PHYS 4315 THE CLASSICAL HALL EFFECT
THE CLASSICAL HALL EFFECT
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PHYS 4315
THE CLASSICAL HALL EFFECT
measured by the lock-in amplifier
Introduction to the lock-in amplifier
Introduction to the classical Hall effect
Measuring the classical Hall effect on n-type and p-type Ge
NOTE: It will be useful to read through this lab guide first. Pay attention to
section 5: Data analysis and content of reports, so that you can plan your data
acquisition and subsequent data analysis and literature research accordingly.
The Lock-in Amplifier
Introduction
It is often necessary to detect a miniscule signal, typically a voltage signal, embedded in a large
amount of background noise. The lock-in was invented for that reason: to detect faint electrical
signals buried in electrical noise. Since practically all measurements are ultimately reduced to an
electrical signal, the lock-in is an important instrument, and is widely used in many forms. Noise
is characterized by a frequency spectrum (figure below), which tells you what the magnitude is
of the noise when measured at a particular frequency, or more accurately in a narrow bandwidth
around a certain frequency. Broadly speaking, noise may be categorized into one of three
groups: interference, 1/f noise, and Johnson noise. Once we understand these three types of
noise, we can see how lock-in detection may be used to reduce them. The idea of the lock-in is
quite simple: from among all the frequencies in a frequency spectrum polluted by noise, listen to
one particular frequency (and phase), in a narrow bandwidth, at which you know your
experiment is driven. It is akin to humans recognizing one voice from among a loud mishmash
of acoustic noise.
Interference
This is likely the first type of noise that comes to mind, and simply put, it is noise due to human-
made sources: 60 Hz, 120 Hz, 180 Hz noise from power lines, television and radio signals in
MHz and GHz, digital switching noise in electronics, switching phenomena in computer and
lighting power supplies, etc. all contribute to interference. This noise tends to occur in a narrow
bandwidth around specific frequencies. Fortunately, interference is relatively easy to avoid
simply by choosing a frequency other than one of these known frequencies and performing the
measurement in a manner that uses knowledge of the interference frequency and of the driving
frequency. The solution could be as simple as using a band pass filter. Or as we will see, the
lock in amplifier provides a solution as well.
1/f noise
It turns out many noise sources have a frequency spectrum that approximately varies as 1/f (large
amplitudes occur relatively infrequently, smaller amplitudes more frequently). This noise
originates from a variety of sources. For our purposes we need not concern ourselves with why
the 1/f relationship exists, but only to accept that it does and choose a frequency for our
measurement so that we minimize this noise. At a glance it appears that this can be
accomplished by choosing a high frequency, but in actuality we need only to use a frequency
such that the 1/f noise is much less than the ‘white’ Johnson noise (below). This happens around
1 kHz in many cases, but is dependent on experimental factors such as temperature and
measurement bandwidth.
Johnson noise
Johnson noise, also called thermal noise, is always present since the apparatus is always at some
temperature above absolute zero. This noise can be derived by considering a long lossless
transmission line in series with a resistor, applying equipartition and equating the power flowing
through the line with the power dissipated by the resistor. This yields the result:
= √4 ∆
with R the resistance of the resistor, k the Boltzmann constant, T the temperature in Kelvin, and
∆ the frequency range (bandwidth) over which the measurement is sensitive. For instance, if
you use a voltmeter with a low-pass filter that measures signals from DC to 30 Hz, then ∆ = 30
Hz. Johnson noise is independent of frequency and is therefore called white noise (the color
white is composed of all frequencies). Note that it is proportional to bandwidth. Note also that
you can normalize to bandwidth by expressing the noise in units of Vrms/√Hz, “volts per root
hertz”, a unit that one often encounters when studying electrical noise. Clearly, we could reduce
T and possibly R to minimize noise, and indeed we should. However often that is not a practical
approach. Another recource then is to reduce the bandwidth of the measurement. This is where
the power of the lock in amplifier comes in handy.
The lock-in amplifier
From the above, it is apparent that to reduce noise in an electrical measurement we should
narrow the bandwidth of frequencies over which we perform the measurement, and we should
pick certain measurement frequencies over others as being less polluted by noise. As we will
see, a lock in amplifier in essence acts as a narrow band-pass filter that tracks a reference
frequency. It automatically centers itself on the reference frequency, even if this frequency drifts
a little throughout the experiment. With a lock-in amplifier, you drive an experiment with an AC
signal at a fixed reference frequency and phase (the reference signal). You then detect the
response of the system at that same frequency or at one of its harmonics, and at a fixed phase
w.r.t. to the reference (you lock into the reference). The technique is versatile and powerful.
