Strong modulation of second-harmonic generation with very large contrast in semiconducting CdS via high-field domain

Dynamic control of nonlinear signals is critical for a wide variety of optoelectronic applications, such as signal processing for optical computing. However, controlling nonlinear optical signals with large modulation strengths and near-perfect contrast remains a challenging problem due to intrinsic second-order nonlinear coefficients via bulk or surface contributions. Here, via electrical control, we turn on and tune second-order nonlinear coefficients in semiconducting CdS nanobelts from zero to up to 151 pm V−1, a value higher than other intrinsic nonlinear coefficients in CdS. We also observe ultrahigh ON/OFF ratio of >104 and modulation strengths ~200% V−1 of the nonlinear signal. The unusual nonlinear behavior, including super-quadratic voltage and power dependence, is ascribed to the high-field domain, which can be further controlled by near-infrared optical excitation and electrical gating. The ability to electrically control nonlinear optical signals in nanostructures can enable optoelectronic devices such as optical transistors and modulators for on-chip integrated photonics.

Here ,i E w (or 2 ,i E w ) depicts the i-component (i=x, y, z) of electric field of FW (or SHG). In terms of electric field induced second-harmonic generation (EFISH), the second-order nonlinear coefficient is rewritten as, where, (2) ,0 ijk c stands for the nonlinear coefficient with no applied field and is zero for structures with mirror symmetry, and F l represents the applied field. Therefore, if the applied field is along the x-axis of CdS (F x ), the new nonlinear coefficients are induced, (2) ( (3) xzz xzzx x F c c = , and (2) (3) A new nonlinear coefficient means it is intrinsically zero and will be produced by applying an external stimulus. In this work, we focus on To determine the cystography of the CdS nanobelt, we performed SHG polarimetry 2 . As shown in Supplementary Fig. 1, the a-axis (or x-axis) is along the long axis of the nanobelt.
To estimate the error of measured d 11 , we looked at the power of the fundamental light, P w , which was measured by the laser power meter (LabMax-TO, Coherent). Its fluctuation is estimated to be D~±0.6%, causing the actual pump power to be P w *(1±D). When using d 33 to estimate d 11  Therefore the error or uncertainty of d 11 is 2´0.6%´151 pmV -1 = ±1.8 pmV -1 , i.e., d 11 =151±1.8 pmV -1 .

Analysis of Schottky contacts in our CdS nanobelt device
For simplicity, we only consider completely ionized donors contributing to the space charge in a classic Schottky barrier, and obtain the potential (V) by solving the Poisson equation 3,4 , where x d is the width of the Schottky barrier, r=qN D is the space charge, N D is the donor density, e 0 is the dielectric in vacuum, and e r is the dielectric constant of CdS. By assuming that the internal field is zero in the bulk semiconductor, ( ) = we obtain the internal electric field and current, where V s0 is the surface potential and is negative for n-type CdS, the built-in potential V in = -V s0 . If the Schottky barrier is positively biased on the metal (V>0), the forward current (J DS ) grows exponentially with respect to voltage whereas the barrier width (x d ) and field at the interface (x=0) decrease. If V<0, the reverse current approaches -J s0 with respect to voltage. In this case, the Schottky barrier increases in its width and internal field at each position.
To interpret the current response to the applied voltage or I-V curve (Fig. 2B) Here DS D S V V V = + , V D is the voltage applied on the contact D whereas V S is applied on the contact S. When the internal electric field in the contact S approaches a critical field, the field induced ionization of acceptor traps becomes significant and then initiates the high-field domain at the contact S (discuss latter), whereby the current saturates, as shown in Fig. 2B.
If a negative voltage (V DS <0) is applied upon the contact D ( Supplementary Fig. 2C), the contact D is reversely biased whereas the contact S is now forward biased. Since DD SS V V >> , the contact D is more significant to control the reverse current (J DS =J s0 (V DD +V D ) <0, reverse current relative to the contact D, direction from S to D) ( Fig. 4B and Supplementary Fig. 2C). In this case, an increase in applied voltage will increase the internal field and width near the contact D.
When the internal field approaches a critical field, the high-field domain can be induced near the contact D (Fig. 4A). The barrier height near the contact S decreases and has little impact on the I-V curve.

