Terahertz Emission via Optical Rectification in a Metal-Free Perovskite Crystal

We report on the emission of high-intensity pulsed terahertz radiation from the metal-free halide perovskite single crystal methyl-DABCO ammonium iodide (MDNI) under femtosecond illumination. The power and angular dependence of the THz output implicate optical rectification of the 800 nm pump as the mechanism of THz generation. Further characterization finds that, for certain crystal orientations, the angular dependence of THz emission is modulated by phonon resonances attributable to the motion of the methyl-DABCO moiety. At maximum, the THz emission spectrum of MDNI is free from significant phonon resonances, resulting in THz pulses with a temporal width of <900 fs and a peak-to-peak electric field strength of approximately 0.8 kV cm–1—2 orders of magnitude higher than any other reported halide perovskite emitters. Our results point toward metal-free perovskites as a promising new class of THz emitters that brings to bear many of the advantages enjoyed by other halide perovskite materials. In particular, the broad tunability of optoelectronic properties and ease of fabrication of perovskite materials opens up the possibility of further optimizing the THz emission properties within this material class.


Calculation of THz Field Intensities 1.1 Linear Dependence of Optical Rectification
For THz fields generated via optical rectification, the field strength of the emitted THz pulse is proportional to the nonlinear polarization of the medium, which is related to the field strength of the excitation pulse ( by:  0 ()) ∝   () = ∑ ,  0  (2)    ()  () (S1) where is the nonlinear polarisation, is the second-order nonlinear susceptibility of the   ()  (2) medium, and is the permittivity of free space. 0 For a pulse of light, the total energy density (fluence) is directly related to the electric field strength via: where is the fluence.A relationship between and follows from the combination of Eqns.S1       and S2: This establishes a linear relationship between the fluence of our pump pulse and the electric field of the THz pulse.

Calculation of THz Field Strengths
To convert the arbitrary values of the THz waveforms to standard SI values, we make use of the formalism outlined by van der Valk et al. 1 .For an electro-optic sampling scheme employing a perfectly transparent and dispersion-free electro-optic sampling crystal, the relative change in intensities at the two photodiodes ( in the limit of low THz fields is given by: ∆   ) where is a vector containing the electric field components of the THz pulse, is the center     angular frequency of the probe pulse, is the refractive index of the detector material at the probe  0 wavelength, is the length of the crystal, and is the electro-optic coefficient.The vector accounts   41  for the direction of propagation of the probe beam, as well as the polarization of the probe beam relative to the crystallographic axes of the electro-optic detection crystal and is described in greater detail elsewhere. 1In practice, the length and direction of can be optimized by rotating the electro- optic crystal; assuming the polarization of the THz field is collinear with that of the probe field and that the crystal orientation has been rotated to maximize the observed THz field, reduces to a single  numerical factor that depends solely on the crystallographic orientation of the electro-optic crystal, which we denote .Thus, the magnitude of the THz field ( ) along the direction collinear to the     polarization of probe pulse is given by: This relationship assumes an ideal electro-optic crystal, in which reflection of the THz pulse at the aircrystal interface, along with absorption of the visible and THz beams within the crystal, are negligible.
To account for reflection losses at the air-crystal interface, we include a Fresnel factor ( ) that  12 depends on the refractive index of the electro-optic crystal in the THz region: Moreover, Eqn.S2 correctly returns the electric field strength if and only if is frequency  41 independent.In general, this assumes the the phase velocity of the terahertz pulse at each frequency matches the group velocity of the probing pulse, which may not hold for real materials.For our choice of detection crystal (2 mm (110)-cut ZnTe) within the observed ~2.5 THz detection bandwidth of our detection crystal, varies smoothly from approximately 4×10 -12 m/V to approximately 3.8×10 -12 , as  41 such, we take an average value of 3.9×10 -12 m/V for .We additionally must include a normalized  41 response function ( ) that accounts for the finite pulse width of the probe pulse, along with other effects, such as group and phase velocity mismatch between the THz pulse and the probe pulse.From these parameters, the terahertz field can then be expressed in terms of the relative change in intensity at the balanced photodiode as: Alongside , the response function is also dependent on the frequency of the emitted THz  41 (  ) radiation.To compute the response, we make use of parameterisations employed by Gallot et al. 2 .The resulting un-normalised response function for a selection of ZnTe crystals is given in Fig. S2, below.smoothly and thus for ease of calculation, we replace the frequency dependent response function with an average value , thus: Where is a constant of proportionality that converts the differential signal at the photodiodes to   the electric field strength.For additional ease of calculation, we have replaced the angular frequency ( with a matching expression of the absolute frequency of the probe pulse ( ).The relevant   ) f  constants for the calculation of are given in Table 1:  Finally, it is necessary to account for nonidealities arising from absorption losses and other factors (e.g.: crystal strain, surface imperfections, etc.) within our crystal.In practice, it is impossible to find simple mathematical expressions to account for these non-idealities-instead we account for these factors by comparing the performance of our ZnTe crystal with that of a 200 μm, (111) cut GaP crystal, whose electro-optic parameters are also given in Table 1, above.We then characterize the emission of an LiNbO 3 crystal using both our ZnTe detection crystal and the GaP detection crystal.We control the power of the resulting THz emission using a wire grid polarizer.
The uncalibrated and calibrated THz peak amplitudes are given in Fig. S2 Given that we are characterizing the same THz generation system, we expect the calibrated response of our two detection systems to return identical THz field strengths (to within a factor of the shot-to-shot variability of the 800 nm excitation laser).However, we find that the calibrated THz field strength detected using the GaP crystal is approximately nine times greater than when detecting a virtually identical field using ZnTe.Given the thickness of the GaP crystal is roughly 1/10 th that of the ZnTe crystal, this suggests that parasitic absorption losses within the ZnTe crystal are strongly attenuating the THz pulse as it passes through the material.These internal losses mean that the true electric field of the THz pulse emitted by the MDNI system is roughly 9.4 times greater than the detected by the GaP.Correcting for this effect means that the 'effective' value for our chosen ZnTe crystal is 139.1   kV cm -1 .

