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<title>II. The Hall Effect</title></head>
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<td width="1005"><img alt="Technical Activities" src="../images/tech-banner1.gif" border="0" usemap="#Navigation"><map name="Navigation"><area href="http://www.eeel.nist.gov/812/tech.html" coords="447,63,603,84" shape="rect" alt="Technical Activities"><area href="http://www.eeel.nist.gov/812/research.html" coords="220,63,446,84" shape="rect" alt="Research Projects / Facilities"><area href="http://www.eeel.nist.gov/812/about-sed.html" coords="97,63,217,84" shape="rect" alt="About SED"><area href="http://www.eeel.nist.gov/812/index.html" coords="3,63,95,84" shape="rect" alt="Home"></map><img alt="National Institute of Standards and Technology" src="../images/tech-banner2.gif" usemap="#NIST" border="0"><map name="NIST"><area href="http://www.eeel.nist.gov/812/search.html" coords="125,63,225,84" shape="rect" alt="Search"><area href="http://www.eeel.nist.gov/812/outputs.html" coords="18,63,123,84" shape="rect" alt="Outputs"><area href="http://www.eeel.nist.gov/812/tech.html" coords="2,63,16,84" shape="rect" alt="Technical Activities"><area href="http://www.nist.gov/" coords="2,4,138,60" shape="rect" alt="NIST home page"></map></td>
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<h1>II. The Hall Effect</h1>
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</div>
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<dir>
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<p><a href="#evol">Evolution of Resistance Concepts</a><br>
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<a href="#lore">The Hall Effect and the Lorentz Force</a><br>
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<a href="#vand">The van der Pauw Technique</a></p>
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</dir>
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<h3><a name="evol"></a>Evolution of Resistance Concepts</h3>
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<p><b>Electrical characterization</b> of materials evolved in three levels of understanding. In the early 1800s, the resistance <i>R</i> and conductance <i>G</i> were treated as measurable physical quantities obtainable from two-terminal <i>I-V</i> measurements (i.e., current <i>I</i>, voltage <i>V</i>).
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Later, it became obvious that the resistance alone was not comprehensive
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enough since different sample shapes gave different resistance values. This
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led to the understanding (second level) that an intrinsic material property
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like resistivity (or conductivity) is required that is not influenced by
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the particular geometry of the sample. For the first time, this allowed scientists
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to quantify the current-carrying capability of the material and carry out
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meaningful comparisons between different samples.</p>
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<p>By the
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early 1900s, it was realized that resistivity was not a fundamental material
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parameter, since different materials can have the same resistivity. Also,
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a given material might exhibit different values of resistivity, depending
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upon how it was synthesized. This is especially true for semiconductors,
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where resistivity alone could not explain all observations. Theories of electrical
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conduction were constructed with varying degrees of success, but until the
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advent of quantum mechanics, no generally acceptable solution to the problem
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of electrical transport was developed. This led to the definitions of carrier
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density <i>n</i> and mobility <i><EFBFBD><EFBFBD></i> (third level of understanding) which are capable of dealing with even the most complex electrical measurements today.<a href="http://www.eeel.nist.gov/812/fig1.htm"><img src="../images/fig1s.jpg" align="right" hspace="10" vspace="70" border="0" alt="Schematic of the Hall effect in a long, thin bar of semiconductor with four ohmic contacts. The direction of the magnetic field B is along the z-axis and the sample has a finite thickness d"></a></p>
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<h3><a name="lore"></a>The Hall Effect and the Lorentz Force</h3>
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<p>The
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basic physical principle underlying the Hall effect is the Lorentz force.
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When an electron moves along a direction perpendicular to an applied magnetic
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field, it experiences a force acting normal to both directions and moves
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in response to this force and the force effected by the internal electric
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field. For an <i>n</i>-type, bar-shaped semiconductor shown in <a href="http://www.eeel.nist.gov/812/fig1.htm">Fig.1</a>, the carriers are predominately electrons of bulk density <i>n</i>. We assume that a constant current <i>I</i>
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flows along the x-axis from left to right in the presence of a z-directed
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magnetic field. Electrons subject to the Lorentz force initially drift away
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from the current line toward the negative y-axis, resulting in an excess
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surface electrical charge on the side of the sample. This charge results
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in the Hall voltage, a potential drop across the two sides of the sample.
