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I. Hall Effect in Semiconductor

II. Pre-Labs must be printed separately.SHE Pre-Lab.PRINT, FILL THIS OUT, Get it Signed by 111-Staff, turn in your Signed Pre-Lab Sheet with your report, to the 111-Lab Staff.

III. Van Der Pauw Theorem

IV. Instrument Manuals

Reprints and other information can be found on the Physics 111 Library Site

Acknowledgment and Disclaimer "This material is based upon work supported by the National Science Foundation under Grant No. 9850553 "Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF)."

This manual is intended to provide a general guidance for the lab. It is not designed to explain all the physics necessary for understanding this experiment, nor is it a "cook-book" telling you exactly which buttons to press, etc. It is up to the student to become familiar with the necessary material in the reprints, and to figure out the exact procedure necessary for completion of the assignments in the lab. Talk with an instructor often, but not until you have given the physics and procedures some thought first. We are here to instruct and help, but not to do your thinking for you.

Before The Lab

View the Hall Effect In a Semiconductor Video, discuss the pre-lab questions with an instructor, and get the SHE Pre-Lab signed.

Prerequiste Reading Materials

More than you ever wanted to know about semiconductors. "Semiconductor Physics and Devices, Basic Principles". D. A. Neamen, (1992)
The "Semiconductor Materials, Lecture Notes for MSE 223" by E. E. Haller, Department of Materials Science and Engineering, University of California Berkeley.
Hall Effect Article by E. Haller
"The Art of Experimental Physics" by Preston and Dietz (1991). In particular, "Electrical Conductivity and The Hall Effect," pg. 303-315.
Excellent introduction to conductivity in solids. Fourth edition, "Fundamental of Physics" extended, D. Halliday, R. Resnick, J. Walker, (1993).
The Hall effect and properties of semiconductors. "Experiments in Modern Physics, A. C. Melissinos, Academic Press (1996).
"Proof of the Van Der Pauw Theorem"?
Four Wire Measurements

See reprints in the laboratory folder: (there are two from Haller)

Useful Definitions

Conductor: is a type of material in which charge carriers (effectively electrons or holes) can be made to flow with arbitrarily small voltages. A conductor is distinguished by having a partially filled conduction band, and, hence, no gaps between the highest occupied and the lowest unoccupied electronic states. Common examples of conductors are copper and iron.

Electron Concentration: is the number of electrons per unit volume in the conduction band of the material.

Extrinsic material: is a material whose number of electrons is not equal to the number of holes (positive carriers).

Hall coefficient: is a parameter that measures the magnitude of the Hall Effect in the sample. It has units of $\frac{\Omega m}{Tesla}$ . (Contrast this with resistivity, which has units of $Ω m$ .) The Hall coefficient is defined as $R_H=\frac{E}{JB}$ where E, J, and B are the magnitudes of the electric field, the current density, and the magnetic field, respectively. In the experimental setup to determine the Hall coefficient, these three vectors are mutually perpendicular.

Hall effect: is a phenomenon that occurs when a conductor or semiconductor is placed in the magnetic field and a voltage (i.e. the electric field in the above definition) is applied through the material perpendicular to the magnetic field. While this voltage induces current flow along the electric field direction, the charge carriers also experience a magnetic deflection from their path. This results a separation of positive and negative carriers, and thus the generation of an electric field perpendicular to the direction of current flow. Note that, at sufficient temperature, the net current in a semiconductor is made up of counteracting currents of p-type and n-type carriers

Hall field: is an electric field perpendicular to the direction of current flow generated by the Hall effect.

Hall voltage: is the potential difference across the semiconductor that is produced by the Hall field. This is the voltage which is exactly enough to compensate for the deflection of charge carriers by the magnetic field, so that the net current perpendicular to the applied voltage is zero.

Hole: is an effective positively charged "empty state." The description of current in terms of the motion of positively charged holes rather than negatively charged electrons is convenient and accurate when describing a material with an almost filled conduction band.

Hole concentration: is the number of holes (positive carriers) per unit volume in the conduction band of the material.

