Effective-Hamiltonian reconstruction through Bloch-wave interferometry in bulk GaAs driven by strong THz fields

TL;DR

The paper tackles reconstructing an effective three-band electron-hole Hamiltonian in bulk GaAs from bulk-sensitive high-order sideband generation under strong THz fields. It develops a Bloch-wave interferometry framework based on dynamical Jones matrices to map sideband polarizations to electron-hole propagators, then derives a detailed saddle-point model that connects these propagators to Hamiltonian parameters such as $E_g$, $oldsymbol{eta}$-like masses, and Luttinger parameters via the composite quantity $\xi=rac{ ext{g}_2 ext{ex}}{m_0}$. By combining HSG polarimetry with low-temperature absorbance, the authors extract the two bandgaps $E_{g, ext{E-HH}}$ and $E_{g, ext{E-LH}}$, the two dephasing constants $\xi$, and the dephasing constants $\Gamma_{ ext{E-HH}}$ and $\Gamma_{ ext{E-LH}}$, achieving unambiguous Hamiltonian reconstruction for GaAs and finding a modest bandgap blue shift relative to absorbance. The results also reveal that THz-field-modulated Fröhlich interactions can renormalize e-h energies and suppress LO-phonon thresholds, suggesting extensions to include Coulomb and phonon coupling for an even more complete EH picture with potential implications for polaron physics under strong fields.

Abstract

Effective Hamiltonians (EHs) are powerful tools for understanding the emergent phenomena in condensed matter. Reconstructing an EH directly from experimental data is challenging due to the intricate relationship between EH parameters and observables. Complementary to ARPES, which probes surface electronic properties, bulk-sensitive techniques based on HHG and HSG have shown strong potential for EH reconstruction (EHR). We reconstruct an effective three-band electron-hole (e-h) Hamiltonian in bulk GaAs based on HSG induced by quasi-cw NIR and THz lasers. We perform polarimetry of high-order sidebands while varying the wavelength and polarization of the NIR laser and the strength of the THz field, to systematically explore the information encoded in the sidebands. Based on previous understanding of HSG in GaAs in terms of Bloch-wave interferometry (BWI), an analytic model is derived to quantitatively connect the EH parameters with the measured sideband electric fields under strong, low-frequency THz fields. Assuming that the exciton reduced mass and the parameter that defines the hole Bloch wavefunctions in GaAs are known from existing experiments, we show that the GaAs bandgap, two dephasing constants associated with two e-h species, and an EH parameter that determines the e-h reduced masses, can be simultaneously and unambiguously determined through BWI. We demonstrate that full EHR can be achieved by combining HSG measurements with absorbance spectroscopy. We find that the extracted bandgap of GaAs is about 10\,meV higher than the value inferred from previous absorbance measurements. Quantum-kinetic analysis suggests that the e-h energy in HSG may be renormalized through Fröhlich interaction that is modified by a strong THz field. We show that the energy threshold for optical-phonon emission can be suppressed by applying a strong THz field, leading to nearly constant dephasing rates.

Figures (18)

