CHARGE TRANSPORT IN BISMUTH
Jamal Derakhshan, West Virginia University
Arthur F. Hebard, University of Florida
This paper explains preliminary measurements and analysis of magnetoresistance
(change in resistance induced by a magnetic field) studies on two pieces of single crystal
bismuth. The bismuth is cut such that the current flows parallel to the binary axis. Both samples
have purities better than 99.999%. The measurements include temperature sweeps between 2K
and 300K and field sweeps between –7T and +7T. We use curve fitting to the data to create a
phenomenological model for the observed magnetoresistance. We find that the
magnetoresistance varies as T2 instead of the predicted T. In addition, the field dependence
approaches the expected quadratic form at low fields.
Introduction
The aim of this project is to gain an understanding of how the charge carriers, both
positive and negative, behave at low temperatures and high magnetic fields in this semimetal.
Bismuth is unique among elements because of its low intrinsic carrier density compared with
metals and because it is compensated (i.e., it has the same number of electrons as holes). Pure
samples (see Table I below) have less elastic scattering of the charge carriers (off vibrating ions
or static impurities) and a longer scattering length (distance traveled between collisions)
compared with their impure counterparts. The long scattering length causes a dramatic decrease
in resistivity
ρ=
RA
.
l
1
R is the resistance, A is the cross-sectional area of the sample, and l is the distance between the
voltage leads (see Fig. 1). Bismuth reacts unusually to the presence of magnetic fields,
exhibiting a change in resistance of 5 orders of magnitude at low temperatures in a field of 7T
(see Fig. 2).
Past studies of bismuth [1-3] have concentrated on resistivity as a function of magnetic
field at a given temperature, with particular attention paid to the Shubnikov-de Haas effect, i.e.,
the existence of magnetic-field-dependent oscillations in the resistivity at high magnetic fields
and low temperatures. Although we have pursued similar analyses, we focused primarily on
measuring resistivity as a function of temperature at fixed magnetic field.
It is beyond the scope of this paper to describe in detail the crystal structure of bismuth;
however, it is necessary to mention that bismuth has one of the most complex crystal structures
of any element. It has both a binary and a bisectrix axes lying in a common plane and a trigonal
axis perpendicular to that plane. We used magnetoresistance measurements to characterize two
samples of crystalline bismuth cut such that the current flowed in the binary direction with
varying magnetic fields (-7T to +7T) applied in the trigonal direction at temperatures between
2K and 300K.
From the magnetoresistance measurements, we determined our own model for resistivity
as a function of temperature and magnetic field. We started by fitting the data to very simple
models, which we then generalized to explain all of our low-field data. The measurements
described above were then compared with the theoretical 2-band model of resistivity in bismuth.
2
Methods
We cut crystals of bismuth in specific orientations and measured their electrical responses
in the geometry shown in Fig. 1. Table I lists the characteristics of the two crystals which we
analyzed.
L
W
I+
V+
V-
V-
I-
Transverse
Longitudinal
FIG. 1. Schematic view from top of sample showing placement of voltage and current leads and the labeling of the
dimensions and directions. The applied magnetic field comes out of the paper.
Sample
Sample 1 (BiRGB2)
Sample 2 (BiXuJd01)
Orientation
binary
binary
Dimensions
1.471mm-1.548mm-1.548mm
2.8mm-3.6mm-4.0mm
Measured using microscope
Measured using calipers
99.9995%
99.9999%
W-H-L (see Fig. 2)
Purity
TABLE. I. Properties of analyzed samples.
The measurements were taken in the Physical Property Measurement System (PPMS)
manufactured by Quantum Design Inc. The sample space consisted of a 1-inch diameter
cylinder kept at 1 torr pressure of helium exchange gas. This cylinder is enclosed by a liquid
helium Dewar (thermos for cryogenic materials) which itself is enclosed by another Dewar filled
with liquid nitrogen. The sample space is cooled either by the flow of gaseous helium (above
4K) or by pumping on liquid helium (between 2K and 4K).
