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1. Semiconductor Fundamentals > 1-8 Hall Effect

1-8 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 E0 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 n-type or p-type by finding the carrier concentration and calculating the mobility μ, by measuring the conductivity σ.

 

Figure 1-37 Schematic diagram of Hall effect. The carriers (electrons or holes) are subjected to a magnetic force in the negative y-direction.

 

Figure 1-37 Schematic diagram of Hall effect. The carriers (electrons or holes) are subjected to a magnetic force in the negative y-direction.

 

Let us consider a rectangular bar of n-type semiconductor of length L, width W and thickness d, as shown in Fig. 1-37.

If a current I is applied in the positive x-direction and a magnetic field B is applied in the positive z-direction, a force will be exerted in the negative y-direction of the current carriers. The current is carried by electron from side 1 to side 2, if the semiconductor is n-type. Therefore Hall voltage VH appears between surfaces 1 and 2. The electric field developed in y-direction Ey is given by:

 

C01E123

 

where, d is the distance between surfaces 1 and 2. In the equilibrium state the electric field Ey due to the Hall effect must exert a force on the carrier, which just balances the magnetic force, and we can write:

 

C01E124

 

where, e is the magnitude of the charge of the carriers, v0 is the drift speed.
Therefore current density J is given by:

 

C01E125

 

where, ρ is the charge density and W is the width of the specimen in the direction of the magnetic field. Combining Eq. (1-123), Eq. (1-124) and Eq. (1-125) we find:

 

C01E126

 

If VH, B, W and I are measured, the charge density ρ can be measured from Eq. (1-126). If the polarity of VH is positive at terminal 1 then the carrier must be an electron and ρ = n0e where n0 is the electron concentration. If the terminal 2 becomes positively charged with respect to terminal 1 the semiconductor must be of p-type and ρ = p0e where p0 is the hole concentration:

 

C01E127

 

where, RH is the Hall coefficient.

 

C01E128

 

Conductivity σ is related to mobility μ by:

σ = n0

 

σ = p0μ

where,

C01E129

 

If the conductivity is measured with Hall coefficient, mobility μ can be determined by:

 

C01E130

 

From Eq. (1-127) and Eq. (1-130) we get:

 

C01E131

 

In the presence of scattering the mobility can approximately be written as:

 

C01E132

Applications of Hall Effect

  1. 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.
  2. By using Hall effect, the type of the semiconductor can be determined as follows: RH > 0 for p-type semiconductors and RH < 0 for n-type semiconductors.
  3. Determination of the mobility: The equation μ = [σRH] determines the mobility by using the experimental values of RH.
  4. Hall effect multiplier: If the magnetic field B produces current I1 then Hall voltage VHII1 The Hall effect multiplier generates an output proportional to the product of two signals. Thus, Hall effect can be used for analogue multiplication.
  5. The power flow in an electromagnetic wave can be measured by the help of Hall effect.
  6. Experimental determination of the magnetic field: By knowing the values of VH, I, ρ and W, we can determine the value of B experimentally.
POINTS TO REMEMBER
  1. Lattice plus basis is equal to crystal structure.
  2. The bravais lattices are distinct lattice types, which when repeated can fill the whole space.
  3. In two dimensions there are five distinct bravais lattices while in three dimensions there are fourteen.
  4. Electrons, which revolve in the outermost orbit of an atom, are called valence electrons.
  5. Electrons, which are detached from the parent atoms and move randomly in the lattice of a metal, are called free electrons.
  6. The range of energies possessed by electrons in a solid is known as energy band.
  7. The range of energies possessed by valance electrons is called the valance band.
  8. The range of energies possessed by free electrons is called the conduction band.
  9. The separation between conduction and valence band on the energy level diagram is called the forbidden energy gap.
  10. 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 III-V, II-VI, IV-VI, ternaries and quaternaries are called compound semiconductors.
  11. There are three types of semiconductors namely n-type semiconductors (when electron concentration is much greater than hole concentration), p-type semiconductors (when electron concentration is much less than hole concentration) and intrinsic semiconductors (when electron concentration is equal to hole concentration).
  12. A semiconductor without any impurities is called an intrinsic semiconductor and a semiconductor with impurities is called extrinsic semiconductor.
  13. 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.
  14. The resistivity of pure semiconductor is of the order of 103 Ωm. The energy gap of pure silicon is 1.12 eV and of germanium is 0.72 eV.
  15. Thermal process is the only process that generates carriers in an intrinsic semiconductor.
  16. The additional velocity acquired by the charge carriers in the electric field is called the drift velocity.
  17. Mobility is defined as the drift velocity acquired per unit electric field strength.
  18. Conductivity of a material is a measure of the material’s ability to allow charge carriers to flow through it.
  19. 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.
  20. An n-type semiconductor is obtained by adding pentavalent impurity to a pure Si or Ge semiconductor. In an n-type semiconductor electrons are majority carriers and holes are minority carriers.
  21. A p-type semiconductor is obtained by adding trivalent impurity to a pure Si or Ge semiconductor. In a p-type 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
  22. An extrinsic semiconductor is electrically neutral.
  23. In an n-type material, the free electron concentration is approximately equal to the density of donor atoms. Thus, nn ≅ NA
  24. In an extrinsic parabolic semiconductors, Fermi level lies exactly midway between valance and conduction bands at T→0.
  25. The product of n and p is a constant and is known as the law of mass action.
  26. The flow of current through a semiconductor material is normally referred to as one of two types: drift or diffusion.
  27. 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.
  28. The directional movement of charge carriers due to their concentration gradient produces a component of current known as diffusion current.
  29. The total current is equal to the drift part of the current plus diffusion part of the current.
  30. The distance that free carrier travels before recombining is called the diffusion length.
  31. 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.
  32. 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.
  33. Hall effect measurements help us in identifying the type of majority carriers in determining carrier concentration and carrier mobility.
IMPORTANT FORMULAE
  1. De-Broglie’s relation of the wave particle is given by: λ = h/p
  2. The effective momentum mass of the carriers is given by: 59_1s
  3. Energy gap between conduction and valence band is: Eg = EcEv
  4. The density-of-state function of the conduction electron in parabolic n-type semiconductors is given by:

     

    59_2

     

  5. The Fermi–Dirac integral of order j is given by:

     

    ali_4

     

    which obeys the following properties:

    1. ali_5
    2. ali_6
    3. ali_7
    4. ali_8
    5. ali_9
  6. The electron concentration in n-type parabolic semiconductors is given by: 59_4
    1. Under non-degenerate electron concentration 59_5
    2. The expression of the critical electron concentration when EF touches EC is given by: 59_6
    3. The variation of Fermi energy with temperature for relatively low values of temperatures in n-type parabolic semiconductors is given by:

       

      59_7

       

    4. Under the condition of extreme carrier degeneracy

       

      59_8

       

  7. The concentration of heavy holes in parabolic p-type semiconductors is given by:

     

    59_9

     

  8. Under the condition of non-degenerate hole concentration we can write the above equation reduces to:

     

    59_10

     

  9. For intrinsic semiconductors EF is given by:t

     

    59_11

     

  10. The law of mass action for non-degenerate parabolic semiconductors is given by:

     

    59_12

     

  11. The intrinsic carrier concentration is given by:

     

    60_1

     

  12. For extrinsic non-degenerate parabolic semiconductors with Nd 0, Na = 0 and 60_2 the Fermi energy can be written as:

     

    60_3

     

  13. For extrinsic non-degenerate parabolic semiconductors with Nd 0, Na = 0 and 60_4 the Fermi energy can be written as:

     

    60_5

     

  14. In the zone of very high temperatures the Fermi energy for non-degenerate parabolic semiconductors when NA = 0 is given by:

     

    60_6

     

  15. The electron mobility (μ) is given by:

     

    60_7

     

    In general, the mobility for isotropic bands is expressed as:

     

    μ = e < τ(E)/m*(E) >

     

  16. Mathiessen’s rule is given by:

     

    60_9

     

  17. The conductivity for mixed conduction is given by: σ = σn + σp = e(n0μn + p0μp)
  18. The electron and hole current densities in the presence of drift and diffusion are given by:

     

    60_10

     

  19. The generalized expression of the Einstein relation for electrons in n-type semiconductors is given by:

     

    60_11

     

    1. Under the condition of extreme degeneracy, the Einstein relation in n-type parabolic semiconductors is given by:

       

      60_12

       

    2. Under non-degenerate electron concentration, the Einstein relation for n-type semiconductors is given by:

       

      60_13

       

