Estimate the fraction of free electrons in copper that are

In the classical model of conduction, the electron loses energy on average during a collision because it loses the drift velocity it had acquired since the last collision. Where does this energy appear?

A metal is a good conductor because the valence energy band for electrons is (a) empty, (b) partly filled, (c) filled, but there is only a small gap to a higher empty band, (d) completely filled, (e) none of the above.

Thomas refuses to believe that a potential difference can be created simply by bringing two different metals into contact with each other. John talks him into making a small wager and is about to win the bet. (a) Which two metals from Table 38-2 would demonstrate his point most effectively? (b) What is the value of that contact potential?

(a) In Problem 3, which choices of different metals would make the least impressive demonstration? (b) What is the value of that contact potential?

When a sample of pure copper is cooled from to its resistivity decreases more than the resistivity of a sample of brass when it is cooled through the same temperature difference. Why?

Insulators are poor conductors of electricity because (a) there is a small energy gap between the filled valence band and the next higher band where electrons can exist, (b) there is a large energy gap between the completely filled valence band and the next higher band where electrons can exist, (c) the valence band has a few vacancies for electrons, (d) the valence band is only partly filled, (e) none of the above.

How does the sign of the change in the resistivity of a sample of copper compare with the sign of the change in the resistivity of a sample of silicon when the temperatures of both samples increase?

True or false: (a) Solids that are good electrical conductors are usually good heat conductors. (b) At T _ 0, an intrinsic semiconductor is an insulator. (c) The Fermi energy is the average energy of an electron in a solid. (d) At the value of the Fermi factor can be either 1 or 0. (e) Semiconductors conduct current in one direction only. (f) The classical free-electron theory adequately explains the heat capacity of metals. (g) The contact potential between two metals is proportional to the difference in the work functions of the two metals.

Which of the following elements are most likely to act as acceptor impurities in germanium? (a) bromine, (b) gallium, (c) silicon, (d) phosphorus, (e) magnesium

Which of the following elements are most likely to serve as donor impurities in germanium? (a) bromine, (b) gallium, (c) silicon, (d) phosphorus, (e) magnesium

An electron hole is created when a photon is absorbed by a semiconductor. How does this hole enable the semiconductor to conduct electricity?

Examine the positions of phosphorus, boron, thallium, and antimony in Table 36-1. (a) Which of these elements can be used to dope silicon to create an n-type semiconductor? (b) Which of these elements can be used to dope silicon to create a p-type semiconductor?

When photons of visible light strike the semiconductor in a junction solar cell, (a) only free electrons are created, (b) only positive holes are created, (c) both electrons and holes are created, (d) protons are created, (e) none of the above.

The ratio of the resistivity of the most resistive (least conductive) material to that of the least resistive material (excluding superconductors) is approximately You can develop a feeling for how remarkable this range is by considering what the ratio is of the largest values to smallest values of other material properties. Choose any three properties of materials, and using tables in this book or some other resource, calculate the ratio of the largest instance of the property to the smallest instance of that property (other than zero) and rank these in decreasing order. Can you find any other property that shows a range as large as that of electrical resistivity?

A device is said to be ohmic if a graph of current versus applied voltage is a straight line through the origin. The resistance of the device is the reciprocal of the slope of this line. A junction is an example of a nonohmic device, as may be seen from Figure 38-21. For nonohmic devices, it is sometimes convenient to define the differential resistance as the reciprocal of the slope of the versus curve. Using the curve in Figure 38-21, estimate the differential resistance of the junction at applied voltages of and

Calculate the center-to-center separation distance between the and the in Do this by assuming that each ion occupies a cubic volume of side The molar mass of is and its density is

The center-to-center separation distance between the \(\mathrm{Li}^{+}\) and \(\mathrm{Cl}^{-}\)ions in \(\mathrm{LiCl}\) is \(0.257 \mathrm{~nm}\). Use that value and the molar mass of \(\mathrm{LiCl}(42.4 \mathrm{~g} / \mathrm{mol})\) to compute the density of \(\mathrm{LiCl}\).