• Figure 1: Effective-Hamiltonian reconstruction through Bloch-wave interferometry in bulk gallium arsenide (GaAs). (a) Experimental setup. A near-infrared (NIR) laser and a terahertz (THz) laser are focused collinearly onto a GaAs epilayer mounted on a sapphire substrate. An indium-tin-oxide (ITO) film on the opposite side of the substrate reflects the THz field to enhance the THz-field strength at the GaAs epilayer through constructive interference. A silicon-dioxide ($\rm SiO_2$) layer deposited on top of the ITO film acts as an anti-reflection coating for the NIR laser and the sidebands. Polarimetry of high-order sidebands is performed by passing the sideband fields through a quarter-wave plate (QWP) and a linear polarizer. A diffraction grating and a charge-coupled device (CCD) are combined to measure the intensities of a series of sidebands simultaneously. (b) The QWP is rotated by $360^\circ$ in $22.5^\circ$ steps. At each QWP rotation angle $\theta_{\rm QWP}$, an intensity spectrum is measured and plotted as a function of the sideband order $n$ (magenta curves), which is defined as the offset of the sideband frequency with respect to the NIR-laser frequency in units of the THz-laser frequency, with the laser linewidths ignored. For each sideband order, the total intensity is calculated as the area under the corresponding sideband peak in an intensity spectrum; its dependence on $\theta_{\rm QWP}$ (green shaded areas) yields the associated Stokes parameters, $S_0(n)$, $S_1(n)$, $S_2(n)$, and $S_3(n)$. (c) The polarization of each sideband is characterized by an orientation angle $\alpha_n$ and an ellipticity angle $\beta_n$, which are defined with respect to the THz electric field that makes an angle $\varphi$ with the [100] crystal axis. The sign of $\beta_n$ is positive (negative) when the sideband electric field rotates clockwise (counterclockwise) as it propagates away from the observer. In the linear regime with respect to the NIR laser, each sideband electric field with two helicity components $E_{\pm,n}$ and the NIR-laser electric field with two helicity components $E_{\pm,\rm NIR}$ are connected through a two-by-two matrix called a dynamical Jones matrix, which contains four complex elements $T_{\pm\pm,n}$. Each dynamical Jones matrix can be determined up to an overall phase factor by the measured Stokes parameters. (d) High-order sideband generation (HSG) in bulk GaAs that is near-resonantly excited by a NIR laser and simultaneously driven by a sufficiently strong linearly polarized THz field can be viewed as a Michelson-like interferometer for Bloch waves. First, the NIR laser is incident on the GaAs, creating an electron-hole Bloch wave. Second, the GaAs acts like a beam splitter, "splitting" the electron-hole Bloch wave, which is a superposition of electron-heavy hole (E-HH) and electron-light hole (E-LH) Bloch waves, into two "arms", one for each electron-hole species (closed circles for the electrons and open circles for the holes). Third, the THz field drives the E-HH and E-LH Bloch waves along different trajectories in their respective energy bands. Fourth, upon sideband emission, the E-HH and E-LH Bloch waves "merge" at the "beam splitter" (GaAs) and interfere with each other. Fifth, the sideband electric field as a function of sideband order $n$ is recorded as a Bloch-wave interferogram. (e) Based on the description of HSG in bulk GaAs in terms of a Bloch-wave interferometer, the measured dynamical Jones matrices are decoded into physical information including the electron-hole propagators. For each sideband, the E-HH (E-LH) propagator $\varsigma^{\rm E-HH}$ ($\varsigma^{\rm E-LH}$) describes a recollision process governed by an effective Hamiltonian $H_{\rm E-HH}$ ($H_{\rm E-LH}$), which contains the parameters of the total effective Hamiltonian $H_{\rm eff}$ for bulk GaAs, $a_1,a_2,\,...$. Here, $\Psi_{\rm E-HH,i}$ ($\Psi_{\rm E-LH,i}$) and $\Psi_{\rm E-HH,f}$ ($\Psi_{\rm E-LH,f}$) represent the initial state and final state respectively for the E-HH (E-LH) pair. (f) By inverting the propagators $\varsigma_{\rm E-HH}$ and $\varsigma_{\rm E-LH}$, the parameters $a_1,a_2,\,...$ are obtained and the effective Hamiltonian $H_{\rm eff}$ is reconstructed.
• Figure 2: Different bandgaps for two electron-hole species. (a) An absorbance spectrum for the GaAs epilayer at 30 K. Two exciton peaks with an energy splitting $\Delta_{\rm ex}\approx2.2$ meV are observed. These peaks are associated with the E-HH and E-LH pairs, respectively. (b) Band structure of bulk GaAs including a lowest conduction band (E band) and two highest valence bands (HH and LH bands). The solid lines represent the energy bands calculated by including a tensile biaxial strain that induces a 2.2-meV splitting between the HH and LH bands at ${\bf k}={\bf 0}$. The dashed lines represent the energy bands with no strain effects included. Here, the dimensionless wavevector $ka$ is used with $a=5.65$ Å being the lattice constant of GaAs soma1982thermaldriscoll1975precision, and $E_{\rm g}$ labels the bandgap.
• Figure 3: Absolute values of the propagator ratio $\varsigma^{\rm E-HH}/\varsigma^{\rm E-LH}$. The data were obtained by using a left-handed circular polarization (helicity -1) for the NIR laser. Panels (a), (b), and (c) show the data collected at three different THz-field strength levels: around 60 kV/cm, 45 kV/cm, and 30 kV/cm, respectively (see Fig. \ref{['FIG:thz_field_strength']} in Appendix \ref{['APP:thz_field_strength']} for the exact THz-field strengths). In each panel, cyan trianges, dark green circles, and magenta squares represent the data obtained at three different NIR-laser wavelengths: 819.5 nm, 818 nm, and 815 nm, respectively. The cyan, dark green, and magenta solid lines represent the corresponding theoretical results. For each set of laser parameters, two solid lines of the same color indicate the one-standard-deviation range resulting from uncertainties in the THz field strengths and the Hamiltonian parameters. The theoretical curves in each panel largely overlap. In (c), a larger $y$ scale is used for the data within the grey box.
• Figure 4: Phases of the propagator ratio $\varsigma^{\rm E-HH}/\varsigma^{\rm E-LH}$ expressed in terms of cosine and sine functions. The data were obtained by using a left-handed circular polarization (helicity -1) for the NIR laser. The first, second, and third rows show the data collected at three different THz-field strength levels: around 60 kV/cm, 45 kV/cm, and 30 kV/cm, respectively (see Fig. \ref{['FIG:thz_field_strength']} in Appendix \ref{['APP:thz_field_strength']} for the exact THz-field strengths). In each panel, cyan trianges, dark green circles, and magenta squares represent the data obtained at three different NIR-laser wavelengths: 819.5 nm, 818 nm, and 815 nm, respectively. The cyan, dark green, and magenta solid lines represent the corresponding theoretical results. For each set of laser parameters, two solid lines of the same color indicate the one-standard-deviation range resulting from uncertainties in the THz field strengths and the Hamiltonian parameters.
• Figure 5: Absolute value of the propagator $\varsigma^{\nu}_n$ ($\nu={\rm E-HH,E-LH}$) relative to its value at the lowest detected sideband order $n_0=12$. The data were obtained by using a left-handed circular polarization (helicity -1) for the NIR laser. The first, second, and third rows show the data collected at three different THz-field strength levels: around 60 kV/cm, 45 kV/cm, and 30 kV/cm, respectively (see Fig. \ref{['FIG:thz_field_strength']} in Appendix \ref{['APP:thz_field_strength']} for the exact THz-field strengths). In each panel, cyan trianges, dark green circles, and magenta squares represent the data obtained at three different NIR-laser wavelengths: 819.5 nm, 818 nm, and 815 nm, respectively. For each set of laser parameters, two solid lines of the same color indicate the one-standard-deviation range resulting from uncertainties in the THz field strengths and the Hamiltonian parameters. The theoretical curves in each panel largely overlap.