298 5 Solid State Physics
nenh= 2. 33 × 1031 T^3 e−Eg/kT
whereEgis the gap width5.38 The effective massm∗of an electron or hole in a band is defined by
1
m∗=
1
^2
·
d^2 E
dk^2
wherekis the wave number (k = 2 π/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 (μn = 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
(Eg= 1 .14 eV) and germanium (Eg= 0 .7 eV) at 400 K.
5.41 Show that at the room temperature (300 K) the electron densities in the con-
duction bands of the insulator carbon (Eg= 5 .33 eV) and the semiconductor
like germanium (Eg= 0 .7 eV) is extremely small.
5.42 A current of 8× 10 −^11 A flows through a siliconp−njunction at temperature
27 ◦C. Calculate the current for a forward bias of 0.5 V.
5.43 Calculate the depletion layer width for apnjunction with zero bias in ger-
manium, given that the impurity concentrations areNa= 1 × 1023 m−^3 and
Nd= 2 × 1022 m−^3 , respectively atT=300 K,∈r=16 and contact potential
differenceV 0 = 0 .8V.
5.44 Consider the Shockley equation for the diode
I=I 0 exp[(eV/kT)−1]
Show that the slope resistancereof theI−Vcurve at a particular d.c bias
is given to a good approximation, at room temperature (T =300 K) by the
expressionre=^26 I Ω(forward bias) whereIis in milliampere, and that for
the reverse biasretends to infinity.5.45 Given that a piece of n-type silicon contains 8× 1021 m−^3 phosphorus impu-
rity atoms, calculate the carrier concentrations at room temperature. It may be
assumed that the intrinsic electron concentration in silicon at room tempera-
ture is 1. 6 × 1016 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?