PHYS 4315 Lab


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.
Other Hall effects. The classical Hall effect forms the subject of this experiment. The
phenomenon carries the name of Edwin Hall, a US physicist who discovered it in 1879.
However, other types of Hall effects have quite recently been discovered, some of deep
significance as topological quantum phenomena, and sometimes still only partially understood.
The Hall effects are part of solid state physics. We know of the integer quantum Hall effect
(Nobel prize in Physics 1985), the fractional quantum Hall effect (Nobel prize in Physics 1998),
the spin Hall effect, the quantum spin Hall effect (measured in 2007), the anomalous Hall effect,
and the quantum anomalous Hall effect (discovered in 2013). The integer quantum Hall effect is
presently used in metrology, as it allows a very precise measurement of the ratio e2/h of
fundamental constants (e2/h has units of electrical conductance and is the quantum of electrical
conductance).
2. Background of the classical Hall effect
The relationship between the current I flowing in an electrically conducting sample (metal,
semiconductor, electrolytic solution, plasma) and the applied voltage V is, in the absence of an
applied magnetic field, approximated by Ohm’s law, V = I R. Here R is the resistance of the
sample. This assumes a linear relationship between voltage and current, which is indeed often
found if voltage and current are sufficiently small. Often two-point current-voltage (I-V)
measurements are used to find R. Two-point means that the electrical contacts to the sample,
used to apply the current are the same as the contacts used to measure the Ohmic voltage drop.
In four-point measurements (which you will use in this experiment), the current and voltage
contacts are not the same, which allows more flexibility in the measurements and allows us to
eliminate the effects of the contact itself.
The resistance R that is measured depends on materials parameters and on sample geometry.
Consider positively charged particles, of charge q = +e (with e the elemental charge) drifting in a
conductor with a drift velocity vd (an average velocity) under a voltage existing over the
conductor. The drift velocity is not the instantaneous velocity of a given particle, rather it is the
average velocity of an ensemble of particles left over after their individual random motion has
been averaged out.
Consider the geometry depicted in the figure on the previous page. For now, let’s omit the
magnetic field B. A sample is oriented with its long side, of length L, along y. It has a width W
along x and a thickness t along z. A current I is applied, such that there is a current density
vector j, mostly along y throughout the sample (see below). The areal cross-section through
which I flows is Wt = A, so j = I / (Wt). Let’s assume t is small so that the current density along
z, jz ≈ 0. Also assume that L / W is very large, so L / W → ∞. Then the current density along x,
jx ≈ 0. Hence the current density vector j will be very predominantly along y and j = jy . The
current density j is associated with the drift velocity vd :
j = n e vd ,
where n is the carrier density. The applied current generates an electric field E along y, given
by:
j = σ E , or E = ρ j ,
where σ is the electrical conductivity and ρ = 1/σ is the electrical resistivity of the material.
Drift velocity and electric field are related by:
vd = µ E ,
where µ is the carrier mobility. Combining some expressions we find:
ρ = 1 / (n e µ) and σ = n e µ .
So we have now defined two parameters, mobility μ and carrier density n.
Let’s relate these parameters to measurable quantities. In the figure we apply a current I as
indicated. We can measure the voltage drop between points 1 and 2, and relate this voltage drop
to the electric field E // y :
V1 - V2 = E L.
Combining some expressions we find:
R = (V1 - V2) / I = E L / (n e µ A E) = L / (n e µ A) ,
R = ρ (L / A).
We can measure V1 - V2 and I, and from the sample geometry, L and A, we know ρ = 1 / (n e µ).
But we don’t know n yet. To know n, we will apply B, oriented along z, as in the figure.
Application of B changes the picture considerably. As before, the current density is along y, j =
jy. B oriented along z will induce a Lorentz force F, transverse to vd and to j , so along x. F will
induce a component of E along x. So now E is no longer purely along y, but will have an x-
component. E and j are no longer parallel, and writing E = ρ j with ρ a scalar will no longer
work. The resistivity has to become a matrix instead of a scalar. In the geometry of the figure,
the so-called magnetoresistivity matrix links the electric field vector E to the current density
vector j as follows:
�



� = � 0 − 0 00���
Here ρ = 1 / (n e µ) is still the scalar resistivity, entering in every (non-zero) element of the
matrix. Note that the relation between E and j is still linear, it just has become 3-dimensional.
With jz = 0 and jx = 0 we find:
Ex = - µ B ρ jy and Ey = ρ jy and Ez = 0.
The expression Ey = ρ jy shows that along the y direction nothing has changed from the case B =
0: we found the same relation above. But Ex = - µ B ρ jy shows that jy has now induced a
component Ex. This component will lead to the classical Hall effect.
We can measure the voltage drop between points 3 and 4, and relate this voltage drop to the
electric field component Ex :
V3 - V4 = Ex W = VH ,
where we have introduced the Hall or transverse voltage VH. Also,
VH = - µ B ρ jy W = - µ B ρ (I/(Wt)) W = - µ B (1/(n e µ)) (I/t) = - (B/(ne))(1/t) I
RH = VH / I = - (B/(ne))(1/t) ,
which defines the Hall or transverse resistance RH. We can call this quantity a resistance
because the relation between Ex and jy is still linear. Note that RH is linear in B and that VH is
linear in both B and I. You will experimentally verify these linearities during this lab. The
expression Ey = ρ jy shows that we can still define what we called “R , the resistance of the
sample”, above. Except, to be careful with words, we will now call R the longitudinal
resistance, and use the symbol RL. And we will call V1 - V2 = VL, the longitudinal voltage. As
above, we have:
RL = (V1 - V2) / I = VL / I = ρ (L / A) = (1/(n e µ)) (L / A) .
The longitudinal voltage drop is dissipative in energy, because Ey • jy ≠ 0. But the Hall voltage
drop is non-dissipative, because Ex • jy = 0.
Note that in our sample geometry, we found Ex = - µ B ρ jy , with a minus-sign. This means that
the actual electric field component Ex points opposite to what is depicted in the figure. You can
draw a Lorentz force diagram using F = q vd x B, to figure out that indeed, for positive charges
flowing through the sample, there will be a net positive charge accumulation as indicated in the
figure, on the right side of the sample. There will be a net deficit of positive charge (negative
charge) on the left side. Ex points from positive to negative charge, hence to the left. So, the
Lorentz force diagram and the magnetoresistivity matrix come to the same conclusion.
In the experiment corresponding to the figure and assuming positive charge carriers, we would
find V3 - V4 < 0. For negatively charged charge carriers, the sign would be the opposite, and we
would find V3 - V4 > 0. In your particular setup, depending how you wire up the circuits, you
should figure out which Hall voltage sign you expect for positive or negative carriers. You
can use the direction of the conventional current, the direction of vector vd and the direction of
vector B plus you knowledge of the Lorentz force F to figure this out. Remember that for
holes in p-Ge (positive charge carriers) the direction of the conventional current and of vd are
the same, but for electrons in n-Ge (negative charge carriers) the direction of the conventional
current and of vd are opposite. The sign of the carriers can be determined from the sign of the
Hall voltage.
By plotting RH vs B, and having knowledge of t, and using RH = VH / I = ± (B/(ne))(1/t), you can
determine the carrier density n.

