JOURNAL OF GEOPHYSICAL RESEARCH: SOLID EARTH, VOL. 118, 915–925, doi:10.1002/jgrb.50111, 2013
Activation of hole charge carriers and generation of electromotive
force in gabbro blocks subjected to nonuniform loading
Akihiro Takeuchi1 and Toshiyasu Nagao1
Received 16 July 2012; revised 28 January 2013; accepted 29 January 2013; published 27 March 2013.
[1] This study verifies the activation of hole charge carriers in gabbro under nonuniform
loading, which has been proposed as the possible source of the stress-induced electromotive
force in dry rocks without piezoelectric effect. When one end of vacuum-dried gabbro
blocks was subjected to uniaxial loading, the unloaded end became electronically
positive (+80 mV at 50 MPa). The Seebeck coefficient of the loaded volume decreased
from ~15.8 mV/K to ~14.9 mV/K when loaded, while the Seebeck coefficient of the
unloaded end did not change remarkably (~15.6 mV/K). This means that this gabbro
originally included a small number of hole charge carriers and the carriers in the loaded
volume increased when loaded. From the viewpoint of the fundamental band model of
solid state, the most reasonable mechanism of the increment is the decrease of the energy
gap between the acceptors and the valence band top. Based on this idea, a generation model
of the stress-induced electromotive force is proposed. Since this model is expected to be
universally applicable to various types of rocks, similar electromotive forces in the crustal
scale may be induced by seismic, volcanic, and tidal activities.
Citation: Takeuchi, A., and T. Nagao (2013), Activation of hole charge carriers and generation of electromotive force in
gabbro blocks subjected to nonuniform loading, J. Geophys. Res. Solid Earth, 118, 915–925, doi:10.1002/jgrb.50111.
1.
Introduction
[2] Olivine and gabbro are most often used to understand
the electrical conductivity structure of the Earth’s lower
crust and mantle. To simulate deep crustal conditions, a rock
sample is generally placed under a specific confining hydrostatic pressure at a specific temperature. In these studies,
as depicted in Figure 1, it is accepted that the small polarons
associated with FeMg• sites act as positive charge carriers
predominant in the temperature range less than 1470 K,
while VMg00 sites behave as negative ones at temperature
range for more than 1660 K [e.g., Poirier, 2000 and references therein]. Therefore, we can infer that the electrical
conduction property of olivine is apparently “p-type” at
low temperatures and apparently “n-type” at high temperatures (different from the original definition of the electrical
conduction property for semiconductors). In addition, the
pairs of H• ions and HM’ sites provide acceptor energy levels
in the energy range of 1–2 eV from the valence band top at
0 K in the case of hydrated olivine [e.g., Wang et al., 2012].
This will also influence the electrical conductivity property
to be apparently p-type. Nowadays, based on the thermodynamics of solid state, it is well known that the electrical
1
Earthquake Prediction Research Center, Institute of Oceanic Research
and Development, Tokai University, Shimizu-ku, Shizuoka, Japan.
Corresponding author: Akihiro Takeuchi, Earthquake Prediction
Research Center, Institute of Oceanic Research and Development, Tokai
University, 3-20-1 Orido, Shimizu-ku, Shizuoka 424-8610, Japan. (atakeuchi@
sems-tokaiuniv.jp)
©2013. American Geophysical Union. All Rights Reserved.
2169-9313/13/10.1002/jgrb.50111
conductivity follows the Arrhenius equation with an activation enthalpy factor [e.g., Xu et al., 1998, 2000], and the
electronic states of polarons and ions at certain pressuretemperature conditions are being clarified gradually.
[3] When we turn our attention to large-scale rock
volumes in nature, i.e., crustal rocks, the pressure-temperature
condition is not homogeneous. The stress/strain distribution in
the crust changes statically and dynamically with seismic,
volcanic, and tidal activities. Therefore, the electronic state
of a definite point is different from that of a specific crustal/
global distance. This may generate a new electric phenomenon on the crustal/global scale, but at present, there is no
literature that identifies such phenomena driven by the
electronic state difference. However, Freund et al. [2006]
and Takeuchi et al. [2006] expected that this mechanism
would link to the electromagnetic phenomena prior to the
occurrence of major earthquakes, often reported as changes
of self-potentials and geomagnetic fields, abnormal emissions
of electromagnetic and thermal-infrared radiations, and
plasma disturbances in the ionosphere [e.g., Eftaxias et al.,
2004; Culter et al., 2008; Uyeda et al., 2009].
[4] In order to study the mechanisms of such stress/straininduced electromagnetic phenomena, Freund et al. [2006]
and Takeuchi et al. [2006] conducted laboratory experiments
of air-dried rocks including gabbro at room temperatures.
A part of the sample volume was uniaxially loaded within
their elastic deformation range (Figure 2). An electrometer,
which connected to the loaded and unloaded ends, detected
an electric current flowing from the unloaded end to the
loaded end via the electrometer (i.e., the loaded end ! the
unloaded end ! the electrometer ! the loaded end). Thereafter, Takeuchi et al. [2011] conducted similar tests using
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TAKEUCHI AND NAGAO: EMF IN GABBRO UNDER NONUNIFORM LOAD
(a)
(b)
then the electric conduction property of gabbro under this
condition is like that of a p-type semiconductor though it
has a high resistance. However, the sign of the charge carriers
and the activation of hole charge carriers are not proven yet.