3
The measurements were computer-controlled and did not require constant intervention by
an operator. This set-up enabled us to take many data points and analyze them as soon as they
became available. We used two types of sequences to analyze the samples. The first was a
temperature sweep between 2K and 300K holding the magnetic field constant at a value between
0T and +7T. Then the magnetic field was set to another constant value and another temperature
sweep was performed. The second type of sequence was a magnetic field sweep from –7T to
+7T while holding the temperature constant. The temperature was set to another constant value
and the measurement repeated. We chose the following temperatures for field sweeps: 2, 5, 10,
15, 20, 30, 50, 80, 130, 210, and 300K.
Although this paper is limited to the measurement, analysis, and results of the
longitudinal resistivity, measurements of the longitudinal and transverse (or Hall) voltages were
simultaneously taken (see Fig. 1). These measurements were necessary in order to understand the
principles underlying the behavior of the longitudinal resistivity. The Hall voltage is a potential
difference caused by accumulation of charge at the sides of the crystal due to currents flowing in
the presence of a magnetic field perpendicular to the longitudinal current.
Results
Figure 2 shows the results of temperature sweeps of the longitudinal resistivity taken for both
samples.
4
0-0.1T in 0.01T steps
0-0.1T in 0.01T steps
1E-5
resistivity (ohm-m)
resistivity (ohm-m)
1E-5
1E-6
1E-7
sample: BiRGB2
1E-8
1E-6
1E-7
sample: BiXuJd01
1E-8
0
50
100
0
temperature (K)
50
100
temperature (K)
FIG. 2. Log-linear plot of longitudinal resistivity of each sample as a function of temperature at various magnetic
fields. Data were taken from 2K to 300 K; however, they are shown only up to 100 K for clarity. In each graph, the
bottom curve corresponds to B=0.
We modeled these data in the following way. The normalized resistivity [resistivity –
resistivity (B=0)] at some finite field was plotted against temperature. These curves were then
individually fit to the following simple equation:
ρ − ρ ( B = 0) =
P1
.
1 + P2T 2
The fitting program iterated until chi squared was no longer reduced. In this way, the best values
for P1 and P2 were determined. Our values of R2 varied from 0.9911 to 0.99829 for sample 1 and
from 0.99339 to 0.99986 for sample 2 (R is the correlation coefficient). The data were fitted
only from 5K-300K in this part of the experiment due to the flattening out of the data below 5K.
The T2 term in the denominator was chosen because it surprisingly fit the temperature sweeps
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much better than the T-dependence predicted by 2-band conduction theory (See discussion
section).
To determine the dependence of P1 and P2 on the applied magnetic field, we plotted these
coefficients vs. B on log-log axes and found that both parameters had a power-law dependence
on B. The slope of the resulting line determined the power-law exponent for each parameter.
This procedure is illustrated for both samples in Figs. 3 and 4. Interestingly, the coefficients
obey relationships of the form P1 ∝ B n−η and P2 ∝ B −η , where η is different for sample 1 and
C o e ffic ie n t
2 .5
2 .0
1 .5
1 .0
P 2
P 1
C o e ffic ie n t
sample 2; however, n = 1.95 ≈ 2 in both cases.
0 .5
0 .0
- 0 .5
0 .0 2 6
0 .0 2 4
0 .0 2 2
0 .0 2 0
0 .0 1 8
0 .0 0 0
0 .0 0 6
0 .0 1 2
0 .0 1 4
0 .0 1 8
(T ^ 1 .7 1 )
1 .6
1 .8
2 .0
2 .2
2 .4
C o e ffic ie n t
1 / F ie ld ^ 0 .2 3 6
1
2 .6
2 .8
3 .0
3 .2
(1 / T ^ 0 .2 3 6 )
0 .0 5
0 .0 4
0 .0 3
0 .0 2
P 2
C o e ffic ie n t
0 .0 2 8
0 .0 1 6
F ie ld ^ 1 .7 1
P 1
0 .0 3 0
0 .1
0 .0 1
0 .0 2 5
F ie ld
0 . 0 5 0 . 0 7 50 . 1
0 .0 1
0 .2 5
(T e s la )
FIG. 3. Coefficients for sample 1. P1 is proportional to
0 .0 1
0 .0 2 5
F ie ld
B1.71±0.01 and P2 to B −0.24±0.01 .
