In the Laboratory
The Dependence of Resistance on Temperature for Metals, Semiconductors, and Superconductors W R. A. Butera and D. H. Waldeck Department of Chemistry, University of Pittsburgh, Pittsburgh, PA 15260 The electrical resistance of a material and its temperature dependence is a traditional way of classifying solids (1, 2). A material with a very low resistivity (typically < 10 {4 Ω-cm) that increases as the temperature increases is classified as a metal. A material with an intermediate-range resistivity (typically 10{3 to 1011 Ω-cm) that decreases with increasing temperature is classified as a semiconductor. A material with a very high resistivity (typically > 10 12 Ω-cm) that is rather independent of temperature is classified as an insulator. This classification scheme is largely consistent with the more modern approach, which uses band theory. A final type of material is a superconductor. A superconductor is a material whose resistance is similar to that of a metal at high temperatures, but whose resistance drops to zero below a particular temperature, which is associated with a phase transition. In this laboratory exercise, students measure the temperature dependence of a metal, a semiconductor, and a superconductor. The experiment has three major goals. The first goal and principal aim of the exercise is to observe the temperature dependence of the resistance for samples that exhibit three different types of electrical conductivity: metallic, semiconducting, and superconducting. The second goal is to compare the experimental results with theoretical models that connect the experimental observable, which is a macroscopic property, to the microscopic properties of the system. For the metal (Cu) they determine the characteristic time over which electrons collide in the material. For the semiconductor (InSb) they determine the band gap energy and the energy of the donor levels relative to the band edge (see below). Lastly, this experiment demonstrates the use of the Joule–Thomson effect to produce a stable low-temperature environment (i.e., the physical basis of the refrigeration unit used in the experiment). It is most instructive to utilize this experiment in a laboratory course that also performs a more traditional experiment involving the Joule–Thomson effect (3). Background
Metals When the atoms of a metal are brought together in a periodic lattice to form the solid, the energy levels associated with each atom are changed because of the electronic interactions between the atoms. With the addition of more atoms, this process proceeds to the formation of a band of energy levels, instead of a discrete energy for the atomic levels (1, 2, 4–9). In this process, the valence electron’s energy levels are perturbed to a larger degree than the energy levels of the core electrons. Furthermore, the valence band is only partially filled and therefore conducts electrons. Even near 0 K the material is conductive. When the temperature is increased, the electrons are excited within the valence band but the number of electrons within the band is largely unchanged. (Some change occurs in the W Supplementary materials for this article are available on JCE Online at http://jchemed.chem.wisc.edu/.
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number of carriers that may be excited above the Fermi level, however [1, 2]). Hence one expects the influence of temperature on the conductivity to be small. Nevertheless, the temperature does modify the conductivity. The increase in temperature causes the atoms to vibrate about their equilibrium position; this breaks the symmetry of the lattice and resistive loss arises. In a perfect periodic lattice (one in which the atoms are rigidly fixed and impurities or voids are absent), an electron wave traveling in a given direction would travel indefinitely without being deflected. An electron wave encounters resistance only when it is deflected or scattered by (i) impurities or imperfections; (ii) thermal vibrations of the atoms in the lattice, or (iii) other contributions that disturb the periodic potential energy, which is established by the periodic atomic positions. The first mechanism, impurities or voids that may be lodged interstitially in the lattice, gives rise to the resistivity term ρ0, the residual resistance. This residual resistance is not temperature dependent, but contributes to the measured resistance. In contrast, the second mechanism is temperature dependent. A phonon (i.e., lattice vibration) scattering term ρT is introduced to account for this second contribution to the resistivity. The third category contains any other contributions to the resistivity that may occur (such as “magnetic”, “spin-flop”, or “electron– electron” interactions). These latter resistive mechanisms will not be considered here. The experimentally determined resistivity of a metal like Cu may be treated as a combination of the first two terms discussed above. If these terms are independent, one may write ρ = ρ0 + ρT
