Monday, 6 June 2016

The Spacecraft That Requires no Fuel: Graphene Photoelectric Solar Sails

Graphene is much lauded for its remarkable properties like robustness, high tensile strength while maintaining flexibility, low weight along with high electric and heat conduction in its atomic-scale thickness. However sometimes ignored is the potential of graphene's electrical properties to transform light energy directly into kinetic energy and motion, an interesting visual phenomenon but also one of importance when considering the concept of using the power of the stars to traverse the immense distances of space.

Photo-kinetic Properties of Levitated Graphene

During the process of using lasers of graphene, we have found that under strong magnetic fields we can create propulsion of graphene by light illumination alone.

Interesting still is the amount of propulsion achieved when exposed to focused sunlight.

We can focus solar radiation very effectively using a Fresnel Lens.

Graphene itself has very high resistance to the damage caused by the heat of focused solar radiation. Graphene has a melting point of 4700 Kelvins.

We can see the advantage of graphene-based solar sail technology when compared with aluminium, which has a much lower melting point of 934 Kelvins. This alone is reason to consider constructing solar sails from graphene, particularly if focused solar or laser radiation is being used to propel the craft.

In the experiment, like using the laser, we set up a chessboard grid of ~1.2 Tesla permanent NdFeB magnets. Graphene has excellent photothermal properties, so that the focused solar radiation can heat up the graphene film in one region instantly, which affects its magnetic susceptibility, making it tilt and hence move.

The fact that graphene also dissipates heat rapidly, in the form of quantised lattice vibrations (phonons), allows the process to be instantly reversible, which is what allows the film to move so responsively. Hence, the graphene film does not simply collapse onto the magnet as it loses its magnetic susceptibility, it simply tilts in one region and stays levitating on the region opposite from the laser focus as the heat dissipates before it reaches there.

This effect of propulsion would be substantially more under high vacuum and also in the vacuum of space itself where there is no air resistance or weight of atmosphere added to the craft. However without a magnetic field to create the effect of diamagnetism we will lose the effect of motion, at least in the way demonstrated.

However, that shouldn't stop us from experimenting with such films on Earth. The fact that the graphene film levitates, in effect ignoring the effect of gravity, is a good set-up in testing the effect of light on graphene films. Moreover, the fast heat transfer capability and the speed at which the magnetic dipoles in the graphene film undergo reversal is also a strong indication that the electrons are moving very fast in graphene which is an important fact in exploring new applications and features in solar sail technology.

New applications include concentrating not only on the affect of photon momentum on the sail, but on the possible mechanism in which electrons can be emitted from the sail and create additional thrust to propel the sail much faster than conventionally possible by photon momentum alone.

Photon and Electron Momentum-driven Solar Sails

In quantum mechanics, light is described as being made up of particles called photons, each of which has its own momentum. Each photon has an energy E and travels at the speed of light, c. It also has a momentum p, given by: p = E/c

In the vacuum of space, the inherent momentum of photons from the sun can either be absorbed or reflected off of a solar sail. If absorbed we are simply transferring momentum proportionally into the sheet, causing it to move with the same momentum as the photon. if reflecting off of the sheet, the effect of momentum conservation would impart twice the momentum onto the surface of the sheet.

Either way, we cause a reflective solar sail to move in zero gravity and since we cannot have a perfectly reflecting surface, both effects of absorption and reflection would cause the sail to move in space.

However there may be other effects that would contribute to the motion of the solar sail in space. An interesting fact is that when we expose the graphene sheet to the focused laser or sunlight, as described in the article here, by drawing up the light energy the graphene in the magnetic field creates a deflagration effect in the spin of electrons in the region illuminated which transmits itself as a quantised spin-wave.

In space, with no significant magnetic field, this importation of energy should instead create a net movement of electrons, an electrical current, which could be released from the film by the photoelectric effect. This would, in effect, propel the sheet in space with a defined momentum.

