Physics · Atomic and quantum physics · Year 3

Quantum dots and the potential well

Why do quantum dots made of the same material glow in different colours depending on their size?

The situation

In an electronics shop in Tashkent the colours of a QLED television are especially bright and pure. Inside the screen are tiny semiconductor crystals a few nanometres across — quantum dots: when blue light falls on them they emit green or red light. Most interesting of all, the green and red dots can be made of the very same material — the only difference is the size. In the laboratory, test tubes of a solution of CdSe dots that differ only in size glow in all the colours of the rainbow under an ultraviolet lamp. Why does a crystal's light get bluer the smaller the crystal is?

Topic

Potential well and quantum oscillator. Quantum-dot display: the energy of a particle in a box is quantised

Key ideas

Energy quantisation
Wave function
Zero-point oscillations

Formulas

\hat{H} ψ = E ψ, \hat{H} = - \frac{ℏ^{2}}{2 m} \frac{d^{2}}{d x^{2}} + V (x)

The time-independent Schrödinger equation: $\hat{H}$ is the Hamiltonian (energy) operator, $E$ is its eigenvalue and $ψ$ is its eigenfunction.

E_{n} = \frac{π^{2} ℏ^{2} n^{2}}{2 m L^{2}} = \frac{n^{2} h^{2}}{8 m L^{2}}, ψ_{n} = \sqrt{\frac{2}{L}} \sin \frac{n π x}{L}

The infinite well (walls at $x = 0$ and $x = L$ ). For an electron with $L = 1$ nm, $E_{1} = 0.376$ eV.

z \tan z = \sqrt{z_{0}^{2} - z^{2}}, - z \cot z = \sqrt{z_{0}^{2} - z^{2}}, z_{0} = \frac{L}{2} \frac{\sqrt{2 m V_{0}}}{ℏ}

The finite well: even and odd states ( $z = k L / 2$ ). The number of bound states is $N = ⌈ 2 z_{0} / π ⌉$ .

E_{n} = ℏ ω (n + \frac{1}{2}), \hat{H} = ℏ ω ({\hat{a}}^{†} \hat{a} + \frac{1}{2}), {\hat{a}}^{†} | n ⟩ = \sqrt{n + 1} | n + 1 ⟩

The harmonic oscillator: the lowest energy is $ℏ ω / 2$ (zero-point oscillations), and $\hat{a}, {\hat{a}}^{†}$ are the lowering and raising operators.

E \approx E_{g} + \frac{ℏ^{2} π^{2}}{2 R^{2}} (\frac{1}{m_{e}^{*}} + \frac{1}{m_{h}^{*}}) - \frac{1.786 e^{2}}{4 π ε ε_{0} R}

The quantum dot (Brus model): the electron and the hole are confined in a sphere of radius $R$ ; the last term is their Coulomb attraction.

Theory

The wave function and operators

$| ψ (x) |^{2}$ is the probability density of finding the particle at the point $x$ , so $\int | ψ |^{2} d x = 1$ . States can be added (the superposition principle). Every measurable quantity corresponds to a Hermitian operator: $\hat{x} = x$ , $\hat{p} = - i ℏ d / d x$ , $\hat{H} = {\hat{p}}^{2} / 2 m + V$ . The eigenvalues of a Hermitian operator are real, and its eigenfunctions are orthogonal and form a complete set — in “Well” mode you can see $⟨ ψ_{n} | ψ_{n + 1} ⟩ \approx 10^{- 13}$ . The mean value is $⟨ A ⟩ = \int ψ^{*} \hat{A} ψ d x$ , and the variance is $Δ A^{2} = ⟨ A^{2} ⟩ - {⟨ A ⟩}^{2}$ . Because $[\hat{x}, \hat{p}] = i ℏ$ , $x$ and $p$ cannot be measured exactly at the same time: $Δ x Δ p \geq ℏ / 2$ (in the well, ≈ 0.568ħ for n = 1). For a free particle ( $V = 0$ ), $ψ = e^{i k x}$ and $E = ℏ^{2} k^{2} / 2 m$ — the spectrum is continuous, whereas the spectrum of bound states is discrete.

The potential well

In an infinite well a whole number of half-waves must fit between the walls: $L = n λ / 2$ , so $E_{n} \propto n^{2} / L^{2}$ . If you make the well 2 times narrower, the levels rise 4 times — this is why the colour of a quantum dot changes. In a finite well $ψ$ “leaks” into the wall as $e^{- κ | x |}$ (the classically forbidden region; through a thin wall this leads to quantum tunnelling and alpha decay — see the “Schrödinger and quantum tunnelling” and “Radioactivity, half-life” experiments), so the levels are lower than in the infinite well and their number is finite. Here the levels are found by a numerical method: $\hat{H}$ on a grid becomes a tridiagonal matrix, the eigenvalues are found with a Sturm sequence and the functions with inverse iteration; the result is compared with the transcendental equation. One and the same state can be written in the position ( $ψ (x)$ ), momentum ( $| φ (k) |^{2}$ — two peaks near $\pm n π / L$ (for $n \geq 2$ )) and energy ( $c_{n}$ coefficients) representations.

The harmonic oscillator

In a parabolic potential $E_{n} = ℏ ω (n + \frac{1}{2})$ — the levels are equally spaced, and the lowest level is not zero: the zero-point energy is $ℏ ω / 2$ . The eigenfunctions are Hermite–Gauss functions. The algebraic solution: $[\hat{a}, {\hat{a}}^{†}] = 1$ , ${\hat{a}}^{†}$ raises the level by one, $\hat{a}$ lowers it, and $\hat{a} | 0 ⟩ = 0$ . In the energy representation only the elements $x_{n, n \pm 1}$ of the matrix of $\hat{x}$ are non-zero — so in the harmonic approximation an HCl molecule absorbs infrared radiation only in transitions with $Δ n = \pm 1$ (the selection rule). For a molecule the reduced mass $μ = m_{1} m_{2} / (m_{1} + m_{2})$ is used in place of the mass (0.98 u for HCl) — the motion of the nucleus in hydrogen-like atoms is taken into account in the same way. For a three-dimensional isotropic oscillator $E = ℏ ω (n_{x} + n_{y} + n_{z} + \frac{3}{2})$ ; the centrally symmetric field, ${\hat{L}}^{2}$ , ${\hat{L}}_{z}$ and hydrogen-like atoms are in the “Orbitals of the hydrogen atom” experiment.

Approximation methods

Perturbation theory: if $\hat{H} = {\hat{H}}_{0} + \hat{W}$ , then to first order $Δ E_{n} = ⟨ n | \hat{W} | n ⟩$ ; add $g ξ^{4}$ to the oscillator and compare this with the exact numerical solution — it is good for small $g$ and gets worse for larger values. For degenerate levels (for example, the states $(1,2)$ and $(2,1)$ in a square box) the perturbation first requires the states to be mixed correctly, and the level splits in two — this is how the Stark effect arises (see the “Spin, the Zeeman effect and MRI” experiment). The variational method: for any trial function $⟨ H ⟩ \geq E_{0}$ — the $E (β)$ curve on the graph never falls below the exact $E_{0}$ . The WKB (quasi-classical) method and the Bohr–Sommerfeld condition $\oint p d x = (n + \frac{1}{2}) h$ are exact for the oscillator, and for an anharmonic potential they are more accurate at large $n$ . For a time-dependent perturbation the probability of a transition is proportional to $| ⟨ f | \hat{x} | i ⟩ |^{2}$ (Fermi's golden rule) — this accounts for emission and absorption (see the “Laser physics and holography” experiment).

Change with time

$i ℏ \partial ψ / \partial t = \hat{H} ψ$ . A stationary state acquires only a phase $e^{- i E t / ℏ}$ — $| ψ |^{2}$ does not change. In a superposition of two states $| ψ |^{2}$ sloshes about inside the well, and the period of $⟨ x ⟩$ is $T = h / (E_{2} - E_{1})$ . The probability current $j = \frac{ℏ}{m} Im (ψ^{*} \partial_{x} ψ)$ satisfies the continuity equation $\partial_{t} | ψ |^{2} + \partial_{x} j = 0$ (the number of particles is conserved; in the panel $d P_{l e f t} / d t = - j (0)$ ). The evolution operator $e^{- i \hat{H} t / ℏ}$ is unitary — the norm stays equal to 1. The mean value of an operator changes according to $d ⟨ A ⟩ / d t = \frac{i}{ℏ} ⟨ [\hat{H}, \hat{A}] ⟩$ ; from this comes Ehrenfest's theorem: $m d ⟨ x ⟩ / d t = ⟨ p ⟩$ (in the panel the two are equal). Quantities that commute with $\hat{H}$ (energy, parity) are constants of motion. This is the Schrödinger picture (the state changes); in the Heisenberg picture the operators change, $x_{m n} (t) = x_{m n} e^{i ω_{m n} t}$ ; the interaction picture is convenient in perturbation theory.

Quantum dots and nanotechnology

Nanotechnology deals with structures 1–100 nm in size, whose properties depend on their size. According to the number of directions in which the electron is confined, there are the quantum well (a thin layer, for example GaAs between layers of AlGaAs — in laser diodes), the quantum wire and the quantum dot (confined on all sides, an “artificial atom”); going from 3D to 0D, the density of states changes from continuous to discrete lines (on the screen). The Brus model gives the trend correctly ( $E \propto 1 / R^{2}$ ) but overestimates the energy for small dots: the walls are in fact not infinite, and the bands are not parabolic. So the colours of the test tubes are taken from a measured curve (Yu et al., 2003, the first absorption peak for CdSe); the luminescence colour is close to this peak, slightly redder. By the Pauli principle, at most two electrons with opposite spins can occupy one level (see the “Spin, the Zeeman effect and MRI” and “Many-electron atoms” experiments). CdSe parameters: $E_{g} = 1.74$ eV, $m_{e}^{*} = 0.13 m_{e}$ , $m_{h}^{*} = 0.45 m_{e}$ , $ε = 10.6$ .

Did you know?

The 2023 Nobel Prize in Chemistry was awarded to Moungi Bawendi, Louis Brus and Alexei Ekimov for the discovery and synthesis of quantum dots. In the early 1980s Ekimov showed that colour depends on size in copper chloride nanocrystals in coloured glass, and Brus showed it for particles in a liquid.
Helium stays liquid at ordinary pressure even near absolute zero: it takes about 25 atmospheres to solidify it. The reason is that the zero-point energy of the light atoms is large, while the attraction between them is very weak.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Elementary particles — Grade 11
Spin, the Zeeman effect and MRI — Year 1
Many-electron atoms — Year 1
Bands in solids — Year 1
Schrödinger and quantum tunnelling — Year 3
The molecule: bonding and spectra — Year 3
Superconductivity — Year 3