Rocks, especially igneous rocks, behave as semiconductors under conditions of temperature and pressure that occur in the Earth's crust . Semiconductor behavior has also been demonstrated in laboratory settings using rocks at room temperatures .

Although the magnetic fields produced by small semiconductors are often negligible, semiconductors on geophysical scales may produce significant magnetic fields. This is of particular interest since these fields can pass through the crust, potentially conveying information about the state of rocks deep below.

Ultra-low

In this context, “ultra-low” refers to electromagnetic waves having frequencies from millihertz to a few Hertz, in contrast to the International Telecommunications Union (ITU) definition of ultra-low, which would correspond to waves having frequencies of 300 Hz–3 kHz.

The conductivity of crustal rocks in fault zones has been measured by magnetotellurics as having conductivity values too low to be a conductor but too high to be an insulator. Such a material is called a semiconductor . Rocks having conductivities in the range 0.1–1 Sm-1 have been observed in magnetotelluric surveys, e.g. from deep crustal rocks along the San Andreas fault . Although there is no generally accepted reason for such high levels of conductivity, several hypotheses have been proposed, such as fluid saturation , partial melting , intergranular carbon films , and the presence of mobile charge carriers referred to as positive holes .

We will show that unipolar pulses can emerge simply from the electrical drift and random diffusion of charge carriers in a semiconducting volume of rock. There are several reasons why this is a plausible mechanism for the observed pulses. Large electrical currents are known to accompany earthquakes, occasionally so large that luminous effects known as earthquake lights become apparent. There is experimental evidence indicating that, during stressing, electrons and holes are freed in igneous rocks and become available to populate states in the conduction and valence bands, respectively.

One proposed source of charge carriers in rock is the break-up of peroxy defects as a result of the increase in tectonic stresses. The oxygen sublattice of a wide variety of silicate minerals can form peroxy defects that act as sources of electron/hole pairs , causing these minerals to exhibit semiconductivity. Once activated, highly mobile electronic charge carriers diffuse through the minerals.

Peroxy defects are point defects, typically introduced through the incorporation of H2O into nominally anhydrous minerals that crystallize in H2O-laden magmas or recrystallize in high-temperature H2O-laden environments . The incorporation of H2O into oxides and silicates leads to OH- pairs that subsequently undergo redox conversion. The two H+ of the OH- pairs combine to form H2, and the O- ions bind to form a peroxy bond. The formation of these peroxy bonds has been extensively studied in laboratory experiments and treated by computational chemistry .

When peroxy bonds are energized via stresses in the rock or by heat, they may produce electron-hole pairs. The peroxy bond breaks, forming a transient state with two unpaired electrons. This is followed by a fully dissociated state in which a hole is free to move through the crystal structure. A neighboring oxygen atom donates an electron and becomes a hole, as its valence shell becomes deficient by one electron. The donated electron becomes trapped near the broken peroxy bond in a new state whose energy level is slightly below the upper edge of the valence band. In terms of the valence state, the neighboring oxygen atom, which was previously in an O2- state, becomes O-. This oxygen anion in the 1 - state is effectively a positive hole with an incomplete valence shell and could also be regarded as an unstable oxygen radical .

Holes are capable of propagating through the oxygen lattice, exchanging valence electrons by a phonon-assisted vacancy hopping mechanism . This process effectively constitutes a diffusion of O- states through a lattice of O2- atoms. The trapped electrons are immobile but participate through recombination and electrostatic interactions. Positive hole charge carriers propagate at 100–200 m s-1 , much slower than piezoelectric effects or crack propagation, which propagate at the speed of sound (2000–8000 m s-1), but faster than ion transport which takes place at much slower speeds. The charge carrier velocity allows mechanisms such as piezoelectricity, crack dislocation effects , and the electrokinetic effect to be ruled out as the predominant source of electrical activity in many rocks.