Intrinsic material: is a material in which the number of negative carriers (electrons) is the same as the number of positive carriers (holes).

Insulator: is a material in which electrons cannot be made to flow with low voltages. Using the quantum mechanical description of periodic solids, insulators are characterized by having completely filled valence bands, and large energy gaps to the lowest-lying unoccupied conduction bands.

Drift mobility: is the drift velocity per unit applied electric field, $\mu_d=\frac{\nu_d}{E}$ .

Drift velocity: is the average velocity in the direction of an applied electric field of the all conducting charge carriers in the sample.

Lorentz force: is the force, $\vec{F}=q(\vec{\nu}\times\vec{B})$ which a moving charge experiences when subjected to a magnetic field. Here, $\vec{B}$ is the magnetic field, $\vec{\nu}$ is the velocity of the carrier, and q its charge.

N-type material: is a material in which the negatively (n) charged carriers are mostly responsible for the conduction.

P-type material: is a material in which the positively (p) charged carriers are mostly responsible for the conduction.

Resistivity: is a material parameter that is a measure of the resistance to current. The resistivity is defined as the resistance of the sample times the cross sectional area of the sample divided by the length of the sample, $\rho=\frac{RA}{l}$ . The resistivity has units of $Ω m$ and it is inversely proportional to the conductivity of the sample, $\sigma=\frac{1}{\rho}$ .

Semiconductor: is a material which is, intrinsically, barely insulating in that the filled conduction band lies very close to the valence band. Semiconductors can be deliberately modified through patterned doping to produce complex, compact, and reliable electronic devices.

Introduction

The semiconductor industry relies on mature technologies for the creation of high purity semiconductors and for their quantitative characterization prior to their use in manufacturing. One important characterization tool in the measurement of the Hall effect to measure mobilities and carrier concentrations in a given semiconductor material. In this experiment, we will make such measurements, observing the Hall effect for one semiconductor sample using the common Van der Pauw technique. The temperature dependence of the Hall coefficient and resistivity will allow us to distinguish between p-type and n-type samples, and to calculate the electron and hole mobilities and concentrations. The maximum sample current you should use is 10 mA. You should try a variety of sample currents, probably in order of magnitudes. ( 1 microamp to 10 milliamps)

Preliminary Concepts

The electrical properties of conductors and dielectrics are easy to understand and straightforward to measure (e.g. a simple confirmation of Ohm's law). In contrast, the properties of semiconductors fall in between these two extremes and require some knowledge of condensed-matter quantum physics to understand.

Recall some simple concepts from the quantum description of an isolated atom. Electrons in an atom are regarded as occupying quantized single-electron energy levels (at least in the "indpendent particle approximation"). If an atom is in its ground state, its N electrons will occupy the N lowest energy (i.e. most tightly bound) states available around that atom's nucleus. Electrons can be promoted to one of the infinite number of unoccupied energy levels (bound or unbound) if sufficient energy and a proper means of excitation are provided (e.g. barring selection rules).

Figure 1: Energy-band diagrams for A) Conductor, B) Insulator, C) Intrinsic Semiconductor. The dark shaded gray parts correspond to the filled states. The lightly shaded gray parts corresponded to unocuppied states. EF is the Fermi energy and Eg is the band gap energy.

In a solid, however, the atoms are packed closely together, and the strong mutual influence of neighboring atoms forces one to treat electron quantum levels as being delocalized throughout the solid rather than as being localized around just a single nucleus. In this situation, the energy spectrum now consists of bands of allowed energies, each of which arises gradually from the isolated atomic states as atoms are brought together to their small spacings in the solid. These energy bands consist of a continuum of energies for electronic states which differ in momentum (or, rather, "quasi-momentum"), and are separated from each other by energy gaps which are energy ranges which electrons in the solid cannot possess.

