Musings inspired by Ashoori's article "Electrons in Artificial Atoms"

These notes come primarily from reading an article by R.C. Ashoori of MIT which was published in the journal Nature, Volume 379, February 1996. The article is entitled "Electrons in Artificial Atoms".

In an artificial atom, the effects of electron-electron interaction are more important than in a normal atom! This is because orbital energies are far lower in artificial atoms than in real ones. The opposing effect of the spreading out of the electrons in space is not as large. Thus the relative importance of electron-electron interactions increases.

Energy resolution of new spectroscopic techniques is only limited by the sample's temperature.

Ashoori describes a setup where a QD is close enough to one contact (capacitor plate) that an electron would be able to quantum tunnel between them. The other capacitor plate is too far away to tunnel. When an electron is successfully added to the QD, you can detect a tiny change in charge (typically about half an electron charge) on the surface of the farther capacitor plate.

A neat extension of this idea is to attach an AC current to the DC gate voltage that is driving electrons onto the QD. This makes it possible for an oscillation to occur at specific DC gate voltages. This is when the electron tunnels to and from the QD with each oscillation of the AC part of the gate voltage. This allows for synchronous detection by the farther capacitor plate as its charge changes slightly in time with these oscillations. This is known as single-electron capacitance spectroscopy (SECS).

Gated transport spectroscopy (GTS) seems to be what we do with our double quantum dots. We maintain a voltage difference between the source and drain contacts, and thus we can measure the current flow changing with changing conditions such as a changing gate voltage. When this article was written (1996), no one had yet successfully conducted GTS with fewer than about 10 electrons on the dot.

There are two main effects that make it more difficult to add extra electrons to a QD. The first is electron-electron interactions. They obviously push each other away. This is called the charging energy. Then there is the quantum energy levels. In order for an electron to be present in the dot, it needs to be occupying a quantum level. Due to the Pauli exclusion principle, it is not possible for more than two electrons to occupy the exact same quantum level. The factor of two is due to different possible spins of the electrons. This review claims that the charging energy is about five times the quantum level spacing for the samples described in the paper.

There is a geometric factor that connects the value of the gate voltage with the actual amount of energy needed to add an electron to the dot. This paper seems to be claiming that they simply use the geometry of the sample to estimate this. In the case of their perfectly symmetric doubly-contacted QD, they claim that this geometric factor is 0.5.

Modeling

The author sketches out the basics of the parabolic potential well assumption. They assume that the z-direction is completely constrained, and that the x and y directions are governed by the parabolic potential well. The potential is circularly symmetric.

I have also read elsewhere that this model matches up rather well with observations. Even in 1996 this was apparently already known.

Introduction of a constant magnetic field to the analysis breaks the degeneracy in the quantum number l. Now positive and negative l's have have slightly different energies due to the contributions of the magnetic moment interactions with the magnetic field. Note that we are talking about the magnetic moment created by the evolution of the electron's wavefunction such that the electron can be considered to be moving in a circle around the center of the potential well. Magnetic field applied along the z axis enhances confinement in the dot. The magnetic field also introduces Zeeman (spin) splitting due to the magnetic moments of the electrons.

With the introduction of the magnetic field to the discussion, the author began to refer to the quantum levels as Landau levels. Strong magnetic fields can cause only one side of the level, let's say the side with positive l, to be populated.

There is an interesting plot demonstrating the zig-zag effect as electrons end up populating different levels as the magnetic field strength increases. At some point, the zig zag stops because all electrons are in the lowest Landau level (and on one side of the l range I believe). This is very striking when seen in a plot.

Another important effect of increasing magnetic field is the fact that all of the radii of the different l levels shrink. This makes sense in light of the observation we made above that the magnetic field increases the confinement strength.

Example with 2 electrons in ground singlet state (l = 1, s = +-1). With increasing B-field, it is possible to increase the Zeeman energy enough that one of the electrons gets promoted to l = 1.

Interactions of magnetic moments alone is not sufficient to produce the observed spin flips and l transitions. We must take into account coulomb interaction between the electrons in order to get the right answer. As the magnetic field increases, the l radius decreases and eventually it becomes possible for an electron to jump into the l = -1 state from the l = 0 state.

Once in the lowest Landau level, more changes can still occur. The lowest energy of all the electrons if they are in the lowest Landau level would nominally be when they are paired off with each pair having opposing spins to each other. As the magnetic field increases however, eventually higher-l states become lower-energy than either the spin up or spin down states, depending on the direction of the magentic field. This means that we will continue to see bumps in the spectrum as electrons will flip spins and move to more distance l values. Self-consistent calculations can reproduce some of these effects, but they seem to overstate the number of flips that happen at low field, and underestimate the number of flips at high field. Something is obviously still missing.

The Hartree-Fock technique takes into account the repulsion between many different electron wavefunctions. It reproduces much of the correct behaviour. An interesting result is that there is actually a short-range attraction force acting on electrons that are in different l bands. Adjacent bands tend to be preferentially populated. This can be compared to the fact that electrons in the same l band tend to be on opposite sides of the dot from one another. By moving to the higher l bands, it seems that the electrons can both be in a lower energy state and be closer to one another, an apparent paradox.

When the magnetic field becomes even higher, eventually it becomes energetically favourable for gaps to form in the l spectrum. The lowerest energy states involving gaps typically involve the gaps being adjacent to one another. Thus we don't end up with a smattering of l gaps throughout our states. We end up with one big block of empty l's.

It is known however that the Hartree-Fock calculations leave something out. They do not take into account the known electron correlation. In this area, some other techniques such as "exact diagonalization" seem to be better. However, the authors do not mention a successful combination of these factors into one theoretical model.

Their final section mentions some exciting further work that is being pursued, or will likely be pursued, in the field of quantum dots. One of the more interesting points for me is that a single-electron transistor obviously has a 'fan-out' problem. That is, normally the result of a piece of digital logic can be used to set off a cascade of other logic operations. This is obviously difficult if the result of your digital logic operation is the movement of a single electron. It seems however that people are finding ways around this. Perhaps I will find out more about this in the future.

 

Leaks, Pumps, and Gauges in Vacuum Science

This post is an experiment. I missed the second class of Experimental Condensed Matter, but I have access to the slides used during the class. My efforts here will be to investigate the topics raised on the slides and to understand them to some degree. The results of my investigation I will write into this post. Curious how well this will work? So am I. Here goes.

This work is based on slides developed by Peter Grütter.

Gas Load and Leaks

Plastic and Metal Diffusion

Gases can diffuse within plastic seals towards their surface. Then these gases can outgas into the vacuumed space, creating a gas load. Similarly for metals. However, metal diffusion gas load of this sort can be well-approximated by a linear decrease with time while the plastic gas load varies with approximately the square-root of time. Therefore after a long time, even the small area of plastic seals can dominate the gas load of this sort.

Permeation

Permeation leaks will always happen. The best we can do is minimize them by using less permeable materials for the construction of our vacuum apparatus.

Vacuum catalogues include a lot of useful information about the properties of materials that might be used in a vacuum system. Among the information available would be things such as permeability, outgassing rates, and the ability to be baked (raised to high temperature to get out annoying things like water).

Virtual Leaks

Virtual Leaks (Trapped Volumes). There are vented screws with holes drilled through their core, so the trapped air can escape more quickly.

Double Seals

Double O-ring systems employ two seals to guard against leaks through the seals. This technique is extended to the idea of differential pumping. Differential pumping is the technique of pumping out the volume between the two seals down to a relatively low pressure. Grütter gave the example of 1mBar for the in-between pressure.

Pump-down curves

Pump-down curves are a valuable troubleshooting technique. Different sorts of gas load have different pumpdown signatures. We went through this in more detail in the previous article on vacuum systems.

Troubleshooting Leaks: The pump-down curve for a virtual leak is different than for a real leak.

Pressure ranges for leak detection techniques

Big leaks are probably near seals or joints. Grütter gives the number of p > 200 mBar. This is pretty big, about .2 Atm.

If p ~ 1-40 mBar, he says that a good technique is to spray the outside of the system with a ‘volatile organic fluid of low vapour pressure and low flammability such as methanol’. When the spray covers the hole, a substantial pressure change should be observable.