3. Aim of the experiments
In this lab, you will measure the Hall resistance of p-doped Ge (p-type Ge or p-Ge, where the
charge carriers are majority positive) and n-doped Ge (n-type Ge or n-Ge, where the charge
carriers are majority negative). In both cases, you will be able to determine the sign of the
charges and the charge carrier density. You will also verify the linearity of RH on B and I.
We briefly review the physics of charge carriers in solids to understand the observations that you
will make. The theory of electrical conduction in metals, semiconductors and insulators relies on
the energy band theory of solids. The presence of the lattice of atoms in the solid leads to
“bands” of allowed energy levels and forbidden bands (energy gaps). The allowed energy levels
are filled up with electrons, until you run out of electrons. The highest filled energy level is
called the Fermi level, if this level falls in a band of allowed energy levels (metal). It may also
happen that the highest filled energy level occurs exactly at the edge of a forbidden band, so that
the highest energy band is completely full (insulator). The electrical behavior of a material is
critically dependent on which case happens.
If the highest filled energy level falls in a band of allowed energy levels, such that that band is
only partially full, then you only need a tiny amount of energy to raise electrons to a higher
empty energy level with an electric field, and you have a metal (a good conductor). Also, the
number of available electrons for conduction is just the number of electrons in the highest energy
band (and this number is independent of temperature).
In contrast, in an insulator or semiconductor the highest energy band (referred to as the valence
band) is completely full. There is an energy gap (typically ~ 0.3 – 2.5 eV) between the full
valence band and the (almost) empty conduction band. To raise an electron to the next higher
empty energy level, you need to overcome this energy gap. At T → 0 K, the conduction band
would be empty. At finite temperature, some of the electrons in the valence band get thermally
excited to the conduction band. They leave behind positive “holes” in the valence band. The
current in an “intrinsic” undoped semiconductor consists of the flow of the electrons in the
conduction band and the holes in the valence band. The carrier densities of electrons and holes
are very sensitive to temperature.
Doped semiconductors are created through the controlled addition of impurities to intrinsic
semiconductors. They result in an excess of one type of charge carrier over that present in the
intrinsic semiconductor. They come in two types: n-type semiconductors have an excess of
negative electrons, while p-type semiconductors have an excess of positive holes. In p-type Ge
you will find that at room temperature, it has a Hall resistance that is consistent with a positive
charge carrier. At room temperature, the density of holes created by the impurities is larger than
that of the intrinsic electrons and holes, but as temperature increases the density of intrinsic holes
and electrons can exceed that of the impurity holes. The electrons have a higher µ than the holes
in Ge, and the intrinsic electrons have effectively more contribution to the Hall resistance than
the holes. Thus, at a certain temperature you may see the Hall resistance change sign. The
energy gap in Ge is 0.67 eV.
The picture above is somewhat simplified. The energy band structure in solids can give rise to
positive charge carriers in metals as well (still called holes), for instance in aluminum.
Ultimately, the dynamics of particles in solids are different from vacuum dynamics due to the
fact that whereas outer space (vacuum) has complete translational invariance, translational
(spatial) symmetries in crystals are more restricted. Translational invariance is associated with
momentum conservation. The lower symmetry inside crystals leads to modified energy vs
momentum relations, and from these follow interesting properties.