Further, under typical conditions, the electrons in a solid are deep in the regime of quantum degeneracy, which we can determine as follows. At zero temperature, electrons in the solid would occupy sequentially all the lowest energy states up to an energy, the Fermi energy. At temperatures T such that $E F = k B T$ , where $k B$ is the Boltzmann constant, the distribution of electrons is modified slightly, with some electrons right near the Fermi energy actually being thermally promoted to energy levels above the Fermi energy, with this redistribution all happening within about $k B T$ of the Fermi energy.

Now we may understand the quantum-mechanical distinction between conductors, insulators, and semiconductors (Fig. 1). In a conductor, the Fermi energy lies within an allowed energy band. Because of this, electrons can respond to minutely small electric fields by small redistributions among nearby energy levels so that an electrical current flows. By contrast, in an insulator, the valence band (the highest energy band which is occupied) is completely filled, and is separated by a large energy gap from the conduction band (the lowest energy band which is not fully occupied). Since large energies are now needed to redistribute electrons in such a material, small electric fields will no longer induce electronic motion, and hence no current will flow.

As shown in Figure 1c, the situation in a semiconductor is between that of a conductor and an insulator. While a semiconductor is, strictly speaking, an insulator, it is insulating only due to a very small energy gap. By "small," we mean the gap is on the order of the thermal energy $k B T$ . For such a material, therefore, there exists a small concentration of empty states (holes) in the valence band and of filled states (electrons) in the conduction band due to thermal excitation. As such, a semiconductor has a small, strongly temperature dependent, conductivity.

Furthermore, the small energy gap makes the electronic properties of a semiconductor highly susceptible to "doping," i.e. the controlled introduction of a small density of electronic levels at energies within the intrinsically void energy gap by the mixing of impurity atoms within a semiconductor. These impurity atoms serve either to add extra filled electron levels at energies near the conduction band edge, thereby providing a surplus of electrons into the conduction band, or to add extra unfilled levels at energies near the valence band edge, thereby providing a surplus of holes into the valence band. The former situation creates an n-type semiconductor, in which the majority of charge carriers are negatively charged, while the latter creates a p-type semiconductor, in which the majority of charge carriers are positively charged. Undoped semiconductors are called intrinsic, while doped ones are called extrinsic. In an intrinsic semiconductor, the densities of free electrons and holes are equal, while in extrinsic semiconductors, as explained above, these densities are unequal.

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Figure 2: Energy band diagrams for extrinsic type semiconductors. The N-type semiconductor in a) contains occupied donor levels located at energy Ed below the conduction band. Electrons in these levels are easily excited into the conduction band. P-type semiconductors b) have vacant acceptor levels at energy Ea above the valence band. Electrons from the top of the valence band can be easily excited into these levels. The dark shaded gray corresponds to filled states and lightly shaded gray corresponds to unoccupied levels.

Differences between conductors and semiconductors can be quantified through the resistivity $ρ$ of each material. Resistivity is a bulk quantity defined as $\rho=\frac{E}{J}$ , where E is the electric field (measured in Volt/meter) and J the current (measured in Ampere/meter²). The SI unit of resistivity is thus the Ohm-meter. As an example of properties for a metal, the resistivity of copper is near 2 x 10^-8 $Ω m$ at room temperature, increasing slowly (at a relative rate of 4 x 10^-3 /K in copper) as temperature increases. A remarkable thing about conduction in metals is that the electrons forming the current move at an extremely slow net drift velocity (~10 micron/s) on top of a random thermal motion at much higher speeds (~10⁶ m/s). The value of the slow drift velocity can be estimated by considering how fast electrons at a density of ~10²⁹/m³ (say one per copper atom) would have to move in a typically-sized conductor to give a typical current (e.g. 1 ampere in a mm² cross section).

By contrast, a semiconductor such as silicon has fewer charge carriers, (10^-16/m3), a larger resistivity (3 x 10³ ohm-meters), and a negative temperature coefficient of resistivity (-10^-3/K). The resistivity can be changed drastically by doping with other materials – typically decreasing from 3 x 10³ to 10^-3 ohm-m!

