PPT

1. Electrical Science-2 (15B11EC211)
Unit-4
Introduction to Semiconductor
Lecture-4
Dr. Akansha Bansal
OLED
Technology
Solar Cells

2. Conductivity and Mobility
• The charge carriers in a solid are in constant motion, even at
thermal equilibrium.
• At room temperature, for example, the thermal motion of an
individual electron may be visualized as random scattering from
lattice vibrations, impurities, other electrons, and defects.
Fig. 1(a) Random thermal motion of an electron in a solid [1]

3. Conductivity and Mobility
If an electric field Ex is applied in the x-direction, each electron experiences a net force
-qEx from the field.
This force averaged over all the electrons, however, is a net motion of the group in x-
direction.
As a result of this electrostatic force , the electrons would be accelerated and the
velocity increases indefinitely with time, however due to the inelastic collision with
ions electrons loses energy and a steady state condition is reached where a finite value
of drift speed vd is attained.
This drift velocity is in the direction opposite to that of the electric field.
𝑣𝑥 = −𝜇𝑛𝐸𝑥
Ex
Field
Fig. 1b Well directed drift velocity with an applied electric field.

4. Conductivity and Mobility
• The quantity 𝜇𝑛, called the electron mobility, describes
the ease with which electrons drift in the material.
• Mobility is a very important quantity in characterizing
semiconductor materials and in device development.
• The mobility can be expressed as the average particle
drift velocity per unit electric field. We have,
μ𝑛 = −
𝑣𝑥
𝐸𝑥
• The units of mobility are cm2/V-s

5. Effect of Temperature on Mobility
The two basic types of scattering mechanisms that
influence electron and hole mobility are:
Lattice scattering
Impurity scattering
Fig.2 Temperature dependence of mobility with both lattice and impurity scattering

6. Effect of Temperature on Mobility
• Lattice scattering: A carrier moving through the crystal is scattered by a
vibration of the lattice, resulting from the temperature. The frequency of such
scattering events increases as the temperature increases, since the thermal
agitation of the lattice becomes greater. Therefore, we should expect the
mobility to decrease as the sample is heated.
• Impurity scattering: Scattering from crystal defects such as ionized
impurities becomes the dominant mechanism at low temperatures.
Since the atoms of the cooler lattice are less agitated and a slowly
moving carrier is likely to be scattered more strongly by an interaction
with a charged ion than is a carrier with greater momentum, impurity
scattering events cause a decrease in mobility with
decreasingtemperature.
The mobilities due to two or more scattering mechanisms add inversely:
1
μ
=
1
μ1
+
1
μ2
… … … …

7. Effect of Doping on Mobility
As the concentration of impurities increases, the effects of impurity scattering
are felt at higher temperatures. For example, the electron mobility 𝜇𝑛 of
intrinsic silicon at 300 K is 1350 cm2/(V-s).With a donor doping
concentration of 1017 cm-3, however, 𝜇𝑛 is 700 cm2/(V-s). Thus, the presence
of the 1017 ionized donors/cm3 introduces a significant amount of impurity
scattering.
Fig.3 Variation of mobility with total doping impurity concentration (N0 + Nd) for Si at 300 K.

8. High Field Effects
Upper limit is reached for the carrier drift velocity in a high field and
this limit occurs near the mean thermal velocity (107 cm/s) and
represents the point at which added energy imparted by the field is
transferred to the lattice rather than increasing the carrier velocity.
The result of this scattering limited velocity is a fairly-constant
current at high field.
Fig.4 Saturation of electron drift velocity at high electric fields for Si

9. Example
Example 1: How long does it take an average electron
to drift 1 μm in pure Si at an electric field of 100V/cm?
Repeat for 105V/cm. Assume μn = 1350 cm2/V-s
Solution:
Low Field:
𝑣𝑑 = μ𝑛𝐸 = 1350 x 100=1.35 x 105 cm/s
t = L/ 𝑣𝑑 = 10-4/1.35 x 105 = 0.74ns
High Field: Scattering limited velocity 𝑣𝑠 = 107 cm/s
t = L/ 𝑣𝑠 = 10-4/107 = 10ps

10. Current Density
• If n electrons are contained in a length L of conductor, and if it takes an
electron a time t seconds to travel distance of L meter in the conductor.
A
𝐼 =
𝑁𝑞
𝑡
=
𝑁𝑞𝑣
𝐿
As L/t is the average, or drift speed, v m/s of the electrons.
Current Density denoted by symbol J, is the current per unit area of the
conducting medium.
• Assuming a uniform current distribution
𝐽 =
𝐼
𝐴
Where, J is Amperes/Square meter
A- Cross-sectional area of the conductor in meters
L

11. Conductivity
𝐽 =
𝑁𝑞𝑣
𝐿𝐴
n =
𝑁
𝐿𝐴
n=no. of electrons per cubic meter
𝐽 = 𝑛𝑞𝑣
𝐽 = 𝑛𝑞𝑣 = 𝑛𝑞μ𝐸 = σ𝐸 (1)
Where,
σ = 𝑛𝑞μ
is the conductivity of the metal in (ohm-meter)-1]
Eq. 1 is recognized as Ohm’s law, namely, the conduction current is
proportional to applied voltage.