Imagine that you want to detect the weak return signal of a radar signal bounced off our moon.
You can detect the returning electromagnetic wave, reflected off the moon, over the noise if you
chop the outgoing electromagnetic wave with a given frequency and with a known phase (you
can do that by rotating a metal disc with a cutout in front of the emitter horn), and then listen in
your detector for a return signal at that frequency and at an appropriate phase. A signal you
detect at another chopping frequency is not likely to originate from the chopped electromagnetic
wave you sent out. Lore has it that a similar experiment was one step towards the invention of
the lock-in, when Bob Dicke, a young physicist working on the radar effort during WW2 at the
Radiation Lab at MIT, decided to try an experiment and invented or refined the lock-in in the
process (he went on to make major contributions to atomic physics, cosmology, astrophysics,
etc.). If everything is set up correctly with a lock-in, the meaningful signal is the one that is
modulated at the reference frequency, or at one of its harmonics. The lock-in will then allow you
to reject a lot of the noise that would otherwise creep into the measurement.
How does a lock-in work?
Modern lock-in amplifiers are sophisticated pieces of equipment. However, at their core, they
consist of 3 main parts. These are a preamp to amplify the incoming signal to be detected (along
with the noise at this point), a signal mixer to multiply the input signal and the reference
waveform, and a low-pass filter.
Let’s imagine we have a signal of the form () = (1) and a reference of the form
() = (2 + ).
Then after the signal mixer (which multiplies the inputs) we have:
= (1)(2 + ) = 2 {[(1 + 2) + ] + [(1 − 2) − ]}
The sum (1 + 2) corresponds to a high frequency. The low-pass filter is designed to filter out
this high frequency. The low-pass filter will pass the low-frequency signal at (1 − 2)
however. So we can drop the first term, and the voltage output will be:
= 2 [(1 − 2) − ]
At this point you may notice the importance of the low-pass filter. If a simple one-stage low-pass
RC filter is used, we see a bandwidth of 1/RC with a drop off of 6 dB per octave of frequency. If
a two-stage low-pass filter is used, the drop off is 12 dB per octave, and so on for more stages
(with modern lock-ins you typically can choose 6 dB, 12 dB, 18 dB, 24 dB). Concerning the
difference frequency (1 − 2) , this signal is transmitted as long as the signal frequency is
sufficiently close to the reference frequency. Let’s suppose ω1 = ω2. Notice that the phase
difference, if any, attenuates the signal,
= 2 [(1 − 2) − ] = 2 [(1 − 2)][] + 2 [(1 − 2)][]
But since [(1 − 2)]~0 and [(1 − 2)]~1 we can describe the output as
= 2 ()
After scaling by a trivial factor 2, the output has the familiar form of a phasor. A lock-in can
simultaneously detect both the signal component in-phase (φ = 0) with the reference signal,
and the signal component 90o out-of-phase (φ = π/2) with the reference signal. These can be
called resp. “X” and “Y”, like the components of a complex number X + iY. The output of
a lockin is a phasor X + iY. In the complex plane also, i denotes a π/2 phase shift. A modern
lock-in also allows you to phase-shift the reference signal, so to change φ. For instance you know
that φ = 0 (as written above) if the X-signal is maximized and the Y-signal minimized.
Quantifying noise reduction
A one-stage low-pass filter with time constant RC will pass frequencies below = 1
. In fact
frequencies at = 1
will be attenuated by 3 dB, so by a factor 0.71 (since 20 log10(0.71) = −3),
and higher frequencies will be attenuated more. So with the lock-in scheme described above, we
are allowing signal frequencies from 1 = 2 − 1 to 1 = 2 + 1 .
This yields ∆ = 2
and hence the bandwidth of the lock-in measurement is ∆ = 1
.
We now see that the effect of e.g. Johnson noise on the output signal of a lock-in depends on the
time constant of the low-pass filter as:
∝ �
1
It is apparent that one can reduce the effects of noise by increasing the time constant of the low
pass filter. However, this comes at the expense of increasing the time the measurement takes to
stabilize since you are effectively integrating the signal over a longer period of time.
Alternatively, you can think of increasing the time constant as decreasing the bandwidth of a
band pass filter with center frequency ω2 that tracks the reference signal.