4. Experiments
4.1. Description of the setup
The setup uses banana cables, often in twisted pairs to mitigate electrical noise pickup, and BNC
coax cables to enable lock-in detection of the small Hall voltage signals. The setup, depicted in
the figure below, consists of:
• An electromagnet consisting of two coils (each 600 windings, 2.5 Ω, 9 mH) over a high-
permeability yoke, with two magnet pole pieces (also of high permeability). The two coils
are wired in series, so the same current passes through them.
• A power supply to provide the DC magnet current Im (so Im flows through the
electromagnet). This is a Kepco bipolar operational power supply / amplifier. The power
supply adjusts itself to whatever voltage is necessary to provide the magnet current you dial
in (up to 2 A), up to the maximum voltage available from the supply, in this case 50 V. You
can read the current (in A) and voltage (in V) from the display. The instrument sources
current of the sign displayed from its red terminal, and sinks it into its black terminal. The
Kepco should initially be set to “Mode: Current”, with “Current Control: on” and “Voltage
Control: on”. This setting allows you to adjust the current manually using the current knob.
When you make manual adjustments to the magnet current, you should only vary the current
knob and leave the voltage knob alone. This is called running the power supply in “current-
controlled” mode. For current programming using a programming voltage, you set “Current
control: off”, and feed the programming voltage into the banana jacks, “Current
programming input”. The Kepco power supply is bipolar and operates as an inverting
amplifier: +10 V programming yields –2 A, –10 V programming yields +2 A. It is
important to only switch “Current Control: on ↔ off” after you make sure 1) the manual
adjustment knob is set to zero output current, 2) the programming voltage is zero, 3) the
small box with a switch (see below) is set to the “I=0” position. The reason to be careful is
that abruptly changing the current through the electromagnet will generate high inductive
voltages.
• Small box with a switch. This box is connected to the power supply at one end and at the
other end to banana cables supplying the current to the electromagnet. The box contains a
voltage snubber and a capacitor to mitigate the effects of possible destructively high
inductive voltages (from the electromagnet’s high inductance) on the power supply. The box
has a switch. In the “I=0” position, the switch shorts out the power supply. In the “I ≠ 0”
position the switch allows current to flow through the electromagnet. Operation: with the
switch to “I=0” make sure the power supply reads zero current. With the power supply
reading zero current, put the switch to “I ≠ 0”. Only then change the magnet current to a
non-zero value. If the power supply reads non-zero current, don’t change the switch
position, because that will abruptly change the current through the electromagnet, generating
high inductive voltages.
• A lock-in amplifier, the SR830 made by Stanford Research Instruments. The lock-in
provides an AC sample current I through the Ge samples at a fixed frequency. The lock-in
will also measure the resulting AC Hall or transverse voltage VH (V3 - V4 = VH, above) over
the Ge samples. Ask the TA to show you the lock-in’s operation if you are not yet familiar
with it from the lock-in write-up. The lock-in has an analog voltage output at the CH1 BNC
that will give a signal ±10 V at full scale (means e.g. that if you set the lock-in at the 2 mV
sensitivity scale, and your signal has a magnitude of 2 mVRMS and zero phase, the CH1 BNC
will output +10 V, etc.). This output will be used to obtain lock-in readings by the Arduino
Uno board.
• A bench multimeter, Keithley 2100. This can be used to measure the various voltages, such
as e.g. the programming voltage controlling the current output of the Kepco.
• A magnetic field meter, the Model 425 Gaussmeter made by Lake Shore Cryotronics, with a
transverse Hall probe. This measures B generated between the pole pieces of the
electromagnet. Its probe functions on the basis of the Hall effect. The probe is very delicate,
be careful with it! B is a vector, not a scalar, but the Gaussmeter obviously can only display
a scalar. The probe hence measures the magnetic field vector component normal to its flat
face, at the end of the probe. The sign of the display is such that if the B vector enters the
face where on the body of the probe you can see a label, then the sign of the display will be
positive. It will be important to keep track of this sign. Put the gaussmeter on 3.5 T scale
(“Range” key, use arrows until 3.5 T shows up and press enter), and on DC (press
“DC/RMS” key). The Model 425 has an analog voltage output cable connected to its rear
panel, that will give ±0.5 V output corresponding to B = ±0.5 T (same sign and magnitude)
if the Model 425 range is set to ±3.5 T full scale. The analog voltage output actually ranges
over ±3.5 V corresponding to ±3.5 T but we will not need this full scale.
• Two sample boards, one with a p-Ge plate, one with a n-Ge plate, both with integrated
heaters and thermometers. For both, thickness t = 1.0 mm.
• A Hall effect electronic circuit on a powered breadboard. This circuit interfaces between the
Arduino and the instruments. The Arduino and this breadboard circuit are connected to each
other using jumper wires at the appropriate breadboard pins. The circuit and the instruments
are connected to each other by an assortment of cables. Refer to the circuit diagram on one
of the next pages. The circuit performs three conversion functions:
1) Via the Arduino’s Pulse Width Modulation (PWM, see below) output signal fed into IN1,
it controls the Kepco BOP power supply (in current output mode) which supplies the ±2 A
DC magnet current Im to create B. The Kepco power supply outputs a +/–2 A current
proportional to a –/+10 V programming voltage at its programming input. The relevant
output of the circuit, at OUT1, is a DC voltage, –10 V to +10 V. There is a small-amplitude
50 mV p-p, triangular wave superposed on the DC signal, remnant of the PWM signal. The
scaling is such that 0% duty cycle of the Arduino PWM gives + 10 V DC output and hence –
2 A, 50% duty cycle gives 0 V DC output and hence 0 A, 100% duty cycle gives –10 V DC
output and hence +2 A. Knowing the duty cycle scaling will be important to build your
Arduino program (called sketch). The Arduino PWM output from one of its digital pins is
fed into the relevant circuit input pin, at IN1. OUT1 is connected to the Kepco’s “Current
programming input” banana jacks.
2) It converts the lock-in amplifier (SR830) analog voltage output of ±10 V DC fed into IN2,
to 0-5 V DC readable by the Arduino, at OUT2. The lock-in amplifier analog voltage output
ranges over ±10 V DC, with ±10 V DC corresponding to ± full scale of the AC signal
measured at the lock-in inputs (whatever full scale the lock-in sensitivity is set at, 2 nV RMS
to 1 V RMS). The relevant output of the circuit, at OUT2, is a DC voltage such that – 10 V
DC from the lock-in gives + 5 V DC circuit output, 0 V DC from the lock-in gives + 2.5 V
DC circuit output, +10 V DC from the lock-in gives 0 V DC circuit output. The relevant
circuit output at OUT2 is fed into one of the Arduino’s analog readout pins A0, A1, A2, A3,
A4, or A5.
3) It converts the gaussmeter (Model 425) analog voltage output of ±0.5 V DC fed into IN3,
to 0-5 V DC readable by the Arduino, at OUT3. Here ±0.5 V output from the gaussmeter
corresponds to B = ±0.5 T if the gaussmeter range is set to ±3.5 T full scale. The relevant
output of the circuit, at OUT3, is a DC voltage such that – 0.5 V DC from the gaussmeter
gives + 5 V DC circuit output, 0 V DC from the gaussmeter gives + 2.5 V DC circuit output,
+0.5 V DC from the gaussmeter gives 0 V DC circuit output. The relevant circuit output at
OUT3 is fed into one of the Arduino’s analog readout pins A0, A1, A2, A3, A4, or A5.
The diagram of the Hall effect electronic circuit is given on one of the next pages. Your
group should analyze the circuit and understand how it works. The circuit consists of
three mostly independent subcircuits performing the functions described above (hint: U1, U2
and U4 are opamps configured as inverting amplifiers / summing circuits. LM741 are used
because they can readily handle the ±10 V DC). The circuit is powered by +15 V and –15 V
vs line ground (= circuit ground, taken as 0 V) provided by the powered breadboard. There
is also a subcircuit using a Zener diode Z1 that provides a fixed reference voltage of –5.1 V
vs circuit ground, indicated as level 5N1V. The circuit has three inputs: IN1 for the PWM
input from the Arduino, IN2 for the lock-in analog voltage output of ±10 V DC, IN3 for the
gaussmeter analog voltage output of ±0.5 V DC. The circuit has three outputs: OUT1 is the
–10 V to +10 V programming input for the Kepco power supply (connect to the Kepco’s
“Current programming input” banana jacks), OUT2 is the 0-5 V DC output fed to an Arduino
analog readout pin to read the lock-in analog voltage output (Hall voltage signal), OUT3 is
the 0-5 V DC output fed to another Arduino analog readout pin to read the gaussmeter analog
voltage output (B signal).
Here follows a brief explanation of PWM. For the control of the magnet current, we need
some sort of a DC analog voltage output. But the Arduino Uno does not have a DAC (digital
to analog converter, outputs an analog voltage corresponding to a digital input command).
Note that the Arduino Uno does have ADC (analog to digital conversion, outputs a number
corresponding to an analog voltage input), at pins A0, A1, A2, A3, A4, A5, which is how we
can read analog voltages. Yet the Arduino Uno has PWM output capabilities. A PWM
output is a voltage square wave (here 0-5 V, 980 Hz at digital pin 5 or 6) with a duty cycle
that can be specified. See the figure below. If you heavily low-pass filter the square wave so
that practically only the DC component is left, then the result is a DC voltage with only a
small AC ripple. A high duty cycle results in a high DC output, a low duty cycle results in a
low DC output. So PWM can to some extent replace a DAC.
The circuit converts the PWM square wave signal of 5 V p-p from the Arduino, going from 0
V to +5 V vs circuit ground, at IN1, to a PWM square wave signal of 20 V p-p, going from –

