Physics · Atomic and quantum physics · Grade 9

Semiconductor physics

Why does a solar panel work poorly in the heat, and what does the colour of an LED depend on?

The situation

At midday in summer in Tashkent the Sun delivers about 1000 W of radiant energy to every square metre, which is why solar panels are appearing on the roofs of more and more houses. The silicon wafers inside a panel convert light directly into electric current, and at night light-emitting diodes with the same kind of p–n junction light the street. Why does a panel give less power on a scorching day, what does the colour of an LED depend on, and under what load is it most useful to run a panel?

Topic

Semiconductor physics. Solar panel, transistor and smartphone processor

Key ideas

p–n junction
Band gap
Solar cell

Formulas

I = I_{s} (e^{q V / n k T} - 1)

The Shockley diode equation: in the forward direction the current grows exponentially.

I = I_{p h} - I_{0} (e^{q (V + I R_{s}) / n N_{s} k T} - 1) - \frac{V + I R_{s}}{R_{s h}}

Solar panel: photocurrent minus diode current ( $N_{s} = 60$ cells in series).

F F = \frac{P_{m a x}}{V_{o c} I_{s c}}, η = \frac{P_{m a x}}{G S}

Fill factor and efficiency ( $G$ — illuminance, W/m², $S$ — panel area).

λ = \frac{h c}{E_{g}} \approx \frac{1240 eV·nm}{E_{g}}

The wavelength of the light from an LED is determined by the width of the band gap.

V_{b i} = \frac{k T}{q} \ln \frac{N_{A} N_{D}}{n_{i}^{2}}

The built-in potential difference across a p–n junction (≈ 0.7 V in silicon).

Theory

Bands and doping

In a crystal the energies of electrons merge into bands: between the filled valence band and the empty conduction band lies the band gap $E_{g}$ (1.12 eV in silicon). In a metal the band is partly filled and there are many free electrons — resistance increases on heating (vibrating ions scatter the electrons more). In a semiconductor, free charge carriers appear under the action of heat or light, so resistance decreases sharply on heating or illumination. Pure silicon has very few free electrons ( $n_{i} \approx 10^{10}$ cm⁻³). If phosphorus is added, its fifth electron sits at a donor level only ≈ 0.045 eV below the conduction band, and at room temperature almost all of these become free — n-type; if boron is added, an acceptor level just above the valence band takes an electron and leaves a hole — a positively charged vacancy: p-type. The Fermi level $E_{F}$ is the energy at which the probability of being occupied by an electron is 1/2: in n-type it is near $E_{c}$ , in p-type near $E_{v}$ (the dashed line in the band diagram). The effective mass of the electron and hole in a crystal and the structure of the bands are covered in the “Bands in solids” experiment.

The p–n junction and the diode

When p and n regions are brought together, electrons diffuse into p and holes into n and recombine with each other. At the boundary a depletion layer (≈ 0.4 µm) of fixed ions and an internal electric field form — in the band diagram this is the “hill” ( $q V_{b i}$ ). A forward voltage lowers the hill and the current grows exponentially; a reverse voltage widens the layer and only a very small reverse current remains (nanoamperes in silicon) — this is a diode. The reverse current is carried by minority charge carriers produced by heat, so it roughly doubles every 7–10 °C. At a very large reverse voltage breakdown occurs (Zener diodes work this way). A barrier can also form at a metal–semiconductor contact — the Schottky diode; the contact potential difference equals the difference of the work functions divided by $e$ .

The solar cell and solar engineering

A photon of energy $h ν > E_{g}$ creates an electron–hole pair (the internal photoelectric effect). The internal field separates them: the electron goes to the n side and the hole to the p side — a current flows in the external circuit. If the load is close to zero there is no voltage; if it is very large there is no current — the power $P = V I$ is at a maximum (MPP) in between. The MPPT controller in a solar station's inverter keeps the load matched to this point. Heating sharply increases $I_{0}$ , so $V_{o c}$ falls by ≈ 2 mV/K in every cell: on a hot day the panel's power drops. A solar panel is a current source running on alternative energy: its emf ( $V_{o c}$ ) and short-circuit current $I_{s c}$ are measured by choosing the load. Uzbekistan has many sunny days: the Large Solar Furnace in the Parkent district of the Tashkent region, commissioned in 1987 (thermal power 1 MW), gathers light with 62 heliostats and a paraboloid concentrator and reaches temperatures of up to 3000 °C; in 2021 the 100 MW “Nur Navoi” solar power station started operating in the Navoi region. The main material of panels is ultra-pure silicon: solar power cannot exist without the semiconductor industry.