The photoelectric effect is explained in terms of the quantum model where the energy of each photon is the frequency, v, multiplied by Planck's constant h.

In the photoelectric effect, the kinetic energy (Ekin) of the photoelectrons to the energy of the absorbed photons (hv) and the potential energy (ϕ) of the surface:

The work function, W, is defined as the energy necessary to remove an electron from the surface of a metal: 

Here,  is the energy of the electron just outside of the surface of the metal and is the Fermi energy. 

The work function is the energy difference of the electron between being just outside of the metal and being at the Fermi Fermi energy.

The Fermi energy itself is the maximum energy occupied by an electron at 0K (degrees Kelvin). By the Pauli exclusion principle, we know that the electrons will fill all empty energy levels before they will share energy levels, and the top of that "Fermi sea" of electrons is called the Fermi energy. 

In the metal there are electrons with energies higher than EF and there are electrons with lower energy than EF. We say the electrons with lower Fermi energy are easier to remove and are, in effect, near the energy surface in the "Fermi sea", wheras the electrons with a higher Fermi Energy are "deeper" in the "Fermi sea" and are harder to remove as they are effectively screened by the lower energy electrons at the surface.

This formula works with metals under illumination of light in vacuum. The metal target has a certain work function, where the electrons are localized in a "sea" at the top of the metal's energy surface and are detected as "free electrons" under illumination.

If a reverse field potential is applied across the metal energy surface, by and introduced electric current say, we can reduce the kinetic energy of the electrons by a certain degree, in effect the induced photoelectric current in the material being reduced.

If the reverse potential is large enough, the photocurrent will be stopped, and this stopping potential is determined. The maximum kinetic energy, Kmax, of the photo-electrons is related to the stopping potential by: Emax = e0Vs

In semiconductor materials, the reverse field potential that creates the characteristic stopping potential of the released photo-electrons is the inherent band-gap energy of the material. 

The band-gap (Eb) is essentially the quantised energy levels of the electrons in the material. In a similar way to how electrons are quantised in energy levels in the atomic model, in semicondincdutors materials such as silicon, germanium and of course our graphene the electrons within are quantised in specific energy bands.

In semiconductors therefore, the energy of the photoelectrons must be reduced by the band gap energy (Eb) of the electrons trapped in the material. 

Another way to think about this is that the band gap energy must be delivered in order to allow the electrons to "quantum jump" from their characteristic band.

Band structure of materials can be constructed where we include the concept of the electrons being confined within bands in the "Fermi sea", as described earlier, with the top of the sea being the "Fermi Level"

We also include the definition of the momentum,

k||  of electrons moving throughout the surface of the semiconductor material. This allows us to properly describe the conservation of momentum in the system of the photon, with intrinsic momentum, impacting the confined energy bands and releasing an electron, which now carries the characteristic momentum of the photon K|| , from one of those bands at a displaced angle, θ .

The conserved momentum of the released electron (of mass m) is equated as:

In monolayer graphene, the unit cell consists of two carbon atoms - A and B.

The band structure of graphene exhibits two bands intersecting at two inequivalent points K and K' in the reciprocal space (kx,ky).

Near these points, the electronic dispersion resembles that of relativistic Dirac electrons. For this reason, K and K' are commonly referred to as the “Dirac points” at the tips of "Dirac Cones". A simulation of monolayer graphene shows this clearly, using the tight binding model.

The band structure of graphene is as follows:

With excitation energy (γ < 0), obtained within a nearest neighbour tight binding model, with a distance a between nearest neighbour carbon atoms. Here, we have set E = 0 at the K–points, where the valence band and the conduction band touch each other.

From this we get the Graphene energy dispersion (showing valance and conduction bands) from the π-bonding in graphene.

The valence and conduction bands are degenerate at the Dirac points. Put another way, the electrons are confined in one cone and the holes are confined in another and hop over the Dirac points under excitation.

