Temperature Dependence of Resistance (last edited 12/20/2006 )

Dr. Larry Bortner

To investigate how the resistances of a conducting sample and a semiconducting sample vary with temperature.

If we apply a voltage V across an object that has a length L and cross-sectional area A, there is a flow of charges from one end to the other
measured by a current I. One of the characteristics of the material that
makes up this object is the *resistivity *ρ. As a measure of an object’s
resistance to a charge flow, we define its *resistance *as

(1)

If ρ does not depend on V, then V and I are directly proportional to each
other and the resistance is said to obey Ohm's law (the verification of which
we leave to another experiment). The question we consider today is how R
changes with temperature. We already know that the length and cross-sectional area changes as the sample expands and contracts. But these
variations are inconsequential compared to how the resistivity changes and
we ignore them in our investigation. This means that the resistance
dependence on temperature is directly proportional to the resistivity
dependence. Values of the resistivity or its inverse, the *conductivity*, have
been tabulated for many materials.

These values can vary wildly, from 10^{16} Ω·m for diamond to 10^{-8} Ω·m for
copper. One way to classify materials is based on their electron transport, or
resistive, properties. Materials with low resistivities (high conductivities) are
*conductors*. Most metals fall under ths category. Those materials that have
high resistivities, where it takes a lot of energy to get an electron from one
end to the other, are *insulators*. And the ones that have intermediate values
are *semiconductors*.

Temperature and Molecular Energy Levels

How do we explain the movement of charge through solid materials? And
how do we account for temperature? Beyond the macroscopic notion of
temperature relating to hot and cold, on the molecular and atomic level,
temperature is a measure of the average kinetic energy per particle. The
most probable kinetic energy of an electron at an absolute temperature T is
2kT, where k=8.61739×10^{-5} ^{ }eV/K
is Boltzmann’s constant. Any model for
resistance we come up with must be based on the principles of quantum
mechanics that reflect the interaction of matter and energy at the atomic
level. A quantum mechanical description (a mathematical model) of solids
focuses on what happens to the discrete electronic energy levels of an
isolated atom as more and more atoms are brought together, bond to each
other, and form a lattice. Calculations show that if N identical atoms are
brought together, each electronic level of the isolated atom gets split into a
group of N distinct levels. The energy spacing between these levels is
extremely small, decreasing as N increases. (See Fig. 1.)

Figure 1 Allowed energy levels in an atom and in a solid composed of N identical atoms.

For a macroscopic sample, N is so large (~10^{23}) that the levels in a
particular group are essentially continuous. The group is then called an
*energy band.* The various bands are separated by *energy gaps*; that is,
differences between the highest allowed energy in one band and the lowest
allowed energy in the next highest band. *An electron cannot have any
energy between these two energies*.

As with atomic levels, the bands in solids may be filled, partially filled, or empty.

• The
lowest energy band that is not completely filled is called the
*conduction band.*

• The highest energy band that has at least one electron (at absolute zero)
is the *valence band*.

Whether the valence band is filled and, if it is, the size of the energy gap between it and the conduction band determine the charge transport properties of the solid.

Band Theory of Electron Transport

The kinetic energy of an electron in the conduction band can easily be raised
by an applied electric field, even a fairly small field. Because there are so
many empty levels nearby, the electrons are mobile within the solid.
Moreover, metals that have the band structure shown in Fig. 2 have *many
*electrons that can participate in the conduction process; the valence band
and the conduction band are the same. The large number of mobile electrons
give rise to very low values of ρ.

Figure 2 Typical band structures of the three resistive types of materials.

In an insulator, the valence band is filled and the conduction band is separated from it by a relatively large energy gap of value ε. When an electric field is applied across the sample, the electrons in the valence band cannot respond to it because no higher energy levels are readily accessible. The energy needed by an electron to reach the nearest unoccupied energy level (in the conduction band) is larger than the energy provided by the field and no electrons are able to participate in the conduction process. Consequently, insulators have very high values of resistivity. (Of course we can always crank up the field so that it provides enough energy to reach the next level, but this is called the dielectric strength and is the point where the insulator breaks down.)

