Maxwell Society Lecture

The Maxwell Society was established in 1935 by Sir Edward Victor Appleton, Wheatstone Professor of Physics at the University of London, 1924-1936, and was named in honour of the pioneering physicist and King College’s own genius of electromagnetism, James Clerk Maxwell, Professor of Natural Philosophy at King’s College London, between 1860 and 1865. It was established to promote knowledge of physics among students of King’s. Events include lectures delivered by staff at King’s or by distinguished guest speakers on a wide variety of subjects including nuclear physics, ultrasonics, radiobiology, quantum dynamics and aspects of applied science including the development of the computer and television. Members also undertake study visits to research laboratories and technical and manufacturing facilities, and organise other, more occasional, events and social activities. The Society is very active in arranging talks and other events, including the Monday Seminar Series, open to the public, The Cumberland Lodge Weekend, and a number of social gatherings throughout the year.

The Maxwell Society introduces students to areas of Physics beyond the scope of their degree courses. It is also dedicated to providing a more relaxing and enjoyable side to the life of those studying the physical sciences.

Last November I was very lucky to be able to attend a Maxwell Society lecture.

The Science and Sociology of Solar Cells

Mark Van Schilfgaarde

King’s College London


Photovoltaics convert light directly into electricity. The lecture was about arguing that if we are to reach a sustainable economy, photovoltaics are essential. Q: Why might this be?

A:The world consumes 15 Terra Watts of power. 10-15% of the world’s economy is spent on energy production. At the moment fossil fuels are the primary engine driving the world economy.

The above internet address gives some idea about the amount of fossil fuels remaining.


The world has consumed large amounts of coal (>100 years’ worth at current consumption). Until recently, natural gas was believed to be running out. Recent discoveries and technology advances (hydraulic fracturing of rocks deep underground, largely developed in the US) will greatly extend the supply. But-


There is also the effect on the environment: Coal has been an environmental disaster; Gas is 2x cleaner so there is a strong argument to switch to gas based on fracking. But …. fracking is not benign! A rapidly rising list of aquifers becoming heavily contaminated near fracking sites. the_United_States

Professor Schilfgaarde continued his lecture by looking at something called Blackbody Radiation (something that physics A level and degree students learn about).

Objects in thermal equilibrium with their surroundings at a certain temperature (Tb) will emit electromagnetic radiation. An object is called a blackbody if it absorbs all frequencies/wavelengths of the electromagnetic spectrum perfectly and at a certain temperature radiates all frequencies/wavelengths (radiates photons). A black body is therefore a perfect absorber and emitter of all frequencies/wavelengths. Blackbody radiation (spectrally) resolved by frequency is called Planck’s Law of radiation with a very scary power spectrum equation


Where B is the power and ω is the angular frequency.

Planck’s law is a universal function of temperature and frequency/wavelength. The spectrum has a “maximum” point where radiation is strongest, which shifts to higher values with increasing temperature.


As the temperature decreases, the peak of the black-body radiation curve moves to lower intensities and longer wavelengths/shorter frequencies.

Many things in nature approximate to black bodies. Infrared sensors work by detecting radiation from bodies hotter than the environment. Radiation from the Earth and Sun approximate quite well to blackbodies.


Solar irradiance spectrum above atmosphere and at surface (the blackbody spectrum is there for comparison). Extreme UV and X-rays are produced (at left of wavelength range shown) but comprise very small amounts of the Sun’s total output power.

The visible spectrum is close to where the Sun’s radiation is most intense.


K stands for temperature in kelvin (freezing point of water would be 273K),


Radiation from the night sky is almost perfectly described as a blackbody at a temperature of 4K.

