Hot Electrons Regain Coherence in Semiconducting Nanowires

The higher the energy of a particle is above equilibrium, the faster it relaxes because of the growing phase space of available electronic states it can interact with. In the relaxation process, phase coherence is lost, thus limiting high-energy quantum control and manipulation. In one-dimensional systems, high relaxation rates are expected to destabilize electronic quasiparticles. Here, we show that the decoherence induced by relaxation of hot electrons in one-dimensional semiconducting nanowires evolves nonmonotonically with energy such that above a certain threshold hot electrons regain stability with increasing energy. We directly observe this phenomenon by visualizing, for the first time, the interference patterns of the quasi-one-dimensional electrons using scanning tunneling microscopy. We visualize the phase coherence length of the one-dimensional electrons, as well as their phase coherence time, captured by crystallographic Fabry-Pèrot resonators. A remarkable agreement with a theoretical model reveals that the nonmonotonic behavior is driven by the unique manner in which one-dimensional hot electrons interact with the cold electrons occupying the Fermi sea. This newly discovered relaxation profile suggests a high-energy regime for operating quantum applications that necessitate extended coherence or long thermalization times, and may stabilize electronic quasiparticles in one dimension.

Popular Summary

Semiconducting nanowires—wires with diameters measured in billionths of a meter—offer potential for use in many innovative applications, from flexible solar panels to miniaturized sensors. But not much is known about how electrons behave in these wires, which are thin enough for quantum effects to come into play, because nanowires are difficult to study. We constructed a “portable vacuum suitcase” that allows nanowires to be transferred intact from the chamber where they are grown to a scanning tunneling microscope, which allows us to perform the first exhaustive spectroscopic mapping of a semiconducting nanowire.

Using a scanning tunneling microscope, we observe interference patterns along the nanowire in order to study the evolution of electron phase coherence. Typically, electron phase coherence in mesoscopic solid-state materials depends on a low rate of scattering among electrons, and Landau’s principle tells us that the higher the energy of a hot electron, the faster it relaxes and therefore loses its phase coherence. We find that in semiconducting nanowires, where electrons are confined to one dimension, dispersion and Coulomb interaction interact in a way that revives phase coherence of ultrahot electrons; above a certain energy threshold, phase coherence grows with increasing energy.

These results are made possible by keeping the nanowires under ultrahigh vacuum at all times, from molecular beam epitaxy growth to subsequent measurement. This paves the way for further spectroscopic studies, such as spatial mapping of putative Majorana fermions—particles that are their own antiparticle—that might appear at the ends of nanowires once they become topologically superconducting.

Figure 1

Visualizing 1D subbands in semiconducting InAs nanowires. (a) STM topography of two InAs nanowires intersecting on top of the Au surface. Inset: Resolved atomic structure of the $\{11–20\}$ facet expected of a Wurtzite nanowire grown in the $⟨0001⟩$ direction. (b) dI/dV measurement (black) and ab initio calculation (orange), both showing the semiconducting gap and onset of the quantized conduction band in the form of van Hove singularities in the LDOS. (c) Topographic image showing a clean facet at a nanowire end. Inset: Topographic image of the catalyst Au droplet terminating the nanowire. The clean facet location is marked by the dashed frame. (d) dI/dV measurement along a line cut on the clean facet [dashed arrow in (c)] showing a dominant dispersing QPI pattern above the semiconducting gap alongside nondispersing van Hove peaks. A mean spectrum, $⟨\mathrm{dI}/\mathrm{dV}⟩$, taken far from the nanowire end [similar to (b)] was subtracted. The residual strength of dI/dV modulations is at most 30% of that mean value. (e) Fourier transform of (d) showing a dominant mode associated with the lowest subband of the quantized conduction band in full agreement with ab initio calculations (dotted line). Faint dispersing QPI of higher subbands are observed as well.

Figure 2

Nonmonotonic phase coherence length $L_{\phi }$ visualized by QPI decay. (a) The dI/dV map from Fig. 1 after applying a “spatial lock-in” filtration (see Appendix pp5-s1), used to highlight the decay profile of the QPI patterns. The extended standing-wave patterns of the hot holes turn into progressively decaying QPI patterns above $E_F$ but then revive above about 80 meV. (b) Representative decay profiles from (a) (dotted lines), each fitted to $A|\sin (qx+\mathrm{\Phi })|e^{-2x/L_{\phi }}$ (gray lines). At the highest measured energies, the decay is too slow to fit reliably within the measured distance, which is limited by the distant surface impurities (hence the large error bars). (c) The fitted phase coherence length at different energies, displaying a nonmonotonic evolution with energy (open circles). The calculated phase coherence length from the model (solid line) agrees well with the measured nonmonotonic trend and stands in sharp contrast to the $E^{-2}$ behavior expected at the low-energy regime (dashed line).

Figure 3

Nonmonotonic phase coherence time extracted from Fabry-Pèrot resonators. (a) Topographic image showing a series of adjacent stacking faults on the top facet of the nanowire. (b) High-resolution TEM micrograph demonstrating that a single stacking fault induces atomic displacement. (c) TEM micrograph of a nanowire from the same growth batch showing a similar distribution of stacking faults. (d) Typical dI/dV line cut along a terrace terminated by adjacent stacking faults (mean spectrum subtracted) showing quantized resonances. Up to 20% of the mean LDOS is modulated. (e) Energy broadening of the quantized resonances $\mathrm{\Gamma }(E)$ as a function of their energy (black dots). The error bars originate in a two-stage fitting procedure described in Appendix pp5-s2. At the Fermi energy, the broadening ${\sigma }_{\mathrm{tot}}$ is attributed to instrumental contributions of temperature and probing ac amplitude, ${\sigma }_0=3.9\mathrm{meV}$ (up to the dotted line) and the transmissivity of the stacking faults, about 1.1 meV, setting a lower bound on the reflection coefficient of ${|R|}^2\ge 0.8\pm 0.1$. The excess energy-dependent broadening (above ${\sigma }_{\mathrm{tot}}$) is attributed to the finite coherence time of the electrons ${\tau }_{\phi }$. Both the model (solid line) and the coherence length data from Fig. 2 (circles) agree on the nonmonotonic evolution of the electronic phase coherence and deviate from the low-energy evolution (dashed line).