For semiconductors, the valence band is also filled, but the energy gap to the
next higher band is small compared to an insulator, small enough that an
appreciable number of electrons cross the gap because their thermal energy
is high enough. Electrons that are in the conduction band can now participate
in the conduction process because, as in a metal, many unfilled energy levels
are available. This gives semiconductors a reasonable conductivity (but
weaker than a normal conductor). Moreover, those electrons thermally
promoted out of the valence band leave behind vacant levels, referred to as
*holes*. An external electric field enables electrons in the next lower band to
be promoted into these vacant levels and this in turn leaves behind other
holes. The net result is an induced migration of holes through the lattice
which can be treated as a flow of positive charge carriers. Negative
(electron) and positive (hole) currents in semiconductors are fundamentally
important to the design and operation of many modern electronic devices. In
addition, the additional mode of electrical conduction (the movement of the
holes) further enhances the conductivity of semiconductors.

Note that an electron thermally excited into the conduction band does not remain there indefinitely. Such an electron remains in the upper band for an average lifetime before losing some of its thermal energy and “falling” back to the valence band. But by that time a different electron from the valence band will have gained enough energy to move up. Given the enormous number of electrons in any real sample, there is a fairly constant number of electrons in the conduction band at any instant. The number of electrons thermally excited into the conduction band (and an associated number of holes in the valence band) is orders of magnitude less than a same-sized conducting sample, so the electrical resistivity of semiconductors is more than that of metals and less than that of insulators. For both semiconductors and insulators, as the temperature and hence the available thermal energy to valence band electrons increases, the number of electrons in the conduction band increases. Thus their resistivity decreases as the temperature increases.

For conductors, the number of conduction electrons doesn’t really change with temperature. What does happen as the temperature increases is that the ions in the lattice vibrate more and it becomes more likely that a conduction electron collides with an ion. We expect the resistivity of a metal to increase with temperature.

Over a wide range of temperatures (far from absolute zero and the melting point) we find empirically that the resistance of a metal varies linearly with temperature. The functional form of the resistance R at a temperature T in Celsius is

(2)

where R_{ref}
is the resistance at the reference temperature T_{ref}
and
α is the
*temperature coefficient of resistance *(or *resistivity*). Note that α
is constant
and positive for a given reference temperature. It is also the fractional
change in resistance with temperature:

(3)

(Eq. 3 is a definition and is not the way to calculate α.) Values of α at room temperature for various metals are shown in the following table:

metal |
α (°C |

antimony |
0.0036 |

brass |
0.002 |

copper |
0.00393 |

gold |
0.0034 |

iron |
0.005 |

nichrome |
0.0004 |

nickel |
0.006 |

palladium |
0.0033 |

platinum |
0.003 |

silver |
0.0038 |

tantalum |
0.0031 |

tin |
0.0042 |

tungsten |
0.0045 |

zinc |
0.0037 |

For semiconductors with energy gap ε the expression for the resistance is

(4)

where R_{0} is a constant and T is the absolute temperature in Kelvins. Applying
the calculus part of Eq. 3 we find that the temperature coefficient of
resistance is

(5)

This is a negative value, as we reasoned it should be in our earlier discussion.

The energy gaps for a few semiconductors at room temperature are listed here:

semiconductor |
ε (eV) |

Si |
1.11 |

Ge |
0.66 |

GaAs |
1.43 |

PbTe |
0.29 |

You need the following equipment:

• Science Workshop interface

• temperature sensor (u{T}=0.5°C)

• Fluke 45 multimeter (u{R}/R= 0.1%)

• Fluke 8050A multimeter (u{R}/R= 0.2%)

• coil with leads

• thermistor with leads

• Bunsen burner and ring stand

• ice water

• beakers

• banana-banana leads, banana-alligator clip leads

1. Light the Bunsen burner and put one of the beakers with about an inch of water on the ring stand. Fill another beaker with ice.

2. Click on these menu items on the Windows desktop: Start> Science Workshop Experiments> Second Quarter> SSThermometer. A DataStudio window opens up with a digital display of the temperature in degrees Celsius from the sensor connected to the interface box. Click on the Start button in DataStudio to start the display.