Find the value of n in Equation 38-6 that gives the measured dissociation energy of \(741 \mathrm{~kJ} / \mathrm{mol}\) for \(\mathrm{LiCl}\), which has the same structure as \(\mathrm{NaCl}\) and for which \(r_0=0.257 \mathrm{~nm}\).

(a) Use Equation 38-6 and calculate for calcium oxide, where Assume (b) If increases from 8 to 10, what is the fractional change in U(r0)?

A measure of the density of the free electrons in a metal is the distance which is defined as the radius of the sphere whose volume equals the volume per conduction electron. (a) Show that where is the free-electron number density. (b) Calculate for copper in nanometers.

(a) Given a mean free path and a mean speed for the charge flow in copper at a temperature of calculate the classical value for the resistivity of copper. (b) The classical model suggests that the mean free path is temperature independent and that depends on temperature. According to this model, what would be at

Silicon has a molar mass of and a density of Each atom of silicon has four valence electrons and the Fermi energy of the material is (a) Given that the electron mean free path at room temperature is estimate the resistivity. (b) The accepted value for the resistivity of silicon is (at room temperature). How does this accepted value compare to the value calculated in Part (a) ?

Calculate the number density of free electrons in (a) and (b) assuming one free electron per atom, and compare your results with the values listed in Table 38-1.

The density of aluminum is How many free electrons are present per aluminum atom?

The density of tin is How many free electrons are present per tin atom?

Calculate the Fermi temperature for (a) (b) and (c)

What is the speed of a conduction electron whose energy is equal to the Fermi energy for (a) (b) and (c)

Calculate the Fermi energy for (a) (b) and (c) using the number densities given in Table 38-1.

Find the average energy of the conduction electrons at in (a) copper and (b) lithium.

Calculate (a) the Fermi temperature and (b) the Fermi energy at for iron.

(a) Assuming that each gold atom in a sample of gold metal contributes one free electron, calculate the free-electron density in gold knowing that its atomic mass is and its density is (b) If the Fermi speed for gold is what is the Fermi energy in electron volts? (c) By what factor is the Fermi energy higher than the energy at room temperature? (d) Explain the difference between the Fermi energy and the energy.

The bulk modulus B of a material can be defined by (a) Use the monatomic ideal-gas relation where is the average kinetic energy, Equation 38- 22 and Equation 38-23 to show that where is a constant independent of (b) Show that the bulk modulus of the free electrons in a solid metal is therefore (c) Compute the bulk modulus in newtons per square meter for the free electrons in a sample of copper and compare your result with the measured value of

The pressure of a monatomic ideal gas is related to the average kinetic energy of the gas particles by where is the number of particles and E is the average kinetic energy. Use this information to calculate the pressure of the free electrons in a sample of copper in newtons per square meter, and compare your result with atmospheric pressure, which is about (Note: The units are most easily handled by using the conversion factors and )

Calculate the contact potential between (a) and (b) and and (c) and

Gold has a Fermi energy of 5.53 eV. Determine the molar specific heat at constant volume for gold at room temperature.

The resistivities and Fermi speeds of and at are and and and respectively. Use those values to find the mean free paths for the conduction electrons in these elements.

The resistivity of pure copper increases by approximately with the addition of 1.0 percent (by number of atoms) of an impurity distributed throughout the metal. The mean free path depends on both the impurity and the oscillations of the lattice ions according to the equation where is the mean free path associated with the thermal vibrations of the ions and is the mean free path associated with the impurities. (a) Estimate using Equation 38-14 and the data given in Table 38-1. (b) If is the effective radius of an impurity lattice ion seen by an electron, the scattering cross section is Estimate this area, using the fact that is related to by Equation 38-16.

Electromagnetic radiation is incident on the surface of a semiconductor. The maximum wavelength of this light that is required if electrons are to cross the energy gap between the valence band and the conduction band is 380.0 nm. What is the energy gap, in electron volts, for the semiconductor?

