Landau Levels in 2D

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Hello! Welcome to the fourth lesson in our module on topological matter.

In our last lesson, we defined the Chern number, $C_{n}$ , as the integral of the Berry curvature over the Brillouin zone. We established that for a gapped 2D lattice, $C_{n}$ is a quantized integer that serves as a topological invariant for the band. The physical significance of this was captured by the remarkable TKNN formula, $σ_{x y} = (e^{2} / h) \sum C_{n}$ , which connects the sum of Chern numbers of occupied bands directly to the quantized Hall conductivity.

The TKNN formula provided a deep theoretical explanation for the Integer Quantum Hall Effect (IQHE), which is experimentally observed in a two-dimensional electron gas (2DEG) under a strong perpendicular magnetic field. As previewed, we will now take a temporary detour from the abstract topological formalism to analyze the quantum mechanics of this specific physical system. Our goal for today is to solve for the Landau level spectrum of a 2D electron gas in a perpendicular magnetic field. This will provide the crucial microscopic picture of the electronic states that underpin the IQHE.

1. From Classical Cyclotron Orbits to the Quantum Hamiltonian

You're familiar with the classical motion of a charged particle in a uniform magnetic field. The Lorentz force causes the particle to undergo circular motion in the plane perpendicular to the field. This is known as cyclotron motion, and it occurs at a specific frequency, the cyclotron frequency:

ω_{c} = \frac{q B}{m}

where $q$ and $m$ are the charge and mass of the particle, and $B$ is the magnetic field strength.

To describe this system quantum mechanically, we must construct the Hamiltonian. As you know from your study of electromagnetism and QFT, this is done through the principle of minimal coupling, where the canonical momentum $p$ is replaced by the kinetic momentum $π = p - q A$ . The Hamiltonian for a particle in a magnetic field $B = \nabla \times A$ is:

H = \frac{1}{2 m} (p - q A)^{2}

A key subtlety here is gauge invariance. The magnetic field $B$ is the physical quantity, while the vector potential $A$ is a mathematical tool that is not uniquely defined. Different choices of $A$ that give the same $B$ are called different gauges. While our intermediate calculations will depend on the gauge, the final physical results (like the energy spectrum) must be gauge-invariant.

For a uniform magnetic field $B = (0, 0, B)$ , a particularly convenient choice is the Landau gauge. We will use the form:

A = (0, B x, 0)

You can easily verify that $\nabla \times A = (0, 0, B)$ . This gauge choice breaks translational invariance in the $x$ -direction (since $A_{y}$ depends on $x$ ) but preserves it in the $y$ -direction. This asymmetry is a mathematical artifact of the gauge, not a physical property of the system.

2. Solving the Schrödinger Equation: Emergence of Landau Levels

With this gauge choice, the Hamiltonian for an electron (charge $q = - e$ ) confined to the $x y$ -plane becomes:

H = \frac{1}{2 m} [p_{x}^{2} + (p_{y} + e B x)^{2}]

Since the Hamiltonian is independent of the $y$ coordinate, the momentum operator $p_{y}$ commutes with it: $[H, p_{y}] = 0$ . This means we can find simultaneous eigenstates of energy and $y$ -momentum. We can therefore write our wavefunction ansatz as:

ψ (x, y) = e^{i k_{y} y} ϕ (x)

where $ℏ k_{y}$ is the eigenvalue of the operator $p_{y}$ . Substituting this into the time-independent Schrödinger equation $H ψ = E ψ$ , the operator $p_{y}$ is replaced by its eigenvalue $ℏ k_{y}$ , and we get a 1D equation for $ϕ (x)$ :

[\frac{p _{x}^{2}}{2 m} + \frac{1}{2 m} (ℏ k_{y} + e B x)^{2}] ϕ (x) = E ϕ (x)

This equation holds the key to the solution. Let's rearrange the potential term:

\frac{1}{2 m} (e B x + ℏ k_{y})^{2} = \frac{( e B ) ^{2}}{2 m} (x + \frac{ℏ k _{y}}{e B})^{2} = \frac{1}{2} m (\frac{e B}{m})^{2} (x - x_{0})^{2}

where we have defined $x_{0} = - \frac{ℏ k _{y}}{e B}$ and used the definition of the cyclotron frequency $ω_{c} = e B / m$ . The Hamiltonian for $ϕ (x)$ is therefore:

H ~ = \frac{p _{x}^{2}}{2 m} + \frac{1}{2} m ω_{c}^{2} (x - x_{0})^{2}

This is precisely the Hamiltonian for a one-dimensional simple harmonic oscillator centered at $x_{0}$ .

