Landau Levels in Solids

Landau levels in a solid are the discrete magnetic-field eigenvalues of a low-energy band Hamiltonian. Their spacing reflects the band dispersion. A locally parabolic, nondegenerate band does inherit the canonical orbital spectrum after its curvature tensor is coupled to a uniform magnetic field. The spacing then depends on field direction through a cyclotron mass, while motion parallel to the field remains dispersive. Real spin, band orbital moment, valleys, confinement, and multiband mixing can split or reorganize that ladder.

This page uses SI units. The carrier charge $q$ is signed, with $q=-e$ for an electron and $e>0$; $B=|\mathbf B|$ is nonnegative. The orbital spacing depends on $|q|B$, while charge signs still matter in commutators, wave-function circulation, Hall conventions, and magnetic-moment terms. An electron valence-band maximum should be reorganized as a positive-energy hole band before a positive hole mass and $q=+e$ are used.

Required background. Band Theory Overview supplies the meaning of a material band, Effective Mass supplies local curvature tensors and mass distinctions, and Landau Levels supplies the ideal orbital spectrum used here.

Helpful background. Semiclassical Dynamics of Bloch Electrons separates weak-field trajectories from resolved quantization; Landau Levels Revisited supplies the gauge-geometric structure; Quantum Oscillations connects the spectrum to measurements; and Graphene and Dirac Materials develops the material realization of the Dirac ladder.

A Local Band Hamiltonian in a Uniform Field

Choose a smooth band extremum at $\mathbf k_0$ and write $\boldsymbol\kappa=\mathbf k-\mathbf k_0$. For an electron minimum, a controlled quadratic expansion is

$$\varepsilon(\mathbf k_0+\boldsymbol\kappa) = \varepsilon_0 + \frac{\hbar^2}{2} \boldsymbol\kappa^{\mathsf T} \boldsymbol\alpha \boldsymbol\kappa + O(\kappa^3), \qquad \boldsymbol\alpha=\mathsf M^{-1}>0.$$

The inverse-mass tensor $\boldsymbol\alpha$ is the Hessian of the chosen dispersion divided by $\hbar^2$. It may come from an empirical model, a downfolded Hamiltonian, or a coherent quasiparticle pole. Those inputs are not interchangeable: quantizing a Kohn–Sham eigenvalue does not by itself prove that the resulting spacing is the measured quasiparticle spacing.

For a uniform field, the envelope-function substitution is

$$\hbar\boldsymbol\kappa \longrightarrow \boldsymbol\pi = -i\hbar\boldsymbol\nabla-q\mathbf A, \qquad \boldsymbol\nabla\times\mathbf A=\mathbf B,$$

so that

$$H_{\mathrm{env}} = \varepsilon_0 + \frac12 \boldsymbol\pi^{\mathsf T} \boldsymbol\alpha \boldsymbol\pi, \qquad [\pi_i,\pi_j] = i\hbar q\epsilon_{ijk}B_k.$$

Because $\boldsymbol\alpha$ is constant in this local model, no operator-ordering ambiguity occurs. A position-dependent mass, abrupt interface, or matrix-valued multiband Hamiltonian needs its own Hermitian ordering and boundary conditions. Likewise, replacing $\mathbf k$ by $\boldsymbol\pi/\hbar$ is an envelope approximation, not an exact rule for an arbitrary lattice Hamiltonian at arbitrary flux.

The quadratic Hamiltonian is enough to determine the level energies. It is not enough to determine a transport trace, plateau, linewidth, or many-body ground state. Those require occupations and chemical potential, disorder and interactions, sample geometry, and a measurement model.

Anisotropic Mass and the Schur Reduction

Let $\hat{\mathbf b}=\mathbf B/B$ and choose transverse unit vectors $\mathbf e_1,\mathbf e_2$. In this field-adapted basis, block the inverse-mass tensor as

$$\boldsymbol\alpha = \begin{pmatrix} \mathsf A & \mathbf a \\ \mathbf a^{\mathsf T} & \alpha_{\parallel} \end{pmatrix},$$

where $\mathsf A$ is the $2\times2$ transverse block. The momentum $\pi_{\parallel}=\hat{\mathbf b}\cdot\boldsymbol\pi$ commutes with both transverse components in a uniform field. Completing the square gives

$$H_{\mathrm{env}}-\varepsilon_0 ={} \frac12 \left( \boldsymbol\pi_{\perp} + \mathsf A^{-1}\mathbf a\,\pi_{\parallel} \right)^{\mathsf T} \mathsf A \left( \boldsymbol\pi_{\perp} + \mathsf A^{-1}\mathbf a\,\pi_{\parallel} \right) + \frac12 \left( \alpha_{\parallel} - \mathbf a^{\mathsf T} \mathsf A^{-1}\mathbf a \right) \pi_{\parallel}^2.$$