An electron occupies the highest energy level of the valence band in a silicon sample. What is the maximum photon wavelength that will excite the electron across the energy gap if the gap is

An electron occupies the highest energy level of the valence band in a germanium sample. What is the maximum photon wavelength that will excite the electron into the conduction band? In germanium, the energy gap between the valence and conduction bands is

An electron occupies the highest energy level of the valence band in a diamond sample. What is the maximum photon wavelength that will excite the electron into the conduction band? In diamond, the energy gap between the valence and conduction bands is

Aphoton of wavelength has just enough energy to raise an electron from the valence band to the conduction band in a lead sulfide sample. (a) Find the energy gap between the bands in lead sulfide. (b) Find the temperature for which equals that energy gap.

The donor energy levels in an semiconductor are below the conduction band. Find the temperature for which

When a thin slab of semiconducting material is illuminated with monochromatic electromagnetic radiation, most of the radiation is transmitted through the slab if the wavelength is greater than For wavelengths less than most of the incident radiation is absorbed. Determine the energy gap of the semiconductor.

The relative binding of the extra electron in the arsenic atom that replaces an atom in silicon or germanium can be understood from a calculation of the first Bohr radius of the electron in these materials. Four of arsenics valence electrons form covalent bonds, so the fifth electron sees a center of attraction with a charge of This model is a modified hydrogen atom. In the Bohr model of the hydrogen atom, the electron moves in free space at a radius given by (Equation 36-12). When an electron moves in a crystal, we can approximate the effect of the other atoms by replacing with and with an effective mass for the electron. For silicon, is 12 and the effective mass is approximately For germanium, is 16 and the effective mass is approximately Estimate the Bohr radii for the valence electron as it orbits the impurity arsenic atom in silicon and in germanium.

The ground-state energy of the hydrogen atom is given by (Equations 36-15 and 36-16 where is substituted for Modify this equation using information in Problem 45 by replacing with and with an effective mass for the electron to estimate the binding energy of the extra electron of an impurity arsenic atom in (a) silicon and (b) germanium.

A doped silicon sample has electrons per cubic centimeter in the conduction band and has a resistivity of at Find the mean free path of the electrons. Use the effective mass of for the mass of the electrons. (See Problem 45.) Compare this mean free path with that of conduction electrons in copper at

In the Hall effect, the Hall coefficient is the proportionality constant between the transverse electric field and the product of the applied magnetic field and the current density. That is, where the current density, the transverse electric field, and the applied magnetic field are in the , and directions, respectively. (The Hall effect is presented in Chapter 26.) The measured Hall coefficient of a doped silicon sample is at room temperature. If all the doping impurities have contributed to the total number of charge carriers of the sample, find (a) the type of impurity (donor or acceptor) used to dope the sample and (b) the concentration of the impurities.

Simple theory for the current versus the bias voltage across a junction yields the equation Sketch versus for both positive and negative values of using that equation.

The base current in an transistor circuit is If 88.0 percent of the electrons entering the base from the emitter reach the collector, what is the base current?

In Figure 38-28 for the amplifier, suppose and Suppose further that a ac base current generates a ac collector current . What is the voltage gain of the amplifier? SSM

Germanium can be used to measure the energy of incident photons. Consider a gamma ray emitted from (a) Given that the band gap in germanium is how many electron-hole pairs can be generated as this gamma ray travels through germanium? (b) The number of pairs in Part (a) will have statistical fluctuations given by What then is the energy resolution of the detector in that photon energy region?

Make a sketch showing the valence and conduction band edges and Fermi energy of a diode when biased (a) in the forward direction and (b) in the reverse direction.

A good silicon diode has the current-voltage characteristic given by Let (room temperature) and the saturation current (a) Show that for small reverse-bias voltages, the resistance is Hint: Do a Taylorseries expansion of the exponential function about or use the expansion for found in Table M-4 of the Math Tutorial. (b) Find the dc resistance for a reverse bias of (c) Find the resistance for a forward bias. What is the current in this case? (d) Calculate the differential resistance for a forward bias.

Along slab of silicon of thickness and width is placed in a magnetic field The slab is in the plane, where the length of the slab is parallel with the axis and the magnetic field points in the direction. When a current of 0.20Aexists in the sample in the direction, a potential difference of develops across the width of the sample and the electric field in the sample points in the direction. Determine the semiconductor type and the concentration of charge carriers. (The Hall effect is presented in Chapter 26.)

