Capacitance-voltage profiling: Research-grade approach versus low-cost alternatives Neal D. Reynolds, Cristian D. Panda, and John M. Essicka) Physics Department, Reed College, Portland, Oregon 97202 (Received 21 July 2013; accepted 22 January 2014) We describe an experiment that implements capacitance-voltage profiling on a reverse-biased Schottky barrier diode to determine the density of impurity dopants in its semiconductor layer as well as its built-in electric potential. Our sample is a commercially produced Schottky diode. Three different experimental setups, one using research-grade instrumentation, the other two using low-cost alternatives, are given and their results compared. In each of the low-cost setups, phase-sensitive detection required to measure the sample’s capacitance is carried out using an inexpensive data acquisition (DAQ) device and a software program that implements a lock-in detection algorithm. The limitations of the DAQ device being used (e.g., restricted analog-to-digital conversion speed, inadequate waveform generation capabilities, lack of hardware triggering) are taken into account in each setup. Excellent agreement for the value of the doping density obtained by the all three setups is found and this value is shown to be consistent with the result of an independent method (secondary ion mass spectroscopy). VC 2014 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4864162] I. INTRODUCTION II. THEORY Semiconductor physics has proved itself a fertile field for A. Capacitance-voltage profiling of Schottky barrier advanced instructional laboratory developers. The properties of semiconductors engender natural student interest as they lie at In this section, we derive the theoretical relations that are the heart of modern-day technologies such as computers, smart the basis of the capacitance-voltage profiling experimental 20,22,23,25 phones, and the internet, while, pedagogically, semiconductor technique. First, consider a Schottky barrier of phenomena provide engaging applications of basic concepts in cross-sectional area A that consists of a metal layer in contact quantum mechanics, electrodynamics, and thermal physics. with an n-type semiconductor (dielectric constant ; for sili- Hence, many semiconductor-related advanced laboratory con ¼ 11.7) and define an x-axis whose origin is at the experiments have been developed. For example, electrical metal-semiconductor interface with its positive direction to- measurements on forward-biased diodes and Hall devices have ward the semiconductor’s interior (Fig. 1). For the been used to measure the semiconducting band gap,1–6 Schottky barrier height,7 charge carrier density,8–10 carrier transport properties,11,12 and carrier statistical distributions.13 Additionally, optical experiments are available to determine the band gap of bulk,14,15 thin film,16 and quantum-dot17 semicon- ductor samples, the Maxwell-Boltzmann distribution of charge carriers,18 and the vibration properties of nanomaterials.19 In this paper, we describe a newly developed instructional lab experiment that implements capacitance-voltage (CV) profiling on a reverse-biased Schottky barrier diode to deter- mine the density of impurity dopants in its semiconductor layer, as well as its built-in electric potential. In contrast to the more complicated dual-semiconductor structure of a pn- junction diode, a Schottky barrier diode consists of a single semiconductor layer in contact with a metallic layer. Due to the simplicity of its construction, the functioning of a Schottky barrier diode can be explained theoretically using only introductory electrodynamics and semiconductor con- cepts. As with all semiconductor devices, fabrication of a high-quality Schottky diode requires specialized expertise and expensive deposition systems. To avoid this hurdle, we use a commercially produced Schottky barrier diode in this project. Additionally, in an effort to contain the cost of the Fig. 1. Band bending in a Schottky barrier (cross-sectional area A) under capacitance characterization system, we describe several bias of –V0 creates a (shaded) depletion region of width W with space charge options for the setup, ranging from one consisting solely of density þeq due to ionized dopant atoms; q may be position-dependent. When the reverse bias is increased by dVR, an additional charge of stand-alone research-grade equipment to others that use dQ ¼þeq(AdW) is created at the tail of the depletion region. The low-cost op-amp circuitry and affordable computer-based metal-semiconductor interface is at x ¼ 0 and it is assumed each ionized instrumentation. dopant atom has charge of þe. 196 Am. J. Phys. 82 (3), March 2014 http://aapt.org/ajp VC 2014 American Association of Physics Teachers 196 This article is copyrighted as indicated in the article. Reuse of AAPT content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 134.10.101.160 