The total power per area emitted by a blackbody is the sum of contributions over all frequencies/wavelengths. Integrating the power gives the Stefan-Boltzmann law




In actuality the Earth is not quite a blackbody (e.g. gases such as carbon dioxide absorb and re-emit photons at a relatively long wavelength). Because the solar spectrum peaks at a 20x shorter wavelength than the Earth, it disrupts the emissions by the Earth (300K) than it disrupts absorption from the Sun (5800K). The quantity of light carbon dioxide traps and re-emits back to the Earth is not large (although global warming may change that), but small differences drive small changes in the black body temperature. The perturbation scales with the amount of carbon dioxide in the atmosphere.

We have pumped 375 billion tonnes of carbon dioxide in to the atmosphere, apparently at a slightly faster rate than plants can absorb it.



The link above shows emission spectra of some elements. Each line is caused by an energetic electron falling down to a lower energy level.

Electronic states in macroscopic systems

clip_image002[4] Ry stands Rydberg unit of energy.


Fermions include all quarks (some of these make up the proton and neutron) and leptons (this includes the electron). Fermions must obey Pauli exclusion principle.

Role of Energy Band structure

The physical properties of condensed matter — chemical, mechanical, electronic, magnetic, optical, are states especially states not far from the Fermi level. What from the eigenstates take depends on the chemical character of the constituent atoms.

Silicon and aluminium are adjacent in the periodic table, yet they are very different. In silicon, a gap appears in the bands. The gap falls precisely between the highest occupied state and the lowest unoccupied state.

clip_image006[4] clip_image008


If a small electric field (E) is applied, electrons (e-) can make the infinitesimal jump across the Fermi level in aluminium and current flows. But for silicon, the jump is discrete; can’t do it. Electrons in filled bands are inert. It is a bit like cars in a traffic jam.


Motion is not possible until one electron is taken out (excited to unoccupied state). This is how we distinguish metals and insulators: States present at the Fermi level give a metal; States absent give an insulator.


VB = valence band and CB = conduction band.

Fermi-Dirac distribution


Electrons can also be excited across the bandgap by photons:

clip_image016 clip_image018

Light can behave as a wave or a particle. In its wave aspect, a photon has clip_image020

In its particle aspect a photon has energy and momentum: clip_image022


Shine a light on an insulator and the conductivity changes by orders of magnitude.

Key point: the “hole” left behind also acts like a particle. A missing electron (e-) behaves like a positively charged hole (h+) moving in the opposite direction (without the h+ the valence band is inert). The h+ also carries current. Thus two “particles” are created.


clip_image028 clip_image030

For a solar cell, we also need —-

The e- – h+ generation process is the basic mechanism by which solar cells absorb light: the first step into generating electrical power. But creation is not sufficient: we need to separate the pair, collect the e- at one lead and the h+ at the other. This is accomplished by a pn junction — the basic widget that makes computers work.

There are two ways that e- and h+ can be created: Thermal excitation of e- – h+ pairs across the gap; Photoexcitation of e- – h+ pairs across the gap.

These processes create e- – h+ pairs in equal measure. We need another way that creates an imbalance: e- without h+ and vice-versa.


In the dilute case, defects in insulators can form discrete states within the forbidden gap.


Analogous to atomic levels of free atoms transitions between band states and defect states is what gives precious gems their colour.


P is phosphorous and B is boron. P: The extra e- goes into εd at 0K. But at room temperature, entropy causes it to jump into the conduction band, because it contains vastly more states.

B: Remember h+ are the mirror image of e-. e- “float down” to lower energy; h+ lower their energy by “floating up”.

At room temperature holes jump out of εa and populate the valence band.

To summarize, doping Si with P creates excess e- in the conduction band; doping Si with B creates excess h+ in the valence band. Thus, doping creates one type of carrier without the other.

Pn Junctions I

Forget about energy bands E(k) and focus just on the band edges. Band diagrams are drawn in real space. For a homogenous semiconductor in equilibrium the bands are constant in space:


Consider Si doped doped with P. There are fixed P+ ions and free electrons (e-). If the e- were to separate from P+, an electric field would result trying to restore the balance.