If p <  1 mBar, Helium leak detection using a helium source and a mass spectrometer. A helium mass spectrometer is a device for finding small leaks. A volume of helium is ionized and leaked into the chamber. The spectrometer can then collect the gases outside of the vacuum system. Since the ionized helium will take a different spectrometer path than any of the other gas atoms and ions, it is relatively easy to measure the amount of helium being picked up. This means we can detect the helium that is leaking from the vacuum system.

Tracer probe leak detection is another technique. Here we use a tracer gas and a measurement device that is sensitive to the gas. The device is placed inside the vacuum system, and the tracer gas is shot towards various outside parts of the vacuum system. When the measuring device detects a sudden leap in the quantity of that gas, we have found our leak. Similar idea to the helium mass spectrometer but reversed placement of the tracer gas and measuring apparatus.

Why helium? It is non-toxic, cheap, goes through any cracks, and is only present at about 5 ppm in the atmosphere.

Pumps

Pumps have a pressure range over which they operate best. Every design of pump has its own ideal operating conditions. It seems that high vacuum pumps are rarely operated in rough vacuum and that ultra-high vac pumps are generally only employed after high vacuum conditions have been attained.

Roughing pumps do the work from atmospheric pressure down to perhaps 10-3 torr. The ‘foreline’ is between the roughing pump and the high-vac pump. It is not clear to me at this point why the foreline has to exist.

Rotary Vane Pump

This is an oil sealed roughing pump. A good illustration of how it operates can be seen on the wikipedia article. A rotating central piece pushes around gases, causing a pumping action from the input line to the output line. The basic idea is to squeeze gas out of the low pressure side into the high pressure side.

This pump in particular suffers from an oil backstreaming problem. Oil backstreaming is when oil molecules outgas and raise the pressure, travelling backwards from the pump back into the vacuumed space.

Oil backstreaming from the roughing pump. How do we deal with it? We looked at one possibility, the Zeolite trap. Molecular sieves can selectively trap certain sizes of molecules. Zeolite traps are a type of these, and can be designed to trap molecules of specific sizes.

Sorption pump

Adsorption is the sticking of a molecule to a surface. The sorption pump adsorbs molecules on a very porous material. The inner construction of the porous material is many small fins to maximize surface area. Used primarily as a roughing pump. Typically cooled with liquid nitrogen. Can achieve 10-2 mBar to 10-7 mbar (with additional special techniques).

No moving parts, and thus no need for lubrication (with the associated possibilities for backstreaming). However, it cannot be run continuously because it is limited in the total volume of gas that it can effectively pump. Also, it cannot effectively pump hydrogen, helium and neon. More specifically, it cannot effectively pump any material with a lower condensation temperature than (liquid) nitrogen.

Cryocondensation

Use very low temperatures to condense volatile gases out of the vacuum volume. Obviously this can only feasibly work with certain gases. Also, this would impose some limits on the materials used in the construction of the vacuum system and the experiment. If the experiment can be done in extremely cold conditions, this is one way to help achieve ultra high vacuum.

Cryo-soprtion

Cool things down so that gases can be adsorbed. Or this can merely slow down the impinging gases, cooling them down, effectively trapping some of them nearby, improving the vacuum.

Oil diffusion pump

Now we look at the Oil diffusion pump. It seems that the idea is basically to blast a jet of material that will carry with it the gases present in the volume towards the far end of the pump and the exhaust. It is called a diffusion pump because the gas is diffusing back towards the vacuum, but it get carried away by the jet of material.

The oil vapour molecules are accelerated to a speed of more than 750 mph apparently. The hot ejection is into the foreline. There we also get degassing of contaminated oil. These pumps can be constructed in a multi-stage fashion so that different purity of oil is used at different levels of vacuum.

Pumping speed follows a relatively predictable shape that is quite interesting first it increases to a steady speed, then stays there until a ‘critical point’ after which there is a fairly sharp drop-off to about half that pumping speed, and then a more gradual exponential decay.

To deal with foreline pressures getting too high, you can introduce some chilled surfaces using some liquid nitrogen. The chilled surfaces will reduce the pressure.

