Landau Levels and the Integer Quantum Hall Effect

Derivation

Landau Levels and the Integer Quantum Hall Effect

D-260 Home PU-303 Threads fields · matter · symmetry Depends on Electrical Conductivity from the Boltzmann Equation, harmonic-oscillator-ladder-operators

Statement

For a non-relativistic two-dimensional electron gas of areal density \(n_e\) confined to the \(xy\)-plane in a uniform perpendicular magnetic field \(\mathbf{B}=B\hat{\mathbf{z}}\), the single-particle spectrum collapses onto discrete, macroscopically degenerate Landau levels \(E_n=\hbar\omega_c\left(n+\tfrac12\right)\) with cyclotron frequency \(\omega_c=eB/m\); each level carries a degeneracy per unit area \(n_B=eB/h\), so that the filling factor \(\nu=n_e/n_B=n_e h/(eB)\). Combining this counting with the strong-field Hall response fixes the Hall conductance to \(\sigma_{xy}=\nu\,e^2/h\), which locks onto the quantized plateaus \(\sigma_{xy}=i\,e^2/h\) (integer \(i\)) with vanishing longitudinal conductance \(\sigma_{xx}=0\) whenever the chemical potential lies in a gap between filled levels.

Why it matters

The integer quantum Hall effect is the first solid-state phenomenon in which a macroscopic transport coefficient is quantized to metrological precision — \(R_{xy}=h/(ie^2)\) is reproducible to parts in \(10^{9}\), independent of sample geometry, disorder, or material. Since 2019 the SI defines the kilogram, ampere and ohm through fixed values of \(h\) and \(e\), and the von Klitzing constant \(R_K=h/e^2=25\,812.807\ \Omega\) is the practical realization of the ohm.

Conceptually it is the prototype of a topological phase: the plateaus are protected not by a symmetry but by a topological invariant (a Chern number), so the quantization survives arbitrary smooth deformations of the Hamiltonian. Landau quantization also underlies de Haas–van Alphen and Shubnikov–de Haas oscillations, cyclotron resonance, and the fractional quantum Hall effect.

Assumptions

Single-particle, non-interacting electrons.With Coulomb interactions and a partially filled level the ground state can reorganize into incompressible correlated liquids — the fractional quantum Hall effect at \(\nu=p/q\) — which this derivation cannot produce.

Strictly two-dimensional motion.If the confining well admits more than one occupied subband, several magnetic subband ladders overlap and the clean \(n_B=eB/h\) counting per level is spoiled.

Uniform, static, perpendicular field.Field gradients or tilt mix the guiding-centre coordinate with real dynamics and lift the exact \(k_y\)-degeneracy that gives the flat levels.

Parabolic dispersion with effective mass \(m\).For Dirac materials (graphene) the spectrum is \(E_n\propto\operatorname{sgn}(n)\sqrt{|n|B}\) with a level at zero energy, and the plateaus sit at half-integer-shifted values.

Disorder present but weak, producing localized states in the tails.With zero disorder the density of states is a set of delta functions and \(\sigma_{xy}\) jumps discontinuously with no plateau; with too much disorder Landau levels overlap and the gaps — hence the plateaus — close entirely.

Zero temperature, or \(k_BT\ll\hbar\omega_c\).Thermal smearing of order \(\hbar\omega_c\) repopulates adjacent levels, restores a finite \(\sigma_{xx}\), and rounds the plateaus into a classical Hall line.

Derivation

\[ \hat H=\frac{1}{2m}\left(\hat{\mathbf p}+e\mathbf A\right)^2 \]

Minimal coupling for a charge \(-e\) (electron) in a vector potential \(\mathbf A\) with \(\mathbf B=\nabla\times\mathbf A\); spin Zeeman energy is added later as an independent shift. B

\[ \mathbf A=(0,\,Bx,\,0)\quad\Rightarrow\quad \nabla\times\mathbf A=B\hat{\mathbf z} \]

Choice of Landau gauge; gauge freedom does not change the spectrum, only the labelling of states. A

\[ \hat H=\frac{\hat p_x^2}{2m}+\frac{1}{2m}\left(\hat p_y+eB\hat x\right)^2 \]

Insert the gauge. Since \([\hat H,\hat p_y]=0\), \(\hat p_y\) is conserved and we take eigenstates \(\psi(x,y)=e^{ik_y y}\phi(x)\), replacing \(\hat p_y\to\hbar k_y\). B

\[ \hat H_x=\frac{\hat p_x^2}{2m}+\frac{1}{2}m\omega_c^2\left(\hat x-X\right)^2,\quad \omega_c=\frac{eB}{m},\ \ X=-\frac{\hbar k_y}{eB} \]