[5] From the viewpoint of the theory of semiconductors,
loading can change the energy band structure of rocks, which
can lead to the activation of hole charge carriers (and also
electrons in principal) and the generation of the electromotive
force. If this is the case, this mechanism can be described
using the band model. Based on this idea, in this study,
(1) the generation of the electromotive force is again confirmed using gabbro blocks subjected to nonuniform loading,
(2) the change of the electric states in the loaded volume is verified from the change of Seebeck coefficient during unloading/
loading, (3) the activation of hole charge carriers in the loaded
volume and the generation mechanism of the electromotive
force are quantitatively discussed on the basis of the band model
of semiconductors, and (4) this electric phenomenon on the laboratory scale is applied to that on the crustal/global scale.
2.
Figure 1. Models of the electronic conduction in olivine.
(a) Electronic conduction by a positive charge carrier (a small
polaron associated with a FeMg• site). (b) Electronic conduction by a negative charge carrier (a VMg00 site).
10 cm long gabbro blocks whose one end was uniaxially
loaded. The blocks generated an electromotive force during
loading. The unloaded end was electrically positive relative
to the loaded end, which was consistent with the direction
of the self-flow electric currents observed by Freund et al.
[2006] and Takeuchi et al. [2006] as mentioned above. Based
on the number of their experimental results, they expected
that the hole charge carriers were activated in the loaded
volume and diffused into the unloaded volume. If this is true,
Loading
Hole charge carrier?
Air-dried rock block
Conductive tape
Current
Figure 2. A schematic of the self-generation current flow
induced by nonuniform loading of air-dried rock blocks.
Modified from Takeuchi et al. [2011]. The electrometer
detects a current flowing in the clockwise direction, and a
flow of hole charge carriers is expected in the rock block.
Seebeck Coefficient and Hot Point Probe Test
[6] When a semiconductor material includes both
electrons and holes as charge carriers with the MaxwellBoltzmann distribution, its Seebeck coefficient a is given by
a¼
sc ae þ sv ah
sc þ sv
(1)
where sC and sV are the electrical conductivity contributed
by the conduction and valence bands, respectively, and ae
and ah are Seebeck coefficients contributed by electron and
hole charge carriers, respectively, as below:
5
Nc
þ ln
gþ
2
ne
kB
5
Nv
ah ¼
þ ln
gþ
2
q
nh
ae ¼
kB
q
(2)
(3)
where kB is Boltzmann constant, q is the elementary
electric charge, g is a constant depending on the scattering
mechanism of charge carriers, NC and NV are the effective
density of states in the conduction and valence bands,
respectively, and ne and nh are the concentration of electron
and hole charge carriers, respectively. For n-type materials,
the majority carriers are electrons, and the a value is negative. On the other hand, when the majority charge carriers
are holes (p-type), a is positive [e.g., Nolas et al., 2001].
[7] Hot point probe test is a simple and easy method to
determine whether majority of the charge carriers are
electrons or holes. This test uses a pair of probes, “hot” and
“cold”, with a temperature difference ΔT between the probes.
When the probes are attached to a material, the charge carriers
are thermally activated under the hot probe and diffuse toward
the cooler volume. As a result, a potential difference ΔV
develops between the probes. This is called the thermoelectromotive force. From this test, the Seebeck coefficient a
is defined as the ratio of ΔV and ΔT:
a¼
ΔV
ΔV
am ΔT
ΔT
(4)
where am is the Seebeck coefficient of the metal used for
the probes (and connecting wires) [e.g., Nolas et al.,
916
TAKEUCHI AND NAGAO: EMF IN GABBRO UNDER NONUNIFORM LOAD
(a)
2001]. The |am| is usually much smaller than |a| of ceramics and semiconductors and is considered negligible.
For example, |a| ~0.001–0.002 mV/K for stainless steel
[Graves et al., 1991], |a| ~0.45 mV/K for Si semiconductor with the resistivity of 3.5 10–5 Ωm at room temperature [Van Herwaarden and Sarro, 1986], |a| ~0.055–
0.370 mV/K for brine-saturated sandstones [Leinov,
2010], and |a| ~1–3 mV/K for synthetic forsterite at
1000–1500 K [Schock et al., 1989]; thus, the order of a
varies depending on the electric resistance, in other
words, the charge carrier concentration as expected from
equations (2) and (3). Here, we can assume that rock
without pore water is a “multigrain multimaterial highelectric-resistance semiconductor”. Each rock-forming
mineral has its own a value. Therefore, we should repeat
measurements of the thermo-electromotive force many
times between different types of mineral grains, take their
averages, and then determine representative a values for
the rock. We can expect that |a| of such highresistance rocks (almost insulators) to be higher than
those mentioned above.
3.
Vacuum-dried gabbro block
(2.5 x 3.0 x 10 cm3)
Conductive tape
(b)
3.2. Nonuniform Loading Tests
[10] Prior to the hot point probe tests, the electromotive
forces between the loaded and unloaded ends were measured
using a voltmeter-mode electrometer with input impedance
Loading (0 or 50 MPa)
Hot-probe
Experimental
3.1. Sample
[8] The rock used for the experiments in this study
was dense gabbro from South Africa. This gabbro had a bulk
unit weight of 2.93 g/cm3 and a compressive strength of
~200 MPa. Its modal composition was: 67.1% of plagioclase,
24.8% of orthopyroxine, 5.1% of clinopyroxene, 1.6% of
tridymite, and 1.4% of biotite. There was no X-ray diffraction
peak of quartz and tourmaline for this gabbro. Clear preferred
orientation of the grain distribution was not seen in the rock.
The effective porosity of the gabbro used was ~0.04–0.05%.
