International
Tables for
Crystallography
Volume D
Physical Properties of Crystals
Edited by A. Authier

International Tables for Crystallography (2013). Vol. D, ch. 1.8, pp. 223-227

Section 1.8.3. Electrical resistivity

G. D. Mahana*

aDepartment of Physics, 104 Davey Laboratory, Pennsylvania State University, University Park, Pennsylvania, USA
Correspondence e-mail: gmahan@psu.edu

1.8.3. Electrical resistivity

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1.8.3.1. Properties of the electrical resistivity

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The electrical conductivity is usually written as (Ziman, 1962[link]; Goldsmid, 1986[link]; Mahan, 1990[link]) [\sigma = {{n_0 e^2 \tau}\over{m^*}},\eqno(1.8.3.1)]where [n_0] is the density of conduction electrons in units of [m^{-3}], [e] is the charge on the electron, [\tau] is the lifetime of the electron and [m^*] is the effective mass. Here we are assuming parabolic bands, so the energy of the electron is [\varepsilon(k) = \hbar^2k^2/2m^*].

If one measures [\sigma] as a function of temperature, then one has determined [\tau(T)], assuming that one knows [n_0] and [m^*] from other measurements, e.g. Allen et al. (1986[link]). The electron density [n_0] can sometimes be determined from the Hall effect, as discussed in Section 1.8.3.4[link]. The effective mass [m^*] can be determined by a cyclotron resonance experiment or a similar experiment that measures the properties of the Fermi surface. Also, the ratio [n_0/m^*] can be found by measuring the frequency dependence of the dielectric function in the infrared (Sievers, 1980[link]): [\epsilon(\omega) = \epsilon_{\infty} - 4\pi n_0 e^2/(m^*\omega^2)]. Here the factors [n_0e^2/m^*] occur in the same combination as found in the d.c. conductivity. The factors [n_0] and [m^*] can also be determined by numerical calculations of the band structure of the solid. In any case, we assume that these parameters are known. The only difficult parameter to find is the lifetime.

The lifetime of the electrons is usually determined by solving a Boltzmann equation for the distribution function of the electrons. The method of solution is described in many references (Ziman, 1962[link]; Goldsmid, 1986[link]; Mahan, 1990[link]) and will not be repeated here. The Boltzmann equation is itself an approximate equation, since one must do some averaging over the particles in deriving it. This approximate equation can then be solved by a variety of methods: analytical with approximations, variationally or numerically with great accuracy. The latter is done quite easily with today's computers. Here we shall summarize the main contributions to the lifetime.

The electrical resistivity is the inverse of the conductivity, [\rho = {{m^*}\over{n_0e^2}}{{1}\over{\tau}}.\eqno(1.8.3.2)]The scattering rate of the electron can often be calculated using Fermi's golden rule, which is an equation of the form [{{1}\over{\tau_i}} = {{2\pi}\over{\hbar}}\sum_{f}|M_{if}|^2[1-\cos\theta]\delta(E_i-E_f).\eqno(1.8.3.3)]Here the lifetime of an initial state [i] is given by summing over all of the final states [f] that can be reached by a matrix element [M_{if}]. The factor of [[1-\cos\theta]] is included to measure the amount of scattering, where [\theta] is the angle through which the electron scatters. Anything that scatters or interacts with an electron contributes to the lifetime. This includes the thermal vibrations of the ions, which is an intrinsic effect. There are also extrinsic effects such as scattering from impurities, grain boundaries, dislocations and the boundaries of the crystal. The latter is important in thin films or wires.

Matthiessen's rule (Matthiessen & Vogt, 1864[link]) states that the resistivities from each type of scattering process can simply be added. The total resistivity can be written as [\rho = {{m^*}\over{n_0e^2}}\sum_j{{1}\over{\tau_j}},\eqno(1.8.3.4)]where [\tau_j] is the lifetime from one of the scattering mechanisms. There are several important disclaimers regarding this rule. It is far from rigorous. It is often untrue. Yet it works very well 95% of the time. We shall adopt the rule here for our discussion of the resistivity.

