Class of distorted Landau levels and Hall phases in a two-dimensional electron gas subject to an inhomogeneous magnetic field

An analytic closed form solution is derived for the bound states of a two-dimensional electron gas subject to a static, inhomogeneous ($1/r$ in plane decaying) magnetic field, including the Zeeman interaction. The solution provides access to many-body properties of a two-dimensional, noninteracting, electron gas in the thermodynamic limit. Radially distorted Landau levels can be identified as well as magnetic field induced density and current oscillations close to the magnetic impurity. These radially localized oscillations depend strongly on the coupling of the spin to the magnetic field, which gives rise to nontrivial spin currents. Moreover, the Zeeman interaction introduces a unique flat band, i.e., infinitely degenerate energy level in the ground state, assuming a spin $g_s$-factor of two. Surprisingly, the charge and current densities can be computed analytically for this fully filled flat band in the thermodynamic limit. Numerical calculations show that the total magnetic response of the electron gas remains diamagnetic (similar to Landau levels) independent of the Fermi energy. However, the contribution of certain, infinitely degenerate energy levels may become paramagnetic. Furthermore, numerical computations of the Hall conductivity reveal asymptotic properties of the electron gas, which are driven by the anisotropy of the vector potential instead of the magnetic field, i.e., become independent of spin. Eventually, the distorted Landau levels give rise to negative and positive Hall conductivity phases, with sharp transitions at specific Fermi energies. Overall, our work merges “impurity” with Landau-level physics, which provides novel physical insights, not only locally, but also in the asymptotic limit. This paves the way for a large number of future theoretical as well as experimental investigations, e.g., to include electronic correlations and to investigate two-dimensional systems such as graphene or transition metal dichalcogenides under the influence of inhomogeneous magnetic fields.

Figure 1

Radially resolved ${\langle r\rangle }_{n,l,s}$ energy eigenvalues $E_{n,l,s}$ of the electrons for a $A_{\varphi }/r$-decaying magnetic field with $A_{\varphi }=-0.186\mu \mathrm{Tm}$. The black doted horizontal lines indicate the boundaries of allowed energy eigenvalues for our solution. Notice, that higher-lying energies exist for $l+s\le 0$, but the solutions remain unknown. In (a), $B$-field aligned spin configuration are marked with $\downarrow$ and antialigned electrons with $\uparrow$. Small horizontal lines correspond to the radial uncertainties $\mathrm{\Delta }r=\sqrt{\langle r^2\rangle -{\langle r\rangle }^2}$. Overall, it is clearly shown that the presence of the inhomogeneous magnetic field increases the radial electron localization close to the origin in combination with a coarse graining of the discrete energy levels. Further away from the origin the denseness of the energy spectrum becomes apparent. The lowest lying ($E=0$), infinitely degenerate, electronic states can only emerge if the Zeeman interaction between the magnetic field and the spin is considered. In (b), the dipole allowed transitions are indicated by green lines for $\mathrm{\Delta }n=0$ and purple lines for $\mathrm{\Delta }n=1$. For illustrative purpose, we restrict our visualization to the spin-$\uparrow$ states (equivalent results hold for the spin-$\downarrow$ states). The first few distorted Landau levels (purple) are labeled by $\nu \ge 0$ at their respective highest energies. Notice the strong curvature for $\nu \ge 1$. Similarly, distorted Landau levels could also be identified for spin-down electrons starting at $\nu =1$.

Figure 2

From top to bottom: radially resolved charge, current and magnetization densities for $A_{\varphi }=-0.186\mu \mathrm{Tm}$. Notice that we explicitly account for the negative electronic charges, i.e., the more negative the charge density becomes, the more electrons accumulate locally. Thin continuous lines correspond to the analytic solution of the fully filled, infinitely degenerate lowest band at $E_F=0^{+}$. Solid lines, made of discrete triangles, correspond to the numerical solution for the fully filled flat band at $E_F=E_{2,1,1/2}=1.96$ meV. The later case nicely exemplifies the $B$-field induced Friedel oscillations for the charge density around the origin. They are accompanied by para- $j^{\mathrm{para}}$ and diamagnetic $j^{\mathrm{dia}}$, as well as magnetization current $j^{\mathrm{s}}$ oscillations, which result in a total current $j^{\mathrm{tot}}$ oscillating around the origin. Interestingly, the corresponding total magnetization of this band indicates a mostly paramagnetic response to the applied magnetic field, whereas the lowest band does not respond at all, i.e., $j_0^{\mathrm{tot}}=m_0^{\mathrm{tot}}=0$, thanks to the exact cancellation of the orbital and spin contributions for $g_s=2$.

