Richard Mann
This is a report of the experiment SS06 of the Oxford Physics Practical Course. This report describes the details of the experiment, the results obtained, comparisons with expectations from theoretical models and conclusions drawn.
The focus of the experiment was an investigation of the band structure of Bismuth. A large magnetic field was used to quantize two components of the crystal momenta of the electrons in levels obtained as a function of the field. By measuring the resistance of the material as the field was altered in strength or direction it was possible to observe these quantised levels leaving the Fermi surface and thus emptying of electrons by observing peaks in the resistance. As will be shown in the theoretical considerations the frequency at which this occurs with relation to field strength can be shown to be related to the shape of the Fermi surface, allowing the surface to be mapped by a study of these oscillations.
Measurements were able to show a periodic anisotropy in the resistance at constant field strength, corresponding to the two axis of symmetry in the observed plane of the first Brillouin zone (Binary and Bisectrix). Periodic variations were also observed in the resistance with increasing field strength, corresponding to expectations, but experimental conditions did not allow a quantitive treatment of these oscillations to determine the shape of the Fermi surface.
The Shubnikov-de Haas effect was first observed by L.Shubnikov and W.J. de Haas in 1930 in experiments carried out using Bismuth. It has been used to measure the Fermi surfaces of various materials. Observationally it is a oscillatory signal in the magnetoresistance of a sample as a result of allowed energy levels moving out of the Fermi surface.
The effect is a result of the band structure of materials. For this experiment we took as known that Bismuth has overlapping conduction and valence bands such that a population of electrons exists in the conduction band and similarly a population of holes exists in the valence band down to absolute 0. Also taken as assumed knowledge was the shape of the first Brillouin zone, which is shown below in a diagram taken from the introduction of the experiment manuscript (Appendix A)
Figure 1a: The 1st Brillouin zone in Bismuth
Figure 1b: The projection of the electron and hole Fermi surfaces onto the trigonal plane, showing distortions in an applied field.
The magnetoresistance of the sample was studied in a plane perpendicular to the trigonal plane (as indicated on the diagram). The sample was cooled using liquid helium to minimise thermoelectric effects and to allow neglect of thermally excited electrons in the conduction band. Thus we could, to a high level of approximation, be sure that all the electrons were within the Fermi surface at all times.
Fig 1b shows the observed plane of the first Brillouin zone, along with the Fermi surface of the electrons. It also shows how the Fermi surface can be expected to deform when the field is applied along both the bisectrix and binary axes of symmetry. As can be seen in the diagram, application of the field in the binary direction leaves far more electrons in the conduction band than when the field is applied along the bisectrix, therefore we can expect to see minima of resistance when the field lies along a binary axis, and maxima when the field lies along a bisectrix axis.
First we establish the conditions of quantisation in the plane. Considering the Lorentz force:
F = qv × B (1) (Ref. 1)
where v = (vx, vy, vz)
by considering the equations of motion in the x and y directions and making the substitution
η = (x + iy) (2)
we establish a solution for η of identical form to the usual simple harmonic oscillator problem (Appendix B).
η = η0exp(iqB/m*)t (3)
By analogy with the simple harmonic oscillator solutions we thus know that the energy eigenvalues in this system are:
E = (L + ½) ħωc (4)
where ωc = (qB/m*) and L takes integer values
This is the energy due to motion in the x, y plane, so adding the energy of motion along z:
E = (L + ½) ħωc + ħ2kz2/2m* (5)
This condition upon the energy levels restricts the values of k in the x, y plane such that:
(ħ2/2m*)(kx2 + ky2) = (L+ ½) ħωc (6)
So in k space the electrons exist on sets of cylinders defined by (6), with kz retaining its usual closely spaced values due to the finite size of the sample. This is shown in the diagram below taken from the manuscript:
Figure 2: x, y energy quantisation creates cylindrical allowed values of k.
As B is increased these cylinders increase in diameter, as shown by (6). When a given cylinder reaches a point where:
EF = (L + ½) ħωc (7)
It will encounter the edge of the Fermi surface and empty itself of electrons. This evacuation of electrons will produce a change in the resistance of the sample, and as the energy value on a cylinder is proportional to B, this should occur periodically in 1/B. This can be seen by considering the density of states of the electrons. The density of states has a sharp peak at the energy where the cylinders emerge from the Fermi surface, and thus the scattering probability is also sharply peaked, leading to peaks in resistance at this point, and this field (see Appendix A, section 2.3).
The theory above shows we can expect to see periodic variations in the resistance signal as the field is increased, periodic in 1/B, with peaks at field strengths given by (7). We also expect to see an anisotropy in the magnetoresistance due to the particular Brillouin zone structure of the sample, with minimum and maximum magnetoresistances for a given field strength occurring along the bisectrix and binary axis of symmetry respectively, and thus being separated by 30 degree intervals.
As mentioned in the introduction, helium was used to cool the sample to 4.2K. This was to achieve a condition where thermally excited electrons near the Fermi energy were negligible, and where broadening of the cylinders due to the magnetophonon effect (see manuscript of SS05) was small compared to the cylinder separation, maintaining the necessary quantisation in the plane.