Supplementary Note 3 Schottky high-field domain model based on field induced ionization of acceptor traps
The high-field domain has been observed in CdS owing to field induced ionization of traps 3, 5-7 .
As known, CdS is n-type due to a large number of sulfur vacancies or electron traps or donors and the ionized donor traps are positively charged with a density of N D . In our case, there are also a large number of deep-level neutral accepter traps (cadmium vacancies) in CdS which can accept electrons and become negatively charged 8 . If a large electric field is applied, electrons can be excited from the valence band into accepter traps 4 . Therefore acceptor traps will be ionized and negatively charged with the density N A and contribute to the space charge (N D -N A ).
The ionized holes will be free in the valence band to recombine with electrons in the conduction band and can also participate in conduction via delocalized band states. This process is called field induced ionization of acceptor traps. If the density (N A ) of the ionized acceptor traps (1) When V<V c , an increase of the applied voltage (V) will increase the internal field (F) and width (x d ) of the Schottky barrier S (Supplementary Equations 5 and 7). In this case, the (2) When V=V c , F=F c at the metal-semiconductor interface of the contact S. Supplementary   Equation 9 is satisfied and the high-field domain is initialized at the interface. Its width is Since the applied voltage (V=V c ) is also applied upon the formation of the high-field domain, the (3) When V>V c , an increase of applied voltage will mostly be applied upon the Schottky barrier and drive the high-field domain to expand into a new region with the constant field F c following steps (1) and (2). This process extends the high-field domain by 4 ≈ ( − 4 ) 4 and rebuilds the barrier at x=x c .
Similarly, at a negative voltage (V=V DS <0), the HFD is initiated at the contact D which is reversely biased and expands toward to the contact S as applied voltage increases.
Once the high-field domain is initiated, an increase in V=V DS only extends the high-field domain by ( ) , and the electric field at any arbitrary position (x) is given by (Fig. 2D), , the point at x=x 0 has already been within the high-field domain and the field should be -F c .

High-field domain induced SHG
In our experiment, the fundamental wave (FW) is a Gaussian beam, where x 0 is the centre of the laser spot and σ = w/2 with the beam waist of 2w ~3 µm in our case.
Therefore, the SHG field excited from the excitation region (centred at x 0 ) is, where I w is the intensity of FW, the profile of the internal field ( )= ( ) c f x F x F and Where erf(x) is the error-function and defined as, If we consider a simple case without the domain boundary (or Schottky barrier), the field profile is a step function, Then the SHG field is, The SHG intensity is calculated from Supplementary Equation 12, Note that D +××× and g IR µI w , we assume the change of the field strength DF c =F c -F c0 is very small and ignore the high-order terms and obtain, , where a is the coefficient which is independent of the IR intensity and relates to the electric field strength (F c0 ) of the high-field domain without IR excitation. Hence, the normalized conversion efficiency of SHG is, and clearly shows linear response to the IR intensity, as observed in our experiments ( Fig. 4C and Supplementary Fig. 4D). mobility µ n =350 cm 2 Vs -1 , and temperature T=300 K, e r =8.9 for CdS 7 and fitted the voltage dependence of SHG to extract the critical field. As shown in Fig. 2A and Fig. 4A, our calculated SHG is fit to measured voltage dependences of SHG very well. To calculate SHG in Fig. 4D, we first extracted the relationship between the critical field and excitation intensity from Fig. 4A.

Then we substituted the critical field into Equation 1 and Supplementary Equation 19 to perform
calculations given the excitation intensity and the -30 V bias. The fitting is shown to be good in Fig. 4D.

Characteristics of CdS SHG transistor
Typically, one characterizes a transistor using ON/OFF ratio, mobility, subthreshold slope and/or modulation strength. For the CdS SHG transistor device, we do it in a similar way. Analogous to the characteristic of the drain current of metal-oxide-semiconductor field effect transistor (MOSFET) at the subthreshold region 4 , we define the subthreshold slope (k s ) for the SHG transistor, which indicates the transition rate (fast or slow) between OFF (low signal) and ON (high signal) states as a function of bias.
Besides, the ON/OFF ratio is defined as the ratio between SHG intensities in the ON and OFF states. The ON state corresponds to the saturation SHG (SHG max ), for example the SHG intensity at ~57 V in Fig. 1C. The OFF state is the SHG intensity (SHG min ) at ~22 V where SHG starts to appear and can be reliably recorded above the noise floor. The ON/OFF ratio is SHG max /SHG min . Please note that the OFF value will be more accurate and smaller if using a more sensitive detector. If the ON/OFF ratio is very large (approaching infinity), then modulation contrast is a more meaningful metric of the device. It is defined as (SHG max -SHG min )/(SHG max +SHG min ). If SHG max >> SHG min , the modulation contrast approaches 1, which is the case of perfect contrast. If SHG max~ SHG min , the modulation contrast approaches 0 and no modulation is achieved.
By analyzing the fitted parameters in Fig. 2A, Other studies have also been reported regarding electric-field induced SHG 12 and estimation of newly generated second-order nonlinear coefficients 13 .

