Spring 2018

ECE 110

Course Notes

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The Sun radiates energy in the form of electromagnetic (EM) waves across a wide range of wavelengths, from about $300\text{ nm}$ upwards. Just a part of this spectrum is visible light, approximately $400\text{ nm}$ to $700\text{ nm}$.

Figure 1

Fig. 1: Solar irradiance spectrum. The white curve (that starts at about $300\text{ nm}$) shows the solar power available per square meter at each wavelength just above the Earth's atmosphere. The area under the white curve is the solar power available per square meter across all wavelengths (about $1400\text{ W/m}^2$ in total). As solar radiation passes through the atmosphere, some energy is lost due to reflection, scattering and absorption, leaving a total irradiance at the Earth's surface of about $1000\text{ W/m}^2$ or less. Image source.

All solar energy can potentially be converted to electrical energy, no matter the wavelength. So, on the rest of this page, we use the word light to refer to EM radiation beyond just the visible spectrum.

The photoelectric effect is one piece of experimental evidence that shows that light carries energy in discrete amounts. In the photoelectric effect, light of a single wavelength can eject electrons from a material as long as the wavelength is below a threshold wavelength characteristic of that material. Einstein explained this phenomenon by arguing that light is delivered in packets of energy, called photons, such that each photon carries an amount of energy inversely proportional to its wavelength. In this way, light of a shorter wavelength has more energetic photons than light of a longer wavelength. If the light has a wavelength shorter than the threshold wavelength, its photons have enough energy to knock electrons off their material. The relationship between the energy of a photon $E_{\text{photon}}$ and its wavelength $\lambda$ is
\begin{align}
E_{\text{photon}} &= \frac{1240}{\lambda}, \label{PHT-EPH}
\end{align}
where $E_{\text{photon}}$ has units of electron volts (with symbol eV) and $\lambda$ has units of nanometers (with symbol nm). In SI units, $1 \text{ eV}=1.6 \times 10^{-19} \text{ J}$.

Figure 2

Fig. 2: Laser pointer warning sticker. A laser is an intense beam of monochromatic (i.e. single wavelength) light. This sticker is on a laser pointer that produces a red beam of wavelength $\lambda = 650\text{ nm}$ and outputs a maximum power of $5\text{ mW}$.

For the laser described in Fig. 2, the energy of each photon is $E_{\text{photon}} = 1.91\text{ eV} = 3.1\times 10^{-19} \text{ J}$ by equation $\eqref{PHT-EPH}$. If the laser outputs its maximum power of $5\text{ mW}$, the number of photons emitted per second (known as photon emission rate) is calculated as follows.
\begin{align}
\text{Photon emission rate} &= \frac{\text{Output power}}{E_\text{photon}} = \frac{0.005\text{ J/s}}{3.1\times 10^{-19}\text{ J}} = 1.6\times 10^{16} \text{ photons per second}
\end{align}

When exposed to light, a semiconductor material can exhibit the photovoltaic effect. Like in the photoelectric effect, photons that hit the material transfer their energy to electrons as long as the photons carry enough energy. Different to the photoelectric effect, the electrons are not ejected; instead, they move from the material's valence band to its conduction band. In this way, light energy carried by photons is converted to electrical energy carried by electrons.

Figure 3

Fig. 3: Band gap energy. The energy gap between the valence and conduction bands of a semiconductor material is called the band gap energy $E_g$. The semiconductor can convert light energy into electrical energy as long as the energy of the incoming photons is greater than or equal to the band gap energy; that is, $E_{\text{photon}} \geq E_g$.

Since $E_{\text{photon}} = 1240/\lambda$ by equation $\eqref{PHT-EPH}$, light of a single wavelength $\lambda$ (measured in nm) can create the photovoltaic effect in a semiconductor of band gap energy $E_g$ (measured in eV) if the following condition holds.
\begin{align}
\lambda &\leq \frac{1240}{E_g} \label{PHT-LMX}
\end{align}

Figure 4

Fig. 4: Silicon solar cell. Silicon is the most commonly used semiconductor in photovoltaic applications. One reason for silicon's widespread use in solar cells (like the one pictured here) is that it has a band gap energy $E_g$ of $1.11\text{ eV}$ at room temperature. According to equation $\eqref{PHT-LMX}$, the maximum wavelength of light energy that it can convert to electrical energy is $\lambda_{\text{max}}=1120\text{ nm}$. Notice that the solar irradiance spectrum in Fig. 1 indicates that most of the solar energy that reaches the Earth has wavelengths below 1120 nm, which means that silicon can be reasonably efficient at harvesting solar power. Image source.

