18 HALL EFFECT
The motion of carriers in the presence of electric and magnetic fields gives rise to a number of galvanometric effects. The most important of these effects is the Hall effect.
When a semiconductor sample carrying a current I is placed in a transverse magnetic field B, then an electric field E_{0} is induced in the specimen, in the direction perpendicular to both B and I. This phenomenon is called the Hall effect. The Hall effect may be used for determining whether a semiconductor is ntype or ptype by finding the carrier concentration and calculating the mobility μ, by measuring the conductivity σ.
Figure 137 Schematic diagram of Hall effect. The carriers (electrons or holes) are subjected to a magnetic force in the negative ydirection.
Let us consider a rectangular bar of ntype semiconductor of length L, width W and thickness d, as shown in Fig. 137.
If a current I is applied in the positive xdirection and a magnetic field B is applied in the positive zdirection, a force will be exerted in the negative ydirection of the current carriers. The current is carried by electron from side 1 to side 2, if the semiconductor is ntype. Therefore Hall voltage V_{H} appears between surfaces 1 and 2. The electric field developed in ydirection E_{y} is given by:
where, d is the distance between surfaces 1 and 2. In the equilibrium state the electric field E_{y} due to the Hall effect must exert a force on the carrier, which just balances the magnetic force, and we can write:
where, e is the magnitude of the charge of the carriers, v_{0} is the drift speed.
Therefore current density J is given by:
where, ρ is the charge density and W is the width of the specimen in the direction of the magnetic field. Combining Eq. (1123), Eq. (1124) and Eq. (1125) we find:
If V_{H}, B, W and I are measured, the charge density ρ can be measured from Eq. (1126). If the polarity of V_{H} is positive at terminal 1 then the carrier must be an electron and ρ = n_{0}e where n_{0} is the electron concentration. If the terminal 2 becomes positively charged with respect to terminal 1 the semiconductor must be of ptype and ρ = p_{0}e where p_{0} is the hole concentration:
where, R_{H} is the Hall coefficient.
Conductivity σ is related to mobility μ by:
σ = n_{0}eμ
σ = p_{0}μ
where,
If the conductivity is measured with Hall coefficient, mobility μ can be determined by:
From Eq. (1127) and Eq. (1130) we get:
In the presence of scattering the mobility can approximately be written as:
Applications of Hall Effect
 Experimental determination of carrier concentration: From the basic formula of Hall effect, the electron and hole concentrations can be determined by using the experimental values of Hall coefficient.
 By using Hall effect, the type of the semiconductor can be determined as follows: R_{H} > 0 for ptype semiconductors and R_{H} < 0 for ntype semiconductors.
 Determination of the mobility: The equation μ = [σR_{H}] determines the mobility by using the experimental values of R_{H}.
 Hall effect multiplier: If the magnetic field B produces current I_{1} then Hall voltage V_{H} ∞ II_{1} The Hall effect multiplier generates an output proportional to the product of two signals. Thus, Hall effect can be used for analogue multiplication.
 The power flow in an electromagnetic wave can be measured by the help of Hall effect.
 Experimental determination of the magnetic field: By knowing the values of V_{H}, I, ρ and W, we can determine the value of B experimentally.
 Lattice plus basis is equal to crystal structure.
 The bravais lattices are distinct lattice types, which when repeated can fill the whole space.
 In two dimensions there are five distinct bravais lattices while in three dimensions there are fourteen.
 Electrons, which revolve in the outermost orbit of an atom, are called valence electrons.
 Electrons, which are detached from the parent atoms and move randomly in the lattice of a metal, are called free electrons.
 The range of energies possessed by electrons in a solid is known as energy band.
 The range of energies possessed by valance electrons is called the valance band.
 The range of energies possessed by free electrons is called the conduction band.
 The separation between conduction and valence band on the energy level diagram is called the forbidden energy gap.
 The semiconductors are a class of materials whose electrical conductivity lies between those of conductors and insulators. Germanium and silicon are called elemental semiconductors. The IIIV, IIVI, IVVI, ternaries and quaternaries are called compound semiconductors.
 There are three types of semiconductors namely ntype semiconductors (when electron concentration is much greater than hole concentration), ptype semiconductors (when electron concentration is much less than hole concentration) and intrinsic semiconductors (when electron concentration is equal to hole concentration).
 A semiconductor without any impurities is called an intrinsic semiconductor and a semiconductor with impurities is called extrinsic semiconductor.
 When a covalent bond is broken, an electron vacates an energy level in valence band. This vacancy may be traced as a particle called hole.
 The resistivity of pure semiconductor is of the order of 10^{3} Ωm. The energy gap of pure silicon is 1.12 eV and of germanium is 0.72 eV.