  20. The relation between the rate of recombination of electrons and holes and equilibrium concentration of electrons and holes is given by:

     

    ri = krn0p0 = kr n2i = gi

     

  21. The time dependence of excess carrier concentration for a p-type material is given by:

     

    60_15

     

  22. The continuity equation for holes for n-type semiconductors is given by:

     

    60_16

     

  23. The transverse voltage produced in Hall effect is given by:

     

    60_17

     

  24. The Hall coefficient: 60_18
  25. Charge density: 60_19
  26. Mobility: μ = σRH
  27. The transverse voltage produced in Hall effect is given by:

     

    61_1

     

  28. The Hall coefficient: 61_2
  29. Charge density: 61_3
OBJECTIVE QUESTIONS
  1. A silicon sample is uniformly doped with 1016 phosphorus atoms/cm3 and 2 × 1016 boron atoms/cm3. If all the dopants are fully ionized, the material is:
    1. n-type with carrier concentration of 3 × 1016/cm3
    2. p-type with carrier concentration of 1016/cm3
    3. p-type with carrier concentration of 4 × 1016/cm3
    4. Intrinsic
  2. n-type semiconductors are:
    1. Negatively charged
    2. Produced when Indium is added as an impurity to Germanium
    3. Produced when phosphorous is added as an impurity to silicon
    4. None of the above
  3. The probability that an electron in a metal occupies the Fermi-level, at any temperature (> 0 K) is:
    1. 0
    2. 1
    3. 0.5
    4. None of the above
  4. Measurement of Hall coefficient enables the determination of:
    1. Mobility of charge carriers
    2. Type of conductivity and concentration of charge carriers
    3. Temperature coefficient and thermal conductivity
    4. None of the above
  5. If the energy gap of a semiconductor is 1.1 eV it would be:
    1. Opaque to the visible light
    2. Transparent to the visible light
    3. Transparent to the ultraviolet radiation
    4. None of the above
  6. The conductivity of an intrinsic semiconductor is given by (symbols have the usual meanings):
    1. σi = eni2nμp)
    2. σi = eni(μnμp)
    3. σi = eni(μn + μp)
    4. None of the above
  7. Consider the following statements: Compared to Silicon, Gallium Arsenide (GaAs) has:
    1. Higher signal speed since electron mobility is higher
    2. Poorer crystal quality since stoichiometric growth difficult
    3. Easier to grow crystals since the vapour pressure Arsenic is high
    4. Higher optoelectronic conversion efficiency

    Of these statements:

    1. 1, 2, 3 and 4 are correct
    2. 1, 2 and 3 are correct
    3. 3 and 4 are correct
    4. None of the above
  8. In an intrinsic semiconductor, the mobility of electrons in the conduction band is:
    1. Less than the mobility of holes in the valence band
    2. Zero
    3. Greater than the mobility of holes in the valence band
    4. None of the above
  9. 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 m3 coulomb–1. The carrier concentration in sample (A) at room temperature is:
    1. ~1021 m–3
    2. ~1020 m–3
    3. ~1022 m–3
    4. None of the above
  10. In a semiconductor, J, Jp and Jn indicate total diffusion current density hole current density and electron current density respectively, 62_1 and images are the electron and hole concentration gradient respectively in x-direction and Dp and Dn are the hole and electron diffusion constants respectively. Which one of the following equations is correct? (e denotes charge of electron).
    1. ali_10
    2. ali_11
    3. ali_12
    4. None of the above
  11. 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:
    1. 0.05
    2. 0.55
    3. 500
    4. None of the above
  12. The Hall effect voltage in intrinsic silicon is:
    1. Positive
    2. Zero
    3. Negative
    4. None of the above
  13. The Hall coefficient of an intrinsic semiconductor is:
    1. Positive under all conditions
    2. Negative under all conditions
    3. Zero under all conditions
    4. None of the above
  14. Consider the following statements: Pure germanium and pure silicon are examples of:
    1. Direct band-gap semiconductors
    2. Indirect band-gap semiconductors
    3. Degenerate semiconductors

    Of these statements:

    1. 1 alone is correct
    2. 2 alone is correct
    3. 3 alone is correct
    4. None of the above
  15. Assume ne and nh are electron and hole densities, μe and μn are the carrier mobilities; the Hall coefficient is positive when:
    1. nh μh > ne μe
    2. nh μh2 < ne μe2
    3. nh μh > ne μh
    4. None of the above
  16. A long specimen of p-type semiconductor material:
    1. Is positively charged
    2. Is electrically neutral
    3. Has an electric field directed along its length
    4. None of the above
  17. The electron and hole concentrations in an intrinsic semiconductor are ni and pi respectively. When doped with a p-type material, these change to n and P respectively. Then:
    1. n + p = ni + pi
    2. n + ni = p + pi
    3. np = ni pi
    4. None of the above are applicable
  18. If the temperature of an extrinsic semiconductor is increased so that the intrinsic carrier concentration is doubled, then:
    1. The majority carrier density doubles
    2. The minority carrier density doubles
    3. Both majority and minority carrier densities double
    4. None of the above
  19. At room temperature, the current in an intrinsic semiconductor is due to:
    1. Holes
    2. Electrons
    3. Holes and electrons
    4. None of the above
  20. A semiconductor is irradiated with light such that carriers are uniformly generated throughout its volume. The semiconductor is n-type with ND = 1019 per cm3. If the excess electron concentration in the steady state is ∆n = 1015 per cm3 and if τp = 10 μsec. (minority carrier life time) the generation rate due to irradiation is:
    1. 1022 e-h pairs /cm3/s
    2. 1010 e-h pairs/cm3/s
    3. 1024 e-h pairs/cm3/s
    4. None of the above
  21. 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 cm2/volt.sec) is:
    1. 1,000
    2. 2,000
    3. 50
    4. None of the above
  22. The mobility is given by:
    1. ali_13
    2. ali_14
    3. ali_15
    4. None of the above
  23. 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:
    1. A direction normal to both current and magnetic field
    2. The direction of current
    3. A direction anti parallel to magnetic field
    4. None of the above
  24. In a p-type semiconductor, the conductivity due to holes (σp) is equal to: (e is the charge of hole, μp is the hole mobility, p0 is the hole concentration):
    1. p0e/μp
    2. μp/p0e
    3. p0p
    4. None of the above
  25. 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:
    1. 3.3 × 1015 and 9.9 × 1013 Hz
    2. 3.7 × 1016 and 3.5 × 1014 Hz
    3. 6.28 × 1017 and 3.1 × 1013 Hz
    4. None of the above
  26. A sample of n-type semiconductor has electron density of 6.25 × 1018/cm3 at 300 K. If the intrinsic concentration of carriers in this sample is 2.5 × 1013/cm3 at this temperature, the hole density becomes:
    1. 1016/cm3
    2. 107/cm3
    3. 1017/cm3
    4. None of the above
  27. The intrinsic carrier density at 300K is 1.5 × 1010/cm3 in silicon. For n-type silicon doped to 2.25 × 1015 atoms/cm3 the equilibrium electron and hole densities are:
    1. n0 = 1.5 × 1016/cm3, p0 = 1.5 × 1012/cm3
    2. n0 = 1.5 × 1010/cm3, p0 = 2.25 × 1015/cm3
    3. n0 = 2.25 × 1017/cm3, p0 = 1.0 × 1014/cm3
    4. None of the above
  28. In a p-type silicon sample, the hole concentration is 2.25 × 1015/cm3. If the intrinsic carrier concentration 1.5 × 1010/cm3 the electron concentration is:
    1. 1021/cm3
    2. 1010/cm3
    3. 1016/cm3
    4. None of the above
  29. A good ohmic contact on a p-type semiconductor chip is formed by introducing:
    1. Gold as an impurity below the contact
    2. A high concentration of acceptors below the contact
    3. A high concentration of donors below the contact
    4. None of the above
  30. Measurement of Hall coefficient in a semiconductor provides information on the:
    1. Sign and mass of charge carriers
    2. Mass and concentration of charge carriers
    3. Sign of charge carriers alone
    4. Sign and concentration of charge carriers
REVIEW QUESTIONS
  1. Briefly discuss the basic developments in the study of electronics.
  2. What are crystalline materials?
  3. Give three examples of Group III-V semiconductors.
  4. What do you mean by pure crystals?
  5. Why is Si preferred over Ge?
  6. Why is GaAs preferred over Si?
  7. Give an example of the constituent material of Gunn Diode.
  8. What is Bravais lattice? Discuss briefly.
  9. What are unit cells and lattice constants?
  10. Explain the differences among simple cubic, body-centred cubic and face-centred cubic lattices respectively.
  11. Explain Czochralski growth of the semiconductor crystal in detail.
  12. Explain the wave particle duality principle.
  13. State Pauli exclusion principle.
  14. What is degenerate energy level?
  15. What is energy band gap?
  16. What do you mean by covalent and electrovalent bonds? How these are affected by the temperature?
  17. Explain thermal equilibrium.
  18. What is valence band and conduction band?
  19. What are conduction band carriers?
  20. Explain the existence of hole.
  21. What is momentum effective mass of the carriers? What is its difference with acceleration effective mass?
  22. Explain indistinguishability between the particles. What should be the value of the spin of the particles obeying Fermi–Dirac statistics?
  23. Explain the term “occupation probability”.
  24. What is bonding model?
  25. Explain the energy band models of semiconductors.
  26. Write the assumptions behind Fermi–Dirac statistics. Give a very simple proof of the same statistics.
  27. Write the five properties of the Fermi–Dirac function.
  28. Define Fermi energy. How does Fermi energy vary with temperature?
  29. Write the basic criterion for the classification of metals, semiconductors and insulators.
  30. Draw the model energy band structure diagram of a semiconductor in general.
  31. The split-off hole parabola is more flattened than the light hole parabola. Justify your answer very briefly.
  32. What are intrinsic semiconductors?
  33. What are extrinsic semiconductors?
  34. Explain donor ion, acceptor ion, majority carriers, minority carriers, doping and dopants in a semiconductor.
  35. What do you mean by the term “carrier degeneracy” of a semiconductor? Explain in detail.
  36. What are compound semiconductors? Explain the uses of compound semiconductors.
  37. Explain the difference between metals, insulators and semiconductors with the help of band structure model.
  38. Define the term “density-of-state function”. Derive an expression for the density-of-state 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.
  39. Derive an expression of electron concentration in n-type semiconductors. Discuss all the special cases.
  40. Derive an expression of electron concentration in p-type semiconductors. Discuss all the special cases.
  41. What is charge density equation?
  42. Show that:

     

    64_1

     

    where, the notations mean as usual.
  43. Derive the law of mass action. Is it valid for degenerate semiconductors? Give reasons.
  44. Does the band gap vary with temperature? Give reasons.
  45. How does concentration vary with temperature?
  46. What is mobility?
  47. What is relaxation time?
  48. What is conductivity?
  49. What is Mathiessen’s rule?
  50. What is diffusion?
  51. What is recombination?
  52. What is generation?
  53. Prove that:

     

    65_1s

     

  54. Write the assumptions of continuity equation and derive continuity equation.
  55. Explain physically the continuity equation.
  56. What is Hall effect?
  57. Derive the relation between mobility and Hall coefficient.
  58. What are the applications of Hall effect?
  59. Find out an expression for Hall coefficient in a semiconductor when both carriers contribute to the current.
PRACTICE PROBLEMS
  1. The energy spectrum of the conduction electrons of III-V compound semiconductors can be expressed as 65_2 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 n-GaAs. Interpret the results. Explain the results for Eg → ∞.
  2. The dispersion relation of the carriers in a semiconductor is approximately given by

     

    E = E0Acos (αkx) − B[cos (βky) + cos(βkz)]

     

    where E0, A, B, α and β are constants. Develop an expression of the density-of-states function for small k.
    1. Calculate the coordinates of three points on the Fermi–Dirac function and enter the coordinates in this table.
      65_4
      65_5
         
         
         
    2. Plot the same three points in the following axis and sketch the Fermi–Dirac function with reasonable accuracy.

       

      65_6

       