Hall Effect and Van Der Pauw Technique

To determine the electrical resistance of an object, one makes measures two quantities: the current run through the object, and the voltage which arises due to the passage of that resistance. There are two typical ways for performing such measurements, either the simple but somewhat inaccurate two-probe technique, or the more cumbersome by more accurate four-probe technique. The difference between these two techniques is particularly pertinent when one wants to measure very small resistances. Using a two-probe technique, such as one usually does with an ohmmeter, current exits the device and runs in series across many resistive elements: the wires coming out of the ohmmeter, the substantial (fraction of an Ohm) contact resistance between the ohmmeter probes and the resistor, then the resistor, more contact resistance, and more wire. The voltage is then measured in the ohmmeter across the entire chain of resistors, meaning that the small resistance you were trying to measure is swamped by the extraneous contact resistances. This difficulty is overcome by using a four-probe technique. One device is used to generate current, and run that current across contact resistances as well as the resistor you're trying to measure. Then a second high-input-impedance device, i.e. one that draws very little current, is used to sample the voltage only across the resistor, and is thereby insensitive to voltage drops due to contact resistance. Since the contact resistance for the voltage probes is much smaller than the input impedance of the voltmeter, we obtain a genuine voltage measurement to use in our resistance determination. In this experiment, we will use several well-conceived measurement techniques to determine the resistivity and the Hall voltage across our semiconductor samples.

Hall Effect

Figure 3: Hall Effect Geometry

Consider a semiconductor sample placed in the magnetic field B that is in the z-direction (refer to figure 3). Suppose we pass a current I through that sample perpendicular to the magnetic field, say in the x-direction. Then the "free" electrons and "holes" in the sample experience a force given by the Lorentz equation,

$\vec{F}=q(\vec{\nu}\times\vec{B})$ (N) (1)

where Q is the charge of free carriers and $\vec{\nu}$ their velocity. This force deflects any charge carriers towards in the positive y direction. If the charge carriers are predominantly electrons, this creates an excess negative charge on the +y side of the semiconductor sample, and thus a positive charge on the –y side. This charge distribution generates an electric field $E H$ (the Hall field) pointing in the +y direction which (1) balances the Lorentz force so as to keep current flowing along the x-direction, and (2) yields a voltage difference $V H = s E H$ (the Hall voltage, named after E.H. Hall who discovered this effect in 1879) across the sample of width s. Considering the balance of forces on the charge carriers

$\vec{F}=q(\vec{E}+\vec{\nu}\times\vec{B})=0$ (N) (2)

we see that

$E H = ν x B z$ (V/m) (3)

Since $I\propto qv_x$ , we see that the sign of the Hall voltage $V H$ will depend on whether the charge carriers are predominantly negatively (electrons) or positively (holes) charged. Thus, by measuring the Hall voltage we can determine whether we have an n-type or p-type semiconductor.

Let us now consider how the velocity of charge carriers, as measured through the Hall effect, can be related to important parameters of the semiconductor sample. Suppose the charge carriers in our sample are the (negatively charged) electrons, i.e. our semiconductor is n-type. Taking into account the dimensions of the sample, the total current is related to the density of charge carriers n and their drift velocity $ν x$ as

$I = ( - e n v x)(s d)$ (A) (4)

i.e. as the product of the current density $J = - q n ν x$ and the cross-sectional area of the conductor. Recalling the relation of the Hall voltage, we obtain the following expression by which the electron density may be determined:

$V_H=-\frac{I_xB_z}{end}$ (V) (5)

Using this definition, we also obtain the Hall coefficient $R H$ as

$R_H=\frac{E_H}{J_xB_z}=\frac{1}{en}$ ( $m 3 / C$ ) (6)

We use these expressions if the Hall voltage $V H$ is negative. If $V H$ is positive, our sample is a p-type semiconductor, the current through which is carried by holes the density which, p, is determined through

$V_H=\frac{I_xB_z}{epd}$ (V) (7)

and obtain for the Hall coefficient $R_H=-\frac{1}{ep}$ .