2-D image analyses of the gabbro yielded the total porosity
in the order of 0.2–0.6%. The pores were elipsoidical with
its longitudinal axis less than 0.05 mm and dispersed. The
inter-pore connectivity is very low so that it is unlikely to
constitute a flow path network. This was the main reason
for the difference between effective porosity and total
porosity.
[9] The gabbro was cut into three blocks with a size of
2.5 3.0 10 cm3. The surfaces were polished with
#800 abrasives. Thereafter, these blocks were dried in a
vacuum oven (EYELA, VOS-201SD) at 353 K for more
than 2 days. Figure 3a shows the experimental setup. One
end of each block was uniaxially loaded by means of a
manual hydraulic press (RIKEN, CDM-20PA). The load
was measured with a load cell (Minebea, CMM1-2T and
CSD-819C). Copper tapes with graphite-based conductive
adhesive were pasted on the load areas (2.5 2.5 cm2 each)
to allow homogeneous contact with the rectangular aluminum load pistons and for the right electric grounding of
both the load pistons and the loaded surface of the blocks.
Although no electric field would be generated in the press
made of metal at all, this was also grounded.
Loading (up to 60 MPa)
Area
A
Area
B
Cold-probe
Figure 3. Schematics of the experimental setups using
vacuum-dried gabbro blocks. (a) Setup for nonuniform loading tests. (b) Setup for hot point probe tests. The measurements for Area A were not performed simultaneously with
those for Area B.
more than 1013 Ω (ADCMT, 8240). Analog data of the load
and electromotive force were sent to a data logger (HIOKI,
8808-50) and recorded at the 8 Hz sampling rate. Figure 4
shows a typical result of the gabbro sample #1 that was
already loaded several times before this performance. It
was confirmed that the block generated electromotive forces
Vemf when loaded from 0 MPa to 60 MPa (elastic deformation range) with a loading rate of 1 0.3 MPa/s under
manual control. The unloaded end was electrically positive
relative to the loaded end. The Vemf increased with an
increase of loading, and reached ~100 mV at ~50 MPa.
The voltage value was roughly kept at an even level for at
least 10 min after stopping to ramp the load. The electromotive force reappeared when the unloading/loading cycle was
repeated. An initial rapid increase of the electromotive force
was sometimes observed. This may be due to uneven
increase in the loading or a result of the block slightly
moving at the application of the first load cycle from
0 MPa. Although we did not use any sensors to detect
917
TAKEUCHI AND NAGAO: EMF IN GABBRO UNDER NONUNIFORM LOAD
electromotive forces ΔV. The analog data of the load and
ΔV were sent to the same data logger (HIOKI, 8808-50) and
recorded at the 8 Hz sampling rate. Since the hot-probe
contact point included only a few mineral grains, different
points were chosen in repeated measurements to obtain the
averaged ΔV of this gabbro. About 12–24 cycles with and
without the hot probe were measured. The rock was allowed
to cool in between cycles.
[12] Applying and removing the hot probe inevitably
closes and opens the electric circuit. This may cause a shift
of the background level or an apparent change of ΔV.
Therefore, ΔV is revised as
60
(a)
20
120
(mV)
Electromotive Force
0
(b)
80
40
0
0
20
40
60
80
ΔV ¼ ΔVobs ΔVad
100
Time (sec)
Figure 4. A typical result of the nonuniform loading tests
for the gabbro block #1. (a) Loading profile. (b) Electromotive force.
acoustic emissions from microcracking, we did not hear any
acoustic emissions during loading. Two other blocks were
also used for this test, and the electromotive forces (sometimes with the initial peak) were similarly induced by nonuniform loading as shown in Table 1.
3.3. Hot Point Probe Tests
[11] Next, hot point probe tests were conducted to the
same blocks. The cold probe was a twisted wire made of
stainless steel. Its terminal was untwisted, and the points of
the fine wires were attached to a few randomly chosen
mineral grains in Area A or B (the loaded or unloaded end,
respectively) on the backside of the blocks without any
paste as shown in Figure 3b. The cold-probe temperature
TC was room temperature (~293 K). On the other hand, the
hot probe was a stainless steel bolt installed as the tip of a
soldering iron. The hot-probe temperature TH was ~423 K
when six 12 V batteries in a series supplied the current to
heat the soldering iron. Therefore, the temperature difference
ΔT = TH – TC 130 K when the hot probe was heated.
The hot probe was attached to a spot on the opposite
side without any paste. After about 50 s, the voltage became
stable. The probes were connected to each other through
the same electrometer (ADCMT, 8240) to measure thermo-
Range for the Seebeck coefficient calculation
Detachment
a (Area A)
a (Area B)
Sample
Vemf
0 MPa
50 MPa
0 MPa
50 MPa
#1
#2
#3
Average
100
100
60
80
15.9 0.4
15.9 0.5
15.7 0.2
15.8 0.7
15.0 0.2
14.9 0.3
14.8 0.1
14.9 0.5
15.9 0.4
15.5 0.5
15.4 0.8
15.6 1.0
15.9 0.4
15.5 0.1
15.6 0.3
15.6 0.5
a
The Vemf is a representative value obtained from a few times of the nonuniform loading tests, and the a is the average of the standard deviation
obtained from 12–24 times of the hot point probe tests.
Attachment
Detachment
0
(a)
50 MPa
–1000
0 MPa
–2000
0
(b)
0 MPa
–1000
50 MPa
–2000
0
Table 1. Electromotive Force Vemf (mV) Induced by Nonuniform
Loading and Seebeck Coefficient a (mV/K) While Unloaded/
Loaded of Vacuum-Dried Gabbro Blocksa
(5)
where ΔVobs is the observed thermo-electromotive force and
ΔVad is the false thermo-electromotive force because of the
attachment/detachment motion. First, premeasurements for
ΔVad were conducted with the blocks under no load and
without heating the hot probe (TH 293 K and ΔT 0 K).