Graphene is, in this sense, a band-gap semiconductor. How the band-gap can be changed is crucial for its application in making devices. There are two ways to lift the degeneracy of the two bands at the Dirac points.

One is to hybridize the electronic states at K and K' which requires breaking of the translational symmetry. The other is to engineer a strain in the graphene lattice so to break the equivalence between the A and B atoms in the lattice, which does not require any translation symmetry breaking.

Introducing more layers to the graphene will also decrease the band-gap energy, as the movement of electrons will behave as though the the multiple lattices are more equivalent to each other as more of them are added, effectively functioning as electron transport in a metal. If designing graphene as a semiconductor we would like a large band-gap energy to work with, hence few to single layer graphene is the best material to make graphene-based semiconductors with.

On average, bilayer graphene is calculated to have an energy band gap of 0.48 eV around the Fermi level, using our model. In experimental practice, a band-gap of anywhere from 0.1eV to 0.5eV is typical of few-layer graphene.

However, when considering use for a photoelectric solar sail, we want a small band-gap energy so multi-layer graphene is ideal. Multi-layer graphene, with its small band-gap energy should therefore release photoelectrons with a relatively high kinetic energy, according to the formula.

Therefore by Newton's third law the net thrust by the release of photoelectrons released from a solar-illuminated graphene sail should also be relatively high and should provide an additional effect of forward propulsion, along with photon absorption and reflection, which would move a solar sail in the vacuum of space.

We might then envision using this information in the approach in creating a light-powered propulsion system, i.e. a solar-sail spacecraft, for future developments in space exploration.

We can perform an experiment on a graphene solar sail prototype using a small vacuum vessel to observe the motion of a sail under concentrated light.

Using the effect of magnetic levitation, we can in effect negate the effect of gravity on our film. However, as stated in a previous article, the effect of the films motion is influenced by the effect of photo-illuminated magnetic spin deflagration changing the magnetic susceptibility of the film. Hence the ideal way to perform the experiment must be in either free-fall under high vacuum (relatively difficult) or in the vacuum of space itself to be a fully proven technology.

How fast, theoretically, could a Photoelectric Graphene Solar Sail Go?

Photons are massless particles, so the energy–momentum relation for photons is simply E = pc. For electrons in a graphene lattice, we know that the wavenumber, measured relative to the energy band minimum, and multiplied by ¯h, is the crystal momentum. More precisely

Theoretically we can attempt to calculate this using the nature of the electrons in graphene. Electrons in the conduction band of graphene may be considered as massless relativistic particles (”massless Dirac fermions”), moving not with the speed of light, but rather with a Fermi velocity.

To find the Fermi velocity we have to get the excitation energy relation, under an excitation quanta gamma, for the graphene electrons at the Fermi level, i.e. near the conduction band minimum:

The derivation is quite messy (willing to share if requested) and is based on the and structure of graphene we have shown earlier. In the end we get a formula that relates the excitation energy in graphene to the velocity of the electrons.

With the Fermi Velocity itself being

The nearest neighbour distance in graphene is a ≃ 1.4 ˚A,

A photon is characterized by either a wavelength, denoted by λ or equivalently an energy, denoted by E. There is an inverse relationship between the energy of a photon (E) and the wavelength of the light (λ) given by the equation:

where h is Planck's constant and c is the speed of light.

h = 6.626 × 10 -34 joule·s

c = 2.998 × 108 m/s

By multiplying to get a single expression, hc = 1.99 × 10-25 joules-m

When dealing with "particles" such as photons or electrons, a commonly used unit of energy is the electron-volt (eV) rather than the joule (J). An electron volt is the energy of a charged particle when place across a potential difference of 1 Volt (i.e. using 2 capacitor plates) at which the charged particle undergoes a quantised change in momentum.The eV unit of energy is universally used in particle physics. A photon with an energy of 1 eV = 1.602 × 10-19 J.