A recent study has suggested that if crustal semiconductivity is the result of fluid-saturation, the positive hole model might not apply. This conclusion was based on a series of experiments involving dry and fluid-saturated rock samples. However, even if the fluid-saturation hypothesis is valid, analysis of these results

In , the authors conclude that stress-stimulated currents do not occur in fluid-saturated samples of gabbro. Figure 4b in , however, clearly indicates that, given the “ST1” experimental setup which is erroneously cited several times as originating from currents change in response to stress. These are, in fact, stronger than those observed in dry samples, as expected due to their increased conductivity. Figure 5b in , based on the “ST2” experimental setup, shows a 43 % increase in current from an unusually high baseline of 4.85 to about 6.95 nA over the course of seven loading-unloading cycles. In spite of these current changes, conclude that stress-stimulated currents are not observed in fluid-saturated samples of gabbro.

The drift-diffusion equations are the most frequently used model for semiconductor physics, and perform well on scales greater than about 5×10-7 m . They describe current in terms of charge carrier concentrations and an electrostatic field, and this determines the change in charge carrier concentrations via continuity of the current density. The drift-diffusion equations are: ∂tn=-R(n,p)+∇⋅(Dn∇n-μnn∇V)∂tp=-R(n,p)+∇⋅(Dp∇p+μpp∇V)ΔV=1ϵ(n-p-C). Here, n, p, R, V, and C are defined on a domain Ω×(0,T), where Ω is a subset of a 3-D space on which these partial differential equations are to be solved. The functions n and p are concentrations of electron and hole charges, respectively, and C is the charge of any dopant ions that are present. n and p are constrained to be non-negative. R(n,p) is the recombination and/or generation rate of electrons and holes. The third equation is Poisson's law of electrostatics whose solution describes the electric potential V. ϵ is the electric permittivity. The constants μn and μp are the mobilities of electrons and holes, respectively, (not to be confused with the magnetic permeability μ or μ0) and Dn and Dp are the corresponding diffusion coefficients. In the particular instance of the model under consideration, μn and Dn are approximately zero due to electrons becoming trapped in the valence band. The conductivity associated with a particular charge carrier is determined by the product of the number density of the charge carrier, the charge carried (one electronic charge, in this case), and its mobility.

Maxwell's equations describe propagation at the speed of light, which is much faster than the charge carriers diffusing in a typical semiconductor. Rather than modeling propagation on two very different timescales, we make use of a quasi-static (magnetostatic) approximation , assuming that currents do not alternate rapidly or approach the speed of light. Specifically, the Maxwell displacement current appearing in Ampere's law is assumed to be negligible: c-2∂tE≈0. This assumption is justified for semiconductors, and is in fact implicit in the drift-diffusion model due to its use of Poisson's equation for the static electrical potential.

The electric current density J(x′) acts as the source of a magnetic field. It may be expressed as the sum of a drift term, involving the electric field, and a diffusive term, involving the concentration gradient. The rate of change of the concentration then becomes a continuity equation that is a function of current density. Explicitly separating the current and continuity equations facilitates coupling to the magnetic field. In this form, the current densities are: Jn=Dn∇n+μnn∇VJp=-Dp∇p+μpp∇V. The continuity equations that describe the change in electron and hole concentrations are then: ∂tn=-R(n,p)+∇⋅Jn∂tp=-R(n,p)+∇⋅(-Jp). The current densities Jn and Jp are summed to obtain the total current density J that acts as a source for the magnetic field. In a magnetostatic approximation, the solution to the magnetic field on a domain may be efficiently computed by solving a set of Poisson equations for the magnetic vector potential. In this case, however, we calculate the field at an arbitrary point in space, which could be outside the domain. We apply the Biot–Savart law to obtain the magnetic field at the point x: B(x,t)=μ4π∫(Jp(x′,t)+Jn(x′,t))×x-x′|x-x|′3d3x′. Here, |x-x′| is the magnitude of the vector from x to x′ and μ is the magnetic permeability. The velocities of the holes are not sufficiently large for the Lorentz force to be significantly influenced by magnetic fields so we do not consider the effect of the magnetic field on the charge carriers.