Estimation of the response time of device
As described in the main manuscript, the high-field domain is due to field-enhanced ionization of traps. This is the process of driving electrons from the valence band to acceptor traps (i.e. deeplevel Cd vacancies) via electric field. Note that the cross-section of capturing holes is ~10 -17 cm 2 for these acceptor holes 8 . In order to achieve high-field domain, the large cross-section for capturing electrons (>10 -17 -10 -18 cm 2 ) is required to ionize these acceptor traps and can be achieved by electric field. Considering the density of acceptor traps N A ~10 16 -10 17 cm -3 8 , thermal velocity of electron <v> ~10 7 cm s -1 and the cross-section of capturing electrons S c >10 -17 cm 2 , the response time of acceptor traps can be estimated as 8 , However, this estimation can become significantly less, leading to a much faster response from the device due to strong applied fields which strongly modulates carrier trapping/de-trapping rates. Also, improved device geometries, smaller cross-section, operating close to the threshold via applying a d.c voltage below the threshold and a modulating voltage to cross the threshold, better spatial overlap, smaller spot size can significantly improve the switching times.

Heating effect on d 11 -SHG signals
When the current is high (e.g. 4 kA cm -2 in Fig. 2B), the heating effect could be introduced and influence the intensity of d 11 -SHG signals. It will cause the shift of exciton resonances and may lead to changes in third-order nonlinear coefficients. It may also affect the formation of highfield domain by altering the carrier dynamics, such as the decay rate of carriers in acceptor traps.
To evaluate the impact of heating, we performed the following experiments. We tested the device at 77 K and observed d11-SHG signals which behave similar to 300K (e.g. saturation, Supplementary Fig. 7A). Heating will not change the origin of d 11 -SHG. Then we heated the device with a heater (300K to 340K) and did not observe any d11-SHG signals (see below) without a bias voltage, showing the heating cannot produce d 11 -SHG signals. Please see more details in Supplementary Fig. 7B. Moreover, pump power dependence of d 11 -SHG shows that with an increase in pump power will increase the saturation current and d 11 -SHG when the excitation light is focused onto the high-field domain (see Fig. 4C). This shows the heating effect due to high current would have positive impact on d 11 -SHG. Gate voltage dependence of d 11 -SHG shows that increase in gate voltage will increase current and decrease d 11 -SHG (see Fig.   4E). This shows the heating effect due to high current would have a negative impact on d 11 -SHG.
Therefore, combining pump power and gate voltage dependence, it can be seen that heating does not significantly alter d 11 -SHG signals.

Discussion on Phase Mismatch in CdS nanobelts
Typically, the SHG field can be written as 1 , where A is related to the electric field of the fundamental wave, nonlinear coefficient and sample length (l), ∆ = &' − 2 ' = 4 is the phase mismatch between the second-harmonic (SH) wave and fundamental wave, l c is the coherence length, and &' (or ' ) is the wave-vector of the second-harmonic wave (or fundamental wave), ∆ 2 is the factor of the phase mismatch.
When phase mismatch (i.e. ∆ ≠ 0 ) occurs between the second-harmonic (SH) wave and fundamental wave, the SH wave is converted from the fundamental wave in the first coherence length and converted back to the fundamental wave in the second coherence length. This is one cycle and will be repeated if the sample is very long. If the phase matching is satisfied (i.e. ∆ = 0), the SH wave will always be converted from the fundamental light and grow continuously. This is why the phase matching is very important to achieve high conversion efficiency in bulk materials. In our work, the phase mismatch effect was not considered in the theoretical model because it does not induce SHG. It only contributes to the signal intensity after SHG was induced.
This effect was automatically considered in the SHG signal (e.g. d 11 -SHG) measured in our experiments, but it is not significant and has no impact on our reported results. First of all, the sample thickness is only few hundred nanometers (l ~300 nm) and well within one coherence length (l c ~800-900 nm). Given Supplementary Equation 24, the SHG field generated in our case is ~5.5% lower than the assumed case of phase matching. Hence the phase mismatch effect is not significant. Secondly, the effective nonlinear coefficient, d 11 , is obtained by comparing d 11 -SHG with intrinsic d 33 -SHG. Therefore, signal intensity reduction due to phase mismatch is cancelled out (i.e. d 11 -SHG/d 33 -SHG) and has little impact on d 11 . Thirdly, the voltage dependence of SHG is normalized and fitted to the model. The factor of phase mismatch will not have impact on the behaviour or trend. Therefore, it is common to ignore the phase mismatch issue in nanostructures 10 .