Calculation of Angular Dependencies Using Jones Matrix Formalism
To obtain the angular dependence of the emitted THz electric field, it is necessary to understand how the polarization of the pump and THz emission change as they propagate through the optical train displayed in Fig. 2 (Main Text).The polarization of the THz field is related to the polarization of the pumping field by the following matrix equation: where and are respectively the rotation matrices that account for the rotation of the half () () wave plate and the MDNI crystal, is a projection matrix that maps the electric field in the laboratory  frame of reference to the crystallographic co-ordinates of the MDNI crystal, maps the electric field  of the traversing pulse to the polarisation of the crystal, accounts for the phase difference of the  0 pump pulse along the ordinary and extraordinary axes of the half-wave plate, and and are     respectively the vector representations of the THz field and the pump pulse.
We begin with a pump pulse that is linearly polarised along the x-axis in the laboratory frame, the vector representation of the pulse is therefore given by: .
Considering first the traversal of the pump pulse through the half-wave plate, the form of is given  0 by the relevant Jones matrix 3 : .
Importantly, this form only applies when the polarisation of the incoming field is aligned to the  0 optical axis of the waveplate.To account for rotation of the optic, it is necessary to include two rotation matrices that map the field in the lab frame to optical axes of the waveplate and then map the rotating electric field back to the laboratory frame.This is accomplished by the two rotation matrices and .Rather than account for each of these processes separately, it can be ( +) ( -) convenient to take the product of the three matrices and use the resulting parameter instead, this gives: Ignoring prefactors, this gives: Having modelled the rotation of our pulse polarization, we then project the polarization along the crystallographic axes of our MDNI crystal.The relationship between the axes in the lab frame and the crystallographic axes is given in the inset of Fig. 2 (main text).We use these axes to define our transfer matrix, as: Finally, the form of our second-order nonlinear tensor, is defined by the space group of the crystal.  For MDNI, which adopts the R3 space group, takes the form 4 We note that exploiting Kleinman symmetry is only valid when the dispersion of the nonlinear susceptibility of the material is negligible.In cases such as SHG, this can represent a valid assumption; here it is likely inapplicable given the extremely large difference in the frequencies (~10 2 THz) of the 800 nm excitation pulses and the output THz pulse.We further demonstrate this by comparing the experimentally derived dependence on the orientation of the MDNI crystal to the dependence predicted using Eqn.S5, along with the form of given in Eqn.S12.The calculated curve, displayed   ′ in Fig. S3, is in poor agreement with the experimentally obtained data.As a result, we resort to the full form of given in Eqn.S11.We note that the previously obtained coefficients do note specify     a value for , which appears in Eqn.S11 but not in Eqn.S12.We estimate a value for by  31 (  ) 31 comparing the shape of the experimentally obtained and calculated angular dependencies for a variety of values (Fig. S4); a similar technique which has been employed previously obtain values  31 for the nonlinear susceptibility tensor in other materials. 5,6In particular, we find that altering the value of alters the amplitude ratio of the 'large' and 'small' peaks, which we label P 2 and P 1 , respectively. 31 By plotting the ratio of these two peaks and finding the value that best matches the experimentally determined peak ratio, we obtain a value for of approximately 2.48 pm/V.

Characterisation of MDNI crystals 3.1 Dynamic Range of the THz Detection System
To characterize the dynamic range of our THz detection system, we obtained the spectrum of a THz pulse produced by an inverse spin hall effect emitter (Fig. S7, below).To estimate the dynamic range of our instrument, we obtain a noise floor by blocking the THz beam path with a metal block and taking the resulting spectrum.The spectrum of the spin Hall effect system with and without the beam block are given below: Based on these measurements, we estimate that the dynamic range of our THz emission system is roughly 5×10 4 Arb.U (47 dB).