The major contributions to the resistivity of solids are:

  • (1) Impurities. Every crystal has impurities, as it is not possible to make a crystal without some defects. The reasons for this are well understood and here we just assume this fact. The formula for the lifetime contains two factors (Mahan, 1990[link]): (a) the concentration of impurities [n_i] and (b) the phase shifts [\delta_{l}(k)] for scattering an electron of wavevector [k] and angular momentum [l], [{{1}\over{\tau_i(k)}} = {{4\pi n_i\hbar}\over{m^* k}}\sum_{l}l \sin^2[\delta_{l}(k)-\delta_{l-1}(k)].\eqno(1.8.3.5)]Although this expression appears complicated, one can view it as consisting of three parts: [\eqalignno{{{1}\over{\tau_i}} &= n_i v_k \sigma_c(k),&(1.8.3.6)\cr v_k &= {{\hbar k}\over{m^*}},&(1.8.3.7)\cr \sigma_c &= {{4\pi }\over{ k^2}}\sum_{l}l \sin^2[\delta_{l}(k)-\delta_{l-1}(k)].&(1.8.3.8)}%fd1.8.3.8]The three factors are the concentration [n_i] of impurities, the electron velocity [v_k] and the cross section [\sigma_c]. For each impurity, the cross section is a function of [k]. The lifetime is the density of impurities multiplied by a simple function of electron energy and is independent of temperature. A careful analysis shows that there is a temperature dependence to the scattering by impurities. However, this dependence is rather slight, and is dwarfed by the large temperature dependence of the electron scattering by phonons (Bass et al., 1990[link]). It is a common approximation to treat [\tau_i] as a constant independent of temperature. It is easy to determine this constant experimentally: the resistivity in the limit of zero temperature contains just the contribution from impurities and defects.

    It is possible to add a known amount of impurities intentionally. Then a measurement of the impurity resistance provides a measurement of the cross section [\sigma_c], since the Fermi velocity [v_F] is usually known.

  • (2) Phonons in metals. Crystals are composed of atoms, which vibrate. As the temperature increases, they vibrate with larger amplitude. These vibrations provide a noise spectrum for the electrons and cause the electrons to scatter. The scattering of electrons by phonons is an intrinsic process. For most solids, this process is the dominant contribution to the electrical resistivity at temperatures above 100 K.

    Ziman (1962[link]) first derived the following expression for the resistivity due to the scattering of electrons by phonons in a metal: [\eqalignno{\rho(T) &= C'\sum_{\lambda}\int q \,\,{\rm d}^3q |M_{\lambda}({\bf q})|^2(\hat{\boldxi}_{\lambda}\cdot{\bf q})^2 \left [-{{\partial n_B(\omega)}\over{\partial \omega}} \right]_{\omega=\omega_{\lambda}({\bf q})} &\cr &&(1.8.3.9)\cr C' &= {{3\hbar \nu_0}\over{M e^2 16 v_F^2k_F^4}}. &(1.8.3.10)}%fd1.8.3.10]The constant [C'] collects numerous constants including the Fermi wavevector [k_F], the Fermi velocity [v_F], the ion mass [M] and the unit-cell volume [\nu_0]. The phonons have wavevector [{\bf q}] and different phonon bands (e.g. TA, LA, TO) are denoted by [\lambda]. The phonon frequencies are [\omega_{\lambda}({\bf q})] and the matrix element for scattering the electron by wavevector [{\bf q}] is [M_{\lambda}({\bf q})].

    Equation (1.8.3.9)[link] is easy to evaluate using a computer code that generates all of the phonons at different points in the Brillouin zone. It is the formula used most often to calculate the temperature dependence of the resistivity of metals. However, the reader is warned that this formula is not exact, as it represents an approximate solution of the Boltzmann equation. In the only case in which the accuracy of equation (1.8.3.9)[link] has been tested against numerically accurate solutions of the Boltzmann equation, Wu & Mahan (1984[link]) found that (1.8.3.9)[link] had an error of a few per cent. However, the formula is useful because it gives an answer that only errs by a few per cent and is relatively easy to calculate.