Therefore, we can rewrite the above constant for hc in terms of eV:

hc = (1.99 × 10-25 joules-m) × (1ev/1.602 × 10-19 joules) = 1.24 × 10-6  eV-m

Further, we need to have the units be in µm (the units for λ):

hc = (1.24 × 10-6 eV-m) × (10^6 µm/ m) = 1.24 eV-µm

By expressing the equation for photon energy in terms of eV and µm we arrive at a commonly used expression which relates the energy and wavelength of a photon, as shown in the following equation:

We know that most of the light emitted by our star, the Sun, is in the visible portion of the electromagnetic spectrum. Being a yellow star, most of the radiation emitted is centered around the yellow-green region with significant overlap in the red and infrared, blue, violet and ultraviolet portions. By measuring the light received from the Sun we know that its radiation corresponds to a surface temperature of about 6300 K (units of Kelvin).

From the calculation, we find that the range of photon energies for visible light emitted from the sun, from red to violet, ranges from about 1.63 to 3.26 eV respectively. This is important knowledge to have for designing semiconductor devices, such as photovoltaics, to harvest solar energy efficiently. In other words, it makes sense to design solar photovoltaic devices using semiconductors which have band gaps in between these energies.

Lets say the peak wavelength energy for exciting the graphene is around 405nm (0.405μm) (the same wavelength as our near-UV laser), i.e. 3eV.

so if γ ∼ 3 eV,

the velocity of the electrons is then calculated as:

Which is roughly 3% light speed. Assuming a 100% efficient mass action-reaction this is the fastest the sail could theoretically go by photoelectric propulsion.

Remember this is assuming that there is no mass being towed and that the energy of the incoming photons remains at a level of 3 eV. Any additional inertial mass, m0, will retard the rate of change in momentum.

In traversing space, the given gravity is the sum of the accelerations due to gravity that influences the craft near the the planet the craft is orbiting or flying past, g0, along with the total gravity of the host star, gS.

The performances of a spacecraft's thrust are characterized by the specific impulse, Isp, which measures the thrust, T, in Newtons produced by unit of mass of propellant at the given gravity.

The maximum effective specific impulse, Isp, of a photoelectric solar sail drive is equal to:

Where C0 is the photoelectric collimation factor (the fraction of the amount of radiation that hits the sail, from the sun or laser source, which will actually create the emission of a photoelectron), Vexh is photoelectron velocity, and ∑g is the sum of standard accelerations due to gravity (Note: this factor is not necessary if Isp is measured in N·s/kg or m/s).

Away from the earth, the value of g0, the Earth's gravity, on the total sum of gravitational influence on the craft will diminish and the Sun's gravity will be the major force retarding the specific impulse. As the sail gets farther from the sun, the value of C0 will decrease, however so too will the inverse relationship of the Sun's gravity gS. At some point the two effects will balance and conservation of momentum would allow the craft to maintain its maximum velocity as long as it encounters nothing to slow it down.

The solar sail craft could in fact maintain its top speed until it reaches another star system in which the value of C0 will increase and the sail will in fact decelerate, as it gets closer to the star, in the opposite direction as it enters another star system. This would allow for a much more autonomous form of long-distance space exploration, in which the craft could be in a state of dormancy during traversing interstellar space from system to system and becoming active within the star system being explored.

Moreover, a large solar sail transport system would in and of itself provide a large reflector dish for radio or laser communications across interstellar space.

Single figure percentages of light speed may not sound very impressive, but even 1% light speed is many times faster than the top record holders of any spacecraft built so far. It took numerous gravity assists, in planetary arrangements that only happen once every 250 years or so, that has flung Voyager 2 into the edges of the solar system and now into the first frontiers of interstellar space. It is currently travelling at about 55,000 km/h, which is only about 0.0051% the speed of light.

Therefore there might be good reason to develop spacecraft designs that get us to at least single percentage figures of light speed and solar sail technology is one with much potential.