With the lock-in filter setting of 6 dB, the bandwidth of the lock-in measurement is ∆ = 1
. It
follows that with the setting of 12 dB, the bandwidth is half that, with a setting of 18 dB, one
third that, and with a setting of 24 dB, one quarter that. How do you actually change the RC
time constant on a lock-in? The lock-in will have a knob for preselected RC time constants, e.g.
RC = 100 ms, 300 ms, 1 s etc. Note that you should select the low-pass filter cut-off frequency
1/RC to be much lower than the measurement frequency ω1 = ω2 , otherwise the sum frequency
ω1 + ω2 may not be fully filtered out and the output voltage will oscillate.
The SR830 lock-in
Hall effect voltage signals are typically small, in the mV to nV range. Hence often a lock-in
amplifier is used to measure Hall voltages. The lock-in you will use is the SR830, made by
Stanford Research Instruments. Most lock-in amplifiers have a similar panel layout and similar
controls and input and output connectors, so once you know how to use one of them you can be
confident around practically any other lock-in. When you turn on the SR830 lock-in amplifier, it
will perform a self-check.
Lock-in amplifiers measure in V RMS (root mean square), not V peak-to-peak or V amplitude
(half peak-to-peak). That means the sensitivity scale is in V RMS and the Sine Out driver signal
(see below) is in V RMS, etc. Output voltages are referenced to line ground (earth ground).
The Time Constant section sets the RC time constant. We will use 300 ms. There, under
Slope/Oct, you also select the filter that sets the bandwidth of the measurement. We will use 18
dB.
The Signal Input section is where you connect the signal to be measured, using BNC cables. In
the Signal Input section, push the Input button until A-B is lit up. This makes the lock-in
understand it should expect a differential voltage input signal between A and B (“Voltage A –
Voltage B”), which is what we will need for the Hall voltage signal. Further, select AC
coupling, and select Ground (this connects the outers of the BNCs to earth ground, so that they
form a good shield).
The Sensitivity section selects the amplification of the signal. In the Sensitivity section, push
the arrows until the sensitivity is set to 1 V (full scale, least sensitive to start with).
The Reserve section optimizes the distribution of amplification to avoid signal overload in the
case of very high noise. We set it at Normal.
The Filters section gives you the option to filter out the line frequency (60 Hz) or 2x line
frequency (120 Hz). We will not use those filters.
The center Display section with Channel One and Channel Two displays, shows the value of the
phasor components. Remember that the output of a lock-in is a phasor: it simultaneously detects
both the signal component in-phase (φ = 0) with the reference signal, and the signal component
90o out-of-phase (φ = π/2) with the reference signal. These can be called resp. “X” and “Y”, like
in a complex number X + iY. We will set the displays to X and Y. You can also set the displays
to show magnitude R = (X2 + Y2)1/2 and phase θ (that is what we called φ above), given that you
can write X + iY = R eiθ = R cosθ + iR sinθ. The Hall signal of interest will be represented by a
voltage value at the CH1 BNC output, set to output X. So, select X as output at the CH1 BNC
(in fact if you select X as the choice for the display, you can also choose Display as output). The
convention is that the BNC output will give a signal ±10 V at full scale. This means e.g. that if
you set the lock-in at the 2 mV sensitivity scale, and your signal has a magnitude of 2 mV (RMS)
and zero phase, the CH1 BNC will output +10 V. Still with the lock-in at a 2 mV scale, and
supposing your signal has a magnitude of 2 mV and π phase, the CH1 BNC will output -10 V.
Supposing your signal has a magnitude of 1 mV and π phase, the CH1 BNC will output -5 V.
Etc. It will be important to keep this scaling in mind when you electronically acquire the data.
The output voltages are referenced to line ground (earth ground). You should not have to use the
CH2 BNC output.
The Auto section allows you to tune the lock-in more rapidly than doing it manually. We will
not use the Gain and Reserve selections. Yet the Phase selection can be useful. See also more
about the phase under the description of the Reference section. Auto Phase will set φ (called θ
on this lock-in) such that the X signal will be maximally positive. If you know that physically
Lock-in amplifiers typically do not “autoscale” their sensitivity setting.
Autoscaling would not be compatible with their typical use. This means you have
to constantly watch the X and Y readings and choose the proper scale
manually. The proper scale is the setting just larger than the signal, so you
obtain maximum resolution and avoid overload (OVL signal lights up). Before
connecting or disconnecting the A and B inputs, put the sensitivity to 1 V
(full scale, least sensitive) to prevent damage to the delicate input circuitry.