Complete the square: \(\dfrac{(\hbar k_y+eBx)^2}{2m}=\dfrac{(eB)^2}{2m}\left(x+\dfrac{\hbar k_y}{eB}\right)^2=\dfrac12 m\omega_c^2(x-X)^2\), using \(m\omega_c^2=e^2B^2/m\). This is a 1D harmonic oscillator whose centre \(X\) (the guiding-centre coordinate) is set by \(k_y\). C

\[ E_n=\hbar\omega_c\left(n+\tfrac12\right),\qquad n=0,1,2,\dots \]

Apply the ladder-operator spectrum of the harmonic oscillator (assumed prior result); the shift \(X\) is a translation of the origin and does not affect the eigenvalues. B

\[ k_y=\frac{2\pi j}{L_y},\quad 0\le X\le L_x\ \Rightarrow\ N=\frac{L_y}{2\pi}\,\Delta k_y=\frac{L_y}{2\pi}\cdot\frac{L_x}{\ell_B^2},\quad \ell_B^2=\frac{\hbar}{eB} \]

The energy is independent of \(k_y\): every allowed \(k_y\) gives the same \(E_n\), so the level is degenerate. Impose periodic boundary conditions in \(y\) and require the guiding centre to lie inside the sample, \(|X|=\ell_B^2|k_y|\le L_x\). Counting the allowed \(k_y\) gives \(N\) states per level. C

\[ n_B\equiv\frac{N}{L_xL_y}=\frac{1}{2\pi\ell_B^2}=\frac{eB}{h}=\frac{B}{\Phi_0},\qquad \Phi_0=\frac{h}{e} \]

Divide by the area. Each state occupies one flux quantum \(\Phi_0=h/e\): the degeneracy per area equals the density of flux quanta. Define the filling factor \(\nu=n_e/n_B=n_e h/(eB)\). A

\[ \sigma_{xy}\xrightarrow[\ \omega_c\tau\gg1\ ]{}\frac{n_e e}{B} \]

From the Boltzmann/Drude conductivity tensor in a magnetic field (assumed prior result), the off-diagonal component in the strong-field limit is \(\sigma_{xy}=n_e e/B\), independent of scattering time \(\tau\). B

\[ \sigma_{xy}=\frac{n_e e}{B}=\left(\frac{\nu eB}{h}\right)\frac{e}{B}=\nu\,\frac{e^2}{h} \]

Substitute \(n_e=\nu\,eB/h\) from Step 7. The magnetic field cancels: the Hall conductance depends only on the number of filled Landau levels. A

\[ \rho_{xy}=\frac{\sigma_{xy}}{\sigma_{xx}^2+\sigma_{xy}^2}\xrightarrow[\ \sigma_{xx}=0\ ]{}\frac{1}{\sigma_{xy}}=\frac{h}{\nu e^2},\quad \nu=i\in\mathbb Z \]

Invert the 2D conductivity tensor. When the chemical potential sits in a mobility gap between Landau levels, all current-carrying states are filled and the extended states are exhausted, so \(\sigma_{xx}=0\); disorder-localized states in the tails let \(\nu\) drift over a finite \(B\)-range while \(\sigma_{xy}\) stays pinned at \(i\,e^2/h\), producing the plateaus. C

Result

\[ E_n=\hbar\omega_c\!\left(n+\tfrac12\right),\quad n_B=\frac{eB}{h},\quad \boxed{\ \sigma_{xy}=\nu\,\frac{e^2}{h}\ \xrightarrow{\ \nu=i\ }\ i\,\frac{e^2}{h}\ },\quad R_{xy}=\frac{h}{i e^2} \]

Reading. The kinetic energy of a 2D electron in a field is quantized into equally spaced Landau levels separated by \(\hbar\omega_c\), each holding \(eB/h\) states per unit area — one per flux quantum. When exactly \(i\) levels are full the Hall conductance is an integer multiple of the conductance quantum \(e^2/h\) and the longitudinal conductance vanishes: the system carries dissipationless edge current whose quantization is set only by fundamental constants and the integer \(i\).

Units check. \(\hbar\omega_c=\hbar eB/m\) has units \(\mathrm{(J\,s)(C)(T)/(kg)}=\mathrm{J}\) since \(\mathrm{C\cdot T=C\cdot kg\,A^{-1}s^{-2}=kg\,s^{-1}}\), so \(\mathrm{(J\,s)(kg\,s^{-1})/kg=J}\). \(n_B=eB/h\Rightarrow\mathrm{C\cdot T/(J\,s)=(kg\,s^{-1})/(J\,s)=kg\,J^{-1}s^{-2}=m^{-2}}\), an areal density. \(e^2/h\Rightarrow\mathrm{C^2/(J\,s)=C^2\,s^{-1}J^{-1}=A\,V^{-1}=S}\), a (sheet) conductance. Numerically \(h/e^2=25\,812.807\ \Omega\).