6
0 . 0 5 0 . 0 7 50 . 1
(T e s la )
0 .2 5
C o e f f ic ie n t
c o e f f ic ie n t
1
2 .5
2 .0
1 .5
P 1
P 1
1 .0
0 .1
0 . 0 00 . 80 1
0 .0
0 .0 2
0 . 0 4 0 . 0 06 . 0 08 . 1
0 .2
0 .0 0 0 0 .0 0 5 0 .0 1 0 0 .0 1 5 0 .0 2 0 0 .0 2 5
F ie ld ^ 1 . 5 4 7
C o e f f ic ie n t
( T e s la )
0 .0 6
0 .0 5
0 .0 4
0 .0 3
0 .0 2
P 2
C o e f f ic ie n t
F ie ld
P 2
0 .5
( T e s la ^ 1 .5 4 7 )
0 .0 4 5
0 .0 4 0
0 .0 3 5
0 .0 3 0
0 .0 2 5
0 .0 2 0
0 .0 1 5
0 .0 1 0
0 .0 0 5
0 .0 0 0
-0 .0 0 5
0 .0 1
0 . 00 0. 060 0. 0
8 1
0 .0 2
0 . 0 40 . 00 6. 008. 1
F ie ld
0 .2
0 .4
-0 .0 1 0
( T e s la )
FIG. 4. Coefficients for sample 2. P1 is proportional to
-2
-1
0
1
2
3
1 /( F ie ld ^ 0 . 4 0 9 )
4
5
6
7
8
(1 /(T ^ 0 .4 0 9 ))
B1.55±0.01 and P2 to B −0.41±0.01 .
As a final test of our fitting equations, we superimposed our parameterization of the
normalized resistivity on our original data. As can be seen from Fig. 5, the agreement is
excellent over the entire range of temperatures and from 0.01T-0.1T.
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Resistivity - Resistivity (B=0)
R=((130)*(B^1.711)*(10^-5))/(1+0.00963*(B^-0.236)*(T^2))
0.000016
0.01-0.09 T in 0.01 T Steps
0.000014
0.000012
0.000010
0.000008
0.000006
0.000004
sample: BiRGB2
0.000002
0.000000
0
20
40
60
80
100
Temperature (K)
Resistivity- Resistivity (B=0)
R=(103*(B^1.547)*(10^-5))/(1+0.00532*(B^-0.409)*(T^2))
0.000018
0.000016
0.000014
0.01T-0.09T in 0.01T steps
0.000012
0.000010
0.000008
0.000006
0.000004
Sample: BiXuJd01
0.000002
0.000000
0
20
40
60
80
100
Temperature (K)
FIG. 5. Normalized resistivity of each bismuth sample vs. temperature at various magnetic fields. Points show
measured values, red lines plot the result of the fitting equations (shown above each graph). Resistivities are given
in units of ohm-m. In each graph the bottom curve corresponds to B=0.01T.
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Before discussing our results, we should explain why we have parameterized the
normalized resistivity instead of the resistivity ρ (B) itself. The reason for this is that ρ (0) has
significant deviations from a linear-in-temperature behavior. There is enough of a deviation
from a straight line that our experimentally determined equations do not describe the total
resistivity as closely as they fit the normalized data. We therefore chose to fit just the
magnetically induced resistance at each temperature—the quantity we were immediately
concerned with. A better approximation to the resistance at zero field can be calculated by the
Bloch-Grüneisen method; however, this analysis was not performed here.