The LED and the model

In an LED the process is reversed: electrons and holes injected by a forward current recombine in the junction and emit a photon of energy ≈ $E_{g}$ — electroluminescence. GaAs (1.42 eV) emits infrared, GaAsP and AlGaInP red, GaP green and InGaN blue. In silicon and GaP the gap is “indirect” — the emission is very weak. This is part of the optics of solids. A semiconductor hardly absorbs light with $h ν < E_{g}$ but absorbs light with $h ν > E_{g}$ strongly — the absorption edge; the intensity decreases as $I = I_{0} e^{- α x}$ . In direct-gap GaAs above the edge $α \sim 10^{4}$ cm⁻¹ — a layer a few micrometres thick absorbs almost all the light; in indirect-gap silicon absorption is weaker, so a solar-cell wafer is about 0.15–0.2 mm thick. Below the edge, at low temperature, an electron–hole pair bound by the Coulomb force — an exciton — gives hydrogen-like lines; in GaAs the binding energy is only ≈ 4 meV, and at room temperature ( $k T \approx 26$ meV) excitons break up. Luminescence is absorbed energy returned as light: excited by light — photoluminescence, by current — electroluminescence (the LED). Model: 60 cells of 156 mm, a single-diode circuit ( $n = 1.2$ , $R_{s} = 0.3 Ω$ , $R_{s h} = 400 Ω$ , under standard conditions $I_{s c} = 9.7$ A, $V_{o c} = 39.6$ V); for the LED, $I_{s}$ is chosen so that a current of 20 mA flows at $V \approx E_{g} / q$ ; rectifier diodes are calculated with a SPICE-type model. The picture of the p–n junction is schematic.

Intrinsic and extrinsic conduction: thermistor and photoresistor

In a pure (intrinsic) semiconductor the numbers of electrons and holes are equal: $n_{i} \propto T^{3 / 2} e^{- E_{g} / 2 k T}$ — it grows very fast with temperature. In doped silicon (the “Si: n(T)” graph): at low temperature the donors are “frozen” and there are few electrons; at ≈ 150–500 K all donors are ionised and $n \approx N_{d}$ — transistors work in this range; at high temperature $n_{i}$ exceeds the donors and conduction becomes intrinsic again. A thermistor (NTC) is a semiconductor resistor whose resistance falls on heating: $R = R_{25} e^{B (1 / T - 1 / T_{25})}$ , changing by ≈ 4 % per 1 °C, while a copper wire changes by only 0.4 %. Its I–V characteristic is non-linear: the current heats the resistor, its resistance falls, so after the maximum the voltage decreases even as the current rises. Electronic thermometers, temperature sensors in solar panels and batteries, and electric kettles with adjustable temperature work this way. In a photoresistor (CdS) light creates electron–hole pairs — photoconductivity: in the dark the resistance is megaohms, in sunlight hundreds of ohms. The photorelay of street lamps switches the lamp on at dusk (≈ 10–30 lx). Thermoelectric phenomena in semiconductors are covered in the “Thermoelectric phenomena” experiment.

Diode I–V characteristic, rectifier and transistor

Lab work: in the “Diode I–V” mode change the voltage from 0 to 0.8 V and from 0 to −20 V and record the points: in the forward direction the current is in milliamperes, in the reverse direction in nano- or microamperes, so a diode passes current one way. A vacuum diode (heated cathode and anode) works the same way, but it needs power for heating, is large and usually needs hundreds of volts; a semiconductor diode is small, works immediately and opens at 0.7 V. Rectifier: a single diode passes only the positive half-wave of an alternating voltage; a bridge of four diodes turns both half-waves in the same direction; a capacitor then smooths the ripple — inside a phone charger the mains voltage is first rectified in a diode bridge too. A transistor (Bardeen, Brattain, Shockley, 1947) has two p–n junctions: in an n-p-n structure the emitter “shoots” electrons into the thin base, almost all of them cross the base and reach the collector; a small base current controls a large collector current: $I_{C} = β I_{B}$ , $β \approx 50$ – $300$ — an amplifier and a switch. In a vacuum triode too, a grid controls the anode current with a small voltage, but a transistor works without heating and billions fit on a microchip (a smartphone processor). A transistor adds energy to an oscillating circuit at the right moment, forming a generator of undamped oscillations — see the “Oscillating circuit (LC)” experiment.

Did you know?

The transistor was created in December 1947 at Bell Laboratories by John Bardeen and Walter Brattain; together with William Shockley they received the Nobel Prize in Physics in 1956.
The band gap of silicon at room temperature is ≈ 1.12 eV — which is why a silicon solar cell hardly responds to infrared light longer than ≈ 1100 nm.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

The photoelectric effect — Grade 11
Atomic structure and spectra — Grade 11
Radioactivity, half-life — Grade 11
Nuclear reactions — Grade 11
Orbitals of the hydrogen atom — Grade 11
Thermal radiation and Planck — Grade 11
Photon momentum and the Compton effect — Grade 11
De Broglie waves — Grade 11
Rutherford's experiment — Grade 11
The atomic nucleus and binding — Grade 11