As a result, it was found that the averaged ΔVad was roughly
150 mV for sample #1, 60 mV for sample #2, and 50
mV for sample #3. Thereafter, the same measurements were
done with the heated hot probe during the unloading/loading
cycles for Areas A and B.
[13] Figure 5a shows the experimental result for Area A
of the gabbro block sample #1. The sign of the averaged
ΔV was negative, and ΔV increased approximately from
–2070 mV to –1950 mV when loaded. Using equations
(4) and (5), a was obtained for the unloaded and loaded
conditions. In the case of Area A, a decreased from
Thermo-electromotive Force (mV)
Stress
(MPa)
40
20
40
60
80
100
Time (sec)
Figure 5. The results of hot point probe tests for the
gabbro block #1. (a) Thermo-electromotive force of Area
A (the loaded volume). (b) Thermo-electromotive force of
Area B (the unloaded volume). Gray and white areas
indicate the attachment and detachment durations of the hot
probe, respectively. Narrow-blue and wide-red curves are
under the 0 and 50 MPa pressures, respectively. The curves
are the averages obtained from multitimes tests. The influence
of the false thermo-electromotive force because of the
attachment/detachment motion of the hot probe is already
subtracted from the originally observed curves.
918
TAKEUCHI AND NAGAO: EMF IN GABBRO UNDER NONUNIFORM LOAD
(a)
~15.9 mV/K to ~15.0 mV/K when loaded. Here, the
Seebeck coefficient am of stainless steel of the probes is
negligible as mentioned above: am 0.001–0.002 mV/K
for stainless steel [Graves et al., 1991]. The same measurements were also conducted for Area B as shown in
Figure 5b and found that a ( 15.9 mV/K) did not change
remarkably when loaded. The other two blocks also had
the same tendency, i.e., a decrease of a in Area A when
loaded, as shown in Table 1. The results were rather scattered which could probably be due to the complicated
inhomogenous composition of the mineral component/distribution. Since the vacuum-dried gabbro is considered to
be almost an insulator, it is appropriate that the obtained
a was much higher than those of general metals, semiconductors, and saturated sandstones as mentioned above.
4.
< F.B. >
unoccupied 3 u* energy level
degenerate 1 xynb energy level
< V.B. >
(b)
Loading
< F.B. >
< V.B. >
(c)
Discussions
Loading
< F.B. >
4.1. Mechanism of the Electromotive Force
[14] All of the a values obtained in this study were
positive. This indicates that the holes are the majority charge
carriers of this gabbro. This gabbro is only slightly p-type and
nh is very small. Parts of the hole charge carriers may be
polarons associated with FeMg• sites (rather than simply holes)
though their influence is usually not predominant in
room temperature. For cases when the charge carriers are
only holes, equation (1) gives us a ah. From equations
(1) and (3), the decrease of a means there is a decrease in
the NV/nh ratio. As NV can be assumed to be almost independent of load, the decrease of the ratio means an increase of nh.
[15] From the viewpoint of the fundamental band model
of solid states, the most reasonable mechanism to cause an
increase in nh is the decrease of the energy gap between
the acceptors and the valance band top. When the energy
levels of acceptors shift toward the valence band in the
loaded volume, electrons in the valence band are easily
thermally excited to the acceptor levels. On the other hand,
since a of Area B did not change, nh in the unloaded volume
would not change, which means there is no shift in the
acceptor energy levels at this volume. As olivine includes
various kinds of energy levels in the forbidden band [e.g.,
Schock et al., 1989; Poirier, 2000; Wang et al., 2012],
gabbro also includes various energy levels. We can then
infer that there are many possible candidates to become
acceptors. When the load is released and the acceptors
would shift up again, the electrons are released from the
acceptors and recombine with the hole charge carriers. This
mechanism is repeatable and in agreement with the fact that
the electromotive force appeared whenever loading was
repeated in this study.
[16] On the other hand, Freund et al. [2006] and Freund
[2009] expected that the energy levels of peroxy bond
(O3X–OO–YO3, with X, Y = Si4+, Al3+, etc.) act as the
source of the stress-induced activation of hole charge carries.
This lattice defect is present in various kinds of igneous
rock-forming minerals. Their model is based on the
assumption that this defect initially has the unoccupied 3s*u
(or simply s*) energy level at the valence band top and the
occupied degenerate 1pxynb (or simply p*) energy level just
below the valence band top (Figure 6a). When a load is
applied and the structure around this defect is deformed,
the 3s*u level shifts downwards, and the 1pxynb level splits
< V.B. >
(d)
Figure 6. Band models for the activation of a hole charge
carrier. Modified from Freund et al. [2006] and Freund
[2009]. (a) The normal energy level structure of a peroxy
bond near the valence band top. (b) Load-induced move of
an electron from the upward-shifting split 1pxynb energy level
to the downward-shifting 3s*u energy level. (c) Further loadinduced move of an electron from a neighboring O2– site to
the further upward-shifting 1pxynb energy level. (d) A model
of the electronic conduction in metal oxide such as SiO2 by
a polaron associated with an OO• site.