 Thermal process is the only process that generates carriers in an intrinsic semiconductor.
 The additional velocity acquired by the charge carriers in the electric field is called the drift velocity.
 Mobility is defined as the drift velocity acquired per unit electric field strength.
 Conductivity of a material is a measure of the material’s ability to allow charge carriers to flow through it.
 The process of adding impurities to a pure semiconductor is called doping. A doped semiconductor is called an extrinsic semiconductor. The dopants are usually trivalent or pentavalent impurities for a tetravalent (Si or Ge) semiconductor material.
 An ntype semiconductor is obtained by adding pentavalent impurity to a pure Si or Ge semiconductor. In an ntype semiconductor electrons are majority carriers and holes are minority carriers.
 A ptype semiconductor is obtained by adding trivalent impurity to a pure Si or Ge semiconductor. In a ptype semiconductor holes are majority carriers and electrons are minority carriers. The energy gap of extrinsic silicon semiconductor is 0.7 eV and of germanium is 0.3 eV
 An extrinsic semiconductor is electrically neutral.
 In an ntype material, the free electron concentration is approximately equal to the density of donor atoms. Thus, n_{n} ≅ N_{A}
 In an extrinsic parabolic semiconductors, Fermi level lies exactly midway between valance and conduction bands at T→0.
 The product of n and p is a constant and is known as the law of mass action.
 The flow of current through a semiconductor material is normally referred to as one of two types: drift or diffusion.
 The combined effect of the movement of holes and electrons constitute an electric current under the action of an electric field and is called a drift current.
 The directional movement of charge carriers due to their concentration gradient produces a component of current known as diffusion current.
 The total current is equal to the drift part of the current plus diffusion part of the current.
 The distance that free carrier travels before recombining is called the diffusion length.
 The average time an electron or hole can exist in the free state or the average time between the generation and recombination of a free electron is called lifetime.
 If a metal or semiconductor carrying a current is placed in a transverse magnetic field, an electric field is induced in the direction perpendicular to both the current and magnetic field. This phenomenon is called Hall effect.
 Hall effect measurements help us in identifying the type of majority carriers in determining carrier concentration and carrier mobility.
 DeBroglie’s relation of the wave particle is given by: λ = h/p
 The effective momentum mass of the carriers is given by:
 Energy gap between conduction and valence band is: E_{g} = E_{c} – E_{v}
 The densityofstate function of the conduction electron in parabolic ntype semiconductors is given by:
 The Fermi–Dirac integral of order j is given by:
which obeys the following properties:
 The electron concentration in ntype parabolic semiconductors is given by:
 Under nondegenerate electron concentration
 The expression of the critical electron concentration when E_{F} touches E_{C} is given by:
 The variation of Fermi energy with temperature for relatively low values of temperatures in ntype parabolic semiconductors is given by:
 Under the condition of extreme carrier degeneracy
 The concentration of heavy holes in parabolic ptype semiconductors is given by:
 Under the condition of nondegenerate hole concentration we can write the above equation reduces to:
 For intrinsic semiconductors E_{F} is given by:t
 The law of mass action for nondegenerate parabolic semiconductors is given by:
 The intrinsic carrier concentration is given by:
 For extrinsic nondegenerate parabolic semiconductors with N_{d} ≠ 0, N_{a} = 0 and the Fermi energy can be written as:
 For extrinsic nondegenerate parabolic semiconductors with N_{d} ≠ 0, N_{a} = 0 and the Fermi energy can be written as:
 In the zone of very high temperatures the Fermi energy for nondegenerate parabolic semiconductors when N_{A} = 0 is given by:
 The electron mobility (μ) is given by:
In general, the mobility for isotropic bands is expressed as:
μ = e < τ(E)/m*(E) >
 Mathiessen’s rule is given by:
 The conductivity for mixed conduction is given by: σ = σ_{n} + σ_{p} = e(n_{0}μ_{n} + p_{0}μ_{p})
 The electron and hole current densities in the presence of drift and diffusion are given by:
 The generalized expression of the Einstein relation for electrons in ntype semiconductors is given by:
 Under the condition of extreme degeneracy, the Einstein relation in ntype parabolic semiconductors is given by:
 Under nondegenerate electron concentration, the Einstein relation for ntype semiconductors is given by:
 The relation between the rate of recombination of electrons and holes and equilibrium concentration of electrons and holes is given by:
r_{i} = k_{r}n_{0}p_{0} = k_{r} = g_{i}
 The time dependence of excess carrier concentration for a ptype material is given by:
 The continuity equation for holes for ntype semiconductors is given by:
 The transverse voltage produced in Hall effect is given by:
 The Hall coefficient:
 Charge density:
 Mobility: μ = σR_{H}
 The transverse voltage produced in Hall effect is given by:
 The Hall coefficient:
 Charge density:
 A silicon sample is uniformly doped with 10^{16} phosphorus atoms/cm^{3} and 2 × 10^{16} boron atoms/cm^{3}. If all the dopants are fully ionized, the material is:
 ntype with carrier concentration of 3 × 10^{16}/cm^{3}
 ptype with carrier concentration of 10^{16}/cm^{3}
 ptype with carrier concentration of 4 × 10^{16}/cm^{3}
 Intrinsic
 ntype semiconductors are:
 Negatively charged
 Produced when Indium is added as an impurity to Germanium
 Produced when phosphorous is added as an impurity to silicon
 None of the above
 The probability that an electron in a metal occupies the Fermilevel, at any temperature (> 0 K) is:
 0
 1
 0.5
 None of the above
 Measurement of Hall coefficient enables the determination of:
 Mobility of charge carriers
 Type of conductivity and concentration of charge carriers
 Temperature coefficient and thermal conductivity
 None of the above
 If the energy gap of a semiconductor is 1.1 eV it would be:
 Opaque to the visible light
 Transparent to the visible light
 Transparent to the ultraviolet radiation
 None of the above
 The conductivity of an intrinsic semiconductor is given by (symbols have the usual meanings):
 σ_{i} = en_{i}^{2}(μ_{n} – μ_{p})
 σ_{i} = en_{i}(μ_{n} – μ_{p})
 σ_{i} = en_{i}(μ_{n} + μ_{p})
 None of the above
 Consider the following statements: Compared to Silicon, Gallium Arsenide (GaAs) has:
 Higher signal speed since electron mobility is higher
 Poorer crystal quality since stoichiometric growth difficult
 Easier to grow crystals since the vapour pressure Arsenic is high
 Higher optoelectronic conversion efficiency
Of these statements:
 1, 2, 3 and 4 are correct
 1, 2 and 3 are correct
 3 and 4 are correct
 None of the above
 In an intrinsic semiconductor, the mobility of electrons in the conduction band is:
 Less than the mobility of holes in the valence band
 Zero
 Greater than the mobility of holes in the valence band
 None of the above
 The Hall coefficient of sample (A) of a semiconductor is measured at room temperature. The Hall coefficient of (A) at room temperature is 4 × 10^{–4} m^{3} coulomb^{–1}. The carrier concentration in sample (A) at room temperature is:
 ~10^{21} m^{–3}
 ~10^{20} m^{–3}
 ~10^{22} m^{–3}
 None of the above
 In a semiconductor, J, J_{p} and J_{n} indicate total diffusion current density hole current density and electron current density respectively, and are the electron and hole concentration gradient respectively in xdirection and D_{p} and D_{n} are the hole and electron diffusion constants respectively. Which one of the following equations is correct? (e denotes charge of electron).
 None of the above
 If the drift velocity of holes under a field gradient of 100 v/m is 5 m/s, the mobility (in the same SI units) is:
 0.05
 0.55
 500
 None of the above
 The Hall effect voltage in intrinsic silicon is:
 Positive
 Zero
 Negative
 None of the above
 The Hall coefficient of an intrinsic semiconductor is:
 Positive under all conditions
 Negative under all conditions
 Zero under all conditions
 None of the above
 Consider the following statements: Pure germanium and pure silicon are examples of:
 Direct bandgap semiconductors
 Indirect bandgap semiconductors
 Degenerate semiconductors
Of these statements:
 1 alone is correct
 2 alone is correct
 3 alone is correct
 None of the above
 Assume n_{e} and n_{h} are electron and hole densities, μ_{e} and μ_{n} are the carrier mobilities; the Hall coefficient is positive when:
 n_{h} μ_{h} > n_{e} μ_{e}
 n_{h} μ_{h}^{2} < n_{e} μ_{e}^{2}
 n_{h} μ_{h} > n_{e} μ_{h}
 None of the above
 A long specimen of ptype semiconductor material:
 Is positively charged
 Is electrically neutral
 Has an electric field directed along its length
 None of the above
 The electron and hole concentrations in an intrinsic semiconductor are n_{i} and p_{i} respectively. When doped with a ptype material, these change to n and P respectively. Then:
 n + p = n_{i} + p_{i}
 n + n_{i} = p + p_{i}
 np = n_{i} p_{i}
 None of the above are applicable