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(Note that the force on holes is toward the same side because of their opposite
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velocity and positive charge.) This transverse voltage is the Hall voltage
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<i>V</i><font size="2"><sub>H</sub></font> and its magnitude is equal to <i>IB/qnd</i>, where <i>I</i> is the current, <i>B</i> is the magnetic field, <i>d</i> is the sample thickness, and <i>q</i> (1.602 x 10<font size="2"><sup>-19</sup></font> C) is the elementary charge. In some cases, it is convenient to use layer or sheet density (<i>n</i><sub>s</sub> = <i>nd</i>) instead of bulk density. One then obtains the equation</p>
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<p>
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<table border="0" cellpadding="0" cellspacing="2">
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<tbody><tr>
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<td width="40"></td>
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<td width="480"><i>n</i><sub>s</sub> = <i>IB</i>/<i>q</i>|<i>V</i><font size="2"><sub>H</sub></font>|.</td>
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<td width="20"></td>
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<td>(1)</td>
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</tr>
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</tbody></table>
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</p>
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<p>Thus, by measuring the Hall voltage <i>V</i><sub>H</sub> and from the known values of <i>I</i>, <i>B</i>, and <i>q</i>, one can determine the sheet density <i>n</i><sub>s</sub>
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of charge carriers in semiconductors. If the measurement apparatus is set
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up as described later in Section III, the Hall voltage is negative for <i>n</i>-type semiconductors and positive for <i>p</i>-type semiconductors. The sheet resistance <i>R</i><font size="2"><sub>S</sub></font>
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of the semiconductor can be conveniently determined by use of the van der
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Pauw resistivity measurement technique. Since sheet resistance involves both
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sheet density and mobility, one can determine the Hall mobility from the
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equation</p>
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<p>
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<table border="0" cellpadding="0" cellspacing="2" align="left">
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<tbody><tr>
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<td width="40"></td>
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<td width="480"><font face="Symbol"><i><EFBFBD></i></font><i><EFBFBD><EFBFBD></i> = |<i>V</i><font size="2"><sub>H</sub></font>|/<i>R</i><font size="2"><sub>S</sub></font><i>IB</i> = 1/(<i>qn</i><font size="2"><sub>S</sub></font><i>R</i><font size="2"><sub>S</sub></font>).</td>
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<td width="20"></td>
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<td>(2)</td>
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</tr>
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</tbody></table>
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<br>
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<br>
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</p>
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<p>If the conducting layer thickness <i>d</i> is known, one can determine the bulk resistivity (<font face="Symbol"><i>r</i></font> = <i>R</i><font size="2"><sub>S</sub></font><i>d</i>) and the bulk density (<i>n</i> = <i>n</i><font size="2"><sub>S</sub></font>/<i>d</i>). <a href="http://www.eeel.nist.gov/812/fig2.htm"><img src="../images/fig2s.jpg" align="right" hspace="10" vspace="70" border="0" alt="Schematic of a van der Pauw configuration used in the determination of the two characteristic resistances RA and RB"></a></p>
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<h3><a name="vand"></a>The van der Pauw Technique</h3>
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<p>In order to determine both the mobility <i><EFBFBD></i> <20><>and the sheet density <i>n</i><sub>s</sub>, a combination of a <a href="http://www.eeel.nist.gov/812/meas.htm#resi">resistivity measurement</a> and a <a href="http://www.eeel.nist.gov/812/meas.htm#halm">Hall measurement</a>
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is needed. We discuss here the van der Pauw technique which, due to its convenience,
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is widely used in the semiconductor industry to determine the resistivity
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of uniform samples (References 3 and 4). As originally devised by van der
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Pauw, one uses an arbitrarily shaped (but simply connected, i.e., no holes
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or nonconducting islands or inclusions), thin-plate sample containing four
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very small ohmic contacts placed on the periphery (preferably in the corners)
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of the plate. A schematic of a rectangular van der Pauw configuration is
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shown in <a href="http://www.eeel.nist.gov/812/fig2.htm">Fig. 2</a>.</p>
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<p>The objective of the <a href="http://www.eeel.nist.gov/812/meas.htm#resi">resistivity measurement</a> is to determine the sheet resistance <i>R</i><font size="2"><sub>S</sub></font>. Van der Pauw demonstrated that there are actually two characteristic resistances <i>R</i><font size="2"><sub>A</sub></font> and <i>R</i><font size="2"><sub>B</sub>,</font> associated with the corresponding terminals shown in <a href="http://www.eeel.nist.gov/812/fig2.htm">Fig. 2</a>. <i>R</i><font size="2"><sub>A</sub></font> and <i>R</i><font size="2"><sub>B</sub></font> are related to the sheet resistance <i>R</i><font size="2"><sub>S</sub></font> through the van der Pauw equation</p>