    Equation (1.8.3.9)[link] has one feature that is simple and important. At high temperature, the resistivity becomes proportional to temperature. The Bose–Einstein occupation number [n_B(\omega)\simeq k_BT/\hbar\omega] and then the derivative with respect to [\omega] is simple. This gives the expression [\eqalignno{\rho(T) &= {{m^*}\over{n_0e^2}}{{1}\over{\tau(T)}} &(1.8.3.11)\cr {{1}\over{\tau}} &= {{2\pi}\over{\hbar}} \lambda_t k_BT &(1.8.3.12)\cr \lambda_t &= {{m}\over{M}}{{\nu_0}\over{16\pi^2k_F^3}}\int q\,\,{\rm d}^3q{{|M_{\lambda}({\bf q})|^2(\hat{\boldxi}_{\lambda}\cdot{\bf q})^2}\over{[\hbar \omega_{\lambda}({\bf q})]^2}}. &\cr &&(1.8.3.13)}%fd1.8.3.13]At high temperature, which in practice is above half of the Debye temperature, the inverse lifetime of the electron is proportional to the temperature. The coefficient is the dimensionless constant [\lambda_t], which is called the `transport form of lambda' (see Grimvall, 1981[link]). This parameter gives the strength of the interaction between the electrons and the phonons. It ranges from very small values [(\lambda_t \sim 0.1)] for the noble metals to values above 4 for heavy metals such as lead and mercury (see Grimvall, 1981[link]).

Now we give some examples of the resistivity of common metals and show that the above formulas give a good account of the resistivity. Fig. 1.8.3.1[link] shows the intrinsic resistivity as a function of temperature for a simple metal (sodium). The data are taken from Bass et al. (1990[link]). The impurity resistivity has been subtracted away. The resistivity is lowest at low temperature, increases at higher temperature and becomes linear at very high temperatures. In actual crystals, the low-temperature value is determined by scattering from impurities and is different for each piece of metal. If one subtracts the constant value and plots [\rho(T)-\rho(0)], then the curve is the same for each crystal of sodium. This is just the phonon contribution to the resistivity.

[Figure 1.8.3.1]

Figure 1.8.3.1 | top | pdf |

The temperature dependence of the intrinsic electrical resistivity of sodium at constant density. The data are taken from Bass et al. (1990[link]).

At very high temperatures, the resistivity is found to deviate from being linear with temperature. This deviation is due to the thermal expansion of the crystal at high temperature. This can be supressed by taking measurements at constant volume, as is the case for the results shown in Fig. 1.8.3.1[link]. If the crystal is put under pressure to maintain constant volume, then the high-temperature resistivity is highly linear with temperature.

Also interesting is the behaviour of the resistivity at very low temperatures, say less than 1 K. For the alkali metals, the temperature dependence was found by Bass et al. (1990[link]) to be [\rho(T) = \rho_i(1 + BT^2) + AT^2. \eqno(1.8.3.14)]The term [\rho_i] is the constant due to the impurity scattering. There is also a term proportional to [BT^2], which is proportional to the impurity resistance. This factor is due to the Koshino–Taylor effect (Koshino, 1960[link]; Taylor, 1964[link]), which has been treated rigorously by Mahan & Wang (1989[link]). It is the inelastic scattering of electrons by impurities. The impurity is part of the lattice and phonons can be excited when the impurity scatters the electrons. The term [AT^2] is due to electron–electron interactions. The Coulomb interaction between electrons is highly screened and makes only a small contribution to [A]. The largest contribution to [A] is caused by phonons. MacDonald et al. (1981[link]) showed that electrons can interact by exchanging phonons. There are also terms due to boundary scattering, which is important in thin films: see Bruls et al. (1985[link]).

Note that (1.8.3.14)[link] has no term from phonons of [O(T^5)]. Such a term is lacking in simple metals, contrary to the assertion in most textbooks. Its absence is due to phonon drag. For a review and explanation of this behaviour, see Wiser (1984[link]). The [T^5] term is found in the noble metals, where phonon drag is less important owing to the complexities of the Fermi surface.

