Recently, the Drude scattering rate τ−1(p, T )has been measured by Armitage et al. [12]. This al-
lows us to attempt an estimate of the low-temperature carrier density, and compare our data to the observation by Armitage et al. that the scattering rate is strongly enhanced at low temperatures. This requires a few significant approximations, which we lay out here. First, recall that in the Drude model, the resistivity is:
ρ(p, T ) = mτ
−1(p, T )
Semimetal-to-semiconductor transition in Bi-I 4.5 Analysis
0
10
20
Pressure (kbar
0
10
20
30
40
50
60
Temperature (K)
Bi #6.1 Bi #7.1Figure 4.33: Pressure dependence of the kink temperature Tk (small open markers) and peak tem-
perature Tmax(large solid markers), for samples in PCC10. Values Tmax = 0imply no fall in the
resistivity was observed down to 2 K. Vertical dashed lines at 18 and 21 kbar indicate the approxi- mate region of the transition from a kink to a peak.
Typically, for the resistivity of metals we assume that the temperature-dependence of the effective mass is negligable - that is, the band structure does not change curvature with temperature. In the case of bismuth this is debatable (see Norin and references therein [56]); in particular, as applied pressure shifts the Fermi level to lower and lower values, the non-parabolic electron conduction band may well have an effective mass that changes with pressure.
Secondly, we are totally neglecting any anisotropy in the resistivity. The data of Armitage et al. do not give different scattering times for the different crystallographic directions. Recent careful mea- surements of the conductivity tensor for bismuth as a function of angle suggest that the mobility is highly anisotropic, but less anisotropic than would be expected based on the anisotropy of m alone [58]. This implies that τ−1itself is also highly anisotropic, but in such a way that the product mτ−1is
more isotropic than m. This anisotropy is irrelevant for our measurements, as we do not have a good understanding of the relation of our current direction to the crystallographic directions, due to twinning in our crystals.
Finally, we are assuming that the Drude scattering rate measured optically by Armitage et al. is the same rate probed by DC resistivity measurements. Armitage et al. argue this is the case, in that the shape of τ−1they observe corresponds quite well to the resistivity results of Balla and Brandt;
additionally, the size of the zero-pressure, high-temperature scattering rate would appear to agree quite well with experimental values of the DC resistivity. Our calculations (e.g. Fig. 4.20) suggest this assumption likely does not hold.
Here, we neglect complexities arising from anisotropy of both m and τ−1, and calculate n(p, T )
from our resistivity data and the Drude scattering rate of Armitage et al. The resistivity ρ(p, T ) is taken from our data; the scattering rate τ−1(p, T )from Armitage et al. [12]. Armitage gives a scat-
Semimetal-to-semiconductor transition in Bi-I 4.5 Analysis
0
100
200
300
Temperature (K)
0
2
4
6
8
10
τ
-1(s
-1)
×10
120.0
6.5
12.0
20.5 kbar
Armitage et al.
Optical scattering
Figure 4.34: Temperature and pressure dependence of the scattering rate, as obtained from the optical conductivity data of Armitage et al. [12]. Solid points are taken from their data; smooth lines are interpolating cubic smoothing splines.
tering rate in meV; we convert to a frequency with: τ−1= 10−3e
~
τA−1 (4.21)
where τA−1 is Armitage’s numerical value in meV (the other factors ensure our scattering rate has dimenions of inverse seconds).
Our values for the scattering rate as measured by Armitage et al. are given in Fig. 4.34. The datapoints are rather sparse (with only ∼ 10 temperatures measured at each pressure point), so we fit a 1000-point cubic smoothing spline through the Armitage data. This allows values of τ−1to be
obtained for 10 < T < 300 K using linear interpolation without unphysical sharp corners appearing in the analysed data.
From this method, we can rearrange Eq. 4.20 to obtain the carrier density n(T ) at a given pres- sure. We use the measured pressures for our resistivity traces, taking pressure points as close as possible to Armitage’s values, and linearly interpolate on the two-dimensional T − p surface taken from the smooth curves fitted to Armitage data to obtain an estimate for τ−1(T )at the pressure
where our resistivity data was taken. To obtain a quantitative ρ we must choose a value for m; we estimate this from Armitage’s τ−1by assuming that, at p = 1 bar, the carrier density increases by a
factor of 8× from 2 to 300 K, based on the data of Issi [8], and that at 1 bar and 300 K the resistivity is 120 µΩ cm. This gives m = 0.011me. As we have discussed previously, different authors give n(T ) in-
creasing by a factor of 4 − 9×, so the correct value of m is quite uncertain. However, this contributes only a single global scaling of n and does not affect the qualitative results.
Fig 4.35 plots the measured resistivity, the measured and interpolated data from Armitage, and the resulting carrier density. Plainly, the carrier density extracted from this process is not physically
Semimetal-to-semiconductor transition in Bi-I 4.5 Analysis
reasonable: it typically shows a dramatic fall as T increases. This can be easily understood by a comparison between the resistivity and Armitage’s data at e.g. p = 6.7 kbar. Armitage observes a fall in τ−1(T )as T increases; however, our resistivity is monotonically increasing. If Armitage’s
results are correct, then (because ρ ∼ τ−1/n) we must have an n that is steeply falling. There does not appear to be a mechanism for this to occur. The high-temperature behaviour of n(T ) extracted from this process, by contrast, is quite reasonable. It typically falls with pressure, by a factor of ∼ 2× from 0 to 20 kbar.
Armitage suggested that their dramatic enhancement in τ−1arose from the formation of some
exotic phases when the carrier density becomes particularly low, and they argue that the shape of their curves for τ−1approximately match the resistivities of Balla and Brandt. Our resistivity data
does not support this hypothesis. Armitage et al. observe an enhancement in τ−1at low T even at only 6.5 kbar, while our own data, and that of Balla and Brandt, only indicates metallic resistivity at this pressure. For the magnitudes of the resistivity and scattering to be consistent, we would need minimal change in the carrier density n(T ), but our calculations indicate this does not hold - at high pressures, the carrier density is strongly temperature-dependent. One possibility could be that the Drude scattering rate estimated by Armitage is very distinct from that probed by the resistivity; why this should be the case is unclear.