(1)
This equation is usually called the Matthiessen equation (5, 6). The phonon scattering process is temperature dependent and goes to zero at 0 K. At high temperatures, the energy change, hν, for an electron–phonon collision is much less than kT. In this limit, the collision can be approximated as elastic and a mean collision time, τ, can be introduced such that
ne τ 1 σ ρT = T = m 2
(2)
where ρT is the temperature-dependent resistivity, σ T is the temperature-dependent conductivity, n is the number of conduction electrons per unit volume, e is the electronic charge, and m is the electron’s effective mass (2). Hence, one may write
R T = ρT
A
=
m 2 Ane τ
(3)
for the resistance, where RT is the temperature-dependent resistance, , is the length of the sample, and A is the crosssectional area of the sample. It has been shown by Meaden (5) that τ is inversely proportional to the probability, P, of an electron collision and that this collision probability is
Journal of Chemical Education • Vol. 74 No. 9 September 1997
In the Laboratory proportional to the mean square amplitude of a lattice atom’s displacement, kX2 l. Thus,
1 ∝ P ∝ X2 τ
(4)
To develop a model for the temperature dependence of R, one must develop a model expression for kX2l. Consider an atom displaced a distance X from its average position. In the harmonic approximation, the force on this atom is {kf X. Classical equations provide the frequency (ν) for this vibration as
ν= 1 2π
kf M
1/2
A
(5)
where M is the mass of the particle and kf is the force constant. The equipartition of energy gives the mean potential energy as
1 k X 2 = 1 kT 2 2 f
2
=
kT 2 4π ν2M
(7)
Thus, ρT ∝ kX 2l ∝ T and the resistivity is proportional to T, so that ρ = ρ0 + aT
(8)
where a is a constant. Combining this result with eq 3 gives
RT =
m = aT A Ane 2τ
(9)
which may be solved for τ to give
τ=
m ne 2aT
Figure 1: Schematic energy diagrams for the electronic bands of a semiconductor. Part A corresponds to an intrinsic semiconductor with an energy gap of Eg, a valence band edge of Ev , and a conduction band edge of Ec. Part B corresponds to an extrinsic (i.e., doped) semiconductor having a donor level at Ed and an acceptor level at Ea .
(6)
where k is the Boltzmann constant. Combining eqs 5 and 6 yields
X
B
(10)
where m is the electron mass, n is the number of conduction electrons per unit volume (cm{3), e is the electronic charge, a is the temperature coefficient of resistivity (ohmcm/K), T is the absolute temperature, and τ is the time between electron–phonon collisions. From a determination of the temperature dependence of the resistance of a metal, the temperature dependence of the time between electron-phonon collisions, τ, can be calculated using this model.
Semiconductors The bonding in semiconductors ranges from covalent to ionic and is quite different from metallic bonding. When the atoms are brought together the energy levels interact and split, but the energy level splitting produces two bands that do not overlap. The conduction band lies higher in energy and the valence band lies lower in energy. For an intrinsic semiconductor, the valence band is full and the conduction band is empty at 0 K. The energy separating the valence and conduction bands is the intrinsic energy gap, Eg = (E c – Ev) (see Fig. 1). When the semiconductor contains dopant (or impurity) atoms, energy states can exist between the conduction band edge, Ec, and the valence band edge, Ev . It is common to refer to these states as lying within the gap. If an atom having one more valence electron than the semiconductor atom (one column to the right in the periodic table) were placed in the lattice, the extra electron
would be localized on the impurity atom and have an energy, E d, just below that of the conduction band (e.g., a phosphorus atom in silicon). This type of impurity is called a donor. In contrast, an impurity atom with one less valence electron than the semiconductor atom will act as a bound hole (i.e., a vacant state that could be filled with an electron) with an energy level, Ea , just above the valence band (e.g., aluminum atom in silicon). This type of impurity is called an acceptor. These dopant atoms dramatically modify the semiconductor’s conductivity. For the specific case of InSb, the crystal contains donor impurities and so a level located at E d is appropriate. The energy necessary to excite the donor electron into the conduction band is ∆E = (Ec – Ed). In the low temperature limit, when kT