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<p>
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<table border="0" cellpadding="0" cellspacing="2">
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<tbody><tr>
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<td width="40"></td>
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<td width="480">exp(-<font face="Symbol">p</font><i>R</i><font size="2"><sub>A</sub></font>/<i>R</i><font size="2"><sub>S</sub></font>) + exp(-<font face="Symbol">p</font><i>R</i><font size="2"><sub>B</sub></font>/<i>R</i><font size="2"><sub>S</sub></font>) = 1</td>
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<td width="20"></td>
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<td>(3)</td>
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</tr>
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</tbody></table>
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</p>
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<p>which can be solved numerically for <i>R</i><font size="2"><sub>S</sub></font>.</p>
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<p>The bulk electrical resistivity <font face="Symbol"><i>r</i></font> can be calculated using</p>
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<p>
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<table border="0" cellpadding="0" cellspacing="2">
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<tbody><tr>
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<td width="40"></td>
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<td width="480"><font face="Symbol"><i>r</i></font> = <i>R</i><font size="2"><sub>S</sub></font><i>d</i>.</td>
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<td width="20"></td>
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<td>(4)</td>
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</tr>
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</tbody></table>
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</p>
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<p>To obtain the two characteristic resistances, one applies a dc current <i>I</i> into contact 1 and out of contact 2 and measures the voltage <i>V</i><font size="2"><sub>43</sub></font> from contact 4 to contact 3 as shown in <a href="http://www.eeel.nist.gov/812/fig2.htm">Fig. 2</a>. Next, one applies the current <i>I</i> into contact 2 and out of contact 3 while measuring the voltage <i>V</i><font size="2"><sub>14</sub></font> from contact 1 to contact 4. <i>R</i><font size="2"><sub>A</sub></font> and <i>R</i><font size="2"><sub>B</sub></font> are calculated by means of the following expressions:</p>
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<p>
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<table border="0" cellpadding="0" cellspacing="2">
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<tbody><tr>
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<td width="40"></td>
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<td width="480"><i>R</i><font size="2"><sub>A</sub></font> = <i>V</i><font size="2"><sub>43</sub></font>/<i>I</i><font size="2"><sub>12</sub></font> and <i>R</i><font size="2"><sub>B</sub></font> = <i>V</i><font size="2"><sub>14</sub></font>/<i>I</i><font size="2"><sub>23</sub></font>.</td>
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<td width="20"></td>
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<td>(5)</td>
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</tr>
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</tbody></table>
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</p>
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<p><a href="http://www.eeel.nist.gov/812/fig3.htm"><img src="../images/fig3s.jpg" align="right" hspace="10" vspace="70" border="0" alt="Schematic of a van der Pauw configuration used in the determination of the Hall voltage VH"></a>The objective of the <a href="http://www.eeel.nist.gov/812/meas.htm#halm">Hall measurement</a> in the van der Pauw technique is to determine the sheet carrier density <i>n</i><sub>s</sub> by measuring the Hall voltage <i>V</i><font size="2"><sub>H</sub></font>. The Hall voltage measurement consists of a series of voltage measurements with a constant current <i>I</i> and a constant magnetic field <i>B</i> applied perpendicular to the plane of the sample. Conveniently, the same sample, shown again in <a href="http://www.eeel.nist.gov/812/fig3.htm">Fig. 3</a>, can also be used for the Hall measurement. To measure the Hall voltage <i>V</i><font size="2"><sub>H</sub></font>, a current <i>I</i> is forced through the opposing pair of contacts 1 and 3 and the Hall voltage <i>V</i><font size="2"><sub>H</sub></font> (= <i>V</i><font size="2"><sub>24</sub></font>) is measured across the remaining pair of contacts 2 and 4. Once the Hall voltage <i>V</i><font size="2"><sub>H</sub></font> is acquired, the sheet carrier density <i>n</i><sub>s</sub> can be calculated via <i>n</i><sub>s</sub> = <i>IB</i>/<i>q</i>|<i>V</i><font size="2"><sub>H</sub></font>| from the known values of <i>I</i>, <i>B</i>, and <i>q</i>.</p>
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<p>There
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are practical aspects which must be considered when carrying out Hall and
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resistivity measurements. Primary concerns are (1) ohmic contact quality
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and size, (2) sample uniformity and accurate thickness determination, (3)
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thermomagnetic effects due to nonuniform temperature, and (4) photoconductive
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and photovoltaic effects which can be minimized by measuring in a dark environment.