Limiting cases

\(B\to0\): \(\omega_c\to0\) and level spacing collapses; the discrete ladder merges into the continuous 2D free-electron spectrum and \(\sigma_{xy}=n_e e/B\) recovers the classical linear-in-\(1/B\) Hall line.
Very strong field \(\nu<1\): only the lowest Landau level is partly filled; the non-interacting picture is degenerate and Coulomb interactions take over, opening the door to the fractional effect.
High temperature \(k_BT\gg\hbar\omega_c\): many levels are thermally smeared, \(\sigma_{xx}\neq0\), plateaus wash out and only the classical Hall slope remains.
Clean limit (disorder \(\to0\)): localized states vanish, plateaus shrink to points, and \(\sigma_{xy}\) rises in sharp steps exactly at integer \(\nu\).
Zeeman-resolved limit: when \(g\mu_B B\) becomes comparable to \(\hbar\omega_c\), spin-split sublevels resolve and plateaus appear at odd integers too, not just even.

Breaks when

Interactions dominate a partially filled level. At fractional filling the non-interacting degeneracy is lifted by Coulomb energy \(e^2/(4\pi\varepsilon\ell_B)\); the single-particle result gives no gap and predicts a metal, whereas experiment shows incompressible fractional plateaus \(\sigma_{xy}=(p/q)e^2/h\).
Landau levels overlap. If disorder broadening \(\Gamma\) exceeds \(\hbar\omega_c\) (low field or dirty samples, \(\omega_c\tau\lesssim1\)), the mobility gaps close, \(\sigma_{xx}\) never reaches zero, and no quantized plateau forms.
Thermal activation across the gap. For \(k_BT\gtrsim\hbar\omega_c\) the plateau widths shrink as \(\sigma_{xx}\propto e^{-\Delta/2k_BT}\) grows; quantization degrades and finally disappears.
Non-parabolic bands. For massless Dirac fermions \(E_n=\operatorname{sgn}(n)\,v_F\sqrt{2\hbar eB|n|}\); the equal-spacing law and the standard plateau sequence both fail, replaced by \(\sigma_{xy}=\pm4(n+\tfrac12)e^2/h\).

Failure modes

Charge-sign slip: writing \(\hat H=(\hat{\mathbf p}-e\mathbf A)^2/2m\) with \(e>0\) for an electron. The magnitude of \(\omega_c\) is unaffected, but the sign of \(\sigma_{xy}\) and the direction of edge circulation flip.
Counting states twice for spin: using \(n_B=eB/h\) while also inserting a factor of 2 for spin, then again splitting by Zeeman. Fix \(n_B=eB/h\) per spin species and treat spin as a separate label.
Confusing \(\sigma_{xy}\) with \(\rho_{xy}\): asserting \(\rho_{xy}=\nu e^2/h\). In 2D the tensor inverts to \(\rho_{xy}=1/\sigma_{xy}=h/(\nu e^2)\) only because \(\sigma_{xx}=0\) on the plateau; off the plateau both components matter.
Using the free-electron mass in GaAs: the effective mass is \(m^*=0.067\,m_e\), so \(\hbar\omega_c\) is ~15× larger than the free-electron estimate; using \(m_e\) underestimates the gap and the required temperature.
Forgetting the \(\tfrac12\) zero-point energy: writing \(E_n=n\hbar\omega_c\). The lowest level sits at \(\tfrac12\hbar\omega_c\), which matters for the total energy and for the field at which the last level empties.
Assuming plateaus need no disorder: claiming the clean system shows plateaus. Without localized states \(\sigma_{xy}\) is a staircase of zero-width steps; the finite plateau width is a disorder effect.

Discussion

The physical heart of Landau quantization is that a magnetic field converts free 2D translation into cyclotron orbits. Classically an electron circles at \(\omega_c=eB/m\) with a radius set by its energy; quantum mechanically the enclosed flux is quantized, so only discrete orbit energies survive. The magnetic length \(\ell_B=\sqrt{\hbar/eB}\) is the quantum orbit scale — the spread of the lowest-level wavefunction — and \(2\pi\ell_B^2\) is the area threaded by exactly one flux quantum, which is why the degeneracy per area is simply \(eB/h\).