Finally, we note that the current through a semiconductor may be simultaneously due both to electrons and to holes. To isolate the density of each, rather than just the overall charge-weighted carrier density, we must examine and interpret the dependence of the Hall voltage on temperature.

Van Der Pauw Technique

Figure 4: Van der Pauw configuration

A clever way of obtaining the Hall voltage and also the resistivity of the sample is the so-called Van der Pauw technique Van Der Pauw Theorem . Suppose we have a semiconductor sample containing four small contacts A, B, C, and D as shown in figure 4. We will use these four contacts to variably drive currents and measure voltage drops, using, for example, the notation that $I A B$ is the current driven from point A to point B, and that $V C D$ is the voltage drop between points C and D (defined positive if the voltage at C is higher than at D). For each configuration of current flow and voltage measurement, we will measure the "trans-resistance" denoted as, for example, $R_{AB,CD}=\frac{V_{CD}}{I_{AB}}$ . Four different measurement configurations will be used, as indicated by the table below:

Current	Voltage	Trans-resistance
$I A B$	$V C D$	$R A B, C D = V C D / I A B$
$I A D$	$V C B$	$R A D, C B = V C B / I A D$
$I A C$	$V B D$	$R A C, B D = V B D / I A C$
$I B D$	$V C A$	$R B D, C A = V C A / I B D$

Measurements indicated by the first two rows (e.g. current driven between adjacent points) will lead us to the resistivity, while those in the last two rows (e.g. current driven between non-adjacent points) to the Hall voltage. Starting with the Hall voltage, we simply use the last two sets of measured quantities to obtain the average trans-resistance $(R A C, B D + R B D, C A) / 2$ . We use this quantity, along with a measurement of the magnetic field $B z$ , and of the sample dimensions to obtain the Hall coefficient, applying the formulae of equations 5 or 7 depending on the sign of the trans-resistance.

Determining the resistivity ρ of the sample is somewhat more subtle. Leaving detailed derivations for Appendix A, we will make use of the Van der Pauw equation

$exp(\frac{-\pi d }{\rho}R_{AB,CD})+exp(\frac{-\pi d}{\rho}R_{AD,CB})=1$ (13)

by which ρ is implicitly determined (here d is the sample depth, referring to Fig. 3). While this equation appears difficult to solve, one readily sees that all solutions with the same ratio $R A B, C D / R A D, C B$ are trivially related to each other. Thus, we may write for the resistivity

$\rho=\frac{\pi d}{ln(2)} \cdot \frac{(R_{AB,CD}+R_{AD,CB})}{2}\cdot f(\frac{R_{AB,CD}}{R_{AD,CB}})$ (14)

The only difficult part remaining is the determination of the function f(x), a task which need be performed only once, and which is indeed already performed for us; the solution is graphed in Figure 5. To a good approximation, we may take f(x)=1/cosh(ln(x)/2.403) (to better than 0.1% for x< 2.2), and better than 1% for x < 4.3). For the derivation of equations (13) and (14) see Appendix A.

Figure 5: The function f versus the resistance ratio. This function is used to determine the resistivity of the sample. Here R_12,34 is equivalent to R_AB,CD and R_23,41 is same as R_AD,CB. (This figure appears in "Semiconductor Materials, Lecture Notes for MSE 223" by E. E. Haller, Department of Materials Science and Engineering, University of California Berkeley.)

For good measure, once the above procedure has been completed, it is worthwhile to verify our experimental technique. Thus, we will make four more sets of measurements (just the reverse of the previous ones), indicated by the table below.

Current	Voltage	Trans-resistance
$I B A$	$V D C$	$R B A, D C = V D C / I B A$
$I D A$	$V B C$	$R D A, B C = V B C / I D A$
$I C A$	$V D B$	$R C A, D B = V D B / I C A$
$I D B$	$V A C$	$R D B, A C = V A C / I D B$

Once again, we use these trans-resistances to determine the Hall voltage and the resistivity. The results of the two sets of measurements should obviously agree.