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is composed of R10 and C2.

4.2. Experimental details
• We use the lock-in’s internal source to provide the AC sample current I to the Ge sample.
This internal source is accessible at the Sine Out BNC in the Reference section of the lock-in.
To make sure the SR830 lock-in knows that it should lock into its own internal source, on the
SR830 panel in the Reference section, press Source until the Internal indicator lights up.
You can set the frequency and the rms amplitude of this source using the Freq and Ampl
keys to display the parameter and using the rotary knob to set the parameter. Choose
frequency = 37.73 Hz and amplitude rms = 5.00 V. The frequency choice is somewhat
arbitrary: it simply should be well removed from 60 Hz and any of its harmonics or
subharmonics, and it should be sufficiently low not to introduce unwanted phase shifts in the
setup with its many cables (later on we will discover another limitation).
• The Sine Out source is an AC voltage source. Yet, we would like to apply an AC current
through the sample. To do that, we use a high-Ω resistor in series between source and
sample. The resistor has value 1.00 kΩ, and is contained in the little blue box labeled “1K”.
The value is chosen to be much larger than the sample resistance, which is about 50 Ω at
room temperature. The outer shell of the Sine Out BNC is earth ground. We connect one
end of the blue box to the Sine Out BNC. We connect the other end of the blue box to a
BNC cable with banana connectors on its other end. The red banana is the BNC inner cable
(connected to the 1 kΩ resistor → Sine Out). The black banana is the BNC shell, and hence
is earth ground. So, we connect the red banana to the sample’s current input jack (see later),
and the black banana to the sample’s current drain jack (which is then also earth ground).
Then the sample current circuit is completed, as in the circuit diagram below. With a
source amplitude of 5.00 V rms, the AC current I through the sample will be 5.00 V / (1 kΩ
+ sample resistance) ≈ 5 mA rms. You can obtain a more precise value of the sample current
by measuring the actual resistance between the sample current input jack and sample current
drain jack using a multimeter in resistance mode, after you disconnect the bananas connected
to the lock-in Sine Out BNC. If you don’t disconnect the bananas the sine signal will disturb
the multimeter and also you would be measuring properties of the Sine Out circuit in parallel.
We have converted an AC voltage source to a relatively good AC current source.
• The lock-in measurements on the Hall samples occur entirely in AC. As you have seen,
the lock-in output is actually a phasor, with magnitude and phase. A measurement which in
DC would yield a positive voltage translates into a phasor with phase zero. A measurement
which in DC would yield a negative voltage translates into a phasor with phase 180o. The
lock-in can display results as the phasor’s magnitude R and phase θ, or display its
components X and Y. It will be more straightforward in this experiment to use the X and Y
display. On the SR830, for Channel one select “X”, and for Channel two select “Y”. Y
denotes the 90o out-of-phase (quadrature) component, but since our measurements are
expected to be entirely resistive in nature (no capacitance or inductance which can shift the
phase), we expect Y = 0. Yet you will read a small non-zero value for Y, mostly due to
parasitic capacitance in the cables. We will need to shift the phase between the excitation
(Sine Out) and the reference signal so that Y reads zero. Once the measurement has started
(not now) do that by pressing Phase in the Auto section of the SR830. Then press Phase in
the Reference section, and note the small phase shift the lock-in applied. With Y = 0, the
sign of X now simply translates into what would have been the sign of the voltage reading in
a DC measurement (e.g. a reading X < 0 translates into what would have been a negative
voltage in a DC measurement). You will need to keep track of the sign of X throughout the
measurements, because the Hall voltage can have either sign. The lock-in has an analog
voltage output at the CH1 BNC that will be used to obtain lock-in readings by the Arduino
and that was described above. So, select X as output at the CH1 BNC.
• All the AC lock-in measurements will be in RMS units. So to calculate a Hall resistance
you can simply divide the RMS voltage readout by the calculated RMS sample current
without worrying about conversion factors. For example, the Ge sample current I will be AC
and the measured Hall voltage VH will be AC, and RH = VH/I = dVH/dI.
• Important: There are two entirely separate current circuits at work in this experiment.
The first current circuit provides a DC current Im through the electromagnet (the magnet
current) and is powered by the Kepco power supply. This first current circuit is high-
current DC. The second current circuit provides a current through the Ge samples (the
“sample current I”) and is powered by the lock-in amplifier by converting the lock-in’s built-
in AC voltage source to an AC current source. This second current circuit is low-current
AC and is expressed in RMS units (e.g. mARMS). Make sure you thoroughly understand the
setups and don’t confuse the two current circuits.