into two levels. Further loading causes the 3s*u level to further
shift downward and intersect with one of the upward-shifting
1pxynb levels. At this time, the electron on the 1pxynb level
jumps in the 3s*u level (Figure 6b). Now, the 1pxynb level
919
TAKEUCHI AND NAGAO: EMF IN GABBRO UNDER NONUNIFORM LOAD
becomes the new acceptor and can activate a hole charge carrier (Figure 6c). This hole charge carrier is in an O-site and
can spread as the electric state of O– hopping in the matrix
of O2–. If the interaction with the lattice is strong, this would
be called a polaron associated with OO• site (Figure 6d) rather
than purely a hole charge carrier. On the other hand, some
early studies on peroxy bonds in SiO2 and HfO2 suggest
that the s* level is located at ~5 eV above the valence band
and the p* level is located at ~1 eV above the valence band
[e.g., O’Reilly and Robertson, 1983; Nishikawa et al.,
1990; Robertson, 2004; Xiong and Robertson, 2005]. If
this initial structure is true and adaptable to various kinds
of igneous rock-forming minerals, the s* and p* levels
have to shift a long distance and cross each other in the
forbidden band to transfer an electron. In this study, we
try to modify Freund’s model to incorporate this in our
model as explained below.
[17] The electromotive force in this study is the electric
potential difference between two spots in a solid material.
Therefore, the potential difference is related to the difference
of the “absolute” Fermi energy levels between the loaded
and unloaded ends. Now, we consider a simple 1-D model
of the gabbro block at room temperature. As indicated in
Figure 7, we define the x-axis to lie along the block length.
The loaded end is at x = 0 cm, and the unloaded end is at x
= L (= 10 cm). Figure 7a depicts the band structure under
normal conditions without any subjection of loading. Since
this gabbro under normal conditions is slightly p-type, there
are actually acceptors in the lower part of the forbidden
band, but the energy levels are so high that they are not so
effective as acceptors. As a result, this gabbro is almost an
insulator. The Fermi energy level lies between the acceptors
and the valence band top, additionally, there are unoccupied
energy levels (e.g., the s* levels of peroxy bonds) over the
acceptors.
[18] As depicted in Figure 7b, when the left range
(0≤ x ≤ 2.5 cm) is subjected to loading, the acceptor energy
levels shift downward, and hole charge carriers are easily
thermally activated. In addition, the s* levels may also shift
downward and become effective acceptors. Or, the upwardshifting p* levels, which split from the original location, may
release their electrons to the downward-shifting s* levels
and become effective acceptors. Since the Fermi energy level
is strongly related with the energy level structure, it is also
changed in the loaded range. Here, its “relative” energy level
eF(x), which is the energy level difference from the valence
band top, has a relation with the concentration of hole charge
carriers nh(x):
eF ðxÞ
nh ðxÞ ¼ Nv ðxÞ exp kB T qVF ðxÞ
¼ Nv ðxÞ exp kB T
(6)
where VF(x) is the potential of the Fermi energy level from
the valence band top. From equations (3) and (6), we obtain:
kB
5
T
VF ðxÞ ¼ aðxÞ gðxÞ þ
2
q
x=0
x = L (= 10 cm)
(a)
Unoccupied energy level
Acceptor
Trapped electron
Hole charge carriers
< C.B. >
< F.B. >
Fermi
< V.B. >
(b)
Loading
< C.B. >
< F.B. >
Fermi
< V.B. >
(c)
Loading
< C.B. >
< F.B. >
Fermi
< V.B. >
Drift
Diffusion
x=0
x = L (= 10 cm)
Figure 7. Band models of a 1-D gabbro block whose one
end (0 ≤ x ≤ 2.5 cm) is uniaxially loaded. C.B., F.B., and
V.B. represent the conduction band, forbidden band, and
valence band, respectively. The dotted pink lines are the
Fermi energy level. (a) The normal state without subject of
loading. (b) Downward shift of acceptor energy levels and
unoccupied energy levels in the loaded volume, leading to
the activation of hole charge carriers and a downward
shift of the “relative” Fermi energy level. (c) Formation of
the electric field in the band (the upward shift of the band
structure) because of the diffusing hole charge carriers and
the trapped electrons, resulting in a slight upward shift of
the “absolute” Fermi energy level as the summation of two
competing shifts.
(7)
[19] As seen in Figure 7c, there appears to be a difference
of nh(x) between the loaded and unloaded ranges, and hole
charge carriers activated in the loaded range diffuse into the
unloaded range. This is called the diffusion current, and its
current density Idiffusion(x) is given by
920
TAKEUCHI AND NAGAO: EMF IN GABBRO UNDER NONUNIFORM LOAD
Idiffusion ðxÞ ¼ qDh ðxÞ
d
fnh ðxÞg
dx
(8)
where Dh(x) is the diffusion coefficient of hole charge
carriers as defined by
Dh ðxÞ ¼
kB T
m ðxÞ
q h
(9)
where mh(x) is the mobility of hole charge carriers. At the
same time, hole charge carriers are electrically attracted
toward the loaded range because of the self-generated electric
field Edrift(x) formed by hole charge carriers diffusing and
electrons trapped in the loaded range. This is called the drift
current and its current density Idrift(x) is given by
Idrift ðxÞ ¼ qnh ðxÞmh ðxÞEdrift ðxÞ
kB T d
½ lnfnh ðxÞg
q dx
(11)
[21] This field makes a slope in the band structure as
depicted in Figure 7c. The Fermi energy level shifts upward
along the slope. As a result, there is a difference in the
“absolute” Fermi energy level between the loaded and unloaded
ends, which is the summation of two competing effects: (1) the
downward shift of acceptor energy levels in the loaded range
and (2) the upward shift of the band structure by the drift
electric field Edrift(x) in the loaded range.