Pn Junctions II

Consider a silicon wafer doped on the left with boron (acceptor) and on the right with phosphorous (donor). Imagine at first a membrane that isolates the B-doped and P-doped regions.

Pull out the membrane: excess e- in the P-Si spill out to the B-Si; excess h+ move the opposite way.

As e- and p+ move past each other, they leave behind an electric dipole field (simplest explanation for electric dipole field is the electric field around two point charges) R which acts as a restoring force.

clip_image042 clip_image044

The restoring E field slows R—>L current because electrons (e-) have to climb a hill (work against E) to reach the low-density region on the LHS.

Equilibrium is reached when L –>R and R—>L currents match. But the L—>R and R—>L currents are subject to different driving forces.

L—>R current originates from a large number of excess e- trying to spill to the LHS where few e- exist. Many attempts to cross the barrier, but only a few succeed because they have to work against the electric field.

An analogue: Imagine a gas in a bottle under pressure, in a very strong gravitational field. Uncork the bottle and gas flows out, but the field makes it hard for the particles to exit and a few particles outside the bottle find their way back in.

Process reaches equilibrium when flow from the few particles balances the flow from the many particles inside.

Rectification of a pn Junction

Now imagine that you could suddenly change the gravitational field. Equilibrium would be disrupted and gas would flow. Exactly this situation applies to a pn junction.

clip_image046 clip_image048

Electrons on the LHS are scarce. An e- meandering around the junction is caught like a water droplet encountering a waterfall. It rolls down, regardless of the size of the fall. Thus this current is independent of the voltage (V).


It must be that the current-voltage characteristics look like


Classical nonlinear rectification behaviour of a pn junction (diode) forms the basis of practically all electronic devices.

Q: What happens when we shine a light on a diode?

clip_image054 clip_image056

Efficiency and bandgap

The efficiency of real solar cell depends on many factors. Consider just an ideal pn junction. Q: how should we choose the gap?

clip_image058 cannot create an e- – h+ pair (energy conservation). So we should not make the gap larger than the mean energy of the Sun’s photons.

clip_image060 clip_image062


Efficiency and bandgap

Real devices are complicated and there are many issues, but the main principles are: Maximise absorption; Minimise recombination (and resistive) losses; Make optimum use of spectrum.

clip_image066 GaSs clip_image068 Si


The above list includes some of the leading candidate materials for PV devices. Right now Silicon is by far the most dominant technology.

About 90% of commercial PV cells are made from silicon, but they are expensive!

Indirect gap and absorbing layer must be thick.

The silicon used is by melting the silicon, inserting a seed crystal and slowly pulling out a crystal.


The crystal is cut into 100 micro metre wafers with a wire saw (how chips in computers are made).

The cells must run for about 5 years before energy invested in cell synthesis is paid back.

Higher efficiency cells can be made of GaAs. Thin films (about 1 micro metre) are thick enough to absorb the vast majority of photons. But, As can occupy a Ga lattice site at relatively low energy cost. As “anti-site defects” generate a mid-gap level.

clip_image074 clip_image076

Mid-gap levels are very efficient recombination centres which cause e- – h+ pairs to recombine before collection. However expensive growth methods must be employed —


Multi-junction cells

The efficiency of a pn junction can be improved by a device with multiple bandgaps in series. Incident light first hits wide-gap, absorbing high-energy photons. Smaller gap material deeper within the film absorb lower energy photons. High voltage from a wide gap + low voltage from small gap add in series higher efficiency.



(a) The structure of a MJ solar cell. There are six important types of layers: pn junctions, back surface field (BSF) layers, window layers, tunnel junctions, anti-reflective coating and metallic contacts. (b) Graph of spectral irradiance E vs. wavelength λ over the AM1.5 solar spectrum, together with the maximum electricity conversion efficiency for every junction as a function of the wavelength.

Films are complicated and very expensive. Use with concentrators.


2 thoughts on “Maxwell Society Lecture

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