The plateaus are a two-part story. The counting \(\sigma_{xy}=\nu e^2/h\) is exact but by itself gives only isolated points at integer \(\nu\). Disorder does the rest: it splits each Landau level into a narrow band of extended states surrounded by localized states that carry no current. As \(B\) (or density) is swept, the Fermi level moves through localized states without changing the number of filled current-carrying channels, so \(\sigma_{xy}\) is pinned while \(\nu\) drifts — the flat plateau. The remarkable fact, established by Laughlin's gauge argument and by the TKNN formula, is that the localized states which broaden the plateau contribute nothing, and the extended states contribute exactly the missing integer, so quantization is preserved to all orders.

Deeper still, \(\sigma_{xy}\) equals \((e^2/h)\) times a Chern number — the integral of the Berry curvature of the filled bands over the magnetic Brillouin zone (the TKNN invariant). This integer is a topological property of the occupied states and cannot change under any smooth, gap-preserving deformation, which is the true origin of the metrological robustness. By the bulk-boundary correspondence, a Chern number \(i\) forces exactly \(i\) chiral edge modes; the quantized transport is most transparently the ballistic conductance of these one-way edge channels, immune to backscattering because there are no counter-propagating states to scatter into. This is the earliest and cleanest example of the general principle that a bulk topological invariant dictates protected boundary transport.

Common misconceptions. The quantization is not a property of a single perfect orbit — it requires the full many-state counting plus disorder-induced localization to make a finite plateau. \(\sigma_{xx}=0\) does not mean the sample is a perfect conductor in the ordinary sense; it means there is no dissipation in the bulk because current flows only along the edges. And the effect is not fragile: unlike most quantum interference phenomena it strengthens with disorder (up to a point), because disorder is what pins the Fermi level in the localized-state gap.

Worked examples

Magnetic length and cyclotron gap for free-mass electrons at \(B=10\ \mathrm{T}\).

Compute \(\ell_B=\sqrt{\hbar/eB}\) and \(\hbar\omega_c=\hbar eB/m_e\); state the temperature below which levels are resolved, \(k_BT\ll\hbar\omega_c\). A

\[ \ell_B=\sqrt{\frac{1.055\times10^{-34}}{(1.602\times10^{-19})(10)}}=\sqrt{6.59\times10^{-17}}=8.1\times10^{-9}\ \mathrm{m} \]

Substitute \(\hbar=1.055\times10^{-34}\,\mathrm{J\,s}\), \(e=1.602\times10^{-19}\,\mathrm{C}\). A

\[ \hbar\omega_c=\frac{(1.055\times10^{-34})(1.602\times10^{-19})(10)}{9.109\times10^{-31}}=1.86\times10^{-22}\ \mathrm{J}=1.16\ \mathrm{meV} \]

Use \(m_e=9.109\times10^{-31}\,\mathrm{kg}\); convert with \(1\ \mathrm{eV}=1.602\times10^{-19}\ \mathrm J\). A

\[ \ell_B\approx8.1\ \mathrm{nm},\qquad \hbar\omega_c\approx1.16\ \mathrm{meV},\qquad T\ll\frac{\hbar\omega_c}{k_B}\approx13\ \mathrm{K} \]

Reading. At 10 T the lowest-level orbit is ~8 nm across and the Landau gap for free electrons is barely 1 meV, so clean quantization needs liquid-helium temperatures. In GaAs (\(m^*=0.067\,m_e\)) the same field gives \(\hbar\omega_c\approx17\ \mathrm{meV}\), which is why real devices work at accessible temperatures.

Units check. \(\ell_B\): \(\sqrt{\mathrm{J\,s/(C\,T)}}=\sqrt{\mathrm{m^2}}=\mathrm m\). \(\hbar\omega_c\) in J, converted to meV. \(\hbar\omega_c/k_B\) in K with \(k_B=1.381\times10^{-23}\,\mathrm{J/K}\).

Filling factor and Hall resistance for \(n_e=2.4\times10^{15}\ \mathrm{m^{-2}}\) at \(B=5\ \mathrm{T}\).