[22] In this case, Vemf is given by
Z
L
L
0
0
Vemf ¼
Z
d
fVF ðxÞg dx þ
dx
Edrift ðxÞdx
(12)
[23] The first term of the right side of equation (12) is the
contribution of the effect (1), and the second term is of the
effect (2). Inserting equations (7) and (11) to equation (12)
gives us
kB T
fgðLÞ gð0Þg
Vemf ¼ faðLÞ að0ÞgT q
kB T
nh ðLÞ
ln
þ
q
nh ð0Þ
Vemf
(10)
[20] We assume that these currents are balanced (Idiffusion
Idrift = 0) during loading at a slow loading rate or during
keeping a certain load level. In this case, equations (8)–(10)
give us
Edrift ðxÞ ¼
is originally very small, we expect the load-induced change
in the pore volume to be negligible for the solid volume
where hole charge carriers flow. The major influences on
g(x) will be still the crystal lattice, impurities, and grain
boundaries even during loading. Though we do not know
any appropriate values for g(x) yet, we can expect that
the load-induced change of g(x) is so small for this gabbro.
If we use porous rock samples, this change may not be
negligible and affect the change in the Seebeck coefficient (see equations (2) and (3)). In any case, the following
calculations do not account for the load-induced changes
in g(x):
(13)
[24] In this study, we treat multicrystal gabbro in which
grain boundaries and pores also scatter the flow of hole
charge carriers in addition to the crystal lattice and impurities. We can represent this effect by giving a value to g(x)
(and also to mh(x) in equations (7) and (8)) in this model.
As the diffusion coefficient of ions in a saturated porous
media depends on the porosity and tortuosity [e.g., Revil
and Jougnot, 2008], g(x) (and also mh(x)) may also depend
on them. The porosity and tortuosity may change during
loading because existing elipsoidical pores will partially or
fully close, and closed microcracks parallel to the load axis
might open. However, because the porosity of this gabbro
kB T
nh ðLÞ
ln
¼ faðLÞ að0ÞgT þ
q
nh ð0Þ
(14)
[25] If a obtained in the hot point probe tests at ΔT = 423
K – 293 K is the same at room temperatures T ( 300 K),
we can adopt a(L) 15.8 mV/K and a(0) 14.9 mV/K.
In this case, the first term of the right side of equation (14)
is ~270 mV. When we use Vemf 80 mV, the second
term of the right side of equation (14) is approximately –
190 mV. In this case, we obtain the nh(0)/nh(L) ratio
~1540. Thus, we can say that a number of hole charge
carriers were activated in the loaded volume. The scatter of
a(x) can result in the scatter of the nh(0)/nh(L) ratio in
some degree. However, it will be appropriate because this
gabbro involves various types of mineral grains distributed
inhomogenously with each Seebeck coefficient as noted in
Introduction. Therefore, we expect there is actually a
variety in the nh(x) in the loaded range on the grain scale.
[26] Even if mh(x) is partially dependent of the loading
level, we can expect that the electrical conductivity of the
loaded volume became so high under the local 50 MPa
pressure. However, according to early studies, the electrical
conductivity of gabbro needs pressure of the order of GPa
to become several tens of times, or, the electrical conductivity of some minerals/rocks to be almost independent of
pressure though it depends on the temperature, crystal axis,
ion contents, oxygen fugacity, and so on [e.g., Brace and
Orange, 1968; Parkhomenko, 1982; Xu et al., 2000; Yoshino
et al., 2012]. The difference between this study and early
studies is in the method of pressing the rock samples, i.e.,
nonuniform uniaxial loading and confining hydrostatic
pressure. As shown in Figure 8, finite-element analyses
using ANSYS (ANSYS Inc.) indicate that such a uniaxial
loading causes the shear stress/strain in the loaded volume
of this gabbro block. We expect that the shear stress/strain
make energy levels shift effectively in the forbidden band
though it is uncertain how the energy levels shift during
stress activation. Further studies are needed to verify this
hypothesis, for example, using first principal calculations.
[27] Since this model is formed by two competing effects
as mentioned above, the negative sign of Vemf can also be
predicted when the influence of the effect (1) is stronger than
that of (2). The effect (2) may depend on the size and/or
shape of block samples as well as the rock/mineral type.
Moreover, if the donor energy levels of an n-type rock
shift toward the conduction band bottom during loading,
the inverse mechanisms will function and the sign of Vemf
will be negative as a result. Although we did not find
921
TAKEUCHI AND NAGAO: EMF IN GABBRO UNDER NONUNIFORM LOAD
negative signs in Vemf in the nonuniform loading tests so far,
such rocks/minerals or loading geometries may exist.
(a)
4.2. Competing Mechanisms
[28] Here, we discuss some of the competing mechanisms for the electromotive force induced by loading to
compare with our mechanism mentioned above. First, the
piezoelectric effect of quartz (and also tourmaline) is most
often discussed for electric and electromagnetic phenomena
in laboratory experiments using rock samples subjected to
loading up to collapse [e.g., Finkelstein et al., 1973; Nitsan,
1977; Yoshida et al., 1997; Yoshida and Ogawa, 2004].
However, we used quartz-free gabbro in this study. Even
if small portions of quartz/tourmaline grains with random
piezo-axes were included in this gabbro, it could not explain
why the sign of Vemf was the same for the three gabbro
blocks. In other words, if the influence of this effect were
strong, the sign of Vemf would have been different for each
block.