(a) Use Equation 38-37 to calculate the superconducting energy gap for tin, and compare your result with the measured value of (b) Use the measured value to calculate the minimum value of the wavelength of a photon that has sufficient energy to break up Cooper pairs in lead at

(a) Use Equation 38-37 to calculate the superconducting energy gap for lead, and compare your result with the measured value of (b) Use the measured value to calculate the minimum value of the wavelength of a photon that has sufficient energy to break up Cooper pairs in tin at

The number of electrons in the conduction band of an insulator or intrinsic semiconductor is governed chiefly by the Fermi factor. Because the valence band in these materials is nearly filled and the conduction band is nearly empty, the Fermi energy is generally midway between the top of the valence band and the bottom of the conduction band; that is, it is at where is the band gap between the two bands and the energy is measured from the top of the valence band. (a) In silicon, Show that in this case the Fermi factor for electrons at the bottom of the conduction band is given by exp and evaluate this factor. Discuss the significance of the result if there are valence electrons per cubic centimeter and the probability of finding an electron in the conduction band is given by the Fermi factor. (b) Repeat the calculation in Part (a) for an insulator with a band gap of

Approximately how many energy states that have energies between and are available to electrons in a cube of silver measuring on a side?

Show that at the expression for the Fermi factor (Equation 38-24) is equal to 0.5.

(a) Using the equation (Equation 38-22a), calculate the Fermi energy for silver. (b) Determine the average kinetic energy of a free electron and (c) find the Fermi speed for silver.

What is the difference between the energies at which the Fermi factor is 0.9 and 0.1 at in (a) copper, (b) potassium, and (c) aluminum.

What is the probability that a conduction electron in silver will have a kinetic energy of at

Show that (Equation 38-43) follows from Equation 38-41 for and from Equation 38-22a for

Carry out the integration to show that the average energy at is

The density of the electron states in a metal can be written where is a constant and is measured from the bottom of the conduction band. (a) Show that the total number of states is (b) Approximately what fraction of the conduction electrons are within of the Fermi energy? (c) Evaluate that fraction for copper at

What is the probability that a conduction electron in silver will have a kinetic energy of at

Using the expression (Equation 38-41) for the density of states, estimate the fraction of the conduction electrons in copper that can absorb energy from collisions with the vibrating lattice ions at (a) and (b)

In an intrinsic semiconductor, the Fermi energy is about midway between the top of the valence band and the bottom of the conduction band. In germanium, the forbidden energy band has a width of Show that at room temperature the distribution function of electrons in the conduction band is given by the MaxwellBoltzmann distribution function.

The root-mean-square (rms) value of a variable is obtained by calculating the average value of the square of that variable and then taking the square root of the result. Use that procedure to determine the rms energy of a Fermi distribution. Express your result in terms of and compare it to the average energy. Why do and differ?

The density of potassium is How many free electrons are there per potassium atom in a crystal of potassium?

Calculate the number density of free electrons for (a) magnesium, which has a density of and (b) zinc, which has a density of For the calculations assume there are two free electrons per atom, and compare your results with the values listed in Table 38-1.

Estimate the fraction of free electrons in copper that are in energy states above the Fermi energy at (a) (about room temperature) and (b)

A certain free-electron energy state of manganese has a 10.0 percent chance of being occupied when the temperature of the manganese is What is the energy of the state?

The semiconducting compound CdSe is widely used for light-emitting diodes (LEDs). The energy gap in CdSe is What is the frequency of the light emitted by a CdSe LED?

A wafer of pure silicon is irradiated with electromagnetic radiation having a wavelength of The intensity of the radiation is and every photon that strikes the sample is absorbed and creates an electron-hole pair. (a) How many electron-hole pairs are produced in one second? (b) If the number of electron-hole pairs in the sample is in the steady state, at what rate do the electron-hole pairs recombine? (c) If every recombination event results in the radiation of one photon, at what rate is energy radiated by the sample?