Supplemental Material Carrier dynamics of rubrene single-crystals revealed by transient broadband terahertz spectroscopy H. Yada 1, R. Uchida 1, H. Sekine 1, T. Terashige 1, S. Tao 1, Y. Matsui 1, N. Kida 1, S. Fratini 2,3, S. Ciuchi 4, Y. Okada 1, T. Uemura 1, J. Takeya 1, and H. Okamoto 1 1 Department of Advanced Materials Science, University of Tokyo, Kashiwa, Chiba 277-8561, Japan 2 Univ. Grenoble Alpes, Inst NEEL, F-38042 Grenoble, France 3 CNRS, Inst NEEL, F-38042 Grenoble, France 4 Dipartimento di Scienze Fisiche e Chimiche Universit_a dell'aquila, CNISM and Istituto Sistemi Complessi CNR, via Vetoio, I-67010 Coppito-L'Aquila, Italy Contents S1. Optical-pump terahertz-probe spectroscopy S2. Analyses of time evolutions of OD S1
S1. Optical-pump terahertz-probe spectroscopy We used two setups (system I and II) of the optical-pump terahertz-probe spectroscopy. Figure S1 shows a schematic of the experimental setup of system I. The light source of the system is a regenerative amplifier with the pulse width of 25 fs, the central wavelength of 800 nm (1.55 ev), the pulse energy of 2.4 mj, and the repetition rate of 1 khz. The output of the regenerative amplifier is divided into two beams. One is frequency-doubled by a type I β-barium borate (BBO) crystal and used as a pump light, whose photon energy (3.1 ev) is large enough to generate directly photocarriers in rubrene. The other output is reflected by a pair of chirped mirrors and further split into two beams, which are used for generating and detecting THz probe pulses. One beam is focused in air by a lens (f = 150 mm) with a frequency doubled pulse (3.1 ev) generated by a BBO crystal, which is placed between the lens and the focal point. THz probe pulses are generated from air-plasma induced by focusing two-color laser pulses in air [24-27]. Those THz pulses are focused in a sample by two off-axis parabolic mirrors. Transmitted THz pulses are further focused on a ZnTe (110) crystal, whose thickness is 10 µm. The other beam is used for the electro-optic (EO) sampling. That beam is focused on the ZnTe crystal and is sent to a balanced detector consisting of a quarter-wave plate, a Glan laser prism, and a balanced photodiode. In this system I, the measurement from 2.5 to 15 THz is possible, except for the region of 4 6 THz. In this region, the strong phonon absorption of the ZnTe crystal makes it difficult to detect terahertz electric fields. The setup of system II is almost the same as that of system I, expect for the specifications of the regenerative amplifier. In system II, the output of the laser source S2
(another regenerative amplifier) has the pulse width of 130 fs, the central wavelength of 800 nm, the pulse energy of 0.8 mj, and the repetition rate of 1 khz. In this system, chirped mirrors were not used. Instead of the thin ZnTe crystal (10 µm) in system I, we use as an EO crystal, GaP(110) (0.3 mm) or a ZnTe(110) (1 mm), with which the measurement from 1 to 4 or 2.5 THz is possible, respectively. For the measurement of the absorption spectrum in terahertz frequency region, an electric-field wave-forms transmitted through the sample, ( ), and without the sample, ( ), were measured. Their Fourier components ( ) and ( ) are related to the complex refractive index = + κ of the sample via the following equation, ( ) ( ) = 4 ( ) ( ( )+1) exp ( ( ) 1), (S1) where is the light velocity, d is the thickness of the sample. In this formula, the effect of the multiple reflections is not taken into account. By solving the equation numerically, we calculate the complex refractive index. From the spectra, the spectrum of the absorption coefficient α can be obtained as shown in Fig. 1(b). For the derivation of transient ( ) spectra, we utilized the following equation [S1], ( )= 2 0 on ( ) off ( ), (S2) off ( ) where ε0 is the permittivity of vacuum, n = 2.29 [14] is the refractive index of rubