Electron-Hole Interference in an Inverted-Band Semiconductor Bilayer

Electron optics in the solid state promises new functionality in electronics through the possibility of realizing nano- and micrometer-sized interferometers, lenses, collimators, and beam splitters that manipulate electrons instead of light. Until now, however, such functionality has been demonstrated exclusively in one-dimensional devices, such as in nanotubes, and in graphene-based devices operating with $p\text{−}n$ junctions. In this work, we describe a novel mechanism for realizing electron optics in two dimensions. By studying a two-dimensional Fabry-Perot interferometer based on a resonant cavity formed in an InAs/GaSb double quantum well using $p\text{−}n$ junctions, we establish that electron-hole hybridization in band-inverted systems can facilitate coherent interference. With this discovery, we expand the field of electron optics in two dimensions to encompass materials that exhibit band inversion and hybridization.

Popular Summary

The main functional principle of optical interferometers is the interference of monochromatic waves that propagate in the same direction. In such interferometers, the interference can be observed as a periodic oscillation of the transmitted intensity upon varying the wavelength of the light. However, the period of the interference pattern strongly depends on the incident angle of the light, and as a result, the interference pattern is averaged out if light of all possible incident angles is sent through the interferometer at once. The same arguments apply to the interference of matter waves as described by quantum mechanics and, in particular, to electronic interferometers in which electrons interfere. Here, we investigate the phenomenon of electronic interference in a solid-state system consisting of two coupled semiconductor layers, InAs and GaSb.

We discover that these systems provide a novel transport mechanism that guarantees nonvanishing interference even under all angles of incidence. The mechanism arises from two properties of the InAs/GaSb system: band inversion, in which the usual ordering of conduction and valence bands is swapped, and hybridization, where the aforementioned bands mix, forming hybrid states. Using a combination of transport measurements and theoretical modeling, we describe the resulting operation of a Fabry-Perot interferometer in which electrons and holes form hybrid states and interfere.

Our results transcend InAs/GaSb systems, as the reported mechanism requires solely the two aforementioned ingredients of band inversion and hybridization. We therefore set the stage for engineering electron optics phenomena in a variety of materials, such as thin mono- and dichalcogenides as well as other 2D and 3D topological insulators.

Figure 1

(a) False-color image describing the device structure and operation. The scale bar represents $2\mu \mathrm{m}$. (b) Schematic device cross section. (c) Exemplary band alignment $E(k)$ in the $p{n}^{\prime }p$ configuration; $\mu$ is the electrochemical potential, indicated by the dotted line, and the solid line indicates the charge neutrality point (CNP) in each region. (d) Color map depicting the differential junction resistance $d{R}_j/d{V}_{\mathrm{LG}}$ as a function of $V_{\mathrm{LG}}$ and $V_{\mathrm{TG}}$ at 1.3 K. The resulting phase diagram is subdivided into quadrants according to the charge-carrier configuration. The horizontal and vertical dashed lines mark the CNPs of the leads and the cavity, respectively. The insets show the parts of the band structure probed to the left and right of the solid line at $V_{\mathrm{LG}}=1.1\mathrm{V}$ in the $p{n}^{\prime }p$ configuration, as highlighted by the shading.

Figure 2

(a) Temperature dependence of the resistance oscillations in $d{R}_j/d{V}_{\mathrm{L}\mathrm{G}}$ in the $p{n}^{\prime }p$ configuration at $V_{\mathrm{T}\mathrm{G}}=-1.8\mathrm{V}$ and in a reduced range in $V_{\mathrm{L}\mathrm{G}}$, chosen so as to highlight the oscillation behavior upon changing the temperature from 50 mK to 8 K. Dotted lines and symbols mark local minima (circles) and maxima (diamonds) used in the analysis presented in (c). (b) Fit of the temperature dependence of the local maximum at $V_{\mathrm{L}\mathrm{G}}=-0.365\mathrm{V}$, indicated by the arrow in (a), with a thermal damping function, Eq. (1). (c) Summary of the mode spacing $\mathrm{\Delta }E$ in the cavity extracted from the local minima and maxima marked in (a). The mean and standard deviation obtained when factoring in all points is inset. The individual points have relative errors of around 5%, such that the error bars are as large as or smaller than the point size.

Figure 3

(a) Sombrero-hat dispersion $E(k_x,k_y)$ of the hybridized conduction band resulting from the theoretical model, Eq. (2), as applied to the InAs/GaSb system. (b) Overview of the scattering processes $ij$ that occur within the $n^{\prime }$-doped cavity when a quasiparticle impinges onto the interface between cavity and lead. The $r_{ij}$ are reflection coefficients. (c) Upper panel: band structure in the three regions of a $pnp$ junction. The dashed horizontal line represents the electrochemical potential $\mu$. Full blue (red) circles represent holelike (electronlike) states moving to the right, while empty blue (red) circles represent holelike (electronlike) states moving to the left. Lower panel: sketch of a Fabry-Perot interferometer for a one-dimensional channel. The colors are coded according to the upper panel, with red (blue) arrows denoting holelike (electronlike) states moving to the right (solid lines) and left (dashed lines).

Figure 4

(a) Two Fermi circles from Fig. 3 with the larger (smaller) circle being the Fermi circle of electronlike (holelike) states. The shaded region marks the momentum transfer $\mathrm{\Delta }{k}_x$ for the $eh$ scattering processes assuming conservation of $k_y$; $\mathrm{\Delta }{k}_x$ does not depend strongly on $k_y$. (b) Momentum transfer $\mathrm{\Delta }{k}_x$ as a function of $k_y$ for the various scattering processes $ij$ from Fig. 3. Here, $\mathrm{\Delta }{k}_{eh}$ and $\mathrm{\Delta }{k}_{he}$ are identical. $k_{F,e}$ ($k_{F,e}^{\prime }$) and $k_{F,h}$ ($k_{F,h}^{\prime }$) are Fermi wave numbers of electronlike and holelike states in the leads (cavity), respectively. (c) Transmission $T$ through the $p{n}^{\prime }p$ junction as a function of the Fermi energy $E^{\prime }$ in the $n^{\prime }$-doped cavity and the momentum $k_y$ of the incident wave, calculated using a scattering matrix approach. The dashed lines denote the constructive interference condition assuming $eh$ and $he$ scattering only [Eq. (5) and inset]. The minimum of the hybridized conduction band is located at 22.1 meV. (d) Reflection amplitudes $|r_{ij}|$ for the various scattering processes $ij$ from Fig. 3 plotted as a function of $E^{\prime }$ and for $k_y=0$.

Figure 5

(a) Differential resistance $dR/d{E}^{\prime }$ as a function of $E^{\prime }$ obtained from Fig. 4 using Eq. (6). Oscillations are present in the shaded energy region in which electrons and holes coexist. (h) Cut through Fig. 1 at $V_{\mathrm{TG}}=-1.7\mathrm{V}$, with the shaded region above the CNP marking electron and hole coexistence.

Figure 6

(a) Color map depicting the differential junction resistance $d{R}_j/d{V}_{\mathrm{LG}}$ as a function of $V_{\mathrm{LG}}$ and $B$ at $V_{\mathrm{TG}}=-1.7\mathrm{V}$ and 1.3 K. (b) Calculated dependence of $dR/d{E}^{\prime }$ on the Fermi energy $E^{\prime }$ in the cavity and $B$. The inset shows schematic trajectories for electronlike and holelike states at the critical magnetic field $B_c$ of the holelike states, represented in both (a) and (b) by the dotted lines. Dashed lines and arrows indicate features common to both experiment and calculation, discussed under points (i)–(iii) in the main text.