Figure S11. Comparison of the ATR spectrum of powdered MDNI (red lines) and DABCO (grey lines).
Assignments of the IR bands of MDNI are based on comparison of the IR bands to those of MDNI.The IR spectrum of DABCO was obtained from the NIST webbook. 7

Structural Characterisation of MDNI
To confirm the orientation of our MDNI crystal in the laboratory frame, we acquired single crystal XRD patterns of MDNI.To characterise the misorientation of our crystal and to establish the primary and secondary miller indices, we compared our obtained 2θ values for the [102] and [200] peaks with their expected positions, using a previously reported structure for the R3 phase of MDNI (a=b=c=7.259A, α=β=γ=84.747°).Based on this analysis, we find the misorientation of the primary and secondary miller indices is less than 4° in both cases.

Comparison of MDNI to Other Crystals
We compare the performance of MDNI to other crystals in Fig. 5 (main text).The cited performances are given in Table S4 below.Where possible, sources were chosen where the fluence, excitation wavelength and peak-to-peak field were known.In cases where the peak-to-peak field wasn't explicitly given, the field was estimated via digital extraction of the time-domain curve from a suitable figure in the reference.The efficiency of the THz emission from a given crystal with a finite length is also dependent on the phase matching between the THz and optical pulses.One metric used to characterize phase matching is the coherence length , which is estimated as follows: 15 reports for the refractive index of MDNI at the 800 nm pump frequency, it can be estimated since the refractive index of perovskites has been shown to be dependent on the bandgap energy. 16The refractive index of the metal-free perovskite DABCO-NH 4 Br 3 has a reported bandgap energy of 5.25 eV similar to the 5.12 eV for MDNI and average refractive index from 375 -650 nm of 1.7. 17Using this value for the , we estimate a coherence length in MDNI as 0.33 mm, which is comparable pump n to or higher than that of other organic crystals including DAST, DSTMS, OH1, and HMQ-TMS at 800 nm. 15,18,19 A we used crystals with thicknesses of approximately 1 mm, this estimate suggests that THz emission from MDNI can be increased using a thinner crystal.Additionally, there is also further opportunity to tune the coherence length of MDNI by pumping at shorter wavelengths where pump n should increase. 18

Figure S1 .
Figure S1.Calculated response function (R(ω THz )) of ZnTe for three crystal thicknessesAs with the electro-optic coefficient, within the bandwidth of the THz emission, varies (  )

Figure S2 .
Figure S2.Referencing of ZnTe crystal against GaP crystal.(left) Peak THz field of an LiNbO 3 emitter measured using the <110> cut ZnTe crystal employed in this study (orange curve), and a 200 µm <111> cut GaP detection crystal.The emittedTHz field was attenuated using a wire-grid polariser.The output of the LiNbO 3 emitter is identical across the two measurements and thus the detected peak THz field should also be identical.(right) Ratio between the peak THz field amplitudes sampled using the ZnTe and GaP crystals; the ratio between the measured fields is consistent as a function of polariser angle (and hence, THz intensity) at around 9.4.

Figure S3 .
Figure S3.THz emission from MDNI and a spintronic emitter taken under the same conditions.

2 Figure S4 .FigureFigure S6 .Figure S7 .
Figure S4.Comparison of calculated THz emission properties with (dashed red line) and without (solid blue line) the application of Kleinman symmetry.The peaks P1 and P2 are used to optimise the calculated curve to obtain agreement with experiment.

Figure S8 .
Figure S8.Spectrum of the THz pulse (blue line) produced using the spintronic emitter in the absence of an MDNI crystal; the noise floor of our detection system (grey line) is also given, with the average noise level demarcated by the dashed line.

Figure S9 .
Figure S9.THz absorbance of MDNI as a function of crystal rotation angle for values of θ between 60 and 150 degrees.

Figure S10 .
Figure S10.Absorbance and refractive index determined for the MDNI powder.
the wavelength, frequency and refractive index of the THz radiation, index of the optical pump, and c is the speed of light.For MDNI, we pump n determined to be approximately 2 at a frequency of 1.5 THz.Although there are no known THz n

Table S1 .
Parameters employed for calculation of THz Conversion Factor 14.8 1032 kV cm -1 1.3 Referencing of ZnTe Crystal

Table S2 .
Parameters used to calculate the peak-to-peak THz emission for MDNI and a spintronic emitter.

Table 3
Values of the nonlinear optical coefficients for MDNI obtained byKasel et al.

Table S4 .
Comparison of peak-to-peak field strengths of organic and perovskite THz emitters.