Dynamical Birefringence: Electron-Hole Recollisions as Probes of Berry Curvature

The direct measurement of Berry phases is still a great challenge in condensed-matter systems. The bottleneck has been the ability to adiabatically drive an electron coherently across a large portion of the Brillouin zone in a solid where the scattering is strong and complicated. We break through this bottleneck and show that high-order sideband generation (HSG) in semiconductors is intimately affected by Berry phases. Electron-hole recollisions and HSG occur when a near-band-gap laser beam excites a semiconductor that is driven by sufficiently strong terahertz-frequency electric fields. We carry out experimental and theoretical studies of HSG from three $\mathrm{GaAs}/\mathrm{AlGaAs}$ quantum wells. The observed HSG spectra contain sidebands up to the 90th order, to our knowledge the highest-order optical nonlinearity reported in solids. The highest-order sidebands are associated with electron-hole pairs driven coherently across roughly 10% of the Brillouin zone around the $\mathrm{\Gamma }$ point. The principal experimental claim is a dynamical birefringence: the intensity and polarization of the sidebands depend on the relative polarization of the exciting near-infrared (NIR) and the THz electric fields, as well as on the relative orientation of the laser fields with the crystal. We explain dynamical birefringence by generalizing the three-step model for high-order harmonic generation. The hole accumulates Berry phases due to variation of its internal state as the quasimomentum changes under the THz field. Dynamical birefringence arises from quantum interference between time-reversed pairs of electron-hole recollision pathways. We propose a method to use dynamical birefringence to measure Berry curvature in solids.

Popular Summary

The study of matter in extreme conditions is one of the most exciting frontiers of science and technology. We have discovered a surprising new phenomenon in the interaction of infrared light with a widely used semiconductor subjected to extremely strong electric fields oscillating nearly one trillion times per second (1 THz). This phenomenon can be used to investigate important but previously unmeasurable quantum-mechanical effects on the motion of charges in semiconductors.

In our experiment, we use a weak infrared laser beam to illuminate thin layers of a semiconductor driven by strong terahertz fields. The infrared beam is polarized either along or perpendicular to the terahertz field. The infrared light transmitted through the semiconductor exhibits a rainbowlike spectrum containing dozens of frequencies, or sidebands. Surprisingly, the sidebands are usually stronger when the infrared beam is polarized perpendicular to the terahertz field than when it is parallel, and the sidebands are also polarized differently than the incident beam.

A new theory explains this phenomenon. The infrared laser creates negatively charged electrons and positively charged holes. The terahertz field first accelerates the electrons and holes away from each other and then back toward each other. During acceleration, Berry curvature—an important quantum-mechanical property of the semiconductor—causes the angular momentum of the holes to rotate. When the electrons and holes re-collide, the emitted sidebands carry information about both the incoming polarization and the Berry curvature.

Our methods open the door to the first direct measurement of Berry curvature in solid matter.

Figure 1

Optical setup and HSG spectrum. (a) Details of the sample. The sample consists of epitaxially grown quantum wells transferred to a sapphire wafer. A thin film of ITO on the back side of the sapphire reflects the THz light back towards the quantum wells while transmitting the NIR laser and sidebands (SBs). See Supplemental Material [40] for more details. For sideband measurements, the NIR and THz lasers are focused collinearly on the same spot on the sample, and propagate normal to the surface. (b) Full HSG spectrum spanning 106 orders from the 5-nm GaAs sample. The sidebands (solid lines) decorate the NIR laser represented by thick dashed red line. The NIR and THz laser polarizations are parallel to each other, and the [011] direction of the lattice makes a 55° angle with the THz polarization.