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Also, the sample lateral dimensions must be large compared to the size of
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the contacts and the sample thickness. Finally, one must accurately measure
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sample temperature, magnetic field intensity, electrical current, and voltage.</p>
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<p><a href="http://www.eeel.nist.gov/812/meas.htm"><img src="../images/continue.jpg" alt="Click for Next Page"></a></p>
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<center>
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<p>
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</p><hr>
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<p></p>
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<p><font size="2">|<a href="http://www.eeel.nist.gov/812/hall.html"> Main Page </a>| <a href="http://www.eeel.nist.gov/812/intr.htm">I. Introduction </a>| <a href="http://www.eeel.nist.gov/812/effe.htm">II. The Hall Effect </a>| <a href="#evol">evolution of resistance concepts </a>|<br>
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| <a href="#lore">the Hall effect and the Lorentz force </a>| <a href="#vand">the van der Pauw technique </a>|<br>
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| <a href="http://www.eeel.nist.gov/812/meas.htm">III. Resistivity and Hall Measurements </a>| <a href="http://www.eeel.nist.gov/812/meas.htm#geom">sample geometry </a>| <a href="http://www.eeel.nist.gov/812/meas.htm#defi">definitions for resistivity measurements</a> |<br>
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|<a href="http://www.eeel.nist.gov/812/meas.htm#resi"> resistivity measurements </a>| <a href="http://www.eeel.nist.gov/812/meas.htm#calc">resistivity calculations </a>| <a href="http://www.eeel.nist.gov/812/meas.htm#defh">definitions for Hall measurements </a>|<br>
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|<a href="http://www.eeel.nist.gov/812/meas.htm#halm"> Hall measurements </a>|<a href="http://www.eeel.nist.gov/812/meas.htm#halc"> Hall calculations </a>| <a href="http://www.eeel.nist.gov/812/work.htm">Sample Hall Worksheet </a>|<a href="http://www.eeel.nist.gov/812/work2.htm"> Worksheet with Typical Data </a>| <a href="http://www.eeel.nist.gov/812/samp.htm">IV. Algorithm </a>|<br>
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|<a href="http://www.eeel.nist.gov/812/references.htm"> V. References </a>| VI.<a href="http://www.eeel.nist.gov/812/hallcom.htm"> Leave </a>or <a href="http://www.eeel.nist.gov/812/hallview.htm">View </a>Comments | <a href="http://www.eeel.nist.gov/812/fig1.htm">figure 1 </a>| <a href="http://www.eeel.nist.gov/812/fig2.htm">figure 2 </a>| |<a href="http://www.eeel.nist.gov/812/fig3.htm"> figure 3 </a>| <a href="http://www.eeel.nist.gov/812/fig4.htm">figure 4 </a>|<br>
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</font></p>
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<p>
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</p><hr>
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<p></p>
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</center>
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<p><font face="Arial,Helvetica,sans-serif" size="1">NIST is an agency of the <a href="http://www.doc.gov/">U.S. Commerce Department's</a> <a href="http://www.ta.doc.gov/">Technology Administration.</a></font></p>
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<p><font face="Arial,Helvetica,sans-serif" size="1">Date created: 12/4/2000<br>
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Last updated: 6/28/2001</font></p></td>
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<td valign="top" width="1" bgcolor="black"><img height="5" src="../images/black_1pix.html" width="1" border="0"></td>
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<td valign="top" width="200" bgcolor="#ffcc66">
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<table cellspacing="0" cellpadding="0" width="95%" border="0">
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<tbody>
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<tr>
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<td width="200"><b><EFBFBD></b>
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<p><a href="http://www.eeel.nist.gov/812/conf.htm"><b>Conferences/Workshops</b></a></p>
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<p><a href="http://www.eeel.nist.gov/812/nano.html"><b>Nanotechnology</b></a></p>
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<p><a href="http://www.eeel.nist.gov/812/inte.htm"><b>Electronic Interconnects/Packaging</b></a></p>
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<p><a href="http://www.eeel.nist.gov/812/comp.htm"><b>Compound Semiconductors</b></a></p>
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<p><a href="http://www.eeel.nist.gov/812/hall.html"><b>Hall Effect Measurements</b></a></p>
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<p><a href="http://www.eeel.nist.gov/812/itrcs.html"><b>Compound Semiconductor Roadmap</b></a></p></td>
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</tr>
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</tbody>
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</table>
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</td>
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