Before turning the lock-in off, also put the sensitivity to 1 V.
you expect your X to be positive, then the Auto Phase operation nullifies the effects of small
stray phases you may have, e,g, due to capacitance of coax cables, inductance of long cables etc.
But Auto Phase can also lead to confusion by artificially make an X signal that you expect to be
negative into a positive signal. For instance, in the Hall effect experiment you likely don’t know
the expected sign of the Hall differential voltage signal, so X could very well be negative in a
physically meaningful way. In that case check that the phase remains fairly close to zero, within
a few degrees (see below, Reference section).
Setup section and Interface section: not used in this experiment.
The Reference section allows control of the phase φ (called θ on this lock-in), the frequency of
both the driver/measured signal (ω1/2π) and the reference signal (ω2/2π), control and output of a
handy driver AC voltage signal, input for an external reference voltage signal etc. The selected
parameter will be shown in the display, and can be changed by the knob. We mentioned that a
lock-in allows you to phase-shift the reference signal, so to change φ. You can change phase φ
by selecting Phase, and turning the knob, or by using the buttons to add or subtract in increments
of π /2. We also mentioned that typically with a lock-in amplifier, you drive an experiment with
an AC signal at a given frequency ω1/2π and phase relative to a reference signal with frequency
(ω2/2π) and phase φ (ω2/2π and φ characterize the reference signal; let’s assign phase zero to the
driver signal; then the reference signal has phase φ). You then detect the response of the system
at the parameters of the reference signal, namely at frequency ω2/2π, which is usually ω1/2π or
one of its harmonics nω1/2π, and at a fixed phase φ relative to the driver signal. Typically the
reference signal either is the driver signal or has the same frequency as the driver signal or the
reference frequency is a multiple (higher harmonic) of the driver frequency. The pure sinusoidal
driver signal is output at the Sine Out BNC. You will use this driver signal to generate a sample
current creating the Hall effect in the semiconductor sample. You control the driver signal RMS
amplitude via the Ampl button, and its frequency ω1/2π via the Freq button. The Harm # button
allows you to set the reference frequency ω2/2π (for the lock-detection process) to the 1st
harmonic ω1/2π or to the 2nd harmonic 2ω1/2π or 3rd harmonic 3ω1/2π etc. We will detect at the
1st harmonic ω1/2π (Harm # = 1). You can also feed the lock-in with an unrelated external
reference signal (not necessarily purely sinusoidal), at the Ref In BNC, and select how the lock-
in will trigger on that signal. That external signal then actually functions as the driver signal: the
Sine Out BNC gives a pure sinusoidal signal at the 1st harmonic frequency (fundamental
frequency) and at the phase (taken as the zero of φ) of the external signal. Hence the shorthand
Ref In is somewhat of a misnomer; it is just that the reference signal is synthesized internally in
the lock-in based on this external signal. If you use the internal reference then you select Source
as Internal. If you use an external reference signal you unselect Internal. If then the lock-in
cannot detect a reference signal, the Unlock sign will light up. We will drive the semiconductor
sample using the internal Sine Out signal, and hence we select Source as Internal. We will detect
at the 1st harmonic. Note that the setting of phase φ rotates the reference coordinate axes by
angle φ (positive CCW) w.r.t the driver coordinate axes. The coordinates X and Y in the Display
section are measured w.r.t. the reference coordinate axes.
Once you actually use the lock-in in your experiment the actual meaning and action of all these
settings will become clearer.
The Hall effect in germanium
1. Introduction
The classical Hall effect. When an electric current I flows through a conductor which is placed
in a magnetic field B, the magnetic field exerts a transverse force F, the Lorentz force, on the
moving charge carriers (see the figure below; we will define the geometry and symbols below).
This force tends to push the carriers to one edge of the conductor. A charge imbalance is built up
between the edges, giving rise to a transverse component of the electric field E. This component
(Ex) is normal to both the current density vector j and the applied magnetic field B. The
transverse electric field component results in a measurable voltage across the width W of the
conductor. The generation of the transverse electric field component and associated voltage is
called the (classical) Hall effect. The Hall effect can be used to measure the charge-carrier
concentration (n), and to determine the sign of the charge carriers. The Hall effect is also used in
sensitive magnetic sensors (the instrument you will use in this lab to measure B is in fact based
on a semiconductor Hall sensor). In this lab, you will measure the classical Hall effect in two
elemental doped semiconductors, n-type and p-type germanium.