  3. In a certain silicon sample at equilibrium, the Fermi level resides at 0.500 eV above the centre of the band gap.
    1. Calculate the occupancy probability for a lone isolated state located right at the centre of the band gap.
    2. This sample contains donor impurities and no acceptor impurities. The donor states are situated 0.045 eV below the conduction-band edge. Find the occupancy probability of the donor states.
    3. Comment on the validity of the assumption of 100 per cent ionization of the donor states in the present situation.
    4. Derive an approximate form of the Fermi–Dirac probability function that could be applied in (b) with reasonable validity.
    5. Use your approximate expression to recalculate the probability in (b) and find the percentage difference in the exact and approximate results.
    6. Sketch the exact probability function accurately by plotting several points. Superimpose on the same diagram an accurate sketch of the approximate expression of (d).
    7. Comment on conditions where the use of the approximate expression is justified.
    8. How much error results when the approximate expression is used to calculate occupancy probability for a state located right at the Fermi level?
  4. The conduction band can be characterized by a state density (number of states per cm3) of Nc = 3.75 × 1019/cm3, with these states assumed to be situated right at the conduction-band edge.
    1. Using this assumption, calculate the conduction-electron density n0 (number of electrons per cm3) for the conditions of Problem 4.
    2. The valence band can be characterized by a state density of NvNc’ with these states assumed to be situated right at the valence-band edge. Using this assumption, calculate the hole density p0.
    3. Calculate the p–n product using the results from (a) and (b).
  5. Determine the approximate density of donor states ND (number of donor states per cm3) for the silicon sample of Problems 4 and 5. Provide reasons.
  6. Derive an expression relating the intrinsic level Ei. to the centre of the band gap Eg/2. Calculate the displacement of Ei. from Eg/2 for Si at 300 K, assuming the effective mass values for electrons and holes are 1.1 m0 and 0.56 m0, respectively.
    1. Explain why holes are found at the top of the valence band, whereas electrons are found at the bottom of the conduction band.
    2. Explain why Si doped with 1014 cm-3 Sb is n-type at 400 K but similarly doped Ge is not.
  7. Calculate the Nc and Nv 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.
  8. A certain uniformly doped silicon sample at room temperature has n0 = 106/cm3 and NA = 1015/cm3.
    1. Find p0.
    2. Find ND.
  9. Using the Boltzmann approximation to the Fermi–Dirac probability function (obtained by dropping the unity term in its denominator), find the Fermi-level position relative to the conduction band edge for a sample having n0 = 3 × 1015/cm3 at room temperature and at equilibrium.
  10. 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.
  11. A silicon sample has NA = 1015/cm3 and ND = 0. Find the:
    1. Majority-carrier density
    2. Minority-carrier density
    3. Conductivity
  12. A certain silicon sample has p0 = 2.5 × 1010/cm3 Find the:
    1. Electron density n0
    2. Resistivity ρ
  13. Calculate the position of the intrinsic Fermi level measured from the midgap for InAs.
  14. Calculate and plot the position of the intrinsic Fermi level in Si between 80 K and 400 K.
  15. 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.
  16. In a GaAs sample at 310 K, the Fermi level coincides with the valence band-edge. Calculate the hole density by using the Boltzmann approximation. Also calculate the electron density using the law of mass action.
  17. The electron density in a silicon sample at 310 K is 1015 cm−3. Calculate EcEF and the hole density using the Boltzmann approximation.