LN2 Trap

Since several types of pumps have backstreaming oil, we would like to stop this oil from getting to the vacuum chamber. Between the pump and the vacuum chamber, we might choose to have a LN2 trap. We use the low temperature of liquid nitrogen to condense out most contaminants, including oil, so that they are not present in the vacuum.

Turbomolecular Pump

The turbomolecular pump relies on a transfer of momentum to gas molecules from spinning rotor blades. Each set of spinning rotors tries to knock the molecule down to the next level of the pump. Between each set of rotors, there is a set of stators that are designed so that the molecules that are hit by the rotors are likely to fly down to the next stage.

All of these pumps are multi-stage, with each stage representing a compression of approximately 10. The rotor blades must be spun very quickly (up to 1500 times per second!), making some sort of bearingl necessary. However, since oil presents problems for the achievement of ultra high vacuum, some of these pumps now use magnetic bearings.

This pump is generally capable of achieving and maintaining high vacuum. It can achieve ultra high vacuum in some circumstances. Typically it is employed in conjunction with a roughing pump since it does not operate well near atmospheric pressure. However, since 2006 models have existed that can exhaust directly to atmospheric pressure.

Larger and heavier molecules are easier to pump than lighter ones in this case. Molecular hydrogen for instance is quite problematic for this pump to move. This is one of the reasons why the maximum vacuum achievable with this pump is only moderately high.

Sputter Ion Pump

The sputter ion pump operates on the principle of ionizing atoms and then using a strong electric potential to move them to a desired surface electrode. The molecules then strike the surface and undergo one of chemisorption, physisorption, or neutralization as they steal an electron and fly away. These neutrals are likely to be ionized again and sent back to the surface. Eventually they may become attached to the surface as neutral molecules.

This pump is capable of achieving 10-11 millibar vacuum under ideal circumstances. It has no moving parts. Grütter calls attention to the fact that titanium is very reactive. Perhaps making it a very good candidate for the electrode material.

Gauges

Depending on the level of vacuum that we are trying to achieve, we need to employ different types of gauges.

Bourdon Gauge

Operates under the principle that a flat tube tends to become more circular when the pressure inside it rises. This effect seems rather small, but there are ways to amplify it. One of these ways is to arrange the tube in a “C” shape where its motion is more noticeable. This motion can then for instance be connected through gearing to a needle that will display the pressure.

These gauges are very linear and can be reasonably sensitive. However, they will not operate in high vacuum or ultra high vacuum.

Pirani Gauge

The Pirani gauge is a type of thermocouple gauge. A metal filament is heated within the vacuumed chamber. When there are many molecules striking it, it loses its heat to them, causing its own heat to decrease. Conversely, when there are very few molecules striking it, as is the case in high vacuum or ultra high vacuum, very little heat will be leaked away. This means that the temperature of the filament will be higher when it is in higher vacuum.

Since resistivity varies with temperature, we can characterize the metal filament at various temperatures so that we understand what temperature it is at when we observe a particular resistance.  When we understand what the temperature is, we have an indicator of the pressure.

Ionization Gauges

Electrons are emitted from a hot filament. These electrons are then attracted by a helical wire (or set of wires) that is held at a positive potential of ~150V. As the electrons fly across the intervening space, they tend to run into any molecules flying around out there. The electron is likely to ionize the molecule, causing it to lose an electron, and thus become positively charged.

The ionized molecule will be pushed away from the spiral electrode and towards a central wire that is held at a negative potential of about -30 V. The ions will strike the central wire, creating a small current. This current is then amplified and measured. Combined with calibration of temperature to pressure, we can then get an accurate pressure reading.

This is the most widely-employed device for measuring vacuum pressure between 10-3 and 10-10 Torr.

Rest Gas Analyzer

These devices are in effect small mass spectrometers. They measure mass-to-charge ratios. They are used for very low pressures. Their highest operating pressure is about 10-4 Torr. Their sensitivity is remarkable, as they can measure partial pressures down to 10-14 Torr.