The second coefficient is the Schur complement of $\mathsf A$. The shifted transverse momenta retain the same commutator, so the canonical result applies without repeating its oscillator construction. Define

$$m_c(\hat{\mathbf b}) = \frac{1}{\sqrt{\det\mathsf A}}, \qquad m_{\parallel}(\hat{\mathbf b}) = \frac{1}{ \alpha_{\parallel} - \mathbf a^{\mathsf T} \mathsf A^{-1}\mathbf a }.$$

The spectrum is

$$E_{n,k_{\parallel}} = \varepsilon_0 + \hbar\omega_c \left(n+\frac12\right) + \frac{\hbar^2k_{\parallel}^2}{2m_{\parallel}}, \qquad \omega_c = \frac{|q|B}{m_c},$$

before internal splittings are added. In coordinate-free form,

$$m_c(\hat{\mathbf b}) = \sqrt{ \frac{\det\mathsf M} {\hat{\mathbf b}^{\mathsf T} \mathsf M \hat{\mathbf b}} }, \qquad m_{\parallel}(\hat{\mathbf b}) = \hat{\mathbf b}^{\mathsf T} \mathsf M \hat{\mathbf b}.$$

For principal masses $m_1,m_2,m_3$ and $\mathbf B$ along axis 3, these reduce to $m_c=\sqrt{m_1m_2}$ and $m_{\parallel}=m_3$. The cyclotron mass is therefore not generally the mass along the field and not generally any one eigenvalue of $\mathsf M$.

Interpreting the Crystal Spectrum

In three dimensions, each integer $n$ labels a one-dimensional subband dispersing with $k_{\parallel}$. For a strictly two-dimensional layer with a perpendicular field, confinement removes the longitudinal continuum and leaves discrete orbital levels. For a tilted field, the ideal zero-thickness orbital problem uses $B_\perp=|\mathbf B\cdot\hat{\mathbf n}|$, where $\hat{\mathbf n}$ is the layer normal; Zeeman coupling still depends on the full field through its tensor. A finite layer thickness can produce additional orbital effects from the parallel component.

The areal degeneracy is $|q|B/h$ per resolved internal branch in a plane perpendicular to the field, or $|q|B_\perp/h$ per physical layer area in the strict two-dimensional limit. Equivalent valleys or spin branches multiply this only when their degeneracy survives the terms included in the Hamiltonian.

The local coordinate $\boldsymbol\kappa$ is measured from a band extremum that may lie far from the Brillouin-zone center. Quantization takes place around that valley. The magnetic length

$$\ell_B = \sqrt{\frac{\hbar}{|q|B}}$$

sets the envelope scale, but it is not the lattice spacing and does not erase the valley label. A continuum envelope is credible only when the magnetic flux through a representative primitive cell is small compared with the flux quantum $h/|q|$ and when the occupied oscillator states remain inside the momentum and energy window over which the local Hamiltonian was fitted.

For a valence-band maximum, negative electron curvature must not be inserted into a positive oscillator frequency. Reorganize the missing-electron excitation using Holes, then quantize the positive hole-energy curvature with $q=+e$. The orbital energy spacing is unchanged by the sign of $q$, while circulation and response signs are not.

The field direction can also select different extremal orbits on a nonparabolic Fermi surface. The local tensor result and the orbit-defined cyclotron mass agree for an ellipsoidal parabolic band. Beyond that limit, use the orbit-area derivative derived in Fermi Surface.

