(50/100) 2. When a phosphorus atom is substituted for a silicon atom in a crystal, four of the phosphorus valence electrons form bonds with neighboring atoms and the remaining electron is much more loosely bound. You can model the electron as free to move through the crystal lattice. The phosphorus nucleus has one more positive charge than does the silicon nucleus, however, so the extra electron provided by the phosphorus atom is attracted to this single nuclear charge +e. The energy levels of the extra electron are similar to those of the electron in the Bohr hydrogen atom with two important exceptions. First, the Coulomb attraction between the electron and the positive charge on the phosphorus nucleus is reduced by a factor of 1/k from what it would be in free space, where k is the dielectric constant of the crystal. As a result, the orbit radii are greatly increased over those of the hydrogen atom. Second, the influence of the periodic electric potential of the lattice causes the electron to move as if it had an effective mass m*, which is quite different from the mass me of a free electron. You can use the Bohr model of hydrogen to obtain relatively accurate values for the allowed energy levels of the extra electron. We wish to find the typical energy of these donor states, which play an important role in semiconductor devices. Assume k = 11.7 for silicon and m* = 0.220me. a) Find a symbolic expression for the smallest radius of the electron orbit in terms of ao, the Bohr radius. b) Substitute numerical values to find the numerical value of the smallest radius. c) Find a symbolic expression for the energy levels E, of the electron in the Bohr orbits around the donor atom in terms of me, m*, K, and En, the energy of the hydrogen atom in the Bohr model. d) Find the numerical value of the energy for the ground state of the electron.
(50/100) 2. When a phosphorus atom is substituted for a silicon atom in a crystal, four of the phosphorus valence electrons form bonds with neighboring atoms and the remaining electron is much more loosely bound. You can model the electron as free to move through the crystal lattice. The phosphorus nucleus has one more positive charge than does the silicon nucleus, however, so the extra electron provided by the phosphorus atom is attracted to this single nuclear charge +e. The energy levels of the extra electron are similar to those of the electron in the Bohr hydrogen atom with two important exceptions. First, the Coulomb attraction between the electron and the positive charge on the phosphorus nucleus is reduced by a factor of 1/k from what it would be in free space, where k is the dielectric constant of the crystal. As a result, the orbit radii are greatly increased over those of the hydrogen atom. Second, the influence of the periodic electric potential of the lattice causes the electron to move as if it had an effective mass m*, which is quite different from the mass me of a free electron. You can use the Bohr model of hydrogen to obtain relatively accurate values for the allowed energy levels of the extra electron. We wish to find the typical energy of these donor states, which play an important role in semiconductor devices. Assume k = 11.7 for silicon and m* = 0.220me. a) Find a symbolic expression for the smallest radius of the electron orbit in terms of ao, the Bohr radius. b) Substitute numerical values to find the numerical value of the smallest radius. c) Find a symbolic expression for the energy levels E, of the electron in the Bohr orbits around the donor atom in terms of me, m*, K, and En, the energy of the hydrogen atom in the Bohr model. d) Find the numerical value of the energy for the ground state of the electron.
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Transcribed Image Text:(50/100) 2. When a phosphorus atom is substituted for a silicon atom in a crystal, four of the phosphorus
valence electrons form bonds with neighboring atoms and the remaining electron is much more loosely bound.
You can model the electron as free to move through the crystal lattice. The phosphorus nucleus has one more
positive charge than does the silicon nucleus, however, so the extra electron provided by the phosphorus atom
is attracted to this single nuclear charge +e. The energy levels of the extra electron are similar to those of the
electron in the Bohr hydrogen atom with two important exceptions. First, the Coulomb attraction between the
electron and the positive charge on the phosphorus nucleus is reduced by a factor of 1/k from what it would be
in free space, where k is the dielectric constant of the crystal. As a result, the orbit radii are greatly increased
over those of the hydrogen atom. Second, the influence of the periodic electric potential of the lattice causes the
electron to move as if it had an effective mass m*, which is quite different from the mass me of a free electron.
You can use the Bohr model of hydrogen to obtain relatively accurate values for the allowed energy levels of
the extra electron. We wish to find the typical energy of these donor states, which play an important role in
semiconductor devices. Assume k = 11.7 for silicon and m* = 0.220me.
a) Find a symbolic expression for the smallest radius of the electron orbit in terms of ao, the Bohr radius.
b) Substitute numerical values to find the numerical value of the smallest radius.
c) Find a symbolic expression for the energy levels E, of the electron in the Bohr orbits around the donor atom
in terms of me, m*, K, and En, the energy of the hydrogen atom in the Bohr model.
d) Find the numerical value of the energy for the ground state of the electron.
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