On: Thu, 16 Oct 2014 23:22:59 semiconducting material, we assume the following. First, boundary conditions: Vð0ÐÞ¼ðVR þ VbiÞ and V(W) ¼ 0. Thus, W positively charged dopant atoms are incorporated in its lat- from VðWÞVð0Þ¼ 0 E Á dx,wefind tice structure with a position-dependent volume number den- eq V þ V ¼ 0 W2 (1) sity (“doping density”) q(x). Second, its temperature is high R bi 2 enough so that these dopant atoms are fully ionized, that is, 0 their extra electrons have all been promoted into the semi- and so the width of the depletion region required to screen conductor’s conduction band. These negative conduction out the reverse bias VR is electrons then perfectly compensate the dopant atom’s posi- sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tive charge so that, in its bulk, the semiconductor is electri- 20ðVR þ VbiÞ W ¼ : (2) cally neutral. Third, at large x the semiconductor is eq0 connected to external circuitry via an ohmic “back contact.” We further assume that the type of metal is properly cho- If the reverse bias VR is increased by a small amount dVR, sen so that when the two materials are joined, semiconductor (remembering Vbi is a constant) we find from Eq. (1) that conduction electrons transfer to the metal’s surface, leaving dVR ¼ðeq0=0ÞWdW, where dW is the increase in the behind a positively charged layer of uncompensated dopant depletion region’s width. This increase in the depletion atoms (called the “depletion region”) in the volume of the region width is due to flow of conduction electrons at the semiconductor nearest the metal. This charge transfer takes edge of the depletion region into the semiconductor’s bulk place until the resulting space charge produces an electric (and out of the back contact), creating the extra space charge potential –Vbi at x ¼ 0(Vbi 0 is termed the “built-in dQ ¼þeq0ðAdWÞ required to screen out the voltage change potential”), which prevents further flow of charge (because –dVR at the metal-semiconductor interface. By definition, the Fermi levels of the two materials have been equalized). this process produces a capacitive response given by Then, if an externally applied voltage –VR (where VR 0 dQ eq ðAdWÞ A is called the “reverse bias”) is applied at the metal contact so C ¼ 0 ¼ 0 ; (3) that the total potential at the metal-semiconductor interface dVR ðeq0=0ÞWdW W is –V0 ¼ –(VR þ Vbi) and the total charge on the metal’s sur- or, using Eq. (2), the Schottky barrier’s capacitance as a face is –Q, movement of semiconductor conduction electrons function of reverse bias VR is away from the interface (and out the back contact at large x) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e q will extend the depletion region to a width W. Beyond the C ¼ A 0 0 : (4) depletion region, the semiconductor is neutral and the effect 2ðVR þ VbiÞ of –Q is not felt (i.e., the electric field is zero there). This effect is called dielectric screening and W is the “screening This expression can be re-written as length.” 1 2 ; (5) Let’s now analyze the special case of uniform doping density 2 ¼ 2 ðVR þ VbiÞ C A e0q0 q(x) ¼ q0 (a constant), where each dopant atom is assumed to have a single charge þe. If the cross-sectional dimensions of the suggesting the following capacitance-voltage characteriza- Schottky barrier are much greater than W,thenbysymmetrywe tion method: For a Schottky barrier with uniform doping 2 canassumetheelectricfieldE in the depletion region points in density, a plot of 1/C versus VR will yield a straight line 2 the (negative) x-direction. Using the rectangular Gaussian sur- with slope m ¼ 2=A e0q0 and y-intercept 2 faceshowninFig.2, with one of its end caps in the neutral bulk b ¼ 2Vbi=A e0q0. Then, the doping density and built-in region (where E ¼ 0) and the other end cap located a distance x potential are found from from the metal-semiconductor interface, Gauss’s law gives 2 EA ¼þeq AðW À xÞ= . For the electric potential V(x)asa q ¼ ; (6) 0 0 0 A2e m function of distance x into the depletion region, we have the two 0 and b V ¼ : (7) bi m The general case of position-dependent doping density q(x) can be solved as follows: We start with the identity d dV dV d2V x ¼ þ x ; (8) dx dx dx dx2 and note from Poisson’s equation that d2V þeqðxÞ 2 ¼ : (9) dx 0 Fig. 2. Gauss’s Law applied to a Schottky barrier (cross-sectional area A) under bias of –V0 ¼ –(VR þ Vbi) to determine the electric field E at location x,assum- Substituting Eq. (9) into Eq. (8) and integrating both sides of ing constant doping density. By symmetry, the electric field E is directed along the resulting expression from x ¼ 0tox ¼ W then yields the (negative) x-axis. A (dashed) rectangular Gaussian surface is chosen with ð W one face in the depletion region, where the space charge density is þeq0;the e other face is in neutral bulk region of the semiconductor, where the electric field V þ V ¼ xqðxÞdx: (10) R bi is zero.