For later experiments the helium was pumped to a low pressure of ~ 40mbar to examine the periodic structure of the oscillations
To investigate the periodic signal in the magnetoresistance the field generating magnet was set at a constant angle, and then the current supply was used to smoothly sweep the field from 0T to ~1T. A voltmeter across the sample then sent a signal proportional to the resistance to a computer for analysis.
To investigate the anisotropy of the magnetoresistance a means of continuous rotation of the field was unavailable, so a continuous plot could not be achieved. Instead measurements of the resistance were taken at 5 degree intervals through a 180 degree range, and the results then plotted. This technique limited the accuracy of determining minima and maxima of the resistance to within a 10 degree range. To determine the bisectrix and binary axes to within a single degree the periodic effect in the increasing field was investigated through the angles in this range, the analysis of which is described in the results.
For the analysis of the results captured on the computer the Origin(TM) graphing and analysis program was used.
The results produced are best shown in the attached graphs. The following is a description of what these graphs detail. All angles recorded correspond to measurements taken against an angular scale attached to the field generating magnet setup, and obviously entail an arbitrary 0. Analysis centres on the differences between measured angles of the two principle axes of the crystal.
During the sweeps of the magnetic field referred to below it was found that to a first approximation the magnetoresistance increased polynomially with field. To examine the oscillatory structure of the signal a second order polynomial fit was made to the plots using Origin(TM) functions. This was then subtracted from the results to leave only the desired oscillations.
Graph 1: Anisotropy of the magnetoresistance in 5 degree steps
This graph was produced by measuring the resistance of the sample in a 180 degree range in 5 degree steps and plotting those points. The minima and maxima as determined to a 10 degree accuracy are shown and labelled as the bisectrix and binary axes of symmetry.
A minimum, corresponding to the binary axis, was located in the range 100-110 degrees, and a maximum, corresponding to a bisectrix axis, was located in the range 130-140 degrees. These extrema were selected for further analysis.
Graph 2: A study of the the periodic effect around the bisectrix axis
Sweeps of the field were examined and the resistance plotted as a function of field for angle in an around a maximum in the angle dependent resistance, indicating the bisectrix axis. The angle corresponding to the sharpest oscillations was identified and recorded as being the angle of the bisectrix axis. It was found to be 133 degrees, the sharp oscillations identified being shown on the graph
Graph 3: A study of the the periodic effect around the binary axis
Sweeps of the field were examined and the resistance plotted as a function of field for angle in an around a minimum in the angle dependent resistance, indicating the binary axis. The angle corresponding to the sharpest oscillations was identified and recorded as being the angle of the binary axis. It was found to be 103 degrees, the sharp oscillations identified being shown on the graph
Graph 4: A study of the periodic structure examined along the binary axis at low temperature and pressure
The liquid helium was pumped down to ~40mbar to increase the oscillatory effect, and a sweep of the field was made, with the magnet fixed along the binary axis. The graph shows the periodic structure against B. The principle maxima have been marked for analysis
Graph 5: Peak number against 1/B for the binary axis
Taking the recorded field positions of the peaks labelled on graph 4 the peak number was plotted against 1/B to examine whether predictions that the peaks are periodic in 1/B were correct.
Graph 1 gives a clear indication of periodic anisotropy in the magnetoresistance, and to the greatest accuracy possible using the graph the period is 20-40 degrees between a minimum and a maximum, which corresponds to predictions that mimina and maxima should occur at 30 degree intervals due to the axis of symmetry in the first Brillouin zone. When examined at 1 degree intervals in the possible ranges of minima and maxima (Graphs 2 and 3) they were found to be 30 degrees apart, though the conditions for measuring this, by judging by eye at which angle the sharpest oscillations occured, is less than totally satisfactory. Nonetheless the available results support the expectations from theory.
Results here were sparse and of dubious merit. On graph 4 the principle peaks are marked, however the choice of what constitutes a peak is at best difficult and arguably arbitrary. With only 4 points chosen there is also the possibility that errors could be of a magnitude to render the interpretation of the results invalid. However, with those 4 points plotted against the inverse field as shown on graph 5 we get a surprisingly good fit to the expected straight line. The conclusion must be that if our peaks have been chosen correctly (and the closeness of the fit suggests they have) then there is experimental support for the theory that the oscillations are periodic in 1/B.
Experimental results were able to show the expected minima and maxima in the magnetoresistance against angle that can be interpretted as the binary and bisectix axes of the trigonal plane of the first Brillouin zone. These were seperated by the 30 degree intervals expected from the six fold symmetry of that plane.
There was also experimental support for the theorectical expectation that the quantisation of the energy levels in the plane according to equation (4) would lead to a oscillatory signal in the magnetoresistance that would be periodic in 1/B. However, the results used to support this were weak and therefore this is not a referable confirmation of the predicted period of the Shubnikov-de Haas effect.
(1) The Feynman Lectures on Physics II (Chap. 1, p 1-2), Feynman, Leighton, Sands
See attached manuscript
Solutions to the simple harmonic oscillator problem
For a potential given by:
V = kx2
The solution to the equations of motion is
x = x0exp(i.(k/m)½t)
and quantum mechanically the energy eigenvalues are
En = (n + ½)ħ(k/m)½