1.8.3.2. Metal alloys

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Alloys are solids composed of a mixture of two or more elements that do not form a stoichiometric compound. An example is Cu[_x]Ni[_{1-x}], in which [x] can have any value. For small values of [x], or of [(1-x)], the atoms of one element just serve as impurities in the other element. This results in the type of behaviour described above. However, in the range [0.2 \,\lt\, x \,\lt\, 0.8], a different type of resistivity is found. This was first summarized by Mooij (1973[link]), who found a remarkable range of behaviours. He measured the resistivity of hundreds of alloys and also surveyed the published literature for additional results. He represented the resistivity at [T=300] K by two values: the resistivity itself, [\rho(T=300)], and its logarithmic derivative, [\alpha = {\rm d}\ln(\rho)/{\rm d}T]. He produced the graph shown in Fig. 1.8.3.2[link], where these two values are plotted against each other. Each point is one sample as represented by these two numbers. He found that all of the results fit within a band of numbers, in which larger values of [\rho(T=300)] are accompanied by negative values of [\alpha]. Alloys with very high values of resistivity generally have a resistivity [\rho(T)] that decreases with increasing temperature. The region where [\alpha =0] corresponds to a resistivity of [\rho^*=150] µΩ cm, which appears to be a fixed point. As the temperature is increased, the resisitivities of alloys with [\rho\,\gt\,\rho^*] decrease to this value, while the resisitivities of alloys with [\rho \,\lt\, \rho^*] increase to this value.

[Figure 1.8.3.2]

Figure 1.8.3.2 | top | pdf |

The temperature coefficient of resistance versus resistivity for alloys according to Mooij (1973[link]). Data are shown for bulk alloys ([+]), thin films ([\bullet]) and amorphous alloys ([\times]).

Mooij's observations are obviously important, but the reason for this behaviour is not certain. Several different explanations have been proposed and all are plausible: see Jonson & Girvin (1979[link]), Allen & Chakraborty (1981[link]) or Tsuei (1986[link]).

Recently, another group of alloys have been found that are called bad metals. The ruthenates (see Allen et al., 1996[link]; Klein et al., 1996[link]) have a resistivity [\rho\,\gt\,\rho^*] that increases at high temperatures. Their values are outliers on Mooij's plot.

1.8.3.3. Semiconductors

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The resistivity of semiconductors varies from sample to sample, even of the same material. The conductivity can be written as [\sigma = n_0 e \mu], where [e] is the charge on the electron, [\mu =e\tau/m^*] is the mobility and [n_0] is the density of conducting particles (electrons or holes). It is the density of particles [n_0] that varies from sample to sample. It depends upon the impurity content of the semiconductor as well as upon temperature. Since no two samples have exactly the same number of impurities, they do not have the same values of [n_0]. In semiconductors and insulators, the conducting particles are extrinsic – they come from defects, impurities or thermal excitation – in contrast to metals, where the density of the conducting electrons is usually an intrinsic property.

In semiconductors, instead of talking about the conductivity, the more fundamental transport quantity (Rode, 1975[link]) is the mobility [\mu]. It is the same for each sample at high temperature if the density of impurities and defects is low. There is an intrinsic mobility, which can be calculated assuming there are no impurities and can be measured in samples with a very low density of impurities. We shall discuss the intrinsic mobility first.

Fig. 1.8.3.3[link] shows the intrinsic mobility of electrons in silicon, from Rode (1972[link]), as a function of temperature. The mobility generally decreases with increasing temperature. This behaviour is found in all common semiconductors. The mobility also decreases with an increasing concentration of impurities: see Jacoboni et al. (1977[link]).

[Figure 1.8.3.3]

Figure 1.8.3.3 | top | pdf |

The intrinsic mobility of electrons in silicon. Solid line: theory; points: experimental. After Rode (1972[link]).

The intrinsic mobility of semiconductors is due to the scattering of electrons and holes by phonons. The phonons come in various branches called TA, LA, TO and LO, where T is transverse, L is longitudinal, A is acoustic and O is optical. At long wavelengths, the acoustic modes are just the sound waves, which can be modelled by a Debye model, [\omega_j({\bf q}) = c_jq], where [c_j] is the speed of sound for the mode [j]. At long wavelengths, the optical modes have a constant frequency which is represented by an Einstein model, [\omega_j({\bf q}) = \omega_{j0}].

The intrinsic mobility is that for a low density of electrons or holes. The existing conducting particles are then confined to the lowest wavevector states near the minimum of the conduction band (electrons) or near the maximum of the valence band (holes). The phonons scatter the particles locally, so that the wavevector changes by small amounts, which can only be done by phonons of long wavelength. The above approximations, of using a Debye model for A modes and an Einstein model for O modes, is accurate. This is because one only needs to consider phonons of long wavelength: the approximations are inaccurate for phonons of short wavelength, but they are irrelevant.