12. Example
Example 2: A Si bar 0.1 cm long and 100 μm2 in cross sectional area is doped
with 1017 cm-3 antimony. Find the current at 300K with 10V applied. Assume
μn = 700cm2/V-s
Solution: σ = 𝑛𝑞μn
= 1.6 x 1019 x 700 x 1017
= 11.2(Ω.cm)-1
ρ = 1/σ =0.0893(Ω.cm)
𝑅 =
ρ𝐿
𝐴
𝐼 =
V
𝑅
=
10
0.0893
= 0.16𝐴

13. Conductivity of semiconductor
Semiconductor contains two types of charge carriers
▫ Negative(free electron) with mobility μn
▫ Positive(hole) with mobility μp
These particles move in opposite direction in an electric field E, but since
they are of opposite sign, the current of each is in same direction.
Hence, Current density
J=(nμn + pμp)qE =σE
n = magnitude of free electron concentration
p = magnitude of hole concentration
σ = Conductivity
Hence,
σ = (nμn + pμp)q

14. Hall Effect
• If a magnetic field is applied perpendicular to
the direction in which holes drift in a p-type bar,
the path of the holes tends to be deflected.
Fig.1 The Hall effect [1]

15. Hall Effect
Using vector notation, the total force on a single hole due to the
electric and magnetic fields is
F = q(E + v × B)
In the y-direction the force is
Fy = q(Ey-vxBz)
• The important result of above equation is that unless an electric
field Ey is established along the width of the bar, each hole will
experience a net force (and therefore an acceleration) in the y-
direction due to the qvxBz product.
• Therefore, to maintain a steady state flow of holes down the length
of the bar, the electric field Ey must just balance the product qvxBz
◦

16. Hall Effect
Once the electric field Ey becomes as large as qvxBz, no net
lateral force is experienced by the holes as they drift along
the bar.
The establishment of the electric field Ey is known as the
Hall effect, and the resulting voltage VAB = Eyw is called
the Hall voltage.
𝐸𝑦 =
𝐽𝑥
𝑞𝑝0
𝐵𝑧 = 𝑅𝐻𝐽𝑥𝐵𝑧
𝑅𝐻 =
1
𝑞𝑝0
𝑅𝐻 is called the Hall Coefficient

17. Hall Effect
A measurement of the Hall voltage for a known current and
magnetic field yields a value for the hole concentration p0
𝑝0 =
1
𝑞𝑅𝐻
=
𝐽𝑥𝐵𝑧
𝑞𝐸𝑦
=
(𝐼𝑥/𝑤𝑡)𝐵𝑧
𝑞(𝑉𝐴𝐵/𝑤)
=
𝐼𝑥𝐵𝑧
𝑞𝑡𝑉𝐴𝐵
If a measurement of resistance R is made, the sample resistivity ρ
can be calculated
ρ (Ω-cm) =
𝑅𝑤𝑡
𝐿
=
𝑉𝐶𝐷/𝐼𝑥
𝐿/𝑤𝑡
Mobility is simply the ratio of the Hall coefficient and the
resistivity
μ𝑝 =
σ
𝑞𝑝0
=
1/ρ
𝑞(1/𝑞𝑅𝐻)
=
𝑅𝐻
ρ

18. Example
Example: Consider a semiconductor bar with w = 0.1mm, t =10 μm, and
L = 5mm. For B=10-4 Wb/cm2 and a current of 1mA, we have VAB = -
2mV, VCD = 100mV. Find the type, concentration, and mobility of the
majority carrier.
Solution
From the sign of VAB, we can see that the majority carriers are electrons:
𝑛0 =
𝐼𝑥𝐵𝑧
𝑞𝑡(−𝑉𝐴𝐵)
=
10−310−4
1.6 𝑥 10−19𝑥10−3𝑥2𝑥10−3=31.25 x 1017 cm-3
ρ =
𝑅𝑤𝑡
𝐿
=
𝑉𝐶𝐷/𝐼𝑥
𝐿/𝑤𝑡
=
0.1 10−3
0.5 0.01 𝑥 10−3 =0.002 Ω.cm
μ𝑝 =
1
ρ𝑞𝑝0
=
1
(0.002)(1.6 𝑥 10−19)(3.125 𝑥 1017)
= 10,000 𝑐𝑚2 𝑉. 𝑠
− 1