A photodiode, like a solar cell, is a photovoltaic semiconductor device. Photodiodes, however, are optimized for light detection while solar cells are optimized for energy conversion efficiency. In this section, we focus on photodiodes because you are likely to use them in the lab.

Photodiodes are essentially identical to diodes since both consist of p-n junctions. The difference is that a diode's p-n junction is usually covered in opaque packaging, while a photodiode's p-n junction is exposed to light through transparent packaging. In this way, photons hitting the p-n junction can stimulate a photovoltaic effect in the photodiode and change its electrical behavior.

Figure 5

Fig. 5: Photodiode symbol and I-V characteristic. The symbol for a photodiode (shown in the top left) is the symbol for a diode with two incoming arrows that represent the influence of light. The labeling of voltage $V_P$ and current $I_P$ follows the convention for diodes. In complete darkness (no illumination), the photodiode behaves exactly like a diode and has the diode's I-V curve (drawn in red). As illumination increases, the I-V curve shifts downwards because an increasing number of photons hit the photodiode and energize electrons from valence to conduction band via the photovoltaic effect. The flow of these electrons create a current offset in the opposite direction to the label for $I_P$.

The datasheet for a photodiode specifies several values including the area of the region that is exposed to light. Also provided are the open-circuit voltage $V_{oc}$ and short-circuit current $I_{sc}$ for a given wavelength of light (matched to the band gap energy) at a given power density. The fill factor $FF$ is a datasheet parameter that quantifies the maximum electrical power that the photodiode can deliver relative to $V_{oc}$ and $I_{sc}$. Since the axes of Fig. 5 match standard labeling for the photodiode, we look at the 4th quadrant (where power is delivered by the photodiode) to determine the fill factor.

Figure 6

Fig. 6: Fill factor. This figure shows the 4th quadrant of a single branch of the photodiode's I-V characteristic for the wavelength and power density specified in its datasheet. The area of the red rectangle is $|V_{oc} I_{sc}|$. The blue rectangle is the largest one that fits within the I-V curve in the 4th quadrant. In other words, the area of the blue rectangle is the magnitude of the maximum power $|P_m|=|V_m I_m|$ that can be delivered by the photodiode. The fill factor $FF$ is the ratio of the blue area to the red area; that is, $FF=|V_m I_m|/|V_{oc} I_{sc}|$. In this way, you can estimate the magnitude of the maximum power delivered from the datasheet as $|P_m|=FF|V_{oc} I_{sc}|$.

A simple way to demonstrate that a photodiode can convert light energy into electrical energy is to connect it in series with a resistor.

Figure 7

Fig. 7: Photodiode circuit for energy conversion. Since the resistor (shown in the circuit in the top left) is in nonstandard labeling with respect to $V_P$ and $I_P$, its I-V characteristic is a negatively sloping line through the origin (shown in green). When there is no illumination, the operating point is at the origin (the intersection of the red and green curves). As illumination increases, the operating point moves positively in $V_P$ and negatively in $I_P$ (as illustrated by the blue arrows); that is, the photodiode delivers more and more power and the resistor absorbs it. The photodiode is converting light energy from photons into electrical energy by forcing a positive current through the circuit in the clockwise direction (opposite to the label for $I_P$).

The circuit in Fig. 7 can also be used to detect light, but it has the drawback that the magnitudes of neither $V_P$ nor $I_P$ scale linearly with the illumination. That is, the blue arrows in the 4th quadrant of Fig. 7 become smaller and smaller in both dimensions as the I-V curve shifts downward. To achieve linear light detection, it is better to place the operating point in the 3rd quadrant, where the branches of the photodiode's I-V characteristic are evenly spaced, by applying a reverse-biased voltage to the photodiode.

Figure 8

Fig. 8: Photodiode circuit for linear light detection. In the circuit shown in the top left, a photodiode is connected in series with a resistor and a voltage source $V_s$ that applies a reverse bias to the photodiode. The I-V characteristic of the Thévenin equivalent subcircuit is a negatively sloping line through $V_P$-intercept $V_s$ (shown in green). As illumination increases from zero, the operating point moves linearly in both $V_P$ and $I_P$ dimensions (as illustrated by the equal sized blue arrows). In fact, the magnitude of $I_P$ is approximately proportional to the amount of illumination.

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