Figure 2

Sideband conversion efficiencies for NIR laser field polarized parallel and perpendicular to THz field. The angles between the THz polarizations and the [011] direction are 85°, 91°, and 93° for the 5-nm GaAs sample, the 10-nm AlGaAs sample, and the 10-nm GaAs sample, respectively. The sideband offset energy is the difference of the sideband photon energy and the NIR laser photon energy. The sideband conversion efficiency is the power in the sideband divided by the power of the NIR laser incident on the sample. The perpendicular-excited sidebands are stronger than the parallel-excited sidebands above an energy offset of roughly 70 meV for the 5-nm GaAs sample and 30 meV for the two 10-nm samples. The perpendicularly excited sidebands fall off more slowly above this offset than below it. In both the 5-nm GaAs and 10-nm AlGaAs samples, the sideband intensities fall off almost exponentially as the sideband order increases. Sideband intensities from the 10-nm GaAs sample, however, show a more complicated relationship with the sideband order, perhaps because of weaker scattering in this sample. The nonexponential decay of negative-order sidebands in this sample may be due to a relatively thick GaAs layer outside of the QWs (see Supplemental Material [40]).

Figure 3

HSG spectra from all three samples for different lattice orientations, and calculated valence subband dispersion relations. The THz polarization was kept horizontal. The angle $\theta$ between the [011] direction and the THz field is shown on the left side, with the propagation direction into the page. Experimental data show sideband offset energy versus sideband (conversion) efficiency. The empty black squares are replotted from Fig. 2. The valence subband dispersion relations are plotted along the directions of the THz electric field to elucidate the relation between the hole subbands and the sideband spectra. 5-nm GaAs QWs: The relatively large avoided crossing between HH1 and HH2 is correlated with a relatively large energy required for the onset of dynamical birefringence (see Fig. 2). The dependence of HSG spectra on lattice orientation is relatively weak, except for the parallel-excited spectrum above 100-meV sideband offset energy. 10-nm AlGaAs QWs: The avoided crossing between HH1 and HH2 is much smaller than for the 5-nm GaAs QW. Broadening caused by alloy disorder is represented as a shaded strip on each curve in the dispersion relation. The sideband conversion efficiencies show little dependence on lattice orientations. 10-nm GaAs QWs: The hole dispersion relations are nearly identical to those for the 10-nm AlGaAs QW, but broadening from alloy disorder is absent in this sample, which has smaller disorder than the other two samples (see Fig. 7 and the related discussions). The sideband spectra depend substantially on angle $\theta$.

Figure 4

Schematic representation of three-step model for high-order sideband generation in position space (left) and momentum space (right). The figures represent a quantum interference process, with the upper (lower) arms associated with creation and dynamics of electron-hole pairs with total spin $-1$ ($+1$). The linearly polarized NIR excitation light is a superposition of a right and a left circularly polarized component, ${\sigma }_{\mathrm{NIR}}^{\pm }$. In step 1, a quantum superposition of an electron-hole pair with total spin $-1$ (electron spin $+1/2$, heavy-hole spin $-3/2$) and an electron-hole pair with total spin angular momentum $+1$ (electron spin $-1/2$, heavy-hole spin $+3/2$) is created. In step 2, the electrons and holes are accelerated by the THz field. In position space, the electron spinors (vertical arrows at four different times along “electron” trajectories, defined in Sec. 3c) do not rotate because Berry curvature is negligible, while the hole spinors rotate substantially under the influence of non-Abelian Berry curvature. In momentum space, the electron state remains confined to a single band during its acceleration, while the hole state mixes into nearby bands during its acceleration. In step 3, electrons and holes re-collide. Upon re-collision, the hole state is a superposition of states (depicted by ovals in the right-hand figure) in several different hole subbands with coefficients determined by the non-Abelian Berry curvature and the initial NIR excitation. Some of these states have allowed dipole transitions with the electron, and so create photons ${\sigma }_{\mathrm{HSG}}^{\pm }$ that then interfere to generate the HSG emission at frequency $f_{\mathrm{HSG}}$ with elliptical polarization. Rotating the NIR polarization changes the relative phase of the ${\sigma }_{\mathrm{NIR}}^{+}$ and ${\sigma }_{\mathrm{NIR}}^{-}$ photons and influences the output by changing how the emitted sideband photon states interfere, resulting in dynamical birefringence.