The exception to this general behaviour is where the conduction band has several equivalent minima and the phonons scatter an electron from one minimum to another. This is called intervalley scattering. If the minima of the two electron pockets are separated by a wavevector [{\bf q}_m], then one needs phonons of energy [\omega_j({\bf q}_m)]. Again, these are a fixed set of constants, so one can assume an Einstein model with these phonons as the frequency. For the calculation of the intrinsic mobility of a semiconductor, one does not need to know all of the phonon modes in the solid, as one does for a metal. Instead, one needs to know only the phonons at selected points in the Brillouin zone.

The inverse lifetimes for each scattering process are simply added: [{{1}\over{\tau(k)}} = {{1}\over{\tau_A(k)}} + {{1}\over{\tau_O(k)}} + {{1}\over{\tau_I(k)}}.\eqno(1.8.3.15)]The three terms are acoustic, optical and intervalley. First, we discuss the scattering by optical phonons. The inverse lifetime is proportional to the density of optical phonons [N_0]: [\eqalignno{{{1}\over{\tau_0}} &= {{N_0}\over{\tau'}},&(1.8.3.16)\cr N_0 &= {{1}\over{\exp({\hbar\omega_0/k_BT})-1}}.&(1.8.3.17)}%fd1.8.3.17]The lifetime [\tau_0 \propto N_0^{-1} = \exp({\hbar\omega_0/k_BT})-1]. This shows that the mobility increases exponentially at lower temperatures according to the factor [\exp({\hbar\omega_0/k_BT})-1]. This feature is common to nearly all semiconductors.

The factor [N_0] occurs because the electrons must absorb an optical phonon in order to scatter. The density of optical phonons in the crystal is proportional to the factor [N_0]. Since usually [k_BT \,\lt\, \hbar\omega_0] at room temperature, the thermally excited electrons have less energy than an optical phonon. In this case, the electrons cannot emit a phonon, since they are unable to lose that much energy: the process has no final state.

Notice that we have not yet discussed the mechanism by which the electron couples to the optical phonons. In general, there are two: the polar interaction and the deformation potential interaction. Polar interactions are found in crystals with different atoms and where there is some ionic bonding. When the charged ions vibrate, it results in oscillating dipoles that create long-range electric fields. Polar interactions are important in III–V and II–VI semiconductors such as GaAs or CdS. Polar interactions are not present in elemental semiconductors such as silicon and germanium, since each ion is neutral. However, the deformation potential interaction is present in these and could scatter strongly.

Next we discuss the intervalley scattering, where an electron moves between equivalent conduction-band minima. Here the phonons have a discrete energy [\hbar\omega_j({\bf q}_m)]. At low temperatures, the electron can only absorb this phonon and the process is proportional to [N(\omega_j({\bf q}_m))]. This behaves, in many ways, like the scattering by optical phonons. However, since [\omega_j({\bf q}_m) \,\lt\, \omega_0], the temperature at which phonon emission can occur is lower. At low temperatures, the intervalley scattering also contributes exponential factors to the inverse lifetime. These contributions are usually lower than the optical phonon scattering. However, in silicon, Rode (1972[link]) showed that the intervalley scattering dominates over the optical phonon scattering.

The scattering by acoustic phonons only is important at low temperature. For most semiconductors, the interaction between electrons and acoustic phonons is due to the deformation potential interaction. The standard calculation gives the inverse lifetime as proportional to [T^{3/2}], which becomes smaller at low temperature. However, since the other phonon contributions become smaller with an exponential dependence upon temperature, at small enough temperatures the acoustic phonon term makes the largest contribution to the inverse lifetime. Therefore, at low temperatures, the scattering by acoustic phonons limits the mobility of the electron. Of course, this presumes that there is no contribution from the scattering by impurities. Since this contribution is a constant at low temperature, it is always the dominant contribution at low temperatures. Only in samples with a small concentration of impurities can one actually observe the limitation by acoustic phonons. For most samples, with a moderate density of impurities, the optical-phonon part forms the limit at intermediate temperatures, the impurity scattering forms the limit at low temperatures and one never observes the limit from acoustic phonons.

