Bipolaron Hopping Conduction in Boron Carbides
T. L. Aselage,1>2D. Emin,2 and S. S. McCready’
1Sandia National Laboratories, Albuquerque, NM 87185-1421,
‘Dept. of Physics and Astronomy, University of New Mexico 87131-
PACS: 71 .38.+i, 72.20. Pa, 72.20.-1, Substance Class: S5
Abstract
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The electrical conductivities of boron carbides, BIZ+XCS.Xwith 0.1< x <1.7, between 300 and
1200K suggest the hopping of a nearly temperature-independent density of small (bi)polarons.
The activation energies of the nobilities are low, = 0.16 eV, and are nearly independent of the
composition. At lower temperatures, conductivities have non-Arrhenius temperature
dependencies and strong sensitivity to carbon concentration. Percolative aspects of low-
temperature hopping are evident in this sensitivity to composition. Boron carbides’ Seebeck
coefficients are anomalous in that 1) they are much larger than expected Ilom boron carbides’
large carrier densities and 2) they depend only weakly on the carrier density. Carrier-induced
softening of local vibrations gives contributions to the Seebeck coefficient that mirror the
magnitudes and temperature dependencies found in boron carbides.
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Boron carbides, B12+XC3.Xwith 0.1< x s 1.7, are refractory solids having distinctive
structures and bonding. Fifteen-atom unit cells
comprise twelve-atom, (B IlC), icosahedral
clusters and three-atom chains. At an idealized stoichiometry of B12C3,x = O,each [CBC] chain
donates an electron, -+ [CBC]+, to fill the internal bonding orbitals of a (B] IC) icosahedron, +=
(BIIC)-. This idealized composition, (B1lC)-[CBC]+, is an electrical insulator. In realizable
boron carbides, x >0, hole-like charge carriers are produced when B atoms substitute for C
atoms. The value of x thereby fixes the carrier density. Because x is large, boron carbides’
intrinsic carrier densities are large:
= 1021/cm3. Measuring low, activated Hall nobilities and
electrical conductivities in boron carbides indicated that these charge carriers move by polaronic
hopping. [1] However, a large paramagnetic susceptibility commensurate with boron carbides’
high carrier densities was not found. [2] In addition, recent measurements of boron carbides’
low-temperature Seebeck coefficients [3] do not find the strong magnetic-field dependence that
is expected of unpaired hopping carriers. These experiments indicate that charge carriers in
boron carbides pair to form singlet bipolarons.
Ernin [4] shows that carrier-induced softening can stabilize singlet pairs occupying
degenerate orbitals, such as the frontier orbitals of icosahedra. Bipolarons in boron carbides are
believed to occupy icosahedral sites. A simple model relates the concentration of bipolarons to
the C concentration. [5] As x increases from zero, structural and vibrational measurements [6]
and free energy considerations [5] suggest that B atoms first replace C atoms within chains, CBC
+ CBB. A CBB chai~ being isoelectronic to a [CBC]+ chain, does not donate an electron to an
icosahedron. Pairing of the resulting holes may be viewed as an icosahedral disproportionation:
2(B11C)0 + (B1lC)- + (B1lC)+, where (B1lC)+ represents a bipolaronic hole. The (Bl lC)+ density
rises with increasing CBB chain substitution, reaching a maximum when each chain is CBB at x
= 1, 13.3 O/OC.The (Bl lC)+ concentration decreases with further reductions in C concentration as
donating &lBB]+ chains progressively replace CBB chains. [6,7] The bipolaron density is thus a
peaked fimction of C concentration, reaching its maximum value (!Aof the (Bl lC) sites) at 13.3
Yoc.
High-temperature (300 - 1775K) conductivities of boron carbides with compositions
spanning the single-phase region are plotted in Fig. 1. These conductivities are proportional to
the densities of bipolaronic holes described by the model; peaking of the conductivity near 13.3
YoCis apparent. Additionally, a nearly Arrhenius mobility is evident in the near linearity of
log(aT) vs. l/T between 300 and 1200K. A slight curvature near 500K in log(uT) of high-
conductivity samples is suggestive of the activation and subsequent exhaustion of some source of
additional charge carriers. The low activation energy of the mobility, = 0.16 eV, is consistent
with the motion of “softening” bipolarons. [4] At very high temperatures, above 1200K, an
abrupt increase in the apparent activation energy, to about 0.6 eV, indicates the onset of another
conduction mechanism.
Fig. 2 displays the conductivities of boron carbides from 300K to below 10K. In this
regime, conductivities have both non-Arrhenius temperature dependencies and strong sensitivity
to C concentration. When compositions are near 13.3 YoC,bipolaronic holes move among
icosahedral sites having nearly uniform surroundings of neutral, CBB, chains. Non-Arrhenius
conductivities under these circumstances result primarily from the freezing out of multi-phonon
processes. [8] As the C concentration departs from 13.3% toward either of its extremes, neutral,
CBB chains are replaced by positively charged, [CBC]+ or [BBB]+, chains. Coulomb repulsion
renders an icosahedral site amid positively charged chains less favorable for a bipolaronic hole
than sites amid neutral chains. The number of favorable icosahedral sites thus becomes