4.3 Guidelines for the Arduino sketch and for using the sketch
The Arduino sketch needs to sweep B in both polarities over as wide a range as experimentally
possible, and while sweeping it needs to acquire 1) data of B vs magnet current (= Kepco
current), and 2) data of Hall voltage vs magnet current at fixed sample current. What is swept is
in fact the magnet current and B follows. Having these two data sets for fixed sample current,
you can deduce the data of Hall voltage vs B. The latter is what is ultimately of interest.
The ±2 A from the Kepco power supply will give about ±0.43 T between the poles of the
electromagnet. The ±2 A variation of Im from the Kepco power supply is controlled by the ±10
V programming voltage provided by the Hall effect electronic circuit at OUT1, as explained in
section 4.1. The duty cycle of the Arduino PWM output pins is programmed by the following
command: analogWrite(pin, pwm); where pin is the digital output pin (3, 5, 6, 9, 10 or 11) and
pwm is an 8-bit integer (0-255). We will refer to pwm as a variable below. The scaling is such
that 0% duty cycle (pwm = 0) of the Arduino PWM gives + 10 V DC output and hence Im = –2
A, 50% duty cycle (pwm = 127) gives 0 V DC output and hence Im = 0 A, 100% duty cycle
(pwm = 255) gives –10 V DC output and hence Im = +2 A. The Arduino Uno has PWM output
capabilities at digital pins 3, 5, 6, 9, 10 and 11. Pins 5 and 6 give a 980 Hz PWM square wave,
pins 3, 9, 10, 11 give a 490 Hz PWM square wave. Because 980 Hz is more easily low-pass
filtered to a low ripple, you should use either pin 5 or 6 for your PWM output.
An Arduino sketch consists of an initialization section, of variables, pins used as inputs or
outputs, baud rate etc., followed by a loop that is executed over and over. Here follow some
conditions about the content of that loop.
The electromagnet is large inductance L. Since the voltage over this inductance L is given by V
= L dIm/dt where Im is the magnet current, an abrupt variation of Im will give a high voltage,
experienced at the output of the Kepco power supply (remember the spark plugs in a car engine).
This can be bad for the power supply and has to be avoided. Hence, your loop must start by
keeping Im = 0 for a while, before the sketch steps Im to obtain the measurements. That is
achieved by keeping pwm = 127 (50% duty cycle, Im = 0) for a while at the start of your loop.
Your loop must also smoothly return to pwm = 127 at its end. At the start and at the end, your
loop’s content should wait for about 3 min at pwm = 127 (Im = 0) to allow you the time to make
the proper connections or secure the power supply (see below). Once the 3 min wait at the end
of the loop has elapsed, the loop will start anew (with a new wait at the start, which is not that
functional but hard to avoid ...). Note that you can also implement this wait with a momentary
pushbutton switch (see Arduino online external resources for information).
Once the initial wait has elapsed, the loop will stepwise ramp up pwm in steps of 1, and will ask
the Arduino to collect the relevant voltages (gaussmeter output and/or lock-in output) at each
pwm, using analogRead() and Serial.print() commands. Note that the lock-in will be set at RC
time constant = 300 ms. The lock-in’s response time will be a few times RC. Indeed, the
smaller the frequency bandwidth the longer the response time. In practical terms that means you
should put a delay time of about 3 s between changing pwm and asking the Arduino to read the
voltage data. That delay allows the lock-in output to adjust to the new B (“... set B – wait –
measure – set B – wait – measure ...”).
According to the ideas above, your loop should go through pwm values in a choice of one of two
sequences:
• “start at pwm = 127 (Im = 0), wait 3 min, stepwise go from pwm = 127 to pwm = 255 (Im
= + 2 A), stepwise go from pwm = 255 to pwm = 0 (Im = – 2 A), stepwise go from pwm =
0 to pwm = 127 (Im = 0), wait 3 min”, or
• “start at pwm = 127 (Im = 0), wait 3 min, stepwise go from pwm = 127 to pwm = 0 (Im =
– 2 A), stepwise go from pwm = 0 to pwm = 255 (Im = + 2 A), stepwise go from pwm =
255 to pwm = 127 (Im = 0), wait 3 min”.
Hence your Arduino loop will most likely consist of 3 while loops or equivalent.
Using the sketch:
Here follows a safe procedure for the measurement, also showing the use of the 3 min waits.
Before running the Arduino sketch, make sure that:
 the box with the switch is set to “I=0” position
 the Kepco power supply is set to “Current control: on” (manual knob enabled)
 Im = 0 (adjust manually if needed)
 the positive (red) connection to the Kepco’s banana jacks “Current programming input”
is disconnected. Note that the black connection is elsewhere connected to line ground, so
it can be left connected to the Kepco.
Start the Arduino sketch (e.g. plug the Arduino into a USB). During the initial 3 min wait:
 connect the positive (red) connection to the Kepco’s banana jacks “Current programming
input”
 set the Kepco power supply to “Current control: off”, and then
 flip the box with the switch to “I ≠ 0” position.
Now the Kepco power supply is ready to set currents according to the Arduino commands. After
the 3 min wait, the sweep of B and data acquisition will start. When the sketch reaches the 3 min
wait at the loop end (likely followed by the 3 min wait at the start of the next iteration) you can
use the wait to safely secure the Kepco power supply and prevent it from sweeping Im again. To
do that during the wait:
 make sure that Im ≈ 0 and then
 flip the box with the switch to “I = 0”
 set the Kepco power supply to “Current control: on”
 disconnect the positive (red) connection to the Kepco’s banana jacks “Current
programming input”.
When the 3 min + 3 min wait is over the loop will restart, but Im will not be swept anymore.
Note that if you don’t go through the steps of securing the Kepco power supply, then the Arduino
will again take valid Hall effect data, sequentially with the data from the first loop. There is no
harm in that, since no data will be overwritten. Yet, when you want to terminate the
measurement you need to secure the Kepco power supply as described above.
Don’t unplug the Arduino from its USB power without securing the Kepco power supply,
because that will cause OUT1 to rise to + 10 V DC and hence Im = –2 A. Also note that starting
Serial Monitor or Serial Plotter or serial data taking on the computer will reset the sketch to start
anew, so start those functions only when Im ≈ 0.
The steps above will become clearer once you understand the logic behind them. The Appendix
contains an example Arduino sketch implementing the ideas above.