The discussion above covers the behaviour in the majority of cases. There are special types of crystal that show special behaviour. One of these is crystals that are strongly piezoelectric. The size of the electron–phonon interaction due to piezoelectricity is governed by the electromechanical coupling constant. In crystals where this number is relatively large, the scattering of electrons by acoustic phonons gives [\tau \propto T^{1/2}] at low temperature, as shown by Mahan (1990[link]). So far, the only class of crystals where this is found is the II–VI semiconductors with the wurtzite structure: ZnO and CdS. These are the most piezoelectric crystals found so far. They both show a dependence of the mobility upon [T^{1/2}] at low temperature.

Finally, we should mention that semiconductors have an intrinsic conductivity that provides an absolute minimum to the conductivity of any sample. The value of this conductivity depends upon temperature. An electron is thermally excited above the energy gap, creating an electron and a hole. The density of electrons or holes is usually determined by the density of the various impurities or native defects, such as vacancies or interstitials. However, in a perfect crystal without defects, there will still be electrons and holes. The density of electrons [(n)] and holes [(p)] obeys the relationship [np = 4N_cN_v\left({{2\pi \hbar^2}\over{m_ck_BT}}\right)^{3/2} \left({{2\pi\hbar^2}\over{m_vk_BT}}\right)^{3/2}\exp({-E_G/k_BT}),\eqno(1.8.3.18)]where [E_G] is the energy gap between the electron and hole bands, and [N_j] and [m_j] are the number of equivalent bands and their effective masses. This formula comes from chemical equilibrium: the recombination of electrons and holes is controlled by phase space and the energy gap. The absolute minimum number of electrons and holes is where [n=p], so that each is equal to the square root of the right-hand side of (1.8.3.18)[link]. If this minimum value is called [n_m=p_m], then the minimum conductivity is [\sigma_m = n_m(\mu_c+\mu_v)], where [\mu_c] and [\mu_v] are the mobilities of the electrons and the holes, respectively. The conductivity is never lower than this value.

1.8.3.4. The Hall effect

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Measurement of the Hall effect is simple and often useful. One takes a crystal and applies a magnetic field [B_z] along the [z] axis. Then one imposes a current density [j_x] along the [x] axis. One finds that the Lorentz force induces a voltage, or the equivalent electric field [E_y], in the [y] direction. The electric field is proportional to both the current and magnetic field. The ratio [E_y/(j_xB_z)] is the Hall constant [R_H]. The inverse of [R_H] is just the charge [e] and the speed of light [c] multiplied by the density of electrons [n_0]: [{{E_y}\over{j_xB_z}}= R_H = {{1}\over{n_0ec}}.\eqno(1.8.3.19)]This provides a simple and accurate method of measuring the density of electrons. It works well when there is only one kind of current carrier and works well in semiconductors with a low density of carriers. A typical experiment for a semiconductor is to measure the conductivity [\sigma] and the Hall constant [R_H]; the mobility is then [\mu = cR_H\sigma]. If the conducting particles are holes in a semiconductor, the Hall constant has the opposite sign, which indicates positive charge carriers.

Measurement of the Hall effect does not work well if the semiconductor contains a mixture of different carriers, such as electrons and holes, or even electrons from different kinds of conduction bands. In these cases, the constant [R_H] is not easily interpreted. Similarly, measuring the Hall effect is rarely useful in metals. It only works well in the alkali metals, which have all of the electrons in the first Brillouin zone on a spherical Fermi surface. In most metals, the Fermi surface extends over several Brillouin zones and has numerous pockets or regions of different curvatures. Regions of positive curvature act as electrons and give a negative Hall constant; regions of negative curvature act as holes and give a positive contribution to the Hall constant. Again, it is difficult to interpret the Hall constant when both contributions are present. In general, the Hall effect is most useful in semiconductors.

1.8.3.5. Insulators

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Insulators are crystals that do not conduct electricity by the flow of electrons or holes. We shall not mention this case. The band gaps [E_G] are sufficiently large that the intrinsic mobility is very small.

1.8.3.6. Ionic conductors

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There are many ionic solids that have an appreciable electrical conductivity from the diffusive motion of ions. Any material in which the conductivity from the motion of ions is very much larger than that from the motion of electrons is useful as a battery material. For this reason, such materials have been investigated extensively, see e.g. Mahan & Roth (1976[link]) or Salamon (1979[link]).

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