice with a single impurity atom (Boron, B) added.
Boron has only three valence electrons, one electron less than the Si atom.
Having only three valence electrons - not enough to fill all four bonds - it creates an
excess hole that can be used in conduction.
This type of impurity is called acceptor.
There is no corresponding free electron created from acceptor impurity
Hole concentration in acceptor - doped semiconductors
If the concentration of acceptor impurity (B atoms) in Si is NA, the hole concentration
pA ≈ NA
For Si and other semiconductors, the typical acceptor doping levels are:
NA = 1015 cm-3 ….1018 cm-3
pA = 1015 cm-3 ….1018 cm-3 (compare to ni = 1.3×1010 cm-3 in intrinsic Si);
pA >> ni
The vast majority of microelectronic devices using hole conductivity,
are based on doped semiconductors
In doped semiconductors, the concentration of intrinsic electrons and holes can be
neglected as compared to those coming from donor and acceptor impurities.
Concentration –temperature dependence in doped
semiconductors
n, cm-3
Impurity electrons
ND
Intrinsic electrons,
intrinsic holes
T
100 K
200 K
300 K
400 K
Typical dependence for n-Si (i.e. donor-doped)
(for p-Si (i.e. acceptor doped) the dependences are similar
Mobile charge carriers energy
In semiconductors, the mobile charge carriers are
the free electrons and holes
Ec
Bound
electron
Ev
Atom
Intrinsic material at low temperature. There are no free
electrons or holes – no free carriers.
The mobile charge energy does not make sense.
valence
band
Conductance band energy
Hole
Ec
conductance
band
Free
electron
Ev
Atom
When the electron in the valence band acquires sufficient extra
energy, it can be detached from its parent atom and reaches
reach the
“conductance band”
The minimum energy of the conduction band is denoted as EC
Energy Band Gap (Eg)
Ec
Band-gap
Ev
Forbidden
Energy
region
Generally no electron can have the energy between Ec and Ev
The “band-gap” is the energy difference between Ec and Ev:
Eg=Ec-Ev
Mobile charge carriers energy
conductance
band
Hole
Ec
Free
electron
Ev
Atom
valence band
Intrinsic material at high temperature. Temperature generates
free electrons and holes in equal concentrations.
The energy of free electrons is close to EC; the energy of holes is
close to EV
Average free carrier Energy – Fermi energy
conductance
band
The average energy of all
the mobile charges in
semiconductor:
Eave Average [(Electron
Average Energy + Hole
Average Energy)] ≈
(EC + EV)/2.
The average energy of all
the mobile charges in
semiconductor is called
Fermi energy EF.
In intrinsic semiconductor:
EF ≈ (EC + EV)/2.
Ec
EF
Ev
valence band
The energy of free electrons is close to EC; the energy of holes is
equal to EV
n-type semiconductor
Extra free electron
Phosphorus (P)
has 5 outer
shell electrons.
In the n-type material most of
the mobile charges are free
electrons.
Therefore, the average energy
of mobile charges is close to
EC:
EF ≈ EC
EC
EFn
EV
p-type semiconductor
Extra electron
vacancy or hole
Boron (B) has 3
outer shell
electrons.
In the p-type material most of
the mobile charges are holes.
Therefore, the average energy
of mobile charges is close to
EV:
EF ≈ EV
EC
EFp
EV
Carrier Concentration and
Fermi level: n-type material
Electron concentration:
nn ≈ N D
Fermi energy level:
Hole concentration
in the n-type
material:
ND - Donor atoms
concentration
EF ≈ EC
ni2
pn =
nn
pn nn = ni2
Carrier Concentration and
Fermi level: p-type material
Hole concentration:
pp ≈ N A
Fermi energy level:
Electron
concentration in the
p-type material:
NA - Acceptor atoms
concentration
EF ≈ EV
ni2
np =
pp
pn nn = ni2
Compensation
If both donor and acceptor are added to an intrinsic
semiconductor then the semiconductor is said to be
compensated
If ND > NA, the free electron concentration:
n = ND-NA
If ND < NA, the hole concentration:
p = NA-ND
Drift Current
The electric current due to electric field is called the
Drift Current.
The electron current density (current per unit area):
J n ,drift = qµ n nE
E
µn is the electron mobility and
n is the electron
concentration.
Similarly the hole current
density:
J p ,drift = qµ p pE
µp is the hole mobility and p is
the hole concentration.
Jn,drift
Jp,drift
…cont… Drift Current and conductivity
The total (electron + hole) drift current density:
J drift = J n ,drift + J p ,drift
= qµ n nE + qµ p pE
J drift = q( µ n n + µ p p ) E
Conductivity:
Resistivity:
σ = q( µ n n + µ p p )
ρ=
1
σ
=
1
q( µ n n + µ p p )
J drift = σ E
Diffusion Current
x
Concentration
Gradual concentration change
Concentration
Abrupt concentration change
Diffusion is due to concentration difference between two
regions of a semiconductor
The carriers will move from higher concentration region to
the lower one.
x
…continued… Diffusion Current
The electron diffusion current density:
J n ,diff
dn
= qDn
dx
Dn is the diffusion coefficient of electrons
J p ,diff
Dp is the diffusion coefficient of holes
Electron
Hole diffusion
diffusion
Jn,diff
x
Hole
Concentration
Electron
Concentration
The hole diffusion current density:
dp
= − qD p
dx
Jp,diff
x
Total Currents in semiconductors with both
electric field and concentration gradients
Electron current density
J n = J n ,drift + J n ,diff
dn
= q µ n nE + qDn
dx
Hole current density:
J p = J p ,drift + J p ,diff
dp
= q µ p pE − qD p
dx
Total current density:
Total electron
current
In = J n × A
Total hole
current
Ip = Jp × A
J = Jn + J p
Total current: I = J × A = ( J n + J p )×A
A is the sample
cross-section area