Project 6 Capacitance of a PN Junction Diode

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Caitlin Thomas
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1 Project 6 Capacitance of a PN Junction Diode OVERVIEW: In this project, we will characterize the capacitance of a reverse-biased PN diode. We will see that this capacitance is voltage-dependent and we will use our measurements to determine the built-in voltage of the diode PN junction. BACKGROUND We have studied the parallel-plate and coaxial capacitors in class using electrostatics (Gauss s Law). Capacitance is geometric quantity. From this study, we found the capacitance is a geometric quantity the depends on the configuration of electrodes and dielectric material used as an insulating medium. Semiconductor diodes (PN junctions) also store charge within their depletion region and, as a result, act as capacitors when reverse biased (i.e, when a voltage with negative polarity is applied to the diode). The capacitance of diode, however, varies with the applied voltage and, consequently, is tunable with bias. This is a useful property and can be exploited to design voltage-tunable phase-shifters, modulators and frequency multipliers (also known as harmonic generators ). You have probably studied PN junctions if you have taken a course on solid-state devices. Below, we present a brief overview of the PN diode and the electrostatics describing the diode capacitance. Current-Voltage Relation for a PN Junction Diode An Overview: The pn junction diode (or just pn diode ) is a two-terminal circuit element made of a material known as a semiconductor. Semiconductors are materials whose conductivity can be engineered by introducing impurities known as dopants. Essentially, dopants are atoms that change the number of mobile electrons in the semiconductor by either lending electrons from their outermost orbitals (such dopants are called donors) or capturing electrons from the semiconductor (these dopants are called acceptors). Semiconductors in which the number of electrons is increased through the introduction of donor atoms are called n-type while semiconductors in which the number of electrons is decreased through introduction of acceptors are called p-type. There are many different materials classified as semiconductors, but the most common are silicon, germanium, and gallium arsenide. Now, let s try to explain briefly how a diode works. Note that the explanation below is essentially correct, but some important details are left out and I m only giving you a bare bones description. Nevertheless, the picture presented below shows fundamentally what is happening inside the device and will be sufficient for this class. For a more complete analysis of diodes and other similar devices, you should take ECE 313. A pn diode is made by bringing an n-type semiconductor into contact with a p-type semiconductor. When this is done mobile electrons will migrate from the n-type material, where they are present in large numbers, to the p-type material where there are fewer electrons. This process is called diffusion and it is the same phenomena that occurs when you add a food coloring to water the dye spreads out from the original droplet until it is more-or-less uniform through the liquid. When electrons diffuse into the p-type material, however, there are unoccupied atomic orbitals associated with the acceptor atoms and these orbitals readily accept the new electrons. As electrons continue this diffusion from the n side to the p-side, negative charge is built up on the p-side (from captured electrons) and positive charge builds up on the n-side (from ionized donor atoms). The net result is that an electric field (or built-in voltage difference) is created in a region near the junction between the n and p-type semiconductors. This potential creates a barrier that grows until it is large enough to block more electrons from diffusing to the p-material. This is process illustrated in the diagrams of figure 1. 1

2 Figure 1. Figures illustrating the physics of the pn junction.(a) Mobile charges from p and n-type semiconductors diffuse across the junction where they recombine and form a depletion region devoid of carriers (b). The built-in electric field in the depletion region opposes further diffusion of charge. (c) An energy band diagram showing the region near the junction. Mobile electrons exist in the bulk n-material and mobile holes exist in the bulk p-material. The built-in electric field (or potential) creates an energy barrier preventing electrons from moving to the p-side and holes from moving to the n-side. (d) Symbol for a diode showing the anode and cathode terminals. The same process described above happens for the p-material. A p-type semiconductor is made by introducing dopant atoms (such as Boron) that only have three electrons in their outer shell, instead of the usual four found in silicon (or other semiconductors). A dopant atom can thus capture an electron from a neighboring semiconductor atom to fill its vacant bonding orbital. This means, however, that the orbital vacancy just moves to that neighboring atom. This process can continue from atom-to-atom with the vacancy (or hole ) propagating through the material just like a free electron. A good analogy is a bubble moving through a liquid. The upshot of all this is that holes can propagate from the p-material across the junction to the n-material where they capture mobile electrons. The overall effect, again, is a build-up of fixed (non-mobile) charge on either side of the junction that creates an electric field that opposes this diffusion of charge carriers from one side to the other. In steady-state, there is no net flow of charge across the junction and a built-in electric field exists in the depletion region where there are very few mobile charge carriers remaining. Now, with this very basic description, we can understand qualitatively how a diode works. When an external voltage is applied to the diode, the energy barrier that blocks electrons from flowing can be either increased or reduced, depending on the polarity of that applied voltage. Figure 2 illustrates this when the external voltage is applied with the opposite polarity as the built-in voltage, the electric field in the depletion region is reduced, the depletion region narrows, and the energy barrier is lowered (figure 2(b)). As a result, charges can more easily cross from one side of the junction to the other, resulting in a large increase in current flow. 2