From measurements of the resistivity, we obtain the mobilities of the charge carriers in our sample as follows:

$1/\rho=e\cdot(n\cdot\mu_n + p\cdot\mu_p)$ (15)

Here, $μ n$ is the electron mobility and $μ p$ the hole mobility. This equation is trivially adapted to samples in which the carriers are exclusively either electrons or holes. The charge carrier densities n (electrons) and p (holes) come from our Hall voltage measurements.

Finally, having experimentally determined the carrier concentrations and mobilities, we obtain their drift velocities, given as $ν d e = μ n E$ for electrons and as $ν d p = μ p E$ for holes. In both expressions, E is the magnitude of the electric field which drives the current.

In this experiment we are going to make measurements of Hall Voltage and sample resistivity as it varies with temperature. We will start with the sample at room temperature, around 320K. Adding liquid nitrogen, we will begin to cool down the sample. Below 240K we will add more liquid nitrogen to replace boil off. Once the system has reached the specified starting temperature, the computer automatically senses it. At this point the heater will start warming up the sample. Once this begins, it is not possible to change the starting and ending temperatures, except by stopping the experiment early. The computer will begin to record data every degree change as the temperature rises at 5K per minute. (These data should be preserved in the data file, even if the program is terminated early or even if the computer crashes.) The temperature controller (running autonomously) will stop heating the sample at 400K (the ending temperature), and computer will stop taking the data. We are going to stop at 400K to prevent the melting of the soldered contacts on the sample. (The contacts will begin to break down at 156° C and we want to stay below that point.) The contact points of the sample consist of Boron dopant covered by a Palladium layer which in turn is covered by a gold layer which is covered by an indium layer. Then, wires are pressed onto the indium contacts (see photo, below, of this process). These contacts are very delicate. Please do not touch them. If you think there is a problem with the mounted sample, ask for help from the staff.

Photo: Hughes Silvestri wires the Germanium sample, using Indium and a wooden rod.

Once all the data have been collected, we are going to reverse the magnetic field and take the same data again. After that is done, we will turn off the magnetic field. (Actually, we will actively drive it to zero field with a small winding current. This overcomes residual hysteretic field.) We will take a third set of data at zero magnetic field (why?).

Apparatus and Procedure

In brief, this experiment involves measuring the resistivity and Hall voltage as functions of temperature on a particular semiconductor sample which has been prepared and mounted into a cryostat (Figure 6). Components of the experiment you will be using are described below.

Temperature control system

To perform these measurements, we will first cool the sample to the lowest temperatures we will be accessing. This cooling of the apparatus is accomplished by adding liquid nitrogen to the cryostat twice: a first batch of liquid nitrogen will get the temperature down to about 240 K, and then we will keep adding nitrogen to lower the temperature further. A computer interfaced thermometric system is used to control the temperature. Once the specified lowest temperature is reached, a heater is turned on and the sample begins to slowly rise in temperature until the specified highest temperature is reached. This controlled temperature ramp can be terminated by stopping the experiment early. The computer will begin to record data every degree change as the temperature rises at 5K per minute. An upper temperature limit of 400 K is imposed so as not to damage the sample (in particular the leads on the sample. The sample we are going to use for this experiment is a doped germanium crystal that is cut out into a small square with thickness of approximately 1 millimeter. The sample is mounted in a cryostat as shown in the Figure 6. The germanium crystal is mounted to the front side of the copper block with connection leads A, B, C, and D. The backside of the copper block contains a silicon diode temperature sensor (DT-470/471/670 SD). This diode measures the temperature of the sample and is a two-lead device with a capability of a four-wire configuration for making more accurate measurements. The temperature range for the diode measurement is about 4K to 475K. Two wires out of four assigned to the diode are current leads, I±. The diode's other two leads are for the voltage measurement, V±. The copper block is mounted to the brass-copper rod containing a 50-watt maximum heating coil through which we will be able to heat up the sample. The brass part of the rod can be thought of as a thermal resistance: we want to limit how fast the sample cools and heats up. The whole assembly is submerged into the chamber in which part of it (copper block with the heating coil) lies in the vacuum and the other part in the chamber where liquid nitrogen will be poured (refer to Fig. 6). Vacuum is needed to keep the Germanium sample from oxidizing when it is heated above room temperature. The cryostat is kept between the poles of the magnet and serves to hold the magnetic field measurement probe in place perpendicular to the magnetic field.