The drift-diffusion equations are solved by expressing the partial differential equations as a system of ordinary differential equations for the time derivatives ∂tn and ∂tp. A finite-difference approximation to this system is then integrated using a fourth-order Runge–Kutta scheme (RK4). Poisson's equation is solved separately at each timestep using successive over-relaxation (SOR) with an adaptive relaxation parameter and open boundary conditions. For the other PDEs, the Dirichlet boundary conditions n=0 and p=0 are applied to the boundary of a grid of uniformly spaced points representing the x, y, and z coordinates over which functions are evaluated. All spatial partial derivatives (∂x, ∂y, ∂z, ∇) of the current and continuity equations are approximated using a fourth-order central difference approximation.

At each timestep, the electric potential is determined by solving Poisson's equation, starting the SOR iteration with the electric potential from the previous timestep. Using the electric potential and the charge carrier concentrations, the components of the current vector fields Jn and Jp are evaluated. From Jn and Jp, the continuity equations are integrated, yielding the concentrations of the charge carriers at the next timestep. The non-negativity constraint on n and p is enforced via truncation.

The magnetic field B is evaluated by applying a discretized Biot–Savart law to the currents. B is calculated at each timestep but since the result does not affect the dynamics, it may be evaluated at a single point.

Since holes are mobile and electrons are immobile, diffusion separates the two species, creating an electric current that acts as an electromagnet. The boundary of a region of excess charge carriers behaves, essentially, like the p-n junction of a diode. Since only holes may flow out of this volume, the initial current diffusing across the boundary is unidirectional, corresponding to forward bias in the diode. However, after a delay period during which recombination reduces the diffusive current, the diode could switch to reverse bias, whereby the p-n junction capacitance generates a reverse recovery current. In the case of a recovery current, holes flow back into the source volume, producing a magnetic pulse that is opposite in polarity to the initial magnetic field.

We use the semiconductor model to calculate an example of a unipolar magnetic pulse. The electric permittivity and magnetic permeability are estimated based on the static properties of MgO (ϵ≈16.75ϵ0 and μ≈μ0) and a temperature of T=673.15K. Since electrons are trapped and immobile in broken peroxy bonds, μn and Dn are set to 0. The mobility and diffusion constant of holes were estimated based on experimental data from an experiment in which a rapid pressure impulse to the center of a gabbro tile injected holes that diffused and drifted away from their source, akin to a Haynes–Shockley experiment. The parameters used are μp=0.063m2(Vs)-1 and Dp=8.5×10-4m2s-1, roughly comparable to their values in pure undoped Silicon.

Charge generation is not explicitly considered in this calculation, and a pre-existing excess concentration of 10-5Cm-3 of both electrons and holes is an initial condition. These dissociated charges are initially present only at grid points inside a piriform teardrop surface of the form (z/4000)4-(z/4000)3+(x/2000)2+(y/1000)2=0. This surface geometry was chosen to avoid cancellation of fields due to spherical symmetry. The recombination rate is proportional to the product of electron and hole concentrations, R=1022np.

The attenuation of the magnetic field as it passes through the Earth is not considered, nor are the effects associated with the surface of the Earth. This is a reasonable approximation at extremely low frequencies. The fact that very slow (nearly DC) variations in the magnetic field can pass through the earth is the basis for magnetotelluric surveys. Ultra-low frequency radio waves are also used for through-the-earth radio communications in mine shafts due to the fact that lower frequency waves can penetrate the earth much more effectively than more conventional radio frequencies. This behavior is typical of most materials: low-frequency waves penetrate more effectively than high-frequency waves. The characteristic depth scale of penetration of an EM wave (the penetration depth or skin depth) is the reciprocal of the imaginary part of the wave vector, which, for a conducting medium, is: d=1ωμϵ21+(σϵω)2-1≈2μσω.

Here, ω is the (angular) frequency, σ is the conductivity, μ is the magnetic permeability, and ϵ is the electric permittivity.

Observed and calculated magnetic pulses are illustrated in Figs.  and . Figure  shows the value of the x component of the magnetic field as a function of time, evaluated 10 km directly above the center of the simulated volume. The amplitude, frequency, and shape of the pulse are similar to pulses that have been observed before earthquakes. Figure  shows several magnetic pulses observed prior to an earthquake near Lima, Peru. These pulses were measured over a period of several days by a pair of magnetometers approximately 25 km away, and the locations of their sources were triangulated. The sources were clustered within a few kilometers of the epicenter of an earthquake that occurred 2 weeks after the onset of the pulses . This analysis has been performed prior to several moderate earthquakes near Lima, with similar results.