[29] Electrokinetic effect of pore water is also most often
discussed for electric and electromagnetic phenomena in
the porous ground where pore water exists in the actual
environmental circumstance [e.g., Mizutani et al., 1976;
Fitterman, 1978; Jouniaux and Pozzi, 1997]. Since the electromotive force induced by nonuniform loading appeared
for more than 10 min even after stopping to ramp the load
in our tests as described in section 3.2, the pore water
must continue flowing from the loaded volume toward the
unloaded volume, also for more than 10 min at least. In
general, it will be possible for fluid to continue flowing/
penetrating in the pores/microcracks networks in rocks for
more than 10 min. However, since we used low-porosity
gabbro blocks, where the pores are dispersed, this would
limit the flow of the pore fluid. Though we believe that the
electrokinetic effect doesn’t work in this gabbro, we considered the creation of microcracks as one possibility to make
such a pass: Closed microcracks parallel to the load axis
might open and the microcrack networks link pores filled
with water. Considering the sign of Vemf, anions must adhere
on the microcrack walls, and cations must flow with the pore
water in the networks from the loaded volume to the
neighboring dilatant volume. If such a flow can really occur
and keep an electric balance with the conduction current
flowing in the inverse direction, then the electrokinetic
coupling coefficient C is given by
(b)
–60
–50
–40
–30
–20
–10
(MPa)
0
10
(c)
(x 10–3)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
C¼
Figure 8. 3-D finite-element analyses of a gabbro block
subjected to nonuniform loading. (a) The finite-element
mesh and boundary conditions. The sample size is 2.5 3.0 10 cm3, and its elastic moduli and Poisson’s ratio
are set as 100 GPa and 0.25, respectively. The mesh is
5 mm hexahedrons with 20 nodes. The bottom edge of the
loaded end is fixed vertically and in the direction of the
block length as indicated with green triangles. A uniformly
distributed 50 MPa traction is applied to the two pistoncontact areas (2.5 2.5 cm2 each) as indicated with red
arrows. (b) The distribution of the third principal stress in
MPa. The positive and negative values denote tension and
compression, respectively. (c) The distribution of the maximum shear strain in 10–3.
Δf ez
¼
ΔP s
(15)
where f is the electrical potential, P is the pore pressure, e is
the dielectric constant of the pore water, z is the zeta
potential, s is the electric conductivity of the pore water,
and is the viscosity of the pore water [e.g., Mizutani
et al., 1976]. Here, we adopt the following values just as
a trial: Δf Vemf = 8 10–2 V, s ~10–1 S/m, e ~80 8.8 10–12 F/m2, ~10–3 Pas. To fit in the range obtained
in early studies, e.g., –6 10–2 V < z < 3 10–2 V for
forsterite [Pokrovsky and Schott, 2000], the ideal minimum
value in the pressure difference must be: ΔP 0 MPa – 50
MPa = –5 107 Pa (z –210–4 V). If |ΔP| is much smaller
(probably, it is more reasonable), z will deviate from
the range.
922
TAKEUCHI AND NAGAO: EMF IN GABBRO UNDER NONUNIFORM LOAD
[30] If dislocations in rock-forming minerals are electrically charged, the movement of which may cause the formation of an electric field in rocks subjected to loading at
higher levels [e.g., Slifkin, 1993]. When the whole volume
of marble blocks was uniaxially loaded, electric currents
were detected with a pair of electrode plates conductively
pasted on the side of the sample perpendicular to the load
axis [e.g., Stavrakas et al., 2003; Vallianatos et al., 2004;
Aydin et al., 2009]. Though the current amplitude was very
small at low stress levels, it increased when the stress level
reached about 60–80% of its failure stress [Vallianatos
et al., 2004]. According to this mechanism, the motion of
charged dislocations will not occur in our loading tests in
which the stress level was only ~25% of its failure stress.
On the other hand, according to our mechanism, loading
the whole volume of rock samples cannot cause the difference of the charge carrier concentration in the volume even
if carriers are activated by the shear stress/strain, so that
the electromotive force will not appear in such a volume.
This is concordant with the fact that the current was not
detected in the whole-volume-loading tests at low stress
levels [Vallianatos et al., 2004; Takeuchi et al., 2011].
4.3. The Electromotive Force in the Crustal/Global Scale
[31] Though only gabbro was treated in this study, we
can expect the mechanism depicted in Figure 7 to occur in
various types of rocks subjected to nonuniform loading.
Actually, the electromotive force induced by nonuniform
loading has been detected in gabbro, granite, and anorthosite, although its amplitude depends on the type of rocks
[e.g., Freund et al., 2006; Takeuchi et al., 2006, 2011].
Therefore, this electromotive force would be considered as
a universal phenomenon. Finally, based on some assumptions, we can apply this electromotive force phenomenon
in the crustal/global scale.
[32] Any major earthquake is one of those natural events
that can dramatically change the distribution of stress/strain
in the Earth’s crust. For example, the analysis of the GPS
network data in Japan indicates a static change of the
maximum shear strain distribution in eastern Japan after
the Mw9.0 Tohoku earthquake in 2011, in which the maximum change reaches ~30 10–6 along the coastline zone
of southern Iwate and northern Miyagi prefectures
[Takahashi, 2011]. Now, we will simply apply our laboratory
results to this situation without taking into account for any
size/shape effect. Considering the finite-element analyses
(Figure 8), the maximum shear strain of the loaded volume
in our experimental setup will reach roughly ~0.5 10–3
when loaded under the 50 MPa pressure. If we assume
that there is a linear relationship between the maximum
shear strain and Vemf (i.e., ~80 mV per ~0.5 10–3) and if
the crustal rocks of the Japanese islands behave similarly to
the vacuum-dried gabbro, we can expect ~5 mV of Vemf as
the shift of the self-potential between the coastline zone and
a far zone after the earthquake. Of course, even if Vemf were
observed between the coastline zone and a zone at a distance
of hundreds kilometers, the actual obtained Vemf will not
reach 5 mV because of differences in the rock type, geometry,
ground water condition, and so on.