 If the temperature of an extrinsic semiconductor is increased so that the intrinsic carrier concentration is doubled, then:
 The majority carrier density doubles
 The minority carrier density doubles
 Both majority and minority carrier densities double
 None of the above
 At room temperature, the current in an intrinsic semiconductor is due to:
 Holes
 Electrons
 Holes and electrons
 None of the above
 A semiconductor is irradiated with light such that carriers are uniformly generated throughout its volume. The semiconductor is ntype with N_{D} = 10^{19} per cm^{3}. If the excess electron concentration in the steady state is ∆n = 10^{15} per cm^{3} and if τ_{p} = 10 μsec. (minority carrier life time) the generation rate due to irradiation is:
 10^{22} eh pairs /cm^{3}/s
 10^{10} eh pairs/cm^{3}/s
 10^{24} eh pairs/cm^{3}/s
 None of the above
 A small concentration of minority carriers is injected into a homogeneous semiconductor crystal at one point. An electric field of 10 V/cm is applied across the crystal and this moves the minority carriers a distance of 1 cm is 20 μsec. The mobility (in cm^{2}/volt.sec) is:
 1,000
 2,000
 50
 None of the above
 The mobility is given by:
 None of the above
 Hall effect is observed in a specimen when it (metal or a semiconductor) is carrying current and is placed in a magnetic field. The resultant electric field inside the specimen will be in:
 A direction normal to both current and magnetic field
 The direction of current
 A direction anti parallel to magnetic field
 None of the above
 In a ptype semiconductor, the conductivity due to holes (σ_{p}) is equal to: (e is the charge of hole, μ_{p} is the hole mobility, p_{0} is the hole concentration):
 p_{0}e/μ_{p}
 μ_{p}/p_{0}e
 p_{0}eμ_{p}
 None of the above
 The difference between the electron and hole Fermi energies of a semiconductor laser is 1.5 eV and the band gap of the semiconductor is 1.43 eV. The upper and lower frequency limits of the laser will be respectively:
 3.3 × 10^{15} and 9.9 × 10^{13} Hz
 3.7 × 10^{16} and 3.5 × 10^{14} Hz
 6.28 × 10^{17} and 3.1 × 10^{13} Hz
 None of the above
 A sample of ntype semiconductor has electron density of 6.25 × 10^{18}/cm^{3} at 300 K. If the intrinsic concentration of carriers in this sample is 2.5 × 10^{13}/cm^{3} at this temperature, the hole density becomes:
 10^{16}/cm^{3}
 10^{7}/cm^{3}
 10^{17}/cm^{3}
 None of the above
 The intrinsic carrier density at 300K is 1.5 × 10^{10}/cm^{3} in silicon. For ntype silicon doped to 2.25 × 10^{15} atoms/cm^{3} the equilibrium electron and hole densities are:
 n_{0} = 1.5 × 10^{16}/cm^{3}, p_{0} = 1.5 × 10^{12}/cm^{3}
 n_{0} = 1.5 × 10^{10}/cm^{3}, p_{0} = 2.25 × 10^{15}/cm^{3}
 n_{0} = 2.25 × 10^{17}/cm^{3}, p_{0} = 1.0 × 10^{14}/cm^{3}
 None of the above
 In a ptype silicon sample, the hole concentration is 2.25 × 10^{15}/cm^{3}. If the intrinsic carrier concentration 1.5 × 10^{10}/cm^{3} the electron concentration is:
 10^{21}/cm^{3}
 10^{10}/cm^{3}
 10^{16}/cm^{3}
 None of the above
 A good ohmic contact on a ptype semiconductor chip is formed by introducing:
 Gold as an impurity below the contact
 A high concentration of acceptors below the contact
 A high concentration of donors below the contact
 None of the above
 Measurement of Hall coefficient in a semiconductor provides information on the:
 Sign and mass of charge carriers
 Mass and concentration of charge carriers
 Sign of charge carriers alone
 Sign and concentration of charge carriers
 Briefly discuss the basic developments in the study of electronics.
 What are crystalline materials?
 Give three examples of Group IIIV semiconductors.
 What do you mean by pure crystals?
 Why is Si preferred over Ge?
 Why is GaAs preferred over Si?
 Give an example of the constituent material of Gunn Diode.
 What is Bravais lattice? Discuss briefly.
 What are unit cells and lattice constants?
 Explain the differences among simple cubic, bodycentred cubic and facecentred cubic lattices respectively.
 Explain Czochralski growth of the semiconductor crystal in detail.
 Explain the wave particle duality principle.
 State Pauli exclusion principle.
 What is degenerate energy level?
 What is energy band gap?
 What do you mean by covalent and electrovalent bonds? How these are affected by the temperature?
 Explain thermal equilibrium.
 What is valence band and conduction band?
 What are conduction band carriers?
 Explain the existence of hole.
 What is momentum effective mass of the carriers? What is its difference with acceleration effective mass?
 Explain indistinguishability between the particles. What should be the value of the spin of the particles obeying Fermi–Dirac statistics?