The following video provides an excellent walkthrough of this derivation. It uses a slightly different convention for the Landau gauge ( $A_{x} = - B y$ ), which results in a harmonic oscillator in the $y$ -direction, reinforcing the idea that the underlying physics is the same.

L14.3 Particle in a constant magnetic field: Landau levels

The video 'Particle in a constant magnetic field: Landau levels' from MIT OpenCourseWare clearly demonstrates how the quantum problem of an electron in a magnetic field maps onto a simple harmonic oscillator.

Please watch from 03:01 to 12:00. The lecturer sets up the Hamiltonian in the Landau gauge and shows step-by-step how it reduces to the familiar harmonic oscillator problem. Focus on the identification of the oscillator frequency and the shift in the equilibrium position.

Since we have mapped the problem to a simple harmonic oscillator, we can immediately write down the energy eigenvalues:

E_{n} = ℏ ω_{c} (n + \frac{1}{2}), n = 0, 1, 2, \dots

These discrete, equally spaced energy levels are the celebrated Landau Levels. The continuous energy spectrum of a free 2D electron gas completely collapses into a series of discrete levels in the presence of a magnetic field.

Landau Levels: Theory and Experimental Observation — This figure illustrates the core concepts of this lesson. On the left, it shows the equally spaced energy spectrum of Landau levels. In the center, it depicts how the constant density of states (DOS) of a 2D electron gas (dashed line) condenses into a series of sharp peaks at the Landau level energies. On the right, it shows experimental evidence of these levels through Shubnikov-de Haas oscillations in the material's resistance as a function of magnetic field.

3. The Degeneracy of Landau Levels

A crucial feature of the Landau level spectrum is that the energy $E_{n}$ depends on the integer $n$ but not on the continuous quantum number $k_{y}$ . This implies that each Landau level must be massively degenerate. Physically, the energy doesn't depend on the center of the harmonic oscillator potential, $x_{0} = - ℏ k_{y} / e B$ , which corresponds to the center of the classical cyclotron orbit.

How large is this degeneracy? Let's quantify it by considering a finite 2D sample of area $A = L_{x} L_{y}$ .

Quantization of $k_{y}$ : If we impose periodic boundary conditions along the $y$ -direction, the allowed values of $k_{y}$ become quantized: $k_{y} = \frac{2 π j}{L _{y}}$ , where $j$ is an integer.
Constraint on $x_{0}$ : The center of the electron's orbit, $x_{0}$ , must physically lie within the sample. This imposes a constraint: $0 \leq x_{0} \leq L_{x}$ .

Substituting our expression for $x_{0}$ , this becomes:

0 \leq - \frac{ℏ k _{y}}{e B} \leq L_{x} ⟹ - \frac{e B L _{x}}{ℏ} \leq k_{y} \leq 0

The total number of allowed states, $N$ , in one Landau level is the number of allowed integer values of $j$ that fall within this range of $k_{y}$ .

N = \frac{Δ k _{y}}{δ k _{y}} = \frac{e B L _{x} /ℏ}{2 π / L _{y}} = \frac{e B ( L _{x} L _{y} )}{2 π ℏ} = \frac{e B A}{h}

This result is profound. Let's introduce the magnetic flux quantum, $Φ_{0} = h / e$ . The total magnetic flux through the sample is $Φ = B A$ . The degeneracy is then:

N = \frac{Φ}{Φ _{0}}

The number of available single-electron states in each Landau level is exactly equal to the number of magnetic flux quanta piercing the sample.

For a detailed, rigorous derivation of the Landau levels and their degeneracy, David Tong's lecture notes are an excellent resource.

Applications of Quantum Mechanics

David Tong's lecture notes on 'Applications of Quantum Mechanics' provide a very clear and standard derivation of Landau levels and their degeneracy.

Please read Section 1.2, 'Landau Levels' (pages 6-8). The text walks through the derivation using the Landau gauge, defines the magnetic length, and arrives at the energy spectrum. Then, pay close attention to Section 1.2.1, 'Degeneracy,' which formally calculates the number of states per Landau level in a finite sample, leading to the \Phi/\Phi_0 result.

Test your understanding!

