Counting Electrons - Challenges at the Micro and Macro Level
tl;dr
Here’s one pass at a summary table with the relevant electron ranges; take it with a grain of salt.
| Number of electrons | Storage mode (capacitor) | Flow mode (current) |
|---|---|---|
| $10^{0} — 10^{3}$ | Quantum dot(s) or traps | Scintillation |
| $10^{4} — 10^{16}$ | Capacitor + transimpedance amplifier | Transimpedance amplifier |
| $10^{16}+$ | Big capacitor, ADC converter | DMM, Magnetic field current clamp |
Introduction
Counting electrons (precisely) is an important engineering tool - this applies to electron imaging, sensing technologies and more! I had recently been thinking about all of the different technologies current in use across industries to study various quantities of electrons (1, 100, 1 mol, etc.) and how the challenge increases at both the large and small ends of the scale.
Counting electrons can be done with or without “storing” the electrons. A bunch of electrons have charge, and a capacitor is essentially a charge storage device. However, anologously to how we measure water, sometimes we must make measurements of transient electrons where we cannot store the full number (or the number is too few that storage would be lossy). Recall that 1 C of charge is $1.602\times10^{19}$ electrons, and that 1 A of current is 1 C/s, so when we think of small currents in the nA or pA range, we are still talking about $10^{10}$ or $10^{7}$ electrons a second!
Single electron detection
The key to single electron detection is tricky because it very much depends on how much energy (kinetic energy) the electron. We know from nuclear physics that beta decay will give us an energetic electron (or positron I guess); the simplest method would be to use a scintillation technique. Scintillation is an indirect technique where we do not observe the electron itself, but we can detect the electron’s interaction with some medium it passes through. This typically occurs with high energy particles (not just electrons) transferring energy to a medium that generates photons which are detected by a photo-multiplier tube (that’s a whole other interesting rabbit hole).
For low energy electrons (i.e. an electron in a semiconductor without a ton of excess kinetic energy), one would have to rely on other techniques to “count”; however, detect may be the more suitable phrase here. This can be done using sophisticated electron traps such as a quantum dot (QD) and looking at some spectroscopic probe (RLC AC resonance, optical transitions, magnetic effects, etc.) response. A quantum dot traps an electron in a 3D potential and has many possible uses, one of which is to create a single electron transistor with a source/drain to control the state of the electron in the QD. This technique can be used for various quantum sensing applications where single charges need to be controlled.
A bushel and a peck of electrons - using amplification
In the previous section, I have outlined two rough methods of counting single electrons - either recording the interaction of high energy electrons via a technique such as scintillation, or by essentially using a QD as a single electron capacitor. For most of the next techniques, we will be looking at either using capacitors to store charge to count electrons, or relying on measuring electron flow precisely.
The main concept at play is going to be using gain to amplify this small quantities to much larger quantities that are easy to detect, i.e. converting a nA signal to a mA signal. However, gain is generally lossy - since energy is not free, we cannot crank the gain on something arbitrarily. This creates noise in the process and depending on the amplification will ultimately set the noise floor for detection.
The main amplifier I will discuss is the transimpedance amplifier (TIA) which relies on an op-amp in closed loop operation. Focusing on current amplifiers, the TIA is essentially a low-noise, high-gain device that can tune the gain delivered by the combination of the feedback resistor and capacitor. This can allow for a small current, say in the ~fA range ($10^{-15}$ A, ~10,000 e/s ) to be amplified by a factor of $10^{3} — 10^{13}$ $V/A$. This essentially can generate a mV signal from a fA input current.
As I mentioned before, the main issue with amplification is always noise - in the TIA this is driven by the noise in the feedback resistor and capacitor. Since electrons are discrete, most noise sources are Poisson or shot noise forms that depend on the square root of the charge/current. In resistors, the thermal noise, or Johnson-Nyquist noise, of a resistor is given by $\sigma_{JN} = \sqrt{4 k_b T R }$; for capacitors the voltage noise is $\sqrt{\frac{k_bT}{C}}$ and the charge noise is $\sqrt{k_b T C}$ (also why this is typically called “KTC noise”). For reference, a 1 pF capacitor at room temperature would have an associated charge noise of ~400 e.
For currents large enough for direct amplification with a TIA, a storage capacitor is not needed unless noise considerations require it. This allows the small current, generated by some device (photodiode, etc.), to be directly amplified for read out. However, smaller currents can be buffered by a fine capacitor (1 F of capacitance is comically large, so think something on the small end of the available capacitances - pF or fF), and once the capacitor has been allowed to fill over time, we can perform a voltage read out on the capacitor. The fundamental equation for a capacitor is that $Q = CV$, so the charge stored over time is converted to an equivalent voltage.
In the realm of small but not impossibly small numbers of electrons ($10^4+$), the main constraints will be set by the application and noise requirements. Detection performance will always be a trade between precision/accuracy and speed/cost/size/etc.
The “Easy” Realm: More than a Coulomb of Electrons
Here, life is easy. We can still detect either charge or current, but now high precision pre-amplification is not needed. We can “just read it out”. This is the realm of most modern digital electronics. With a simple 8-bit ADC and a supercapacitor in the 1000 F range with a 2.5 V rating, one could easily count up to 10 C of electrons or $1.6\times10^{29}$ electrons. Side note - it is weird to mix SI multiples and divisions on a unit like the Farad where units are predominantly on the small divisor range (pF to $\mu$F).
Reading currents on the other hand begin to get tricky - a perfect ammeter should have infinitesimal resistance, but as the current increases, the need for fusing and other safety measures need to be increased to avoid damaging measuring equipment. Switching to a dedicated current meter with a current clamp allows for much higher currents to be read (both AC and DC) by measuring the magnetic field generated by the flowing current of the wire in the clamp. Remebering the Biot-Savart law, the magnetic field generated by a current in a wire is:
$$
B = \frac{\mu_0 I}{2\pi R}
$$
where $R$ is the distance from the wire. For large currents, this involves measuring a magnetic field in the $\mu T$ or $mT$ range.