Spin, Orbital Moment, and Valley Matrices

A material ladder is usually a spectrum of a small internal matrix, not a single repeated scalar level. In a spin, orbital, or valley subspace one may organize the effective Hamiltonian as

$$H_{\mathrm{eff}} = H_{\mathrm{env}}\mathsf I + H_Z + H_m + H_v + H_{\mathrm{mix}},$$

with all matrices expressed in the same declared basis. The terms have distinct physical roles:

Term	Representative form	Physical effect	Relation to the other terms
envelope orbit	$\frac12\pi_i\alpha_{ij}\pi_j$	$m_c$, $m_{\parallel}$, orbital ladder	Was the fitted dispersion already field dependent?
spin Zeeman	$H_Z=\frac{\mu_B}{2}\sigma_i g_{ij}B_j$	spin splitting and avoided crossings	Is $g$ a spin-only parameter or a fitted total splitting?
band orbital moment	$H_m=-\mathbf m^{\mathrm{orb}}\!\cdot\!\mathbf B$	band-, valley-, and momentum-dependent shifts	Is this remote-band contribution already inside $g$ or the multiband model?
valley or layer sector	$[H_v]_{\nu\nu'}$	multiplicity, splitting, intervalley mixing	Was degeneracy already removed by strain, interface, or confinement?
residual mixing	$H_{\mathrm{mix}}(\boldsymbol\pi,\mathbf B)$	anticrossings and nonuniform spacing	Does a scalar one-band projection still exist?

The Bohr magneton is $\mu_B=e\hbar/(2m_e)$ and contains the bare mass. The tensor $g_{ij}$ is an effective parameter with a basis and sign convention; it need not equal 2. If $\mathbf B=B\hat{\mathbf z}$ and $H_Z=(g^*\mu_BB/2)\sigma_z$, the two eigenvalues differ by $|g^*|\mu_BB$. Calling that quantity “the Zeeman energy” while another source uses the half-splitting is a common factor-of-two error.

The orbital moment of a Bloch band arises from virtual interband structure. The same remote bands also renormalize effective masses and effective $g$ factors. Consequently, a $g^*$ obtained by fitting the total observed splitting must not be combined with an additional orbital shift unless the parameterization explicitly separates them. A full multiband $k\cdot p$ Hamiltonian often generates both effects on diagonalization; adding scalar one-band corrections afterward can count the same coupling twice.

Valleys related by time reversal or crystal symmetry may be degenerate at zero field yet carry opposite orbital moments. Strain, interfaces, finite thickness, or a field orientation can reduce that multiplicity. If every retained internal term is block diagonal in Landau index, construct and diagonalize the internal matrix separately at each $n$ and $k_{\parallel}$. A momentum-dependent term containing ladder operators can instead couple $n$ to $n\mathbin{\pm}1$ or more distant indices; then diagonalize the coupled Hamiltonian in a truncated basis $\{|n\rangle\otimes|a\rangle\}$ of orbital and internal states, and increase the cutoff until the eigenvalues of interest converge. In either case, count the resulting eigenvalues—not independent “spin times valley” factors appended by habit.

Dirac and Kane-Band Contrasts

The parabolic ladder is not universal. For a two-dimensional gapless Dirac Hamiltonian,

$$H_D = v \left( \pi_x\sigma_x + \pi_y\sigma_y \right),$$

the levels are

$$E_{0}=0, \qquad E_{n,\lambda} = \lambda v \sqrt{2n\hbar|q|B}, \qquad n\ge1, \quad \lambda=\pm1.$$

The $\sqrt{B}$ spacing, positive and negative branches, and zero mode follow from the two-component Hamiltonian. For this particular chirality and a perpendicular signed field $B_z$, adding $+\Delta\sigma_z$ gives $E_0=\operatorname{sgn}(qB_z)\Delta$ and $E_{n,\lambda}=\lambda\sqrt{\Delta^2+2n\hbar|q|Bv^2}$ for $n\ge1$ in the minimal model. Reversing the Dirac chirality reverses the zero-mode sign; the exercise below derives it. Graphene and Dirac Materials explains how the two valleys and real spin appear in a material. Fitting a constant parabolic mass across the Dirac point destroys precisely the structure one is trying to infer.