3. Connect the coil (your conducting sample, encased in red plastic) to the Fluke 8050A, COMMON and V/kΩ/S. Turn the power on and be sure the kΩ and 200Ω buttons are depressed. The units displayed are Ω.

4. Connect the thermistor (semiconductor, long aluminum tube) to the Fluke 45, COM and VΩ. Color coding is important here: red to red and black to black. Turn on the power and press the Ω button. Hold the tip of the thermistor probe between your fingers. A noticeable change in resistance should occur. If it doesn’t, try switching the leads.

5. Start a data table on your data sheet, three columns with these headings:

T (°C) R_{coil} (Ω) R_{thermistor} (kΩ)

6. Once the water is boiling, turn off the gas. (Do not keep the water boiling vigorously. If you are not quite ready to start taking measurements, turn the flame down.)

7. You want the resistances of the conductor (coil) and the semiconductor (thermistor) at particular temperatures. Put the Science Workshop sensor, the coil, and the thermistor in the hot water. (Do not have the flame on when you do this.) The ends of the sensor and the thermistor have to be in the water and the coil has to be completely submerged.

a. Record the temperature from DataStudio, the resistance from the Fluke 8050A, and the resistance from the Fluke 45. T should be close to 100°C and not less than 95°C.

b. The water will start cooling down immediately after you turn off the heat. If the temperature changes before you can record the second resistance, leave the resistance entry blank for the first temperature. Enter the second temperature on the next line and write down the resistance in the corresponding column.

8. Record T, R_{coil}, and R_{thermistor}
every 5 to 10 degrees as the water cools
down.

a. Add a few pieces of ice and stir the water a little with all three elements in it to speed things up. Use caution: too much ice will cause the water to cool down quicker than you can take the data.

b. You may need to drain some of the water into the empty beaker.

c. You’ll have to add more and more ice when the temperature gets below 20°C.

d. Your last point should be at or near 0°C.

e. The total number of temperature-resistance pairs for each element should be between 11 and 20.

1. Start the Generic Lab Spreadsheet, entering in your names and the name of the experiment. In the blank spreadsheet, enter values of the defined temperature of ice water in kelvins and Boltzmann’s constant in eV/K.

2. For the coil enter in labels for T (°C) and R_{coil} (Ω) then enter in the data
values.

3. For the thermistor, enter in labels of T (°C), T(K), 1/2kT (eV^{-1}), R_{therm}(kΩ),
and R_{therm}(Ω). Enter in the data values of T (°C) and R_{therm}(kΩ).

4. Fill out the table for the semiconductor using appropriate formulas.

5. Plot R vs. T for the coil and include a trendline. Is this a linear relationship? Does the resistance increase or decrease as the temperature increases?

6. Do a linear least squares fit of the coil data to find
αR_{ref} and its
uncertainty. Refer to Eq. 2.

7. From this fit find the resistance at 20°C and its error. This is R_{ref}_{}.

8. Find α and u{α} for your sample. From the table in the Background, what is your best guess as to what metal makes up the coil? Justify your assertion.

9. Plot R in Ω vs. T in K for the thermistor. Include a trendline that best fits the points. Is this a linear relationship? Does the resistance increase or decrease as the temperature increases?

10. Do a semilog plot of R in Ω vs. 1/2kT . Include a trendline that best fits the points. Is this a straight line on the graph?

11. Do an exponential least squares fit of R in Ω vs.1/2kT to find e^{ε}
and u{ε}.

12. Calculate ε. From the table in the Background, what is your best guess as to the semiconductor that makes up the thermistor? Justify your assertion.

1. Given the fact that the resistivity of semiconductors decreases as the temperature increases, is it reasonable to expect there to be some elevated temperature at which the resistance of a semiconducting sample would disappear completely? Why or why not?

2. A fuse is wired in series with a light. If the filament of the bulb is a metal (e.g., tungsten), is the fuse more likely to blow-out immediately after the switch is closed or some later time after the bulb has been on for a while? Explain your answer. What if the filament were made from a semiconducting material (e.g., carbon)?

3. Propagate the uncertainty for R_{ref}_{}.

4. Propagate the uncertainty for α.