 Explain the term “occupation probability”.
 What is bonding model?
 Explain the energy band models of semiconductors.
 Write the assumptions behind Fermi–Dirac statistics. Give a very simple proof of the same statistics.
 Write the five properties of the Fermi–Dirac function.
 Define Fermi energy. How does Fermi energy vary with temperature?
 Write the basic criterion for the classification of metals, semiconductors and insulators.
 Draw the model energy band structure diagram of a semiconductor in general.
 The splitoff hole parabola is more flattened than the light hole parabola. Justify your answer very briefly.
 What are intrinsic semiconductors?
 What are extrinsic semiconductors?
 Explain donor ion, acceptor ion, majority carriers, minority carriers, doping and dopants in a semiconductor.
 What do you mean by the term “carrier degeneracy” of a semiconductor? Explain in detail.
 What are compound semiconductors? Explain the uses of compound semiconductors.
 Explain the difference between metals, insulators and semiconductors with the help of band structure model.
 Define the term “densityofstate function”. Derive an expression for the densityofstate function of the conduction electrons in a semiconductor having parabolic energy bands. Draw the graph of the density of state function versus energy and explain the graph.
 Derive an expression of electron concentration in ntype semiconductors. Discuss all the special cases.
 Derive an expression of electron concentration in ptype semiconductors. Discuss all the special cases.
 What is charge density equation?
 Show that:
where, the notations mean as usual.
 Derive the law of mass action. Is it valid for degenerate semiconductors? Give reasons.
 Does the band gap vary with temperature? Give reasons.
 How does concentration vary with temperature?
 What is mobility?
 What is relaxation time?
 What is conductivity?
 What is Mathiessen’s rule?
 What is diffusion?
 What is recombination?
 What is generation?
 Prove that:
 Write the assumptions of continuity equation and derive continuity equation.
 Explain physically the continuity equation.
 What is Hall effect?
 Derive the relation between mobility and Hall coefficient.
 What are the applications of Hall effect?
 Find out an expression for Hall coefficient in a semiconductor when both carriers contribute to the current.
 The energy spectrum of the conduction electrons of IIIV compound semiconductors can be expressed as where the notations have their usual meaning. Find out the momentum effective mass and the acceleration effective mass of the conduction electrons respectively. Draw the plots in two cases on the same graph paper with the independent variable as energy by taking the example of nGaAs. Interpret the results. Explain the results for E_{g} → ∞.
 The dispersion relation of the carriers in a semiconductor is approximately given by
E = E_{0} − Acos (αk_{x}) − B[cos (βk_{y}) + cos(βk_{z})]
where E_{0}, A, B, α and β are constants. Develop an expression of the densityofstates function for small k.

 Calculate the coordinates of three points on the Fermi–Dirac function and enter the coordinates in this table.
 Plot the same three points in the following axis and sketch the Fermi–Dirac function with reasonable accuracy.
 In a certain silicon sample at equilibrium, the Fermi level resides at 0.500 eV above the centre of the band gap.
 Calculate the occupancy probability for a lone isolated state located right at the centre of the band gap.
 This sample contains donor impurities and no acceptor impurities. The donor states are situated 0.045 eV below the conductionband edge. Find the occupancy probability of the donor states.
 Comment on the validity of the assumption of 100 per cent ionization of the donor states in the present situation.
 Derive an approximate form of the Fermi–Dirac probability function that could be applied in (b) with reasonable validity.
 Use your approximate expression to recalculate the probability in (b) and find the percentage difference in the exact and approximate results.
 Sketch the exact probability function accurately by plotting several points. Superimpose on the same diagram an accurate sketch of the approximate expression of (d).
 Comment on conditions where the use of the approximate expression is justified.
 How much error results when the approximate expression is used to calculate occupancy probability for a state located right at the Fermi level?
 The conduction band can be characterized by a state density (number of states per cm^{3}) of N_{c} = 3.75 × 10^{19}/cm^{3}, with these states assumed to be situated right at the conductionband edge.
 Using this assumption, calculate the conductionelectron density n_{0} (number of electrons per cm^{3}) for the conditions of Problem 4.
 The valence band can be characterized by a state density of N_{v} ≅ N_{c’} with these states assumed to be situated right at the valenceband edge. Using this assumption, calculate the hole density p_{0}.
 Calculate the p–n product using the results from (a) and (b).
 Determine the approximate density of donor states N_{D} (number of donor states per cm^{3}) for the silicon sample of Problems 4 and 5. Provide reasons.
 Derive an expression relating the intrinsic level E_{i}. to the centre of the band gap E_{g}/2. Calculate the displacement of E_{i}. from E_{g}/2 for Si at 300 K, assuming the effective mass values for electrons and holes are 1.1 m_{0} and 0.56 m_{0}, respectively.