Figure 6: Operating Position of the Cryostat

In this experiment we are going to use 5000 gauss electromagnet to observe the Hall effect. The electromagnet is powered by the two 60V DC power supplies connected in series. To prevent the magnet from overheating, water-cooling lines are connected to the magnet. Water interlock circuitry in the gold box monitors the flow of the water through the magnet. When it senses that there is no water circulation, this circuit prevents any significant operation of the magnet by tripping a disconnect relay whenever more than 5 watts are being dissipated into the magnet's coil. Magnet operation resumes as soon as water circulation is restored or the target field is reduced to close to zero. See Figure 7 for the simple block diagram of the experiment. Note that the data collection software still requires water to be flowing, at least at the beginning, even if measurements are being made at zero field.

The magnet is monitored by the LakeShore 450 Gaussmeter. It provides precise measurements of the magnetic fields. To allow the LakeShore 450 to be used in a control loop to reach the target field, its auto-ranging and filtering have both been turned off (these settings persist even when turned off). To pass a current through any two leads of the sample we will use 2400 SourceMeter. To measure the voltage across any two leads of the sample we will use 6514 Electrometer. (These are finicky devices, sometimes needing to be turned off and back on again during startup of data-taking.) 330 Temperature Controller measures and autonomously controls the temperature of the sample using the silicon diode temperature sensor. All these instruments are connected to the computer through a GPIB connection. The operation manuals of these instruments are contained in the Appendix B.

The Gold box is a computer controlled system that automatically triggers the 2400 Source meter to deliver its current to the sample. It also contains a bank of 16 relays to deliver this current to one of four pairs of the sample's leads and measure the corresponding voltage on the remaining pair. By the computer's command, the Gold box has the capability of switching the above, plus the magnetic field current and the sample current directions. It also monitors the status of the experiment and if something goes wrong, it has the capability to turn things off.

To give students some familiarity of what this experiment is about, for the first part of the procedure they will use manually operated Gray box that does essentially the same thing as the Gold box. The Gray box is a home made circuitry that has a great amount of built-in error. It is not designed to make precise measurements, but rather to understand more directly what the Gold box does automatically. The Gray box will let you choose through which leads of the sample you want to send the current and allow you to measure voltage across the other two.

Figure 7: Simple Block Diagram for the Hall Effect in Semiconductors experiment.

Procedure

Below is a "rough" procedure for this experiment.

Manual part of the procedure using the Gray box

Using the Gray box you will be able to find an approximate Hall coefficient and an approximate resistivity of the sample at the room temperature.

Turn the Grey box on
Select the reverse or normal mode current
Run the program just far enough to be able to operate the magnet. Set a target B field.
Using the "Current voltage Selection" select the appropriate combination of current/voltage, i.e. driving current through AC and measuring voltage through BD. On the knob,
1. P0= Off
2. P1= drive current AB, measure voltage CD
3. P2= drive current AD, measure voltage CB
4. P3= drive current AC, measure voltage BD
5. P4= drive current BD, measure voltage CA
Select the appropriate current value on the "Sample Current Range" knob
Using the 6514 System Electrometer measure the voltage coming out of "Sample Output Voltage" from the Gray box.
Do the above procedure for the reversed current to measure all eight data points.
Using the data, calculate the Hall coefficient the and resistivity ρ the of the sample.

