Observation of quantum states without a semiclassical equivalence bound by a magnetic field gradient

Resonant transmission through electronic quantum states that exist at the zero points of a magnetic field gradient inside a ballistic quantum wire is reported. Since the semiclassical motion along such a line of zero magnetic field takes place in the form of unidirectional snake trajectories, these states have no classical equivalence. The existence of such quantum states has been predicted more than a decade ago by theoretical considerations. We further show how their properties depend on the amplitude of the magnetic field profile as well as on the Fermi energy.

Figure 1

(a) Scheme of the lateral sample layout. (b) Atomic force microscope picture of the Dy stripe on top of the oxide lines. (c) Cross section of the device at $y=0$, with the magnetic field profile $B_z\left(x\right)$ (solid blue line) and the energy of the lowest QWR mode $E_1\left(x\right)$ (dashed red line). (d) Conductance of the QWR as a function of the top gate for two applied magnetic fields $B^e$.

Figure 2

Plots of $G(B^e,V_{\text{tg}})$ around the plateaus (a) 2 and (c) 4. In these measurements, $B^e$ was changed in steps of $8\mathrm{mT}$ and the top gate voltage was varied. In (c), the ${\mu }_{0}M$ axis is also shown, which can be determined from the $B^e$ axis, as described in the text. The red dashed line indicates the evolution of the resonance. (b) Temperature dependence of $G\left(B^e\right)$ at fixed gate voltages, as measured in a ${}^4\text{He}$ cryostat with a variable temperature insert. (d) Simulation of the conductance $G(E_F,{\mu }_{0}M)$ as a function of the Dy magnetization and the Fermi energy, in the transition region between conductance plateaus 3 and 4. The peak position is indicated by a red dashed line. Adjacent curves have an offset of $0.024\times 2e^2/h$ for better visibility.

Figure 3

Simulations for a realistic QWR exposed to a magnetic field gradient. (a) The LDOS along the QWR as a function of $E_F$, calculated for ${\mu }_{0}M=0.38\mathrm{T}$. One zero-dimensional state is visible at $x=0$. The red line is the energy of the fourth QWR mode. (b) $\mathrm{LDOS}(x,y)$ for the parameters of (c) at $E_F=1.04\mathrm{meV}$, and a classical electron trajectory (white line). The red lines mark the extremal points of $B_z\left(x\right)$.

Figure 4

Numerical studies of the system over a wide range of magnetizations and low Fermi energies. The calculated conductance is shown in (a). The dashed line marks the magnetization of ${\mu }_{0}M=1.33\mathrm{T}$ used for the LDOS as a function of $E_F$ and the $x$ coordinate shown in (b). Here, $j$ denotes the mode index.