How do they work? Gas molecules are ionized by a beam of electrons. The details of the electron beam creation from a hot filament dictate that this process is best conducted at low pressures. The presence of a lot of oxygen for instance can be bad for a filament.

The next step is a Quadrupole Mass Analyzer (QMA) which is capable of selecting a specific range of mass-to-charge ratios for analysis. The QMA will allow these selected ions to pass through while all others will strike the sides.

The ions are then collected by a detector such as an electron multiplier. The final result is a graph of mass-to-charge ratio with intensities. So now we know what the mass-to-charge ratio is for the materials in our vacuum, and we also understand the relative number of ions of each type. Using our knowledge of mass-to-charge ratios, we can figure out what molecule each signature is for.

The difference in the spectra of two different system states can be striking. Grütter shows an unbaked normal vacuum system as compared to a system with an air leak. In the unbaked system, H20 is the largest overall signal with some H2, CO2, N2 and CO. The air leak system looks very different, with N2 dominating, followed by H20 and 02, and finally CO2 and H2. These differences in composition can be very helpful with ascertaining the nature of a leak or unknown gas source in the system.

Scratching the surface of surface science

The primary resource for this material was a lecture by Peter Grütter.

Why has it been reseached?

The semiconductor industry is probably the primary reason why surface science has received so much attention. It is well established in industry and in science. The development of the tools and techniques has been driven primarily by the semiconductor industry.

Another major driver is catalysis. This is the area of study of how to catalyse reactions/processes so that they can happen more quickly and/or with lower energy requirements. Surfaces can be extremely important for catalysis. With regards to catalysis, the sites of interest on the surface are actually the kinks and defects rather than the flat surface itself.

Small features can be of primary importance in many of these condensed matter fields of study. Dr. Grütter calls attention to the fact that in the semiconductor industry, the doping atoms among the silicon of crucial to the operation of the devices.

Introduction to Surface Science

In this class, we will be talking primarily about solid-vacuum interfaces rather than solid-liquid interfaces. We are building on the knowledge we gained in the introductory sections on vacuum systems.

Surfaces are 3 dimensional. They are not merely two-dimensional planes. They are a layer of transition from bulk conditions to vacuum conditions.

The dipole layer is an interesting physical phenomenon that takes place at the surface of a material. Electron density does not drop off to zero once we are outside the surface atoms. It tapers off, becoming negligible some small distance away from the surface. This distance is on the order of one fermi wavelength, which would vary depending on the material.

So some negative charge ends up outside the surface. The only picture I can find of this effect online is here, even though it is given in terms of electrostatic potential rather than electron density. Rather than a smooth drop in electron density, we end up with a periodic (on the scale of fermi wavelengths) charge density as we look into the surface. Thus, just inside the surface we actually have a higher electron density than we do further into the bulk of the material. This interface between the high internal electron density and the low external electron density is called the dipole layer.

The dipole layer can stop atoms from diffusing out of the surface. As they diffuse towards the surface, they suddenly come up against a larger density of electrons, which push them away. In the image linked above, the diffusion would be taking place from right to left. The lower potential pushes back on the atom’s electrons, causing it to have more difficulty getting through the surface than it had moving throughout the bulk of the material.

A few Observations

As we already know, taking an electron out of the surface will take some energy. The amount of energy depends on several things such as strength of bond to ion core, interaction of electrons with each other, etc. This is known as the work function. There are two versions, one considers the energy needed to move the electron to just outside the solid surface, while the other considers the move of the electron to infinity).

The work function depends primarily on the dipole layer. Can be different work functions for different surfaces (faces) of crystals! Depends on the orientation of the atoms. Work function also depends on step density. What is a step? Consider a perfect planar surface of atoms. Now consider adding another layer to half the surface, so that there is a ‘step’ up to the second layer. There can be many such steps. As a heavy and long-time computer user, one of the first things I visualized was the fact that an angled line on a computer monitor is not smooth, it has ‘steps’ made of straight sections. Similarly with a surface viewed at the nano scale. The closer the steps are to each other, the higher the step density. Step density changes the work function because of the details of the dipole layer at each step.