4.4. Procedures
1. First pick up the sample board that is not installed and identify the connections. Refer to the
picture of the sample board below to identify the connections. Don’t connect the cables to the
instruments yet, that comes later. Identify:
• The thin Ge sample in the middle of the board.
• The connectors and leads providing the AC sample current I. One end is labeled +12….30V,
the other end - (use the “−“ closest to the Ge sample). The setup will work no matter what
the direction is of the applied current, but you should make a note of the direction as it will
affect the sign of the Hall voltage you measure. In the picture we are sourcing the current
into the “+“ lead using the red banana termination of the BNC cable connected to the “1K”
box, and draining the current from the “−” lead using the black banana termination (earth
ground).
• The connectors and leads used to measure the Hall voltage VH (labeled UH). We use cables
with a banana termination on the board end and a BNC termination on the lock-in end. One
cable will connect the top Hall lead to the A connector at the lock-in, the other will connect
the bottom Hall lead to the B connector at the lock-in. The lock-in will measure top −
bottom as A − B (we need to perform this differential measurement because neither the top
Hall lead nor the bottom Hall lead are at line ground).
2. Now look at the electromagnet and Kepco power supply and see how they are connected in
the figure below. In the figure, the gaussmeter probe is in place between the electromagnet
poles, in the configuration you will use to characterize the magnet.
3. Now you can characterize the electromagnet using the Arduino program you wrote for that
purpose. The characterization consists of data and a plot of B vs magnet current Im (Kepco
current). The data and plot will allow you to use your reading of magnet current as a substitute
for an actual measurement of B during the Hall effect measurements. There is indeed no space
for both the sample board and the magnetic field probe in the small air gap of the electromagnet.
The idea is that if you know the relation between B and magnet current and between Hall voltage
and magnet current, then you know the relation between Hall voltage and B. The latter is what
you need to ultimately find. For the characterization, rotate the pole pieces to form the smallest
air gap between them. Put the gaussmeter on 3.5 T range and make sure it is DC mode. Read
the gaussmeter while the probe is sufficiently far from the electromagnet (about a foot), and zero
the instrument if needed. Carefully position the mount with the Hall probe of the gaussmeter so
that the Hall probe tip is centralized between the pole pieces. The height can be set by the
vertical position of the clamp. The probe face should be parallel to the pole piece faces. While
the Im = 0, the gaussmeter will read a small B, because the yoke and pole pieces have hysteresis
and a remanent magnetization. Connect the gaussmeter analog voltage output to IN3 of the Hall
effect electronic circuit. Make sure the Arduino is properly connected. Run your
characterization Arduino sketch, referring to “Using the sketch”. Then carefully remove the
mount and its Hall probe from the air gap. The plot of B vs magnet current Im you obtain will
show a small hysteresis due to the materials used in the yoke and pole pieces. This small
hysteresis does not much affect the results, but we are leaving it up to you if and how you want
to take it into account.
4. Next you can acquire Hall effect data using the Arduino sketch you wrote for that purpose,
first for one Ge sample (n-type or p-type), then for the other Ge sample (p-type or n-type). Make
sure the SR830 is powered on and make sure that in the Sensitivity section it is on the least
sensitive scale of 1 V. Select a Ge sample board. Don’t make the cable connections to the Ge
sample board yet. Carefully, from the top, slide the board into the electromagnet air gap.
5. The SR830 may already be partially set up, but here follows the procedure if it is not set up.
In the SR830’s Reference section, press Source until the Internal indicator is lit. This makes the
lock-in understand it should listen to its internal reference source. Press Freq, and set 37.73 Hz.
Press Ampl and set 5.00 V. Press Harm#, and set to 1. Set Channel one to X and Channel two to
Y. Select X for CH1 BNC. No offsets. In the Filters section, no line filters active. In the
Reserve section, set Normal. In the Sensitivity section, select 1 V (so that it will not overload
initially). In the Time Constant section, select 300 ms time constant, 18 dB filter, and Sync off.
In the Signal Input section, select A−B, AC coupling, and Ground (grounded outer BNC shells to
serve as shields).
6. Connect the “1K” box to the Sine Out BNC of the SR830 if it is not connected yet. Referring
to the picture of the Ge sample board above, make sure the “1 K” box is properly connected to
the sample board at its other end. Connect the cables used to measure the Hall voltage to the
SR830: the cable from the top Hall lead to the A/I BNC connector, the cable from the bottom
Hall lead to the B connector. The SR830 should show small readings on the X and Y displays.
7. We will now minimize the Y reading on the lock-in. For this we need a representative Hall
voltage VH, with VH > 0. Manually (“Current Control: on”) set the magnet power supply to
about Im = ±0.7 A (the sign such that VH > 0; the exact value does not matter). Set the SR830 to
a sensitivity range appropriate for the reading. In the Auto section, press Phase. The Y reading
should be minimized. Then press Phase in the Reference section, check that the applied phase
shift is close to 0o, and make a note of it for future reference. Manually set the magnet power
supply back to Im = 0.
8. Make sure Im = 0. Now you are ready to run your Arduino sketch for measuring Hall data.
Connect the lock-in analog voltage output to IN2 of the Hall effect electronic circuit. Make sure
the Arduino is properly connected. Lock-ins typically don’t autorange, so set the lock-in
sensitivity range about to the value you found above. More precisely, make sure that you will
use as wide a range as possible, without overloading, of the lock-in analog voltage output of ±10
V DC (corresponding to ± full scale of the Hall AC signal measured at the lock-in inputs). Run
your Hall effect Arduino sketch, referring to “Using the sketch”. Once you perform the data
analysis (below) you will see that RH(B) is approximately linear in B (with deviations from
linearity at higher B). Keep in mind that we will ultimately be interested in to plotting RH vs B.
Make a note of what the temperature in the room approximately is, as the Hall effect
measurements are sensitive to temperature.
9. Repeat this Hall effect measurement (RH(B) vs B) for various sample currents I. Remember
that the small sample current I through the Ge sample is powered by the lock-in amplifier by
converting the lock-in’s built-in AC voltage source to an AC current source (e.g. expressed in
mARMS, see above). At 5 VRMS Sine Out on the SR830, I ≈ 4.76 mARMS (check that calculation).
Pick about 4 other values of I, between ~ 1 mARMS to ~ 5 mARMS by varying the amplitude of
Sine Out on the SR 830. You don’t need to hit exact values of I, but you need to calculate and
know I at each step (see above, calculating I). From the data obtained you will plot VH vs I at
fixed B, and verify linearity in I. The linearity (implicit in the magnetoresistivity matrix) is what
allows us to use AC detection methods to begin with and what allows us to define RH.
10. Carefully remove the sample board from the electromagnet gap.
11. Repeat the same Hall effect measurements for the other sample board, so that you have
measurements on both p-Ge and n-Ge.