A compact nonparabolic semiconductor reduction is the Kane-type relation

$$E(1+\alpha_K E) = \frac{\hbar^2k^2}{2m_b}.$$

A zero-field dispersion alone does not uniquely determine its magnetic Hamiltonian: interband orbital terms are additional information. If one specifically assumes the scalar operator model $H(1+\alpha_KH)=\boldsymbol\pi^2/(2m_b)$, define

$$X_{n,k_{\parallel}} = \hbar\frac{|q|B}{m_b} \left(n+\frac12\right) + \frac{\hbar^2k_{\parallel}^2}{2m_b},$$

and solve

$$E_{n,k_{\parallel}} = \frac{ \sqrt{1+4\alpha_KX_{n,k_{\parallel}}}-1 }{2\alpha_K}.$$

The spacing now decreases with energy when $\alpha_K>0$. This scalar formula is a diagnostic, not a substitute for diagonalizing a coupled Kane Hamiltonian when spin–orbit mixing, band inversion, or strong interband coupling is important. In a matrix model, the field couples the components before the eigenvalues are found.

When the levels remain reliable and visible

Two questions must be answered separately. Model control asks whether the chosen Hamiltonian remains a faithful low-energy reduction in the applied field. Observability asks whether its discrete levels can be resolved in a specified measurement. A mathematically sharp ladder can be experimentally invisible, and a visible oscillation can occur near the edge of a questionable one-band model.

Kind	Practical diagnostic	What a pass supports	If it fails
model: local curvature	all occupied $E_n-\varepsilon_0\ll E_{\mathrm{np}}$ and $\kappa_n\sim\sqrt{2n+1}/\ell_B\ll\kappa_{\mathrm{quad}}$	constant mass tensor over the used energy and momentum patch	refit energy dependence or use a multiband Hamiltonian
model: band isolation	$\hbar\omega_c$, moment shifts, and mixing matrix elements $\ll\Delta_{\mathrm{sep}}$	projection to one band or declared multiplet	retain coupled nearby bands
model: lattice continuum	$\lvert q\rvert BA_{\mathrm{cell}}/h\ll1$ and $\ell_B\gg a_{\mathrm{lat}}$	envelope quantization rather than magnetic minibands	use a magnetic-Bloch or Magnetic Translations treatment
model: field texture	$R_n/L_B\ll1$ for every used $n$, with $R_n\simeq\sqrt{2n+1}\,\ell_B$ in the isotropic case	locally uniform-field levels over each retained orbit	solve the inhomogeneous problem
model: orbit integrity	$P_{\mathrm{MB}}\sim\exp(-B_0/B)\ll1$ at each relevant avoided crossing	a single closed-orbit label	include tunneling networks and coupled orbits
observable: orbital resolution	nearest relevant spacing $\Delta_{\mathrm{LL}}\gtrsim\max(k_{\mathrm B}T,\Gamma_q)$	the two branches being compared may be resolved	lower $T$, improve quantum lifetime, or report unresolved quantization
observable: spin resolution	$\lvert g^*\rvert\mu_BB\gtrsim\max(k_{\mathrm B}T,\Gamma_q)$	separate spin branches may be visible	do not infer exact degeneracy from unresolved peaks
observable: valley resolution	valley splitting $\Delta_v\gtrsim\max(k_{\mathrm B}T,\Gamma_q)$	distinct valley branches may be assigned	treat valleys as unresolved and state multiplicity
observable: measurement coupling	predicted level crossings modulate the measured channel above its noise and background uncertainty	a spectrum-to-data comparison	change probe or weaken the inference

Here $E_{\mathrm{np}}$ and $\Delta_{\mathrm{sep}}$ must be extracted from the same band description used to define the mass, while $\kappa_{\mathrm{quad}}$ is the nearest wave-vector scale at which the quadratic fit fails. The estimate for $\kappa_n$ is an isotropic order-of-magnitude diagnostic; a strongly anisotropic valley should be checked in mass-scaled coordinates. Likewise, $L_B$ is the field-variation length, and local uniformity must hold over the largest retained cyclotron orbit, not merely over $\ell_B$. For anisotropic valleys, compare each directional orbit extent with the corresponding field-variation scale. A localized packet additionally needs its full spatial extent to fit inside that locally uniform region. The breakdown field $B_0$ depends on the local gap, velocities, orbit geometry, and convention. In the displayed convention, stronger $B$ generally increases transfer because $P_{\mathrm{MB}}$ approaches unity.

