Resonance-enhanced optical nonlinearity in the Weyl semimetal TaAs

The second-order conductivity of a material, ${\sigma }^{\left(2\right)}$, relating current to the square of electric field, is nonzero only when inversion symmetry is broken, unlike the conventional linear conductivity. Second-order nonlinear optical responses are thus powerful tools in basic research as probes of symmetry breaking; they are also central to optical technology as the basis for generating photocurrents and frequency doubling. The recent surge of interest in Weyl semimetals with acentric crystal structures has led to the discovery of a host of ${\sigma }^{\left(2\right)}$-related phenomena in this class of materials, such as polarization-selective conversion of light to dc current (photogalvanic effects) and the observation of giant second-harmonic generation (SHG) efficiency in TaAs at photon energy 1.5 eV. Here, we present measurements of the SHG spectrum of TaAs, revealing that the response at 1.5 eV corresponds to the high-energy tail of a resonance at 0.7 eV, at which point the second harmonic conductivity is approximately 200 times larger than seen in the standard candle nonlinear crystal, GaAs. This remarkably large SHG response provokes the question of ultimate limits on ${\sigma }^{\left(2\right)}$, which we address by a new theorem relating frequency-integrated nonlinear response functions to the third cumulant (or “skewness”) of the polarization distribution function in the ground state.

Figure 1

Schematic of the experimental apparatus used to study the spectrum of the nonlinear optical response in TaAs. Laser pulses of pulse width 50 fs, at a repetition rate of 5 kHz with photon energy between 0.5 and 1.6 eV are generated by an optical parametric amplifier coupled to an regeneratively amplified laser. The incident radiation passes through a neutral density filter (ND), polarizer $\left(P\right)$, and a low pass filter (LF) to lower the incident intensity and to remove spurious polarizations and wavelengths, respectively. A half-wave plate (HW) is used to rotate the incident linear polarization over ${360}^{\circ }$. The incident laser radiation is then focused on to the TaAs sample (S) using a reflective objective (RO), which is insensitive to the photon energy, in contrast with conventional refractive microscope objectives. The SHG radiation from the sample is collimated by the RO, picked off by a D-shaped mirror (D), and passed through an analyzing polarizer $\left(A\right)$, whose polarization state is determined by the scan being measured (see text). The radiation then passes through a stack of short pass filters (SF) to remove any incident light and then is focused by a lens (L) onto a photomultiplier tube (PMT) for detection. Inset: A single crystal of TaAs with the (112) facet visible.

Figure 2

Second harmonic generation polarimetry. SHG intensity measured in the “parallel” scan as a function of the incident polarization angle, plotted on a normalized scale, for three different incident photon wavelengths. The polar pattern changes from a twofold pattern at shorter wavelengths (800 nm = 1.55 eV) to a six-fold pattern at longer wavelengths (2200 nm = 0.56 eV)

Figure 3

Measured nonlinear optical conductivity as a function of incident photon energy. (a) Spectra of the conductivity components $|{\sigma }_{zxx}|,|{\sigma }_{xzx}|$, and $|{\sigma }_{\text{eff}}|=|{\sigma }_{zzz}+4{\sigma }_{xzx}+2{\sigma }_{zxx}|$. The solid lines are guides to the eye. The nonlinear optical conductivity of GaAs, $|{\sigma }_{xyz}|$, is multiplied by 100 and plotted for comparison. (b) The (expt.) lower and upper bounds of $|{\sigma }_{zzz}|$ are estimated using measured values of $|{\sigma }_{\text{eff}}|,|{\sigma }_{zxx}|$, and $|{\sigma }_{xzx}|$ from the experiment. The dashed black line depicts the best fit to $|{\sigma }_{zzz}|$ using the phenomenological model described in the text.

Figure 4

The Rice-Mele model. Depiction of the model of coupled Rice-Mele chains (center) used to reproduce the experimentally observed second harmonic generation in TaAs (left). An enlargement of the shaded unit cell is depicted on the right specifying the different hopping terms present in the model.

Figure 5

Minimized mean-squared deviation fits (solid lines) to the SHG intensity data (points) as a function of the incident polarization angle $\theta$, in three polarization channels: (a) parallel, (b) vertical, and (c) horizontal, for incident photon wavelength $\lambda =1400\mathrm{nm}$, plotted on a normalized scale. (d) Fits to the SHG intensity in the vertical channel assuming the relative sign between ${\chi }_{zxx}$ and ${\chi }_{zzz}$ to be $(+)$ (dashed line), and $(-)$ (solid line).

Figure 6

Measured FROG spectrum for input fields $\lambda =1100$ nm (nominally) showing the projection of the data along the temporal (above) and spectral (right) axes. Fits of the projected data to Guassian models, as described in the text, produced the observed pulse duration $\tau$, the center SHG wavelength $\lambda$, and the spectral width of the pulse $\mathrm{\Delta }\lambda$.

Figure 7

(a) Representative data from knife-edge measurement of the beam diameter and corresponding fit for ${\lambda }_0=1100$ nm incident light. (b) Experimentally measured transmittance of the 540-nm bandpass filter. (c) The linear optical correction factor $BP\left(\omega \right)$ as a function of incident photon energy.

Figure 8

A schematic of the first three cumulants of the polarization distribution. In each panel, one cumulant is varied while the others are held fixed. Note that the true polarization distribution is periodic in $L$.

Figure 9

(Left) Reconstructed polarization distributions in the extended Rice-Mele model via the maximum entropy method [30]. (Right) The gauge-invariant cumulants and spectral weight in the next-nearest neighbor extension of the Rice-Mele model. The maximum in the normal Rice-Mele model, Eq. (B23), is the black line, and the spectral weight as a function of $\gamma$ is computed via Eq. (C22). All parameters are given in the text.