3 Figure 2. Illustration of diode operation from the viewpoint of electron and hole energy. The steady-state diode with no applied voltage is shown in (a). When an externally-applied voltage opposes the built-in potential, the internal energy barrier is reduced and current flows more easily from the p-side to the n-side (b). When the external voltage has the same polarity as the built-in voltage, the energy barrier is increased and almost no current flows through the device (c). In fact, the distribution of electrons in the conduction band and holes in the valance band are exponential functions of energy and described (a result from statistical thermodynamics) by the Fermi function, ff(ee): ff(ee) = ee (EE EE FF ) kkkk exp (EE EE FF) kkkk, if EE EE FF The expression on the right (known as the Boltzmann approximation) is valid for energy states significantly higher (by about 3kkkk) than the Fermi energy and describes the distribution (probability of occupancy) of electrons in the conduction band. The distribution of vacancies (or holes ) for low energy states (in the valance band) is described by 1 ff(ee) = ee (EE EE FF ) kkkk 1 + ee (EE EE FF ) kkkk exp (EE FF EE) kkkk, if EE EE FF 3

4 Due to the exponential dependence of electron and hole distributions with energy, the increase in current with applied forward bias is exponential, as given by the familiar current (II) -voltage (vv) characteristic of a diode (plotted in figure 3): qqqq II(vv) = II ss ee nnnnnn 1 (a) (b) Figure 3. Current-voltage characteristic of a diode, plotted on a linear (a) and logarithmic (b) scale. In the above expression, II ss is known as the reverse saturation current, nn is called the ideality factor, TT is temperature (measured in Kelvin), and kk is Boltzmann s constant ( J/K), When the externally-applied voltage superimposes over the diode built-in voltage with the same polarity, the electric field in the depletion region increases, as does the width of the depletion region and the energy barrier. This results in a very small reverse current flowing through the device (II ss ). The charge (ionized dopants) stored in the depletion region is described by a junction capacitance and the increase in depletion width with reverse bias voltage results in a change in this capacitance. The junction capacitance dependence on applied voltage is the topic if this project. Electrostatics of PN Junctions: In the depletion approximation it is assumed that there are no mobile charge carriers in the depletion region near the junction of a PN diode. This situation is depicted in figure 4. We can write an expression for the electric field in the depletion region using Gauss s Law and ignoring any variations in the diode geometry in directions parallel to the junction (i.e., assuming we have a one-dimensional problem in which the device and charge distributions only vary in along the x-direction). In this case, Gauss s Law and Poisson s equations give us: dddd dddd = qq (NN DD NN AA ) and dd 2 φφ ddxx 2 = qq (NN εε rr εε DD NN AA ) Where NN DD is the concentration of donors, NN AA is the concentration of acceptors, and the electric field EE is in the x-direction. In the above expressions, we have assumed that all the donors and acceptors are ionized and that there are no mobile electrons or holes within the depletion region. Noting that there are no donors on the p-side of the junction and no acceptors on the n-side, we can write 4

5 dddd dddd = qqnn DD, for xx xx nn dddd dddd = qqnn AA, for xx εε rr εε pp xx and EE is zero in the bulk region (xx > xx nn and xx < xx pp ). Figure 4(a) shows the geometry of the PN junction diode and figure 4(b) the charge density (in the depletion approximation). Integrating the above expressions, we can find the field and the potential difference Figure 4. (a) Geometry of a PN junction showing the bulk and depletion regions. (b) Charge density as a function of position for a PN junction in the depletion approximation. (c) Electric field of the PN junction. (d) Potential (voltage) across a PN junction with the bulk P region as reference potential. Φ bi is the built-in voltage of the junction. 5

6 (voltage) across the junction as a function of position x: EE(xx) EE(xx) = qqnn AA EE(xx) = qqnn DD EE(xx) dddd = qqnn AA xx pp + xx, for xx pp xx dddd = qqnn DD (xx nn xx), for xx xx nn The resulting electric field is shown in figure 4(c). Note that since the PN junction, as a whole, is electrically neutral, we must have NN AA xx pp = NN DD xx nn Integrating the field over a path integral from the (reference) p-side to some position x allows us to find the electrostatic potential as a function of position in the PN diode: xx Φ(xx) = qqnn AA qqnn AA xx εε rr εε pp + xx dddd = xx 2εε rr εε pp + xx 2, for xx pp xx xx pp Φ(xx) = qqnn AA xx 2 2εε rr εε pp + qqnn DD (xx εε rr εε nn xx) dddd = xx qqnn AA 2 xx pp 2 + qqnn DD 2 xx 2, for xx xx nn The potential is plotted in figure 4(d). The total voltage across the junction is given by Φ(xx nn ) = qqnn AA 2 xx pp 2 + qqnn DD 2 xx nn 2 = Φ bi which is known as the built-in voltage. The built-in voltage is a consequence of mobile charges diffusing across the junction when the diode is formed, resulting in an internal electric field arising from ionized dopant atoms. The width of the depletion region, ww, can be expressed in terms of this built-in voltage as shown below. Solving for xx nn using the previous relation: xx 2 nn = 2εε rrεε Φ bi qqnn AA xx 2 pp = 2εε rrεε Φ bi qqnn AA NN 2 DD xx nn qqnn DD 2 qqnn DD 2 NN AA we find the depletion width to be: xx nn = 2εε rrεε Φ bi qq NN AA NN DD (NN AA + NN DD ) 6