We use a Lorentzian half width at half maximum $\Gamma_q=\hbar/(2\tau_q)$, where $\tau_q$ is the single-particle quantum lifetime. It need not equal the transport lifetime $\tau_{\mathrm{tr}}$ inferred from mobility: small-angle scattering can strongly broaden levels while relaxing little current. These comparisons are estimates rather than universal resolution thresholds. A Lorentzian has full width $2\Gamma_q$; a probe broadened by the Fermi derivative has thermal full width $4\operatorname{arcosh}(\sqrt2)k_{\mathrm B}T\simeq3.53k_{\mathrm B}T$. Matrix elements, line shape and instrumental resolution also matter. An unresolved splitting does not restore an exact degeneracy. The Coulomb scale

$$E_C = \frac{e^2}{4\pi\epsilon_0\epsilon_r\ell_B}$$

uses the absolute permittivity $\epsilon=\epsilon_0\epsilon_r$. If this scale is comparable to the retained orbital, spin, or valley splittings, exchange and correlation can reorganize the spectrum; an unqualified one-particle interpretation then fails even if the band-projection tests pass. Static screening, field-induced order, and chemical-potential motion can impose further limits, so $\epsilon_r$ is an environmental or material input rather than a universal constant.

Worked Example: Six Silicon Valleys at 10 T

Treat bulk silicon’s six conduction valleys as ideal ellipsoids. The multivalley anisotropy was established by Dresselhaus, Kip, and Kittel. For the numerical example, choose the illustrative model parameters

$$m_l=0.916m_e, \qquad m_t=0.190m_e,$$

where $m_l$ is along a valley axis and $m_t$ is transverse. These chosen values are not a transcription of that historical experiment’s fitted masses. Take $\mathbf B=10\,\mathrm T\,\hat{\mathbf z}$, ignore strain and intervalley coupling, and use a scalar $g^*=2.00$ only for an illustrative spin estimate.

For the two $\pm z$ valleys, the field lies along the longitudinal axis, so

$$m_c^{(\pm z)}=m_t=0.190m_e, \qquad m_{\parallel}^{(\pm z)}=m_l=0.916m_e.$$

For the four $\pm x,\pm y$ valleys, the transverse orbit samples one longitudinal and one transverse mass:

$$m_c^{(\perp z)} = \sqrt{m_lm_t} = 0.417m_e, \qquad m_{\parallel}^{(\perp z)}=m_t=0.190m_e.$$

Using $\hbar e/m_e=0.1158\,\mathrm{meV/T}$ gives

$$\begin{aligned} \hbar\omega_c^{(\pm z)} &= \frac{1.158\,\mathrm{meV}}{0.190} = 6.09\,\mathrm{meV}, \\ \hbar\omega_c^{(\perp z)} &= \frac{1.158\,\mathrm{meV}}{0.417} = 2.78\,\mathrm{meV}. \end{aligned}$$

Thus the orbital zero-point terms differ by

$$\frac12 \left( 6.09-2.78 \right)\,\mathrm{meV} = 1.66\,\mathrm{meV}.$$

The field therefore reorganizes the valley energies even before an explicit valley moment is added: two valleys and four valleys acquire different orbital ladders. This is an orbital-mass effect, not automatically an intrinsic zero-field valley splitting.

Three checks keep the number meaningful. First, $\ell_B=25.66\,\mathrm{nm}/\sqrt{10}=8.12\,\mathrm{nm}$, much larger than the approximately $0.543\,\mathrm{nm}$ silicon lattice parameter certified in NIST SRM 640d. Second, using the conventional cubic-cell face $a^2$ as a representative microscopic area scale gives $Ba^2/(h/e)\simeq7.1\times10^{-4}$, so the lattice-flux test is favorable; this is not a claim that $a^2$ is a primitive-cell area. Third, at $T=4\,\mathrm K$, $k_{\mathrm B}T\simeq0.345\,\mathrm{meV}$; if the Lorentzian quantum half width were $\Gamma_q=0.50\,\mathrm{meV}$, both orbital spacings would exceed these broadening scales. Actual peak resolution still depends on the probe and line shape.