Figure 4

Modeling the relaxation rate of a 1D electronic subband. (a) Schematic comparison of the slow three-body relaxation of hot electrons on the lowest subband, which excites both copropagating and counterpropagating electron-hole pairs, and the fast two-body relaxation of hot electrons on higher subbands (red and blue arrows, respectively). (b) The calculated relaxation rate of a single subband with quadratic dispersion. The dispersion was chosen to have the same Fermi momentum and Fermi velocity as the nanowire’s lowest subband.

Figure 5

(a) STM topography of a nanowire grown without faceting, exhibiting a round cross section. (b) Zoomed-in topography of (a). The round cross section, in conjugation with the lattice discretization, yields a corrugated stepped surface filled with randomly oriented dangling bonds. (c) A large-scale topography of a wide facet on top of a wire with multiple stacking faults (same as in Fig. 3). (d) Zoomed-in topography of one of the terraces. The average distance between impurities is a couple of nm.

Figure 6

Scanning electron microscopy images of InAs nanowires. (a) The faint shadows along the nanowire axis are atomically flat facets. (b) Nanowires on their (111)B InAs substrate. The tilted nanowire randomly grows in the (100) direction.

Figure 7

AFM topographies and statistical analysis of surface roughness. Topography of clean (a) and nanowire-covered (b) Au surfaces. (c) Height distribution of the topography in (a) (orange) and (b) (purple). The roughness can be characterized by the variance of these distributions.

Figure 8

UHV growth, harvest, and transfer of nanowires. (a) UHV suitcase (blue) and harvesting chamber (red) in which we disperse the grown nanowires on a Au surface and transfer them under UHV from the MBE to STM chamber. (b) Topography of freshly prepared Au(100) substrate. (c) SEM image of vertically growing InAs nanowires. (d) SEM image of harvested nanowires dispersed over the Au substrate.

Figure 9

(a) Electronic spectrum of InAs nanowire. $E_F$ is about 75 meV above the conduction-gap minimum, and the semiconducting gap is about 350 meV. Resonances indicating that the 1D subbands are seen in both the conduction and valance bands. (b) van Hove singularities from spectra taken on different crystal terminations. Locations across the nanowire are shown in the inset by arrows of corresponding color over the topography.

Figure 10

Tight-binding calculation of InAs nanowires’ band structure. Dotted lines mark some of the calculated bands.

Figure 11

Quasiparticle interference imaging of the quantized bands in InAs nanowires. (a) Topography of the nanowire surface with various surface impurities that serve as scattering centers. The horizontal direction is parallel to the direction of the nanowire. (b)–(d) Differential conductance (dI/dV) maps at 80, 70, and 60 meV, showing dispersive QPI patterns manifested in spatial fluctuation around the mean value at each energy. Comparing the spatial standard deviation of the LDOS to the mean spectrum shows that roughly 5% of the LDOS is modulated. (e) Mean spectrum showing van Hove singularities. The spectrum was obtained by spatially averaging the dI/dV maps and subtracting a monotonic smooth background for visibility. (f) Fourier transform of the dI/dV maps with respect to the direction of the nanowire. The red dotted curves are copied from Fig. 10 (the $x$ axis was scaled to represent momentum transfer $q=2k$) and demonstrate the close resemblance between the calculated and the measured band structures. The solid red lines indicate that the bottom of each band originates from a specific van Hove singularity.

Figure 12

Relaxation rate for individual subbands. We show contributions to $\mathrm{\Gamma }$ by exciting the Fermi sea of different subbands: the lowest subband ($-75\mathrm{meV}$ below ${ϵ}_F$, blue), two higher occupied subbands ($-50\mathrm{meV}$ in red and $-25\mathrm{meV}$ in yellow), and the sum of all three contributions (black). The energy denotes the bottom of the subbands, and we use energy spacing and dispersion extracted from the experiment.

Figure 13

(a) The kernel used to calculate the cross correlation, $\cos (q_E{x}^{\prime })\mathrm{rect}(q_E{x}^{\prime })$, is shown for each energy $E$ as a function of $x^{\prime }$. This is simply a single period of a cosine with a wavelength corresponding to the inferred momentum transfer $q_E$. (b) Spatial derivative of the line cut shown in Fig. 1. The profile of the nonmonotonic decay of the standing-wave pattern can easily be traced.

Figure 14

Quantized resonances in different terraces. Panels (a)–(c) show the dI/dV maps from three different terraces, averaged over 12 line cuts across the nanowire ($Y$). A smooth background has been subtracted for visibility. The observed puddles are the quantized resonances used to determine the lifetime of the states.

Figure 15

Measuring the energy broadening of a resonance peak. (a) 3D slices of a resonance in the dI/dV map, showing its energy (i) and spatial (ii) distributions. (b) Gaussian fitting of the spatial distribution of the peak at a given line cut ($Y_0$) and energy. (c) Multiple Gaussian model fitting of the energy distribution of the peak, determining its width and peak energy.

Figure 16

Quantized resonances in the calculated LDOS of a quantum box with ${|R|}^2=0.9$. (a) Resonances of the lowest energy band, calculated from the experimentally inferred dispersion. (b) Same resonances after instrumental broadening due to temperature (about 1 meV) and finite probing amplitude (3 meV).