Abstract
A number of methods used for measuring the semiconductor parameters are illustrated here. Apart from the intrinsic usefulness, the methods are interesting because they show the connection with the theories worked out in other chapters. For example, the measurement of lifetimes exploits the features of the net thermal recombination and of optical generation, that are combined in a simplified form of the continuity equation for the minority carriers. Similarly, the measurement of mobility carried out with the Haynes-Shockley experiment is based on a clever use of the diffusion of optically generated carriers. The Hall effect, in turn, provides a powerful method to extract the information about the concentration and mobility of the majority carriers; the method exploits the effect of a magnetic field applied in the direction normal to that of the current density, and is widely used for determining, e.g., the dependence of concentration and mobility on the concentration of dopants and on temperature. The analysis of the Hall effect is enriched by a detailed treatment of the case where the standard theory is not applicable because the device is not sufficiently slender; a deeper analysis based on the concept of stream function shows that the equations describing the current-density field are in fact solvable for any aspect ratio of the device. This chapter is completed by the illustration of a method for measuring the doping profile in an asymmetric, reverse-biased, one-dimensional junction; the procedure is based on the observation that despite the fact that the relation between the applied voltage and the extension of the space-charge region is nonlinear, the differential capacitance of the junction has the same form as that of a parallel-plate, linear capacitor. Finally, the van der Pauw method for measuring the conductivity of a sample is illustrated, based on the use of a two-dimensional Green function introduced in an earlier chapter and on the conformal-mapping method shown in the Appendix.
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Notes
- 1.
The required thinness of the device can be achieved by growing an n-type epitaxial layer over a p-type substrate, and keeping the layer-substrate junction reverse biased.
- 2.
See also Prob. 21.13.
- 3.
The fact that the size of the capture cross-section is sometimes similar, like in the present case, to the size of an atom may explain the popularity of expressing the capture phenomenon in terms of cross-sections. This, however, oversimplifies the problem: capture cross-sections, in fact, range from 10−25 to 10−12 cm2 [12, Sect. 4.1.2].
- 4.
The generated electrons drift to the left and are absorbed by the left contact.
- 5.
It is useful to remark that in a uniform resistor it is J 3a ′ = q μ 3a  n a  E 3 = 0, since E 3 = 0 due to uniformity; in a MOSFET’s channel it is J 3a ′ = k B  T μ 3a  ∂n a ∕∂x 3 + q μ 3a  n a  E 3 = 0 because the drift and diffusion components balance each other. Also, in a MOSFET’s channel the components of the current density parallel to the x, y plane depend also on z; however, one can neglect such a dependence in (25.33) by considering J 1a and J 2a as averages over h [115].
- 6.
With reference to Prob. 25.2, the second condition in (25.36) is derived by observing that along the side y = −L∕2 the unit vector t is aligned with the side itself and points in the positive direction of x. As a consequence, n points in the positive direction of y, namely, n = j and ∂ψ∕∂y = ∂ψ∕∂n. By the same token, the sign in the second condition of (25.37) is justified by observing that the currents at the contacts sum to zero, and by using the relation ∂ψ∕∂y = −∂ψ∕∂n that holds on the side y = L∕2 because n = −j there. It is assumed that the current density is uniformly distributed along the contact.
- 7.
The difference in sign is related to the orientation of B with respect to the z axis. In the analysis of Sect. 25.5, such an orientation is not specified.
References
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Rudan, M. (2018). Measuring the Semiconductor Parameters. In: Physics of Semiconductor Devices. Springer, Cham. https://doi.org/10.1007/978-3-319-63154-7_25
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DOI: https://doi.org/10.1007/978-3-319-63154-7_25
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