298
5 Solid State Physics
n e n h = 2.33 × 10
31 T
3 e
−Eg/kT
where E g is the gap width
5.38 The effective mass m
∗ of an electron or hole in a band is defined by
1
m ∗ =
1
2 ·
d
2 E
dk 2
where k is the wave number (k = 2π/λ). For a free electron show that
m
∗
= m.
5.39 After adding an impurity atom that donates an extra electron to the conduction
band of silicon (μ n = 0.13 m
2
/Vs), the conductivity of the doped silicon is
measured as 1.08 (Ωm
−1 ). Determine the doped ratio (density of silicon is
2,420 kg/m
3 ).
5.40 Estimate the ratio of the electron densities in the conduction bands of silicon
(E g = 1.14 eV) and germanium (E g = 0.7 eV) at 400 K.
5.41 Show that at the room temperature (300 K) the electron densities in the conduction bands of the insulator carbon (E g = 5.33 eV) and the semiconductor
like germanium (E g = 0.7 eV) is extremely small.
5.42 A current of 8 × 10
−11 A flows through a silicon p − n junction at temperature
27
◦ C. Calculate the current for a forward bias of 0.5 V.
5.43 Calculate the depletion layer width for a pn junction with zero bias in germanium, given that the impurity concentrations are N a = 1 × 10
23 m
−3 and
N d = 2 × 10
22 m
−3 , respectively at T = 300 K, ∈ r = 16 and contact potential
difference V 0 = 0.8 V.
5.44 Consider the Shockley equation for the diode
I = I 0 exp[(eV /kT ) − 1]
Show that the slope resistance r e of the I − V curve at a particular d.c bias
is given to a good approximation, at room temperature (T = 300 K) by the
expression r e =
26
I
Ω (forward bias) where I is in milliampere, and that for
the reverse bias r e tends to infinity.
5.45 Given that a piece of n-type silicon contains 8 × 10
21 m
−3 phosphorus impurity atoms, calculate the carrier concentrations at room temperature. It may be
assumed that the intrinsic electron concentration in silicon at room temperature is 1.6 × 10
16 m
−3 .
5.2.5 Superconductor
5.46 It is required to break up a Cooper pair in lead which has the energy gap of
2.73 eV. What is the maximum wavelength of photon which will accomplish
the job?
5 Solid State Physics
n e n h = 2.33 × 10
31 T
3 e
−Eg/kT
where E g is the gap width
5.38 The effective mass m
∗ of an electron or hole in a band is defined by
1
m ∗ =
1
2 ·
d
2 E
dk 2
where k is the wave number (k = 2π/λ). For a free electron show that
m
∗
= m.
5.39 After adding an impurity atom that donates an extra electron to the conduction
band of silicon (μ n = 0.13 m
2
/Vs), the conductivity of the doped silicon is
measured as 1.08 (Ωm
−1 ). Determine the doped ratio (density of silicon is
2,420 kg/m
3 ).
5.40 Estimate the ratio of the electron densities in the conduction bands of silicon
(E g = 1.14 eV) and germanium (E g = 0.7 eV) at 400 K.
5.41 Show that at the room temperature (300 K) the electron densities in the conduction bands of the insulator carbon (E g = 5.33 eV) and the semiconductor
like germanium (E g = 0.7 eV) is extremely small.
5.42 A current of 8 × 10
−11 A flows through a silicon p − n junction at temperature
27
◦ C. Calculate the current for a forward bias of 0.5 V.
5.43 Calculate the depletion layer width for a pn junction with zero bias in germanium, given that the impurity concentrations are N a = 1 × 10
23 m
−3 and
N d = 2 × 10
22 m
−3 , respectively at T = 300 K, ∈ r = 16 and contact potential
difference V 0 = 0.8 V.
5.44 Consider the Shockley equation for the diode
I = I 0 exp[(eV /kT ) − 1]
Show that the slope resistance r e of the I − V curve at a particular d.c bias
is given to a good approximation, at room temperature (T = 300 K) by the
expression r e =
26
I
Ω (forward bias) where I is in milliampere, and that for
the reverse bias r e tends to infinity.
5.45 Given that a piece of n-type silicon contains 8 × 10
21 m
−3 phosphorus impurity atoms, calculate the carrier concentrations at room temperature. It may be
assumed that the intrinsic electron concentration in silicon at room temperature is 1.6 × 10
16 m
−3 .
5.2.5 Superconductor
5.46 It is required to break up a Cooper pair in lead which has the energy gap of
2.73 eV. What is the maximum wavelength of photon which will accomplish
the job?