Compute \(\nu=n_e h/(eB)\); if \(\nu\) rounds to an integer \(i\), read the plateau resistance \(R_{xy}=h/(ie^2)\). B

\[ n_B=\frac{eB}{h}=\frac{(1.602\times10^{-19})(5)}{6.626\times10^{-34}}=1.21\times10^{15}\ \mathrm{m^{-2}} \]

Degeneracy per area at 5 T, using \(h=6.626\times10^{-34}\,\mathrm{J\,s}\). A

\[ \nu=\frac{n_e}{n_B}=\frac{2.4\times10^{15}}{1.21\times10^{15}}=1.99\approx2 \]

Two Landau levels (one spin-split pair) are filled; the chemical potential sits in the gap. A

\[ R_{xy}=\frac{h}{2e^2}=\frac{25\,812.807\ \Omega}{2}=12\,906.4\ \Omega \]

Insert \(i=2\) into \(R_{xy}=h/(ie^2)\) with \(h/e^2=R_K\). A

\[ \nu\approx2,\qquad R_{xy}=\frac{h}{2e^2}\approx1.29\times10^{4}\ \Omega,\qquad \sigma_{xy}=\frac{2e^2}{h}\approx7.75\times10^{-5}\ \mathrm S \]

Reading. At this density and field the system sits on the \(\nu=2\) plateau; the Hall resistance is exactly half the von Klitzing constant, and the longitudinal resistance drops to zero along the plateau.

Units check. \(n_B\) in \(\mathrm{m^{-2}}\), \(\nu\) dimensionless, \(R_{xy}\) in \(\Omega\), \(\sigma_{xy}=1/R_{xy}\) in S.

Problems

Compute the Landau-level degeneracy per unit area \(n_B\) at \(B=8\ \mathrm{T}\).
Solution
\(n_B=eB/h=(1.602\times10^{-19})(8)/(6.626\times10^{-34})=1.2816\times10^{-18}/6.626\times10^{-34}=1.93\times10^{15}\ \mathrm{m^{-2}}\). Equivalently \(1.93\times10^{11}\ \mathrm{cm^{-2}}\), one state per flux quantum \(\Phi_0=h/e=4.14\times10^{-15}\ \mathrm{Wb}\).
Find the cyclotron energy \(\hbar\omega_c\) (in meV) for GaAs with effective mass \(m^*=0.067\,m_e\) at \(B=6\ \mathrm{T}\).
Solution
\(\omega_c=eB/m^*=(1.602\times10^{-19})(6)/(0.067\times9.109\times10^{-31})=9.612\times10^{-19}/6.10\times10^{-32}=1.575\times10^{13}\ \mathrm{rad/s}\). Then \(\hbar\omega_c=(1.055\times10^{-34})(1.575\times10^{13})=1.66\times10^{-21}\ \mathrm J=10.4\ \mathrm{meV}\). Far larger than the free-electron value (0.70 meV) because \(m^*\ll m_e\).
Compute the magnetic length \(\ell_B\) at \(B=1\ \mathrm{T}\) and state how it scales with \(B\).
Solution
\(\ell_B=\sqrt{\hbar/eB}=\sqrt{1.055\times10^{-34}/(1.602\times10^{-19}\cdot1)}=\sqrt{6.59\times10^{-16}}=2.57\times10^{-8}\ \mathrm m=25.7\ \mathrm{nm}\). Since \(\ell_B\propto B^{-1/2}\), raising the field to 10 T shrinks it to \(25.7/\sqrt{10}=8.1\ \mathrm{nm}\), consistent with Worked Example 1.
On the \(\nu=4\) plateau, find \(R_{xy}\) and \(\sigma_{xy}\).
Solution
\(R_{xy}=h/(4e^2)=R_K/4=25\,812.807/4=6\,453.2\ \Omega\). The Hall conductance is \(\sigma_{xy}=4e^2/h=4/R_K=4/25\,812.807=1.55\times10^{-4}\ \mathrm S\), and \(\sigma_{xx}=0\) so that \(\sigma_{xy}=1/R_{xy}\) holds exactly on the plateau.
For a 2DEG with \(n_e=1.0\times10^{15}\ \mathrm{m^{-2}}\), find the field \(B\) at which \(\nu=1\), and estimate the temperature (free-electron mass) below which this last plateau is thermally resolved.
Solution
\(\nu=n_e h/(eB)=1\Rightarrow B=n_e h/e=(1.0\times10^{15})(6.626\times10^{-34})/(1.602\times10^{-19})=6.626\times10^{-19}/1.602\times10^{-19}=4.14\ \mathrm T\). At this field \(\hbar\omega_c=\hbar eB/m_e=(1.055\times10^{-34})(1.602\times10^{-19})(4.14)/(9.109\times10^{-31})=7.68\times10^{-23}\ \mathrm J=0.48\ \mathrm{meV}\), so \(T\ll\hbar\omega_c/k_B=0.48\times10^{-3}\times1.602\times10^{-19}/1.381\times10^{-23}\approx5.6\ \mathrm K\). In GaAs the ~15× larger gap relaxes this to tens of kelvin.