The illustrative spin splitting is

$$|g^*|\mu_BB = 2(0.05788\,\mathrm{meV/T})(10\,\mathrm T) = 1.16\,\mathrm{meV}.$$

The spin splitting is smaller than either orbital spacing and close to both the disorder full width $1.00\,\mathrm{meV}$ and the Fermi-derivative thermal full width $1.22\,\mathrm{meV}$. Separate spin peaks therefore need a more careful line-shape calculation. The mass model also needs its own nonparabolicity and interband-mixing checks; a large spacing alone cannot establish the accuracy of that model.

The spin-averaged silicon model at $10\,\mathrm T$. Both panels use the same band-edge energy and axes. The two valleys parallel to the field have larger orbital spacing but flatter longitudinal dispersion. Each plotted curve represents two or four equivalent valleys, respectively; real spin is omitted. No additional valley-splitting parameter is needed for the two sets to acquire different zero-point energies.

The next step depends on the experiment. Quantum Oscillations derives the inverse-field period and thermal amplitude: they constrain an extremal orbit area and its cyclotron mass. Integer Quantum Hall Effect explains why occupations, localization and edges are needed to turn a ladder into a Hall plateau. Neither result follows from the mass tensor alone.

Exercises

1. A tilted field and the longitudinal mass

Take $\mathsf M=m_0\operatorname{diag}(1,2,4)$ and $\hat{\mathbf b}=(1,0,1)/\sqrt2$. Find both masses. Why is $m_{\parallel}$ not $1/(\hat{\mathbf b}^{\mathsf T}\mathsf M^{-1}\hat{\mathbf b})$?

Solution

The tensor formulas give

$$m_{\parallel}=\frac52m_0,\qquad m_c=\sqrt{\frac{8}{5/2}}\,m_0=\frac4{\sqrt5}m_0.$$

Choose $\mathbf e_1=(1,0,-1)/\sqrt2$ and $\mathbf e_2=(0,1,0)$. Then

$$\mathsf A=\frac1{m_0}\begin{pmatrix}5/8&0\\0&1/2\end{pmatrix},\quad \mathbf a=\frac1{m_0}\begin{pmatrix}3/8\\0\end{pmatrix},\quad \alpha_{\parallel}=\frac5{8m_0}.$$

Completing the square yields $\alpha_{\parallel}-\mathbf a^{\mathsf T}\mathsf A^{-1}\mathbf a=2/(5m_0)$. The naive value $1/\alpha_{\parallel}=8m_0/5$ fixes transverse momentum at zero; the actual subband adjusts its transverse oscillator center when longitudinal momentum changes. That adjustment lowers the longitudinal kinetic coefficient.

2. The massive Dirac zero mode

For $H=v(\pi_x\sigma_x+\pi_y\sigma_y)+\Delta\sigma_z$, derive the zero-mode energy for both signs of $qB_z$. The two-component index here is a band or sublattice index, not real spin.

Solution

For $qB_z>0$, set $a=(\pi_x+i\pi_y)/\sqrt{2\hbar qB_z}$. The commutator gives $[a,a^\dagger]=1$, and

$$H=\begin{pmatrix}\Delta&c_Ba^\dagger\\c_Ba&-\Delta\end{pmatrix}, \qquad c_B=v\sqrt{2\hbar|qB_z|}.$$

The state $(|0\rangle,0)^{\mathsf T}$ has energy $+\Delta$. For $qB_z<0$, the upper-right entry is $c_Ba$ instead, and $(0,|0\rangle)^{\mathsf T}$ has energy $-\Delta$. Thus $E_0=\operatorname{sgn}(qB_z)\Delta$. For $n\ge1$, each paired oscillator subspace has energies $\pm\sqrt{\Delta^2+c_B^2n}$. A valley with reversed chirality interchanges the sublattice assignment and the zero-mode sign.

3. Nonparabolic spacing

In the scalar Kane model at $k_{\parallel}=0$, expand $E_n$ through first order in $\alpha_KX_n$ and show how adjacent spacings change with $n$.

Solution

Using $\sqrt{1+4\alpha_KX}=1+2\alpha_KX-2\alpha_K^2X^2+\cdots$ gives

$$E_n = X_n-\alpha_KX_n^2+O(\alpha_K^2X_n^3), \qquad X_n = \hbar\omega_{c0} \left(n+\frac12\right).$$

Therefore

$$E_{n+1}-E_n = \hbar\omega_{c0} - 2\alpha_K (n+1) (\hbar\omega_{c0})^2 + O(\alpha_K^2).$$

For $\alpha_K>0$, the ladder compresses with increasing $n$.