The calculated pulse is unipolar and rises faster than it falls, like all but one of the pulses shown in Fig. . The duration of the calculated pulse is roughly 5 s, falling within the range of 0.1–10 s for observed pulses. The intensity of the calculated pulse reaches 34 nT, whereas observed pulses have ranged from the noise floor (around 1 nT) to 200 nT in intensity. Whereas the calculated pulse (which used a diffusion coefficient estimated using gabbro at room temperature) in Fig.  has a comparatively long duration and moderate intensity, the pulses shown in Fig. , have short duration (≈200 ms) but high intensity (80–150 nT). Thus, in addition to having essentially the same form, the total energy of calculated and observed pulses are similar.

It is reasonable to expect that different rocks under different conditions will produce a variety of diffusion coefficient and mobilities, leading to a variety of pulse amplitudes and durations. If the charge carriers are not injected instantaneously, as we have modeled here, then different pulse shapes can result. For example, if charge injection is a continuous process rather than a rapid impulse, the rise of the pulse can be slower than its fall, as is the case for one of the pulses shown in Fig. .

In the model presented here, when a volume of rock enters a semiconducting state, e.g. from heat or stress, excess holes and electrons are injected. The mobile holes begin to diffuse out of the source volume, while electrons are trapped within the source volume and undergo recombination with the holes that have not diffused out. The flux of holes leaving the source effectively creates a p-n diode: the source volume becomes an n type semiconductor and the surrounding rock becomes p type. A depletion region forms between the two layers of the p-n junction and the p-n double layer screens electric fields outside its immediate vicinity.

After charge injection, a diffusive current of holes flows as a result of the concentration gradient across the source boundary. This corresponds to a forward bias state of the diode, dominated by diffusion capacitance rather than junction capacitance. This current creates a transient magnetic field. As the hole concentration gradient decreases, the diffusive current and the magnetic field decay. After holes have diffused outward, creating p-type and n-type regions, a junction capacitance results from layers of positive and negative charge separated by a depletion region at the junction.

After a delay period, the diode may switch to a reverse bias state. In this case, electron-hole recombination consumes the holes remaining within the source volume, leaving mostly electrons inside. The junction capacitance causes a transient reverse recovery current. If the potential drop across the depletion region is sufficiently strong, reverse-bias electrical breakdown may occur as Coulomb attraction pulls holes back into the source volume.

It has been suggested that circuit closure is necessary to produce electric currents and hence sources of magnetic fields. Although this may be a useful rule of thumb in everyday experience, it is not a law of physics, and the model developed here illustrates that continuous closed current loops (i.e. circuits) are not necessary to produce magnetic fields. The outflow current and subsequent return current associated with a large number of charge carriers is capable of producing significant magnetic fields, albeit transient, without formal circuit closure.

A distinctive form and the ability to pass through the earth at ultra-low frequencies make magnetic pulses potentially useful for the observation of pre-seismic shifts in the stress level of rocks that are otherwise inaccessible due to depth. Preliminary observations suggest that by triangulating the source of these magnetic pulses, the increased buildup of stress around future earthquake epicenters might be identified days or weeks in advance of seismicity.

In addition to unipolar pulses, other types of electromagnetic precursors or seismic electromagnetic phenomena might be predicted from semiconductor models. Oscillatory ULF fields, for example, have been observed immediately preceding earthquake activity .

The positive hole semiconductor theory modeled here seeks to unify a wide range of electromagnetic phenomena associated with seismic activity. The direct coupling of semiconductor drift-diffusion currents and electromagnetism produces a model consistent with observations of pre-seismic magnetic pulses as well as laboratory experiments that rapidly load and unload rock samples. This agreement between theory, experiment, and field observations supports the hypothesis that pre-earthquake ULF activity might result from geophysical semiconductor processes.