Question that you must learn to ask yourself: You must ask yourself if what you are studying is affected by small defects in the system. In the history of science this has been overlooked many times. How big of an effect can these things have? Well, it turns out that a 5% difference in work function for Tungsten can be created by step density. Even more astounding, a 1 eV difference in work function can be measured depending on tungsten orientation! 1 eV at the nanometer scale indicates a huge difference in electric field. These hugely different electric fields can help explain why such small defects can often have a large effect on chemical reaction rates via catalysis.

Surface Energy

The simplest way to explain surface energy that I can find is from Wikipedia, where it is stated that surface energy can be defined as the excess energy at the surface of a material compared to the bulk of the material.

In class, the first thing we discuss about surface energy is the jelly model (jellium), which is quite similar to the plum pudding model. It feels almost heretical to be talking about this, since this class is in a building named after the man who proved that the plum pudding model was wrong (Ernest Rutherford).

We can calculate surface energy for jellium quite easily. This tends to agree with experiment at low densities, then eventually becomes very broken at higher densities. The more complicated (and accurate) models are quite difficult to calculate. Additionally, the surface energy is very hard to measure experimentally.

Surface energy is crucial to our understanding of many physical aspects of surfaces. For example, it helps us understand how we can grow materials on other materials. Will we get island growth or layer-by-layer growth?

One of the reasons this is difficult to model correctly is that the electron correlation effect between d-orbitals are difficult to calculate. This is why estimating the surface energy of elements such as gold, iron, etc involves very complicated calculations.

It turns out that finding the minima of surface energy will show us the shape of an equilibrium crystal. Real crystals may not completely agree because our physical crystal growth is not perfect. In closing, surface energy is important for studying crystal shapes as well as understanding what materials we can grow on what substrates and how they grow.

Surface Structure

There are three major ways in which the surface structure can be very different from the bulk structure.

Relaxation

The spacing between surface atoms and second layer is often not equal to the distance between the 2nd and third. Surface atoms tend to get pulled in little bit because they do not have a bond on one side. This is true for both covalent bonds and metals. This relaxation may be up to three layers deep (distances grow towards lattice standard as we go deeper).

Reconstruction

Where the surface structure is different from the bulk. For example, there might be more atoms on the surface layer than in a bulk layer. They may be connected to each other at different angles. Thus, the unit cell of the surface crystal can be very different from the bulk unit cell. We actually cannot calculate some of these structures because they are too complicated.

Related aside, Dr. Grütter began talking about silicon (111). He said, “This was the Guinea Pig or Drosphila of surface science for a number of years.” Apparently about 20 years of work went into understand silicon (111), which has what is called a “7x7 reconstruction” comprised of 64 atoms in 4 layers. The problem was eventually solved by a combination of scanning tunneling microscopy and diffraction studies.

Aside from the aside: This is not the industrially relevant silicon unit cell. That role is filled by silicon (001). Dr. Grütter says that it is very important that one can grow very smooth layers of oxide on silicon (001).

Aside3: Dr. Grütter says that silicon cannot be used as a photon emitting material very well because this would violate momentum conservation. However, gallium arsenide is capable of being a useful photon emitting material.

Composition

Most materials are an alloy, there are multiple constituent elements. Will the surface layer be the same composition as a bulk layer? It turns out that often surface layers are usually completely different than the bulk in terms of composition. Surface might be all of one element. Second layer might be a split of some kind. Third layer might be a different split.

This fact has huge implications for surface characteristics such as the ability to catalyze reactions, corrosion resistance, hardness, etc.

Surface Complexity

We tend to think of surfaces as atomically flat, but they are not. A decent flat surface might have truly flat areas that are 10nm in length. We might be able to get 100nm of nice flat area if we try really hard and employ a lot of tricks.

Some of the forms of imperfections in a surface are:

  1. kinks
  2. terraces
  3. vacancies
  4. adatoms
  5. monoatomic steps
  6. step-adatoms

Curious about what these are? Check out this Wikipedia page which includes some of their definitions.

A fair amount of research has been done on the subject of the effects of these imperfections in surfaces. For example, we have learned that electromigration is affected. Defects can backscatter electrons. This can become important when the surface atoms are a notable number of total atoms in the wire, which happens at the nano scale.