Figure 3

On the left: Radial density distributions reveal the quantization of the energy continuum around the origin, due to the applied inhomogeneous magnetic field with $A_{\varphi }=-0.186\mu \mathrm{Tm}$. On the right: The energy and density structure of the emergent levels is visualized based on the number of electrons $N_{m}ax$ contained within each level up to a radial distance of $r_{\max }=50$ nm. The horizontal yellow lines indicate the standard Landau-levels for a setup with identical magnetic flux, measured through the circular surface area limited by $r_{\max }.$ Dashed horizontal lines identify the most (red) and second most (black) prominent levels, which are given by $\nu =1$ and $\nu =2$. Notice that the observed DOS discretization seems to be a local effect, which vanishes for larger radii $r_{\max }\gg 50$ nm, where the discrete energy spectrum becomes denser (i.e., almost continuous).

Figure 4

Angularly resolved density distribution for $A_{\varphi }=-0.186\mu \mathrm{Tm}$. The color-coding of each row is fixed to ensure horizontal comparability, whereas the displayed color-bars extend over the entire value range, which accounts for the substantial inhomogeneity at the origin. Left column shows the analytic solution of the lowest band, orbital $j^{\mathrm{orb}}$ and magnetization $j^{\mathrm{s}}$ currents cancel exactly and lead to a zero magnetic response $m^{\mathrm{tot}}=0$ to the applied $B$ field. The right column corresponds to the total densities of the system with occupied bands up to $E_F=E_{2,1,1/2}$. It demonstrates that the magnetic induced oscillatory behavior persists for the entire system and is not only restricted to specific bands (e.g., at $E_F=E_{2,1,1/2}$ as displayed in the middle column). However, we find that the joint magnetic response of all filled levels always remains diamagnetic (e.g., $m^{\mathrm{tot}}\ge 0\forall r$ at the bottom of the right column), whereas selected degenerated energy levels may respond paramagnetically (see bottom figure in the middle column).

Figure 5

Sketched Hall conductivity and induced currents for two different perturbation with static electric fields: (a) Angularly averaged Hall conductivity tensor ${\sigma }_{xy}\left(r_x\right)$ at radius $r_x$ for a static, homogeneous electric field perturbation $E_y=E{\mathit{e}}_y$ with an inhomogeneous magnetic field $B_z\left(r\right)$ directed along the negative $z$-axis. Longitudinal electronic current densities induced by $E_y$ are indicated by $j_y(r_x,\varphi )$. Transversal electronic Hall current densities are given by $j_x^{\mathrm{Hall}}(r_x,\varphi )$, which are related to the $E_y$-perturbation by the local transversal Hall conductivity ${\sigma }_{xy}(r_x,\varphi )$. The background color indicates the sign and magnitude of the total current densities $j^{\mathrm{tot}}\left(r\right)$, induced by the inhomogeneous magnetic field, as shown in Fig. 4. (b) Similar setup, where the electric field perturbation is given by $E_y\left(r_x\right)=\delta (r-r_x)E{\mathit{e}}_{\mathit{y}}$ instead, which induces different Hall currents.

Figure 6

(a) Radially resolved Hall conductivity ${\sigma }_{xy}$ for a homogeneous electric field perturbation along the $y$ direction. To reach a locally converged solution within $r_{\max }=50$ nm, a large number of single electron eigenstates with $\langle r\rangle \gg r_{\max }$ is required, which indicates that the locally observed Hall conductivity switch is driven by the constant anisotropic vector potential instead of the localized inhomogeneous magnetic field. (b) Consequently, the integrated Hall conductivity transition at $E_F=E_{A^2}(1-1/4)$ is expected to persist in the asymptotic limit $r_{\max }\to \infty$, which automatically implies that spin contributions are of minor importance for this observable, except for $E_F=0+$.

Figure 7

(a) Radially resolved Hall conductivity ${\sigma }_{xy}$ for a homogeneous electric field perturbation along the $y$ direction for $g_s=0$, i.e., without Zeeman interaction. To reach a locally converged solution within $r_{\max }=50$ nm, a large number of single electron eigenstates with $\langle r\rangle \gg r_{\max }$ is required, which indicates that the locally observed Hall conductivity switch is driven by the constant anisotropic vector potential instead of the localized inhomogeneous magnetic field. Thus, the locally resolved result is almost identical to the spin-dependent $g_s=2$ calculation, confirming that the asymptotic properties are already present at $r_{\max }=50$ nm. (b) Consequently, the integrated Hall conductivity transition at $E_F=E_{A^2}(1-1/4)$ is again considered to persist in the asymptotic limit $r_{\max }\to \infty$.

Figure 8

(a) Radially resolved Hall conductivity ${\sigma }_{xy}$ measured at $r_x$ for a cylindrical electric field perturbation at $r_x$ along $y$. Multiple sharp conductivity transitions (dotted horizontal lines) are observed for this radially localized (!) perturbation, which follow a Rydberg series (labeled by $\lambda$) that alternates with rather smooth transition of opposite sign. (b) Integrated Hall conductivity transition pattern (summation over many local measurements at different radii), which clearly allows to distinguish smooth from sharp transitions with respect to the Fermi energy $E_F$.

Figure 9

Expected single electron currents with respect to their energy and radial position expectation values. They nicely illustrate that every electron leads has a positive total current $J^{\mathrm{tot}}\ge 0$