5. Data analysis and content of report
1) You measured the classical Hall effect in p-Ge and n-Ge, which have carriers of different
signs. Look at the schematic Hall bar in the Introduction and note the direction of the
Lorentz force. Take into account how you wired your setup and the signs of the data you
obtained. Are the signs of the Hall voltages as you expected according to the sign of the
carriers? Are they opposite for both samples? Show your reasoning about the sign of the
carriers using a Hall sample schematic adapted from the schematic used in the introduction.
2) Plot B vs Kepco power supply DC magnet current Im, and use this plot or data to calibrate
your B-values. Note the small hysteresis. Note that from here on the role of Im is fulfilled
and when needed all your plots will use B rather than this DC magnet current.
3) Plot RH (in Ω) vs B (at room temperature) for both samples, for B ranging from ~ –400 mT to
+400 mT (label your plot with the temperature). These plots can use the data you obtained at
the highest I, when the lock-in’s Sine Out was 5 VRMS. If necessary, try to take into account
the hysteresis of the magnet. You may note that RH is linear in B at lower B and deviates a
little from linearity at higher B. We will explain this below. Although values can vary
somewhat depending on temperature and Ge sample, your plots must look like the example
plot below, in that RH is essentially linear in B. The sign of the slope is, as we explained
above, arbitrary and dependent on how the sample was wired up. In the example plots we
have taken the p-Ge to show a positive slope and the n-Ge a negative slope. But, once wired
up the sign of the slope must be consistent with the carrier type (see (5.1) above). If you find
RH vs B to resemble hyperbolic functions then something is terribly wrong. It turns out that
the longitudinal resistance RL (not measured in this experiment) is ~ parabolic in B over the
range of B you used. A dependence of RL on B is called magnetoresistance. The specific
phenomenon at work here is called geometrical magnetoresistance, and it has the same
classical origin as the Hall effect. If you look at the magnetoresistivity matrix and the
expression Ey = ρ jy you may conclude that there should be no magnetoresistance (RL =
constant). That is correct given our assumption that L / W → ∞. But for our samples, L / W
→ ∞ is not such a good approximation. This means the transverse current density jx may be
small but jx ≠ 0. The current density j goes at a slight angle, the Hall angle, with respect to
Ey. This makes the current flow lines slightly longer as B increases, making the effective
sample length longer, increasing the longitudinal resistance RL with B and giving
magnetoresistance. The small transverse current also shorts out VH a little, reducing VH at
higher B. Hence, VH deviates somewhat from linearity. All this happens because L / W → ∞
is not quite true.
4) You must extract carrier density n for each sample by performing numerical linear fits
(e.g. using least-squares method) to the data RH vs B and using the value of the slope dRH/dB
to calculate n (use t = 1.0 mm). In your report, mention the values you find for n. For the fits,
emphasize the sufficiently linear region. See the plots above: the solid lines are fits. Note
that intrinsic Ge has n = 2.4 x 1019 m-3 at 27° C. You have doped samples, so you expect n
above the intrinsic value. This fact allows you to make sure that the values you obtain for n
make sense for each sample. The example plots of RH vs B provide a guide to the
approximate appearance of numerical linear fits and to the values you should find.
5) Show plots of VH vs I at a few fixed values of B for both samples. The plots of VH vs I
should be linear, but will have different slopes RH(B) = dVH/dI . Discuss.
6) Note that in this experiment we regularly use Im > 0 or Im < 0 but we always refer to I as
being positive. Why?
7) In your report you must show all of the plots above, as well as your reasoning and
calculations of n.
8) In your report in addition to the classical Hall effect you must also discuss all six of the
following in order to provide context: integer quantum Hall effect, fractional quantum Hall
effect, spin Hall effect, quantum spin Hall effect, anomalous Hall effect, quantum anomalous
Hall effect.

Appendix: example Arduino sketch
/*
Starts at magnetic field zero, ramps up/down and back to zero magnetic field.
Also blinks LEDs to keep track.