7 ww = xx pp + xx nn = 1 + NN DD xx NN nn = 2εε rrεε Φ bi AA qq (NN AA + NN DD ) NN AA NN DD Capacitance of the PN Diode: The expression for ww derived above can be modified to account for an external voltage applied to the PN diode. If a voltage, VV, with polarity shown in figure 2(b) (i.e., positive polarity on the p-side and negative polarity on the n-side) is applied across the diode, then the PN junction is in forward bias and the depletion width becomes, ww = 2εε rrεε (Φ bi VV) qq (NN AA + NN DD ) NN AA NN DD that is, the depletion width narrows. Note from the expression above, that a voltage with negative polarity ( VV) will increase the width of the depletion region. A PN junction placed in reverse bias stores charge in its depletion region. This charge storage is described by a depletion capacitance that is voltage-dependent. Essentially, the PN junction acts as a parallel plate capacitor of width ww, giving us a capacitance, CC jj (VV), of CC jj (VV) = εε rrεε AA ww = AA 2εε rrεε (Φ bi VV) (NN AA + NN DD ) qq NN AA NN DD where AA is the cross-sectional area of the junction. The above expression for junction capacitance is often expressed (more conveniently) in the form, CC jj (VV) = CC jj 1 VV Φ bi = CC jj 1 VV Φ γγ (1) bi where CC jj is the zero-bias (VV = ) junction capacitance, given by qqnn AA NN DD CC jj = 2 (NN AA + NN DD ) Φ bi AA and γγ is the capacitance modulation factor that describes the variation of capacitance with applied voltage VV. For an abrupt junction PN diode, γγ =.5, and for a linearly-graded junction (in which the dopant concentration varies linearly with distance from the junction), γγ =.33. 7

8 PROCEDURE: In this project, we will measure the capacitance-voltage relation for a silicon varactor diode. A varactor diode is a diode that has been designed specifically to have a large capacitance tuning range with reverse bias and these are used for tunable filters, phase-shifters, and voltage-controlled oscillators in RF (radio frequency) circuit applications. The diode we will use for this project is the MVAM18 silicon varactor and the measurement setup (familiar from the previous project) is shown in figure 5. Place a 1 kω resistor in series with the diode and use the oscilloscope probes to clip the resistor and diode together, as well as to monitor the voltage across the resistor (as shown in figure 5). The layout of the diode (showing the leads corresponding to anode and cathode) are also shown below. Clip the leads from the VirtualBench function generator output across the resistor-diode circuit (between the + terminal and ground shown below). Figure 5. Measurement setup for the diode capacitance measurement, and package layout for the MVAM18 diode. 8

9 1. Once you have connected your resistor and diode up for measurement, set the VirtualBench function generator to output a sinusoidal voltage with amplitude of.1 V and zero DC offset. Set the frequency of the sinusoidal signal to 8 khz. 2. Use the oscilloscope to measure the voltage across the diode (probe B in figure 5) and the voltage drop across the 1 kω resistor (use MATH mode to display Channel A-Channel B). This voltage difference will allow you to monitor the current flowing through the diode (i.e., the voltage amplitude reading will correspond to the current in ma). It is convenient to use the measurement window of the VirtualBench to monitor the rms or peak amplitude values of the diode voltage (channel B) and the MATH display (channel A- channel B). 3. Record the voltage and current amplitudes (or rms amplitudes) of the waveforms for the diode as you vary the voltage across the diode from V to -12 V (in steps of.5 V). Be sure that you are applying a reverse (negative) voltage polarity to the anode with respect to the cathode. You can do this by connecting the diode as shown in figure 5 and applying a positive voltage offset to the + terminal OR equivalently by flipping the diode orientation and applying a negative voltage at the + terminal. Note that you can tell immediately if you have the voltage polarity applied correctly by observing the waveforms on the oscilloscope the current should lead the voltage by 9º ( as it does for any capacitor). If the waveforms are in-phase, then the diode is in forward bias. CONCLUSIONS AND QUESTIONS In the next homework assignment. you will be asked to use the data collected from this project to perform the following calculations: 1. Make a plot of the capacitance versus voltage dependence for the MVAM18 diode. Plot capacitance on the Y-axis and voltage on the X-axis and be sure to label your plot with appropriate units. 2. The diode C-V relation should follow that given in equation (1) above. Using your measured data and this relation, determine the following quantities for the diode: The zero-bias junction capacitance, CC jj The built-in voltage, Φ bi The capacitance modulation factor, γγ 9

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