Bound states on a Quantum Dot and Tunneling on and off the Dot

Question

Question is essentially: How can the states on a quantum dot be bound when we can tunnel onto them? If plane wave states can tunnel onto them, these plane waves states and the 'bound' states will have the same energy. Thus we could find a state containing plane waves on either side of the dot and the 'bound' state. So this overall will not be a bound state. So how can we speak of an integer number of electrons on the dot?

Consider two fermi liquid reservoirs on either side of a quantum dot. When making the Coulomb blockade argument one simply says that, if there are already N electrons on the dot, the $(N+1)^{th}$, electron needs $E_{N+1}-E_{N}+\frac{e^2}{C}$ more energy to be added than the $N^{th}$ electron did. Or rather, this is the amount the chemical potential of the dot jumps once the $Nth$ electron is added. $C$ is some capacitance and this term is related simply to the electrostatic energy of the confined electrons on dot. $E_N$ is the energy of the $N^{th}$ energy level. It may be equal to $E_{N+1}$ depending on spin degeneracy and whether $N$ is odd or even. So, when we then use a gate voltage to effectively lower the energy levels of the dot, the new chemical potential of the dot will eventually align with the reservoirs' chemical potential and if a small bias is applied electrons can hop across between the reservoirs leading to a current.

However, let's step back and consider some simple tunneling transport. Consider a 1d system, where we have free space on the left and right of a generic barrier. One approach is simply to solve the Schrödinger equation. Here we find eigenstates and once this is done, if these states are not bound states, we can straightforwardly read off transmission and reflection coefficients. All of the states which represent an incident electron being reflected and transmitted are not bound states. Furthermore, all bound states have energies less than those of the states which are not bound. This is obvious as the wavevector of a bound state, in free space on either side of the barrier, must be imaginary, so its energy is less than the not bound states, for which this is not true. So, the bound states are not involved in transmission of an electron. It does make sense to consider an incident, plane wave, electron tunneling into one of these bound states; the two states have different energies.

However, this seems to be exactly what we do with the quantum dot. It seems that we do discuss tunneling into quantum dot bound states. But these states cannot be bound, as there exist plane waves in the reservoirs tunneling into them, which thus must be of same energy.

My question is this: How can the states on the dot be bound if we can tunnel onto them. As then the states have same energy as plane waves in the reservoirs and we would form non bound states. And how can we speak of an integer number of electrons on a dot if these states aren't bound?

We haven't included the fact that there is electrostatic energy. But I still don't quite see how this can help us. We simply shift the energy levels up due to this energy and still our states on the dot won't be bound if we can tunnel onto them.

Answer 1

Let me first note that we talk here about a particular type of quantum dots - those obtained by split gate technique, whose main interest in passing current through them. There are plenty of other types of quantum dots, interesting for other type of their properties (e.g., the optical ones) where the problems describe din the question are not relevant.

Weak coupling
Indeed, Quantum dot states are not strictly bound, since they are coupled to the continuum states outside of the dot. However, the tunneling probability is usually small, which means that the tunneling probability through the dot takes form of narrow resonances. The problem is not unlike that of resonant tunneling through a pair of barriers, considered in the basic quantum mechanics. This assumption about weak coupling about the dot and the leads is always implicit in the relevant scientific papers.

Fermi level
If we now consider a one-dimensional problem of resonant tunneling through a double barrier, the transmission probability will have resonances at energies roughly corresponding to those of the energies of the dot, if it were truly isolated from the leads by infinite barriers (for now I do not consider Coulomb interaction). Obviously, the waves can be incident both from the left and from the right, and when we fill the states up to the Fermi level, the the right- and left-moving states come in pairs, so that the net current is zero. The only states that do consider to the current flow are those near the Fermi surface, if the Fermi levels are different (their difference is the applied voltage bias).

Transfer Hamiltonian
Using the true extended states, as in the problem with scattering through a double barrier, is rather impractical, which is why one often resorts to using so-called transfer Hamiltonian, where one separates the system into regions (dot and leads) coupled via weak tunneling. This is somewhat similar to the tight-binding approach to crystal structure. The transfer Hamiltonian is not exact, and there are some mathematical issues with using it, but most of the time it is a good approximation.

Coulomb interaction
Including Coulomb interaction in the dot, but not in the leads is also justified by the weak coupling, as I described above. But there is more to that: the leads are actually two- or even three-dimensional Fermi seas. The Coulomb interaction is present in the leads, but can be absorbed in effective parameters, since Landau Fermi liquid theory applies. When the leads are truly one-dimensional wires, they behave as Luttinger liquid - Coulomb interaction is essential.

Answer 2

How can the states on the dot be bound if we can tunnel onto them?"

Strictly speaking, you are correct that quantum dots do not have any bound states; they have quasi-bound states. This means that the barriers separating the dot from everything else are "big" enough that the tunneling rate is low (i.e. the lifetime of the quasi-bound states are high), and it is useful to analyze the system as if has bound states.

And how can we speak of an integer number of electrons on a dot if these states aren't bound?

This is basically the same question as "How can we talk about a number of U-235 atoms if they're unstable?" You can because altho U-235 is unstable, it is metastable. Bound/stable and quasi-bound/metastable are basically the same thing.

This is not mere hand-waving; you can make it quite rigorous. You can write the Hamiltonian of the whole dot+outside world system as the Hamiltonian of the dot (as if it were isolated) plus the Hamiltonian of the outside world plus a term that couples them together. Then you can do things like find the expectation value for the number of electrons on the dot. You'll find that, as long as the coupling terms are weak, for most purposes you can treat the dot separately from the outside world. So, people generally don't go thru the trouble because it doesn't improve on the simple analysis.