Discussion
We found that the normalized magnetoresistance varies as T-2 –a fact that is completely
unexpected. The theoretical 2-band model for a compensated metal predicts a longitudinal
resistivity
ρ ( B) =
ρ1 ρ 2
R2 B2
,
+
ρ1 + ρ 2 ρ1 + ρ 2
where ρ1, 2 are the resistivities of the electrons and holes, and R is the Hall coefficient.
According to this model, if ρ1, 2 are linearly dependent on T, then in the absence of a magnetic
field the resistivity is linear in T. This result is confirmed in our experiment. However, this
model also predicts a magnetoresistivity of the form
A1 B 2
.
ρ − ρ ( B = 0) =
1 + A2T
Our experiment rules out this functional form; both our samples obey
ρ − ρ ( B = 0) =
A1B 2−η
A1B 2
.
≡
1 + A2T 2 B −η Bη + A2T 2
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Note that as B approaches zero, a B2 dependence is recovered. This is important for three
reasons. First, it agrees with the fundamental symmetry expected for the parabolicity of the
magnetoresistance at low fields. Secondly, it fits all our data. This is surprising because we
could not have seen it without doing our temperature sweeps. If we simply tried to fit the field
sweeps, we saw different exponents of the magnetic field and obtained fits that were not nearly
as good. Finally, this model may explain why previous researchers have seen deviations from
parabolicy at low fields: they may not have looked at low enough fields to see the B2 dependence
[4].
Our results seem to indicate that the magnetoresistance of bismuth is not well described
by the straightforward application of 2-band theory. We speculate that the reason may be a
difference between the temperature-dependent scattering rates of the electrons and of the holes
(i.e., that one type of carrier dominates the conductivity). However, we do not yet have a
complete theory to support this picture. More work is needed in this direction.
Understanding how bismuth acts in moderate magnetic fields ( ≤ 7T ) and at low
temperatures ( T ≥ 2 K ) is the preliminary step before considering quantum effects predicted at
high magnetic fields (30T) and very low temperatures (20mK). At those extremes, the charge
carriers in a low-carrier-density material such as bismuth or graphite are predicted to behave in a
one-dimensional manner because they travel in cyclotron orbits in the presence of large magnetic
fields [5]. At high enough magnetic fields, the diameter of these orbits can become smaller than
the distance between charge carriers. The charged particles still interact via the Coulomb
interaction; however, they can be treated as one-dimensional objects. This phase is analogous to
having separated conducting “wires” within the three-dimensional object. In this state, the
charge carriers may comprise a postulated “Luttinger-liquid phase.” Understanding the response
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of bismuth to magnetic fields and temperatures in a classical regime will enable us to proceed to
the next level of experimentation. We hope to perform measurements at the National High
Magnetic Field Laboratory where it is possible to test samples at magnetic fields up to 30T and
temperatures in the 20mK range.
Acknowledgments
The REU program is sponsored and supported by the National Science Foundation
(NSF). One big thank you to all members of Dr. Hebard’s laboratory: Quentin Hudspeth, Kevin
McCarthy, Nikoletta Theodoropoulou, Rick Piciullo, Jeremy Nesbitt, Dr. Steven Arnason for
explaining much of the material, and Xu Du for preparing the bismuth samples. Dr. Dmitri
Maslov and Gregory Martin worked out the theoretical 2-band model. Drs. Kevin Ingersent and
Alan Dorsey organized and administered the Research Experiences for Undergraduates (REU)
program at the University of Florida.
References
[1] Y. Iye, L. E. McNeal, and G. Dresselhaus, Phys. Rev. 135, 1118 (1964).
[2] R. D. Brown III, Phys. Rev. B 2, 928 (1970).
[3] E. M. Lifshitz, and L. P. Pitaevskii, Physical Kinetics (Nauka, Moscow, 1979).
[4] A. B. Pippard, Magnetoresistance in Metals (Cambridge University Press, New York, 1989).
[5] C. Biagini, D. L. Maslov, M. Yu. Reizer, and L. I. Glazman, preprint (cond-mat/0006407).
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