  18. A GaAs sample is doped n-type at 4 × 1018 cm−3. Assume that all the donors are ionized. What is the position of the Fermi level at 300 K?
  19. Consider a n-type silicon with a donor energy 60 meV below the conduction band. The sample is doped at 1015 cm−3. Calculate the temperature at which 30 per cent of the donors are not ionized.
  20. Consider a GaAs sample doped at Nd = 1015 cm−3 where the donor energy is 5 meV. Calculate the temperature at which 80 per cent of the donors are ionized.
  21. 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.
  22. A Si device is doped at 2.5 × 1016 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?
  23. Estimate the change in intrinsic carrier concentration per K change in temperature for Ge, Si, and GaAs and InSb at 300 K.
  24. A certain silicon sample has ND = 5.30 × 1015/cm3 and NA = 4.50 × 1015/cm3.
    Find the:
    1. Majority-carrier density
    2. Minority-carrier density
    3. Conductivity σ, using μn = 400 cm2/Vs, and μp = 300 cm2/Vs
    4. n0
    5. σ using μn = 500 cm2/Vs and μp = 300 cm2/V.s
    6. ρ
  25. A silicon samples is doped with 2.5 × 1015 phosphorus atoms per cubic centimetre and 0.5 × 1015 boron atoms per cubic centimetre, and is at equilibrium. Find the:
    1. Majority-carrier density.
    2. Minority-carrier density.
    3. Calculate the sample’s conductivity, using μn = 1020 cm2/Vs.
    4. Calculate its resistivity.
  26. A silicon sample has ND = 1016/cm3 and NA = 0.
    1. Find ρ.
    2. What is the conductivity type of the sample in (a)?
    3. Another sample has ND = 0.5 × 1014/cm3 and NA = 1016/cm3. Find p0.
    4. Find n0 in the sample of c.
    5. Another sample has ND = 1.5 × 10l5/cm3 and NA = 10l5/cm3. Find p0.
    6. Find n0 in the sample of (e).
    7. Another sample has ND = 0.79 × 1014/cm3 and NA = 1015/cm3. Find p0.
    8. Find n0 in the sample of (g).
  27. A Silicon specimen in the form of circular cylinder (Length L = 20 mm, area of cross-section A = 2 mm2 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.
    1. Calculate hole current density Jp.
    2. Calculate electron current density Jn.
    3. How Jn is affected by doubling sample length L and keeping A, ρ and applied voltage V the same as in part (a)?
    4. How is Jn affected by doubling cross-sectional area A and keeping ρ, applied voltage V, and sample length L the same as in part (a)?
  28. Ohmic contacts are made to the ends of a silicon resistor having a length L = 0.5 cm. The resistor has a cross-sectional area A that is given by the product of its width W and thickness X, where W = 1 mm and X = 2 μm. The silicon is uniformly doped with NA = 2.5 × 1015 cm3 and ND = 2.5 × 10l5/cm3. For this doptng density, μp= 250 cm2/Vs and μn = 400 cm2/Vs. Find the resistance R.
  29. A silicon sample contains 3.5 × 1016/cm3 of one impurity type and a negligible amount of the opposite type. It exhibits a resistivity of 0.22 Ω–cm at room temperature.
    1. Determine majority-carrier mobility.
    2. Is the sample n-type or p-type? Explain your reasoning.
    1. Given an extrinsic but lightly doped n-type silicon resistor R of length L and cross-sectional area A, derive an expression for its net impurity density.
    2. For the resistor with, R = 1 kilo-ohm, L = 5 mm, and A = 1.5 mm2. Evaluate the expression derived in a. What is the probable majority impurity in the resistor?
  30. Minority carriers in a particular silicon sample drift 1 cm in 100 μs when E0 = 15 V/cm.
    1. Determine drift velocity V0.
    2. Determine minority-carrier diffusivity D.
    3. Determine the conductivity type of the sample and explain your reasoning. A sample of heavily doped p-type silicon has a drift-current density of 100 A/cm2. Hole drift velocity is 50 cm/s. Find hole density p0.
  31. 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:

     

    68_1

     

    where Eg(0), α, β are given by
    Si: Eg(0) = 1.17 eV, α = 4.37 × 10−4 K−1, β = 636 K
    Ge: Eg (0) = 0.74 eV, α = 4.77 × 10−4 K−1, β = 235 K
    GaAs: Eg(0) = 1.519 eV, α = 5.4 × 10−4 K−1, β =204 K
  32. 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 n0 = 8.3 × 1022/cm3.
    1. Given that the current in the wire is 2 A, calculate the current density J.
    2. Find the magnitude of the drift velocity V0 of the electrons in cm/s and cm/hr.
    3. Calculate the resistivity ρ of the wire.
    4. Calculate the electric field E0 in the wire.
    5. Calculate the mobility magnitude |μn| of the electrons in copper.
  33. Assume complete ionization.
    1. Combine the neutrality equation and the law of mass action to obtain an accurate expression for n0 in near-intrinsic n-type silicon.
    2. Use the expression obtained in a. to calculate n0 in a sample having ND = 0.9 × 1014/cm3 and NA = 1.0 × 1014/cm3.
    3. Comment on the accuracy of the approximate equation n0 ≈ ND – NA for ordinary doping values.
  34. Given a hole concentration gradient of –1020/cm4 in a lightly doped sample, calculate the corresponding hole diffusion-current density.
  35. A lightly doped field-free sample of silicon that is 1 μm thick in the x direction exhibits a majority-electron gradient of (dn/dx) = –1015/cm4.
    1. Calculate the diffusion current density for electrons in this case.
    2. If the electron density at the higher-density (left) face is 1014/cm3, what is it at the right face?
    3. In a thought experiment, we supply an additional population of electrons throughout the sample in the amount of 1014/cm3. Repeat the calculation of (a).
    4. Return to (a) and (b). Suppose that without any other changes an electric field of E0 = 0.045 V/cm is superimposed on the sample in the positive x direction. Calculate the diffusion current density due to electrons approximately.
    5. Calculate the total current density for the electrons.
  36. In a certain lightly doped silicon sample there exists a hole current-density value due to drift equal to –1.075 A/cm2. Calculate the corresponding density gradient.
  37. A certain silicon sample has negligible acceptor doping and a donor doping that is linearly graded from 1.5 × 1015/cm3 at the left surface to 2.5 × 1014/cm3 at the right surface. The two surfaces are 100 μm apart.
    1. Assuming that n(x) ≈ ND(x) throughout, calculate diffusion current density due to electrons at the middle of the sample.
    2. In a thought experiment, we add 1.5 × 1016/cm3 of donors uniformly to the sample of (a); recalculate diffusion current density due to electrons.
    3. In a second thought experiment we add 1.5 × 1015/cm3 of acceptors to the sample of (a); recalculate diffusion current density due to electrons.
  38. A thin silicon sample receives steady-state radiation that produces excess carriers uniformly throughout the sample in the amount ∆P0 = ∆n0 l010/cm3. Excess-carrier lifetime in the sample is 1 μs. At t = 0, the radiation source is turned off. Calculate the excess-carrier density and recombination rate at (a) t = 1 μs; (b) t = 1.5 μs; (c) t = 4 μs.
  39. A thin n-type sample of silicon having an equilibrium minority-carrier density p0 is subjected to penetrating radiation with the radiation source turned on at t = 0. At t = ∞, p = p (∞).
    1. Write the differential equation appropriate to this situation.
    2. Find the solution for the differential equation of (a) under the given boundary conditions.
  40. Uniform, steady-state ultraviolet radiation impinges on the surface of a semi-infinite silicon sample in which n0 = 1015/cm3, producing an excess-carrier density at the surface of ∆p0(0) = ∆n0(0) = 101l/cm3. Given further that τ = 1 μs, and that the spatial origin is at the irradiated surface.
    1. Calculate diffusion current density due to holes at x = 0.
    2. Calculate diffusion current density due to electrons at x = 0.
    3. Calculate diffusion current density due to holes and diffusion current density due to electrons at x = Lp. Sketch vectors to scale representing these current-density components.
    4. Since the sample is open-circuited, the total current density at x = Lp 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.
    5. Calculate the magnitude and direction of the field cited in (d) at the position x = Lp.
    6. Does the presence of the field destroy the validity of the current density due to holes calculations that were based upon a pure-diffusion picture? Explain.
    7. Derive an approximate expression for E0. A thin n-type 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 × 1013/cm3/s. Calculate the steady – state generation rate caused by the radiation.
    1. A Si bar 0.5 cm long and 120 μm2 in cross-sectional area is doped with 1017cm–3 phosphorus. Find the current at 300 K with 10 V applied. Repeat for a Si bar 1 μm long.
    2. 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.
  41. A Si sample is doped with 6.5 × 1015 cm–3 donors and 2.5 × 1015 cm−3 acceptors.
    Find the position of the Fermi level with respect to Ei. at 300 K. What is the value and sign of the Hall coefficient?
  42. Find out the expressions of the density-of-state functions for the dispersion relation of the conduction electrons as given in Problem 1 under the conditions (a) E/Eg << 1 and (b) E/Eg >> 1 respectively. Draw the graphs and explain the result physically.
  43. Find the expressions for n0 under the conditions as stated in Problem 47. Draw the graphs and explain the result physically.
  44. 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.
  45. Find out a simple expression of photo emitted current density from semiconductors having parabolic energy bands.
SUGGESTED READINGS
  1. Pierret, R. F. and G.W. Neudeck. 1989. Modular Series on Solid State Devices. Boston M.A.: Addison Wesley.
  2. Singh, J. 1994. Semiconductor Devices: An Introduction. New York, NY: McGraw-Hill.
  3. Millman, Jacob and Christos C. Halkias. 1986. Integrated Electronics: Analog and Digital Circuits and Systems. New Delhi: McGraw-Hill.
  4. Streetman, B.G. and S. Banerjee. 2000. Solid State Electronic Devices. New Delhi: Pearson Education.
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