Control Program procedure using the Gold box

Turn on the computer if not turn it on.
Turn the water on. There are three valves that must be opened; first turn on the lever valve on the far wall, then turn on the one behind the apparatus, and finally push the button on the interlock box.
Make sure the vacuum below 60 milli-torr

Then turn on all the equipment in the Rack. The on/off switches are in the back of some of the units.
Starting from the top most piece; Electrometer, Current Source, Gauss Meter, the Lakeshore Temperature Controller and the GOLD Box. Then go to the two power supplies under bench near the magnet and turn them on.
Now push the button on the GOLD Box just above the on/off switch. The power supplies should turn on with a fan sound.
On your Desktop or in the Support folder on the C: drive is the LabView Program called "Control Program v8_9_5_multifield" excute this program..SHE Control Program Front Panel.
The data the program takes is as follows .SHE Data Taking Table.
Run the Control Program; Then input the experiment parameters; Sample Current, Magnet Field in Gauss ( Max 5000), Start Temperature, End Temperature (in K, Then click the Start Experiment button. Name a file to save with an extension of *.TXT or *.DAT for your data ( dialog Box will pop up).
Test the system by inputing temperature range 120K to 200K.
Then you can use the defaults 95K to 350K the program will take about 1 hour 40 minutes to finish.
Do NOT RUN or take data Overnight.
If you want to STOP early use the STOP button in the program DO NOT Stop LabView or X out the program. This would be like driving your car at 65 miles per hour and you get removed from your car while it is still going. ( this takes about 55 seconds to stop)
Now add liquid nitrogen to the cryostat, so it can begin cooling. Move on to the next step, but periodically check the temperature displayed by the Lakeshore 330 Temperature Controller (check that it is on) and the Control Program display; when it reaches 240K, add more liquid nitrogen.
Check the temperature. The minimum temperature that we are capable of reaching is about 88K, but this is hard to get to; as long as you get bellow approximately 95K you should be fine. (88K takes about 1 hour to get)
The Temperature Ramp Rate is set to increase the temperature by 2K per minute (calculate how much time it will take for you to complete a run, you may want to try a lower rate in order to get more detailed data, but remember, this is NOT and overnight lab: you must finish before closing time).

When Finish for the day Stop Control Program or let it finish data taking.
Turn off GOLD Box, and Heater controller before you leave the 111-Lab.

Control Program Description

Start the Control Program ie; RUN LabView program
Input Experiment Parameters
Click Start Experiment
The data taking will start after the start temperature is reached the second time.
there is over shoot you will see it on the temperature graph.
Temperature goes below set point then heater will come on and when start temperature is reached again data taking will start.
NOTE: if for some reason the program crashes or you accidentily X out the program You must restart the program and just click start experiment button, with no parameters entered, to reset the system.

Check to see that the Heater is OFF.

Turn off GOLD Box, and Heater controller when finished before you leave the 111-Lab.

Report

The following should be included in your report:

Resistivity and Hall Coefficient of the sample at room temperature. At what magnetic field intensity, would the E field due to Hall effect be of the same magnitude (though perpendicular in direction) as the E field due to resistance?
Plot of resistivity ρ of the sample versus inverse temperature. Find at what temperature values the sample is in the extrinsic or intrinsic regions.
Plot of conductivity σ=1/ρ versus inverse temperature.
Plot of Hall coefficient of the sample versus inverse temperature.
Explain what type of material we have: p-type or n-type. (Why don't you need to know the direction of the magnetic field after all?)
Find electron or hole concentrations for the sample versus temperature.
Find the Hall Coefficient $R H$ and the Hall mobility $\mu_H=R_H\cdot\sigma$ versus temperature.
Find electron or hole concentration in the extrinsic region using equation 8 or 7.
Find electron and hole mobilities in the extrinsic region.
Compare the resistance measured for the sample (at zero field) with the magnetoresistance (the resistance measured while the magnetic field is on).

Hall Effect in Semiconductor

From Physics 111-Lab Wiki

Contents

Before The Lab

Prerequiste Reading Materials

Useful Definitions

Introduction

Preliminary Concepts

Hall Effect and Van Der Pauw Technique

Hall Effect

Van Der Pauw Technique

Apparatus and Procedure

Procedure

Manual part of the procedure using the Gray box

Control Program procedure using the Gold box

Control Program Description

Report

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