 Explain why holes are found at the top of the valence band, whereas electrons are found at the bottom of the conduction band.
 Explain why Si doped with 10^{14} cm^{3} Sb is ntype at 400 K but similarly doped Ge is not.
 Calculate the N_{c} and N_{v} for the conduction and valence bands of Si and GaAs at 100 K, 200 K and 400 K respectively by assuming the data you require.
 A certain uniformly doped silicon sample at room temperature has n_{0} = 10^{6}/cm^{3} and N_{A} = 10^{15}/cm^{3}.
 Find p_{0}.
 Find N_{D}.
 Using the Boltzmann approximation to the Fermi–Dirac probability function (obtained by dropping the unity term in its denominator), find the Fermilevel position relative to the conduction band edge for a sample having n_{0} = 3 × 10^{15}/cm^{3} at room temperature and at equilibrium.
 Given that the majority impurity in the foregoing problem is phosphorus, find the occupancy probability at the donor level. Comment on the assumption of 100 per cent ionization in this case.
 A silicon sample has N_{A} = 10^{15}/cm^{3} and N_{D} = 0. Find the:
 Majoritycarrier density
 Minoritycarrier density
 Conductivity
 A certain silicon sample has p_{0} = 2.5 × 10^{10}/cm^{3} Find the:
 Electron density n_{0}
 Resistivity ρ
 Calculate the position of the intrinsic Fermi level measured from the midgap for InAs.
 Calculate and plot the position of the intrinsic Fermi level in Si between 80 K and 400 K.
 Calculate the density of electrons in silicon if the Fermi level is 0.45 eV below the conduction bands at 290 K. Compare the results by using the Boltzmann approximation and the Fermi–Dirac integral.
 In a GaAs sample at 310 K, the Fermi level coincides with the valence bandedge. Calculate the hole density by using the Boltzmann approximation. Also calculate the electron density using the law of mass action.
 The electron density in a silicon sample at 310 K is 10^{15} cm^{−3}. Calculate E_{c} – E_{F} and the hole density using the Boltzmann approximation.
 A GaAs sample is doped ntype at 4 × 10^{18} cm^{−3}. Assume that all the donors are ionized. What is the position of the Fermi level at 300 K?
 Consider a ntype silicon with a donor energy 60 meV below the conduction band. The sample is doped at 10^{15} cm^{−3}. Calculate the temperature at which 30 per cent of the donors are not ionized.
 Consider a GaAs sample doped at N_{d} = 10^{15} cm^{−3} where the donor energy is 5 meV. Calculate the temperature at which 80 per cent of the donors are ionized.
 Estimate the intrinsic carrier concentration of diamond at 700 K (you can assume that the carrier masses are similar to those in Si). Compare the results with those for GaAs and Si.
 A Si device is doped at 2.5 × 10^{16} cm^{−3}. Assume that the device can operate up to a temperature where the intrinsic carrier density is less than 10 per cent of the total carrier density. What is the upper limit for the device operation?
 Estimate the change in intrinsic carrier concentration per K change in temperature for Ge, Si, and GaAs and InSb at 300 K.
 A certain silicon sample has N_{D} = 5.30 × 10^{15}/cm^{3} and N_{A} = 4.50 × 10^{15}/cm^{3}.
Find the:
 Majoritycarrier density
 Minoritycarrier density
 Conductivity σ, using μ_{n} = 400 cm^{2}/Vs, and μ_{p} = 300 cm^{2}/Vs
 n_{0}
 σ using μ_{n} = 500 cm^{2}/Vs and μ_{p} = 300 cm^{2}/V.s
 ρ
 A silicon samples is doped with 2.5 × 10^{15} phosphorus atoms per cubic centimetre and 0.5 × 10^{15} boron atoms per cubic centimetre, and is at equilibrium. Find the:
 Majoritycarrier density.
 Minoritycarrier density.
 Calculate the sample’s conductivity, using μ_{n} = 1020 cm^{2}/Vs.
 Calculate its resistivity.
 A silicon sample has N_{D} = 10^{16}/cm^{3} and N_{A} = 0.
 Find ρ.
 What is the conductivity type of the sample in (a)?
 Another sample has N_{D} = 0.5 × 10^{14}/cm^{3} and N_{A} = 10^{16}/cm^{3}. Find p_{0}.
 Find n_{0} in the sample of c.
 Another sample has N_{D} = 1.5 × 10^{l5}/cm^{3} and N_{A} = 10^{l5}/cm^{3}. Find p_{0}.
 Find n_{0} in the sample of (e).