Consider a two-dimensional electron gas in a GaAs semiconductor heterostructure, where the electron effective mass is $m^{*} = 0.067 m_{e}$ ( $m_{e}$ is the free electron mass). A perpendicular magnetic field of $B = 5$ Tesla is applied.

Calculate the energy separation between adjacent Landau levels, $ℏ ω_{c}$ , in units of milli-electron volts (meV).
For a square sample of size $1 μ m \times 1 μ m$ , calculate the degeneracy $N$ of each Landau level.

(Use $ℏ \approx 1.055 \times 1 0^{- 34}$ J·s, $e \approx 1.602 \times 1 0^{- 19}$ C, $m_{e} \approx 9.109 \times 1 0^{- 31}$ kg, and $1 eV = 1.602 \times 1 0^{- 19}$ J).

Show answer

Energy Separation:
First, calculate the cyclotron frequency $ω_{c}$ :
$ω_{c} = \frac{e B}{m ^{*}} = \frac{( 1.602 \times 1 0 ^{- 19} C ) \times ( 5 T )}{0.067 \times ( 9.109 \times 1 0 ^{- 31} kg )} \approx 1.316 \times 1 0^{13} rad/s$
Now, calculate the energy $ℏ ω_{c}$ :
$E_{sep} = ℏ ω_{c} = (1.055 \times 1 0^{- 34} J\cdots) \times (1.316 \times 1 0^{13} s^{- 1}) \approx 1.388 \times 1 0^{- 21} J$
Finally, convert to meV:
$E_{sep} (meV) = \frac{1.388 \times 1 0 ^{- 21} J}{1.602 \times 1 0 ^{- 19} J/eV} \times 1000 meV/eV \approx 8.66 meV$
Degeneracy:
First, find the sample area $A$ :
$A = (1 \times 1 0^{- 6} m) \times (1 \times 1 0^{- 6} m) = 1 \times 1 0^{- 12} m^{2}$
Now, calculate the number of states $N$ , using $h = 2 π ℏ$ :
$N = \frac{e B A}{h} = \frac{( 1.602 \times 1 0 ^{- 19} C ) \times ( 5 T ) \times ( 1 \times 1 0 ^{- 12} m ^{2} )}{6.626 \times 1 0 ^{- 34} J\cdots} \approx 1209$
So, each Landau level can hold approximately 1209 electrons (per spin state).

4. A Note on the Symmetric Gauge

To solidify the concept of gauge invariance, it's worth noting that we could have chosen a different gauge. A common alternative is the symmetric gauge, $A = \frac{B}{2} (- y, x, 0)$ . This gauge preserves rotational symmetry about the origin.

If you solve the Schrödinger equation in this gauge (as detailed in Section 1.2.2 of Tong's notes), you find that the wavefunctions are eigenstates of the angular momentum operator $L_{z}$ . They look like concentric rings, completely different from the strip-like wavefunctions we found in the Landau gauge. However, the energy spectrum is exactly the same: $E_{n} = ℏ ω_{c} (n + 1/2)$ . This is a beautiful demonstration that the physical observables (like the energy levels) are independent of our mathematical gauge choice.

Conclusion

Today, we have analyzed the fundamental quantum mechanical problem of a 2D electron gas in a perpendicular magnetic field. This is a cornerstone of modern condensed matter physics, with direct relevance to the experimental systems where topological effects are observed.

Key Takeaways:

The continuous energy spectrum of a 2DEG collapses into a series of discrete, equally spaced Landau levels with energies $E_{n} = ℏ ω_{c} (n + 1/2)$ .
The spacing between levels is controlled by the cyclotron frequency $ω_{c} = e B / m$ , making it proportional to the magnetic field.
Each Landau level possesses a massive degeneracy, with the number of states per unit area being $n_{B} = B / Φ_{0}$ , where $Φ_{0} = h / e$ is the magnetic flux quantum.
The physical energy spectrum is independent of the chosen gauge, although the form of the wavefunctions depends on it.

Preview of the Next Lesson

We have now established the "container" for the electrons: a ladder of highly degenerate energy levels. In the next lesson, we will start filling this container. We will introduce the crucial concept of the filling factor, $ν$ , which describes how many Landau levels are occupied by electrons. We will see how this filling factor directly leads to the plateaus observed in the Integer Quantum Hall Effect, finally connecting our Landau level picture back to the TKNN formula and the topological Chern number.