Uno has no DAC, but analogWrite() can be used on pins
3, 5, 6, 9, 10, 11 (all labeled ~) to put out 256 different PWM duty cycles.
Pins 5, 6 are 980 Hz, 3, 9, 10, 11 are 490 Hz. We use pin 5, 980 Hz.

- PWM output pin 5 fed to scaling circuit and from there to Kepco.
- Gaussmeter Lakeshore LS 425 read at pin A0 through scaling circuit.
- Lock-in SR830 read at pin A3 through scaling circuit.
- Both converted from 0-1023 reading to volt.
- Prints results to the Serial Plotter or Monitor with 4 digits after decim. pt.
(data can then also be stored a computer using Python program).

Labels for top of Serial Plotter must use colon, and no space.
Delay between a PWM write and an ADC is not needed.
Wait 20 ms after ADC and between ADC of different pins for ADCs to settle;
depends on output impedance of source or sensor.
*/

// these constants won't change, gives names to the pins used.
// note analog in pins don't have to be called A3, just 3 works
// if they are used in analog mode only.
const int analogmf = 0; // ADC analog in pin from scaled gaussmeter LS425, pin A0
const int analogli = 3; // ADC analog in pin from scaled lock-in SR830, pin A3
const int PWMoutK = 5; // PWM out 980 Hz for Kepco, pin 5
const int LEDpin = 7; // digital out pin 7 to drive ext. LED
const int stepdelay = 3000; // delay in ms between steps
const float agauss = -0.0009775171;
const float alockin = -0.01955034;

// these variables will change:
int voltgauss = 0; // ADC integer 0-1023 from A0, gaussmeter
int voltlockin = 0; // ADC integer 0-1023 from A3, lock-in
int pwmout = 0; // 0-255 to PWM pin 5 (analog out)
float gaussm = 0; // gaussmeter voltage out in volts
float lockin = 0; // lock-in voltage out in volts
unsigned long millisstart = 0; //timer start
unsigned long millisnow = 0; // timer

void setup() {
pinMode(PWMoutK, OUTPUT); // not actually needed to set as output before analogWrite
pinMode(13, OUTPUT);
Serial.begin(9600); // initialize serial communications at 9600 bps
}

void loop() {
pwmout = 127; // initialize to Kepco current at zero
analogWrite(PWMoutK, pwmout); // output PWM value
Serial.println("waiting 3 min at start ...");
millisstart = millis(); // delay 3 min to set up
millisnow = millis();
while (millisnow - millisstart < 180000) {
millisnow = millis();
}
pwmout = 126; // initialize to Kepco current at zero
digitalWrite(13, LOW);
digitalWrite(LEDpin, LOW); // turn onboard LED pin 13 and ext. LED off

// ramp PWM up and down in loop to drive Kepco, and take data
while (pwmout <= 254) {
pwmout++;
analogWrite(PWMoutK, pwmout); // output PWM value
delay(stepdelay); // wait for field and lock-in to stabilize
voltgauss = analogRead(analogmf); // read scaled gaussmeter 0-1023 at A0
delay(20);
voltlockin = analogRead(analogli); // read scaled lock-in 0-1023 at A3
gaussm = 0.50 + agauss * voltgauss; // gaussmeter voltage out
lockin = 10.0 + alockin * voltlockin; // lock-in voltage out

Serial.print("PWM-Counter:");
Serial.print(pwmout, 1);
Serial.print(" ");
Serial.print("gaussV:");
Serial.print(gaussm, 4);
Serial.print(" ");
Serial.print("lockinV:");
Serial.println(lockin, 4);
}
digitalWrite(13, HIGH);
digitalWrite(LEDpin, HIGH); // blink LEDs for 2 s
delay(2000);
digitalWrite(13, LOW);
digitalWrite(LEDpin, LOW);
// now pwmout = 255, ramp pwmout down

while (pwmout >= 1) {
pwmout--;
analogWrite(PWMoutK, pwmout); // output PWM value
delay(stepdelay); // wait for field and lock-in to stabilize
voltgauss = analogRead(analogmf); // read scaled gaussmeter 0-1023 at A0
delay(20);
voltlockin = analogRead(analogli); // read scaled lock-in 0-1023 at A3
gaussm = 0.50 + agauss * voltgauss; // gaussmeter voltage out
lockin = 10.0 + alockin * voltlockin; // lock-in voltage out

Serial.print("PWM-Counter:");
Serial.print(pwmout, 1);
Serial.print(" ");
Serial.print("gaussV:");
Serial.print(gaussm, 4);
Serial.print(" ");
Serial.print("lockinV:");
Serial.println(lockin, 4);
}
digitalWrite(13, HIGH);
digitalWrite(LEDpin, HIGH); // blink LEDs for 2 s
delay(2000);
digitalWrite(13, LOW);
digitalWrite(LEDpin, LOW);
// now pwmout = 0, ramp pwmout up

while (pwmout <= 126) {
pwmout++;
analogWrite(PWMoutK, pwmout); // output PWM value
delay(stepdelay); // wait for field and lock-in to stabilize
voltgauss = analogRead(analogmf); // read scaled gaussmeter 0-1023 at A0
delay(20);
voltlockin = analogRead(analogli); // read scaled lock-in 0-1023 at A3
gaussm = 0.50 + agauss * voltgauss; // gaussmeter voltage out
lockin = 10.0 + alockin * voltlockin; // lock-in voltage out

Serial.print("PWM-Counter:");
Serial.print(pwmout, 1);
Serial.print(" ");
Serial.print("gaussV:");
Serial.print(gaussm, 4);
Serial.print(" ");
Serial.print("lockinV:");
Serial.println(lockin, 4);
}
digitalWrite(13, HIGH);
digitalWrite(LEDpin, HIGH); // blink LEDs for 2 s
delay(2000);
digitalWrite(13, LOW);
digitalWrite(LEDpin, LOW);
// now pwmout = 127, starting point

Serial.println("waiting 3 min to restart ...");
millisstart = millis(); // delay 3 min to stop if wanted
millisnow = millis();
while (millisnow - millisstart < 180000) {
millisnow = millis();
}

}