4. Accuracy versus resolution

At $B=12\,\mathrm T$, a band has $m_c=0.25m_e$, $E_{\mathrm{np}}=18\,\mathrm{meV}$, $\Delta_{\mathrm{sep}}=40\,\mathrm{meV}$, $T=8\,\mathrm K$, and $\Gamma_q=1.2\,\mathrm{meV}$. Compare the lowest three orbital levels $n=0,1,2$ with the broadening and band-curvature scales.

Solution

The spacing is

$$\hbar\omega_c = \frac{0.1158\,\mathrm{meV/T}\times12\,\mathrm T}{0.25} = 5.56\,\mathrm{meV}.$$

Also $k_{\mathrm B}T\simeq0.689\,\mathrm{meV}$, so the nearest spacing $\Delta_{\mathrm{LL}}=\hbar\omega_c$ exceeds $\max(k_{\mathrm B}T,\Gamma_q)$ and adjacent levels may be resolved. Their orbital energies are $2.78$, $8.34$, and $13.9\,\mathrm{meV}$ above the edge. All are below $E_{\mathrm{np}}$ and $\Delta_{\mathrm{sep}}$, but $n=2$ is not parametrically below $18\,\mathrm{meV}$. Resolution passes more convincingly than quadratic-band control; the highest level needs a nonparabolicity check rather than an unqualified parabolic fit.

Band isolation cannot be certified from $\Delta_{\mathrm{sep}}$ alone because the problem supplies no off-diagonal field-induced mixing matrix elements. It also supplies neither a microscopic cell area nor a magnetic-breakdown scale $B_0$, so lattice-flux and orbit-integrity tests cannot be assessed from these inputs.

References

• Ando, T., Fowler, A. B., and Stern, F., “Electronic properties of two-dimensional systems,” Reviews of Modern Physics 54, 437–672 (1982), doi:10.1103/RevModPhys.54.437 — disorder broadening, quantum lifetimes, and magnetotransport in two-dimensional electron systems.
• Black, D. R., Windover, D., Henins, A., Gil, D. L., Filliben, J. J., and Cline, J. P., “Certification of NIST Standard Reference Material 640d,” Powder Diffraction 25, 187–190 (2010), doi:10.1154/1.3409482 — certified silicon lattice parameter and uncertainty.
• Dresselhaus, G., Kip, A. F., and Kittel, C., “Cyclotron resonance of electrons and holes in silicon and germanium crystals,” Physical Review 98, 368–384 (1955), doi:10.1103/PhysRev.98.368 — anisotropic masses and multivalley cyclotron resonance.
• Goerbig, M. O., “Electronic properties of graphene in a strong magnetic field,” Reviews of Modern Physics 83, 1193–1243 (2011), doi:10.1103/RevModPhys.83.1193 — Dirac Landau levels and graphene.
• Kane, E. O., “Band structure of indium antimonide,” Journal of Physics and Chemistry of Solids 1, 249–261 (1957), doi:10.1016/0022-3697(57)90013-6 — multiband nonparabolic semiconductor structure.
• Kohn, W., “Theory of Bloch electrons in a magnetic field: the effective Hamiltonian,” Physical Review 115, 1460–1478 (1959), doi:10.1103/PhysRev.115.1460 — effective band Hamiltonians in magnetic fields.
• Luttinger, J. M., “Quantum theory of cyclotron resonance in semiconductors: general theory,” Physical Review 102, 1030–1041 (1956), doi:10.1103/PhysRev.102.1030 — multiband cyclotron resonance.
• Winkler, R., Spin–Orbit Coupling Effects in Two-Dimensional Electron and Hole Systems, Springer, 2003, doi:10.1007/b13586 — matrix effective Hamiltonians, Zeeman terms, and spin–orbit coupling.
• Yu, P. Y., and Cardona, M., Fundamentals of Semiconductors: Physics and Materials Properties, 4th ed., Springer, 2010, doi:10.1007/978-3-642-00710-1 — semiconductor parameters, multivalley bands, and magneto-optical response.