 Another sample has N_{D} = 0.79 × 10^{14}/cm^{3} and N_{A} = 10^{15}/cm^{3}. Find p_{0}.
 Find n_{0} in the sample of (g).
 A Silicon specimen in the form of circular cylinder (Length L = 20 mm, area of crosssection A = 2 mm^{2} and resistivity ρ = 15 ohm cm is placed in series with an ideal battery of 2 V in a complete circuit. Answer the following questions.
 Calculate hole current density J_{p}.
 Calculate electron current density J_{n}.
 How J_{n} is affected by doubling sample length L and keeping A, ρ and applied voltage V the same as in part (a)?
 How is J_{n} affected by doubling crosssectional area A and keeping ρ, applied voltage V, and sample length L the same as in part (a)?
 Ohmic contacts are made to the ends of a silicon resistor having a length L = 0.5 cm. The resistor has a crosssectional area A that is given by the product of its width W and thickness X, where W = 1 mm and X = 2 μ_{m}. The silicon is uniformly doped with N_{A} = 2.5 × 10^{15} cm^{3} and N_{D} = 2.5 × 10^{l5}/cm^{3}. For this doptng density, μ_{p}= 250 cm^{2}/Vs and μ_{n} = 400 cm^{2}/Vs. Find the resistance R.
 A silicon sample contains 3.5 × 10^{16}/cm^{3} of one impurity type and a negligible amount of the opposite type. It exhibits a resistivity of 0.22 Ω–cm at room temperature.
 Determine majoritycarrier mobility.
 Is the sample ntype or ptype? Explain your reasoning.

 Given an extrinsic but lightly doped ntype silicon resistor R of length L and crosssectional area A, derive an expression for its net impurity density.
 For the resistor with, R = 1 kiloohm, L = 5 mm, and A = 1.5 mm^{2}. Evaluate the expression derived in a. What is the probable majority impurity in the resistor?
 Minority carriers in a particular silicon sample drift 1 cm in 100 μs when E_{0} = 15 V/cm.
 Determine drift velocity V_{0}.
 Determine minoritycarrier diffusivity D.
 Determine the conductivity type of the sample and explain your reasoning. A sample of heavily doped ptype silicon has a driftcurrent density of 100 A/cm^{2}. Hole drift velocity is 50 cm/s. Find hole density p_{0}.
 Calculate the intrinsic carrier concentration of Si, Ge and GaAs as a function of temperature from 4 K to 600 K. Assume that the band gap is given by:
where E_{g}(0), α, β are given by
Si: E_{g}(0) = 1.17 eV, α = 4.37 × 10^{−4} K^{−1}, β = 636 K
Ge: E_{g} (0) = 0.74 eV, α = 4.77 × 10^{−4} K^{−1}, β = 235 K
GaAs: E_{g}(0) = 1.519 eV, α = 5.4 × 10^{−4} K^{−1}, β =204 K
 The resistance of No. 18 copper wire (having a diameter d = 1.15 mm) is 6.5 ohm / l000 ft. The density of conduction electrons in copper is n_{0} = 8.3 × 10^{22}/cm^{3}.
 Given that the current in the wire is 2 A, calculate the current density J.
 Find the magnitude of the drift velocity V_{0} of the electrons in cm/s and cm/hr.
 Calculate the resistivity ρ of the wire.
 Calculate the electric field E_{0} in the wire.
 Calculate the mobility magnitude μ_{n} of the electrons in copper.
 Assume complete ionization.
 Combine the neutrality equation and the law of mass action to obtain an accurate expression for n_{0} in nearintrinsic ntype silicon.
 Use the expression obtained in a. to calculate n_{0} in a sample having N_{D} = 0.9 × 10^{14}/cm^{3} and N_{A} = 1.0 × 10^{14}/cm^{3}.
 Comment on the accuracy of the approximate equation n_{0} ≈ N_{D} – N_{A} for ordinary doping values.
 Given a hole concentration gradient of –10^{20}/cm^{4} in a lightly doped sample, calculate the corresponding hole diffusioncurrent density.
 A lightly doped fieldfree sample of silicon that is 1 μm thick in the x direction exhibits a majorityelectron gradient of (dn/dx) = –10^{15}/cm^{4}.
 Calculate the diffusion current density for electrons in this case.
 If the electron density at the higherdensity (left) face is 10^{14}/cm^{3}, what is it at the right face?
 In a thought experiment, we supply an additional population of electrons throughout the sample in the amount of 10^{14}/cm^{3}. Repeat the calculation of (a).
 Return to (a) and (b). Suppose that without any other changes an electric field of E_{0} = 0.045 V/cm is superimposed on the sample in the positive x direction. Calculate the diffusion current density due to electrons approximately.