Figure 5

Polarization state of the sidebands in terms of the two ellipticity angles. (a) Definition of $\alpha$ and $\gamma$ for the polarization ellipse. (b) Experimental and theoretical values for the polarization angles. Experimental measurements are filled scatter points, theoretical calculations are empty scatter points. The shaded region is the error in the calculation propagated from error in the measured NIR laser polarization state. Polarization state measurements were performed on both 10-nm samples for both excitation geometries, and the nearly perpendicular measurements were performed twice, labeled trials 1 and 2. For sideband polarimetry for the 10-nm GaAs sample, see Supplemental Material [40]. The angle of the [011] direction relative to the THz field is 93°. The NIR laser polarization state is plotted as the order zero sideband and circled in black for each measurement. Overall, both theory and experiment agree that the polarization state of a given sideband is extremely sensitive to the NIR laser polarization state.

Figure 6

Comparison of dynamical birefringence in experiment and theory. The ratio $I_{\perp }/I_{\parallel }$ is compared in all three quantum wells at different lattice angles. The nonzero Berry connection is responsible for deviations from unity. The quantum theory of sidebands includes only macroscopic dephasing, where the only $k$ dependence arises when the electrons (or holes) have enough energy to be elastically scattered into other bands (i.e., the dephasing rates are step functions). Comparing intensity ratios instead of absolute intensities largely cancels out the spin-independent scattering effect, which is assumed to be the dominant mechanism of scattering. The blue line is the result if the Berry connection is assumed to be zero.

Figure 7

Optical absorption spectra of the three samples measured by differential transmission. In all three samples, the heavy-hole exciton lines (starred peaks) and the light-hole exciton lines (slightly weaker, blueshifted by 10 or 25 meV depending on the well widths) are both clearly resolved. The measurements are performed at 15 K. In the 5-nm GaAs sample, the onset of the heavy-hole 2D band gap is apparent at about 1.630 eV. The onset of the light-hole 2D band gap is also evident (not pictured). The smaller heavy-hole–light-hole splitting in the 10-nm samples masks the heavy-hole 2D band gap in them, but, in the 10-nm GaAs sample, the onset of the light-hole 2D band gap is apparent at 1.570 eV. The NIR laser wavelength used for the HSG experiments are shown as dark red arrows.

Figure 8

Semiclassical trajectories for the $n=20$ and $n=60$ sidebands in the 10-nm AlGaAs sample. Upper left: Full perpendicular HSG spectrum experimentally measured in the 10-nm AlGaAs sample oriented at 47°. Upper right: Time trace of the THz electric field. The arrows point to the time instants of ionization $i_n$ and re-collision $r_n$ for the two sidebands considered here. Middle left: The real-space trajectories of the electron and hole for the 20th-order sideband. The spin state is ${\overrightarrow{\eta }}_e=[\uparrow ,\downarrow {]}^T$ for the electron and ${\overrightarrow{\eta }}_h=[|\mathrm{HH}1⟩,|\mathrm{HH}2⟩{]}^T$ for the hole. Each arrow represents a spin or pseudospin in $xz$ plane for each of seven instants throughout the trajectory. Lower left: The location of the hole in the valence subbands at each of those seven instants with the area of each maroon circle representing the relative weight in either subband. Middle right: The real-space trajectories of the electron and hole for the 60th-order sideband, with the spinor directions drawn for eight instants. The hole spinor almost entirely flips to the HH2 state at re-collision. Bottom right: The location of the hole in the valence subbands at each of the eight instants. For both cases, the electron is always in spin-down state and the $y$ component of the hole pseudospin is approximately zero. Comparing the two cases shows the scale of the effects of non-Abelian Berry curvature.

Figure 9

Theoretical calculation of sideband intensity ratio $I_{\perp }/I_{\parallel }$ for the 5-nm GaAs sample at 55° with two different well widths for the conduction band, 5 nm (red curve) and 8.13 nm (black curve replotted from Fig. 6). The blue line is the result if the Berry connection is assumed to be zero.

Figure 10

Dynamical birefringence ratio $I_{\perp }/I_{\parallel }$ from two independent measurements. The black squares are from the intensity measurements (center left, Fig. 6). The red squares are obtained from the unitary Jones matrices ${\mathcal{J}}_n$ calculated from the polarization state angle measurements from Fig. 5.