Hall Effect NavigationMain PageI. IntroductionII. The Hall EffectIII. Resistivity and Hall MeasurementsIV. AlgorithmV. ReferencesLeave CommentsView CommentsFigure 1Figure 2Figure 3Figure 4
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## II. The Hall EffectEvolution of Resistance Concepts The Hall Effect and the Lorentz Force
Electrical characterization of materials evolved in three levels of understanding. In the early 1800s, the resistance R and conductance G were treated as measurable physical quantities obtainable from two-terminal By the early 1900s, it was realized that resistivity was not a fundamental material parameter, since different materials can have the same resistivity. Also, a given material might exhibit different values of resistivity, depending upon how it was synthesized. This is especially true for semiconductors, where resistivity alone could not explain all observations. Theories of electrical conduction were constructed with varying degrees of success, but until the advent of quantum mechanics, no generally acceptable solution to the problem of electrical transport was developed. This led to the definitions of carrier density
The basic physical principle underlying the Hall effect is the Lorentz force, which is a combination of two separate forces: the electric force and the magnetic force. When an electron moves along the electric field direction perpendicular to an applied magnetic field, it experiences a magnetic force - acting normal to both directions. The direction of this magnetic force can be determined by using the right hand rule convention. With an open hand, the fingers are pointed along the direction of the carrier velocity and curled into the direction of the magnetic field. The magnetic force direction on an electron is then determined by the opposite direction that the thumb is pointing. The resulting Lorentz force B is therefore equal to - Fq( + E x v) where Bq (1.602x10^{-19} C) is the elementary charge, is the electric field, E is the particle velocity, and v is the magnetic field. For an Bn-type, bar-shaped semiconductor such as that shown in Fig.1, the carriers are predominately electrons of bulk density n. We assume that a constant current I flows along the x-axis from left to right in the presence of a z-directed magnetic field. Electrons subject to the Lorentz force initially drift away from the current direction toward the negative y-axis, resulting in an excess negative surface electrical charge on this side of the sample. This charge results in the Hall voltage, a potential drop across the two sides of the sample. (Note that the force on holes is toward the same side because of their opposite velocity and positive charge.) This transverse voltage is the Hall voltage V_{H} and its magnitude is equal to IB/qnd, where I is the current, B is the magnetic field, d is the sample thickness, and q (1.602 x 10^{-19} C) is the elementary charge. In some cases, it is convenient to use layer or sheet density (n_{s} = nd) instead of bulk density. One then obtains the equation
Thus, by measuring the Hall voltage
If the conducting layer thickness In order to determine both the mobility The objective of the resistivity measurement is to determine the sheet resistance
which can be solved numerically for The bulk electrical resistivity
To obtain the two characteristic resistances, one applies a dc current
The objective of the Hall measurement in the van der Pauw technique is to determine the sheet carrier density A second type of geometry that is sometimes used includes the parallelepiped or bridge-type sample. These may be more desirable in the case of anisotropic material properties. the restrictions on shape and size are more rigid than those of the van der Pauw specimen, but measurements can be made using either a six or an eight contact configuration. The bridge-type specimen differs from the parallelepiped in that the contacts are placed on arms that branch off the main parallelepiped base. The details of this method can be obtained from the ASTM document listed in the references. There are practical aspects which must be considered when carrying out Hall and resistivity measurements. Primary concerns are (1) ohmic contact quality and size, (2) sample uniformity and accurate thickness determination, (3) thermomagnetic effects due to nonuniform temperature, and (4) photoconductive and photovoltaic effects which can be minimized by measuring in a dark environment. Also, the sample lateral dimensions must be large compared to the size of the contacts and the sample thickness. Finally, one must accurately measure sample temperature, magnetic field intensity, electrical current, and voltage. |