 Calculate the total current density for the electrons.
 In a certain lightly doped silicon sample there exists a hole currentdensity value due to drift equal to –1.075 A/cm^{2}. Calculate the corresponding density gradient.
 A certain silicon sample has negligible acceptor doping and a donor doping that is linearly graded from 1.5 × 10^{15}/cm^{3} at the left surface to 2.5 × 10^{14}/cm^{3} at the right surface. The two surfaces are 100 μm apart.
 Assuming that n(x) ≈ N_{D}(x) throughout, calculate diffusion current density due to electrons at the middle of the sample.
 In a thought experiment, we add 1.5 × 10^{16}/cm^{3} of donors uniformly to the sample of (a); recalculate diffusion current density due to electrons.
 In a second thought experiment we add 1.5 × 10^{15}/cm^{3} of acceptors to the sample of (a); recalculate diffusion current density due to electrons.
 A thin silicon sample receives steadystate radiation that produces excess carriers uniformly throughout the sample in the amount ∆P_{0} = ∆n_{0} l0^{10}/cm^{3}. Excesscarrier lifetime in the sample is 1 μs. At t = 0, the radiation source is turned off. Calculate the excesscarrier density and recombination rate at (a) t = 1 μs; (b) t = 1.5 μs; (c) t = 4 μs.
 A thin ntype sample of silicon having an equilibrium minoritycarrier density p_{0} is subjected to penetrating radiation with the radiation source turned on at t = 0. At t = ∞, p = p (∞).
 Write the differential equation appropriate to this situation.
 Find the solution for the differential equation of (a) under the given boundary conditions.
 Uniform, steadystate ultraviolet radiation impinges on the surface of a semiinfinite silicon sample in which n_{0} = 10^{15}/cm^{3}, producing an excesscarrier density at the surface of ∆p_{0}(0) = ∆n_{0}(0) = 10^{1l}/cm^{3}. Given further that τ = 1 μs, and that the spatial origin is at the irradiated surface.
 Calculate diffusion current density due to holes at x = 0.
 Calculate diffusion current density due to electrons at x = 0.
 Calculate diffusion current density due to holes and diffusion current density due to electrons at x = L_{p}. Sketch vectors to scale representing these currentdensity components.
 Since the sample is opencircuited, the total current density at x = L_{p} must be zero, J = 0. In fact, a tiny electric field accounts for the “missing” current density component. Speculate on the cause of the electric field.
 Calculate the magnitude and direction of the field cited in (d) at the position x = L_{p}.
 Does the presence of the field destroy the validity of the current density due to holes calculations that were based upon a purediffusion picture? Explain.
 Derive an approximate expression for E_{0}. A thin ntype silicon sample with τ = 5 μ sec is subjected to infrared radiation for a long period. At t = 0, the radiation source is turned off. At t = 020 μ sec, excess – carrier recombination rate is observed to be 4.5 × 10^{13}/cm^{3}/s. Calculate the steady – state generation rate caused by the radiation.

 A Si bar 0.5 cm long and 120 μm^{2} in crosssectional area is doped with 10^{17}cm^{–3} phosphorus. Find the current at 300 K with 10 V applied. Repeat for a Si bar 1 μm long.
 How long does it take an average electron to drift 1 μm in pure Si at an electric field of 500 V/cm? Repeat for 10 V/cm.
 A Si sample is doped with 6.5 × 10^{15} cm^{–3} donors and 2.5 × 10^{15} cm^{−3} acceptors.
Find the position of the Fermi level with respect to E_{i}. at 300 K. What is the value and sign of the Hall coefficient?
 Find out the expressions of the densityofstate functions for the dispersion relation of the conduction electrons as given in Problem 1 under the conditions (a) E/E_{g} << 1 and (b) E/E_{g} >> 1 respectively. Draw the graphs and explain the result physically.
 Find the expressions for n_{0} under the conditions as stated in Problem 47. Draw the graphs and explain the result physically.
 Find the expression of the average energy of electrons in semiconductors having parabolic energy bands for the conditions of both non degenerate and degenerate electron concentrations.
 Find out a simple expression of photo emitted current density from semiconductors having parabolic energy bands.
 Pierret, R. F. and G.W. Neudeck. 1989. Modular Series on Solid State Devices. Boston M.A.: Addison Wesley.
 Singh, J. 1994. Semiconductor Devices: An Introduction. New York, NY: McGrawHill.
 Millman, Jacob and Christos C. Halkias. 1986. Integrated Electronics: Analog and Digital Circuits and Systems. New Delhi: McGrawHill.
 Streetman, B.G. and S. Banerjee. 2000. Solid State Electronic Devices. New Delhi: Pearson Education.