[33] Seismic waves dynamically change the local distribution of stress/strain in the Earth’s crust. For example, a large
plate electrode in a mine gallery detected electric signals at
the arrival of S-waves [Okubo et al., 2006]. These signals
were equivalent to the generation of upward-pointing
electric lines of force. One possible mechanism is a positive
electrification of the dry gallery floor. Therefore, this was
proven by laboratory experiments [Takeuchi et al., 2010]:
When one end of an air-dried andesite block, which was
quarried from the same gallery floor, was loaded up to
~8 MPa, the unloaded surface was electrically charged
positive up to ~10–12 C/m2. The electrification can be
explained by the mechanism proposed in this study; the
diffusion of activated hole charge carriers formed an electric
polarization in the block, which charged the unloaded
surface positive. If the local loading/unloading were
repeated quickly, the positively electrified surface would
undergo complicated charging and discharging. Similarly,
seismic waves—especially S-waves that have the component of shear stress/strain—will cause the positive electrification of the ground surface during propagation if the
surface zone is dry.
[34] Earth tides also dynamically change the distribution
of stress/strain in the Earth’s crust. The amplitude of the
Earth tidal shear strain is roughly ~10–8 in the medium longitudes with the period of ~12 h. Now, we again simply apply
our laboratory results to this situation without taking into
account for any size/shape effect. If we can assume a
linear relationship between the maximum shear strain and
Vemf (i.e., ~80 mV per ~0.5 10–3) and if the Earth’s
surface behave similarly to that of vacuum-dried gabbro,
we can expect ~2 mV of Vemf as the shift of the self-potential
with the period of ~12 h between two zones at a distance of
~90 in the longitude direction. Such a small Vemf will be
hidden in the background variations because of other effects.
However, considering the rough estimation of the nh(0)/nh
(L) ratio conducted above, the tidal change in the concentration of hole charge carriers will also occur in the Earth’s
crust. Therefore, very delicate magneto-telluric observations
may detect it as a change of the apparent resistivity.
[35] Both earthquakes and earth tides also change the
states of ground water in the actual ground and of seawater
on the actual Earth’s crust. Therefore, as far as we can
observe the geoelectric field on or under the ground surface,
the electric influence of the mechanism proposed in this
study will be partially or completely canceled by the other
mechanisms such as streaming potential coupled with vibration of the (partially) saturated porous ground [e.g., Pride,
1994; Bordes et al., 2008; Takeuchi et al., 2012], selfpotential by electric dipoles existing vertically at the
water table [e.g., Revil et al., 2003], and electromotive force
by tidal streams of seawater [e.g., Longuet-Higgins and
Deacon, 1949; Olsson, 1953]. But, when we turn our attention to the Moon’s crust where there is no ground water, we
can expect a major contribution of our mechanism to electric
phenomena (e.g., changes in the self-potential and apparent
resistivity) accompanied with moonquakes and moon tides.
If this is the case, we can say that the Moon may be as
electrically dynamic as the Earth.
5.
Conclusions
[36] We confirmed in laboratory that the electromotive
force was generated in vacuum-dried gabbro blocks during
nonuniform loading. Thereafter, we conducted hot point
923
TAKEUCHI AND NAGAO: EMF IN GABBRO UNDER NONUNIFORM LOAD
probe tests of the samples under the same loading condition.
Based on the load-induced change of the Seebeck coefficient
in the loaded volume, we found that hole charge carriers
were activated in the loaded volume. Using the band model
of solid state, we proposed the mechanism of the electromotive force to be induced by nonuniform loading. The shear
component of the load-induced stress/strain, which was
simulated by finite-element analyses, will effectively shift
the acceptor energy levels (and also the “relative” Fermi
energy level) toward the valence band top. This downward
shift makes electrons easily thermally excite to the acceptors. The electromotive force induced by loading is related
to the difference in the “absolute” Fermi energy levels
between the loaded and unloaded edges as the result of two
competing effects: (1) the downward shift of the “relative”
Fermi energy level in the loaded volume and (2) the upward
shift of the band structure (including the Fermi) because of
the electric field caused by diffusing hole charge carriers
and trapped electrons. This mechanism is repeatable in the
range of elastic deformation because the activated hole charge
carriers recombine with the trapped electrons when unloaded.
It is in agreement with the fact that the electromotive force
appeared whenever loading was repeated in this study.
Because this mechanism is expected to work in various types
of minerals, similar electromotive force may have appeared in
the Earth’s crust where the stress/strain changes sub-statically
and dynamically.
[37] Acknowledgments. The authors would like to thank Dr. Izumi
Sakamoto (Tokai University, Japan) for his help on the mineral component
analysis and XRD analysis of the gabbro sample. We would also like to
thank Dr. Ömer Aydan (Tokai University, Japan) and Dr. Naohiko Tokashiki
(Ryukyu University, Japan) for their help on the porosity measurements of the
samples. Comments and suggestions from a reviewer Dr. Friedemann Freund,
an anonymous reviewer, an associated editor, and the editor Dr. Andre Revil
improved our manuscript. Dr. Menchie Montecillo helped proofreading our
manuscript. This study is partially supported by “Observation and Research
Program for Prediction of Earthquakes and Volcanic Eruptions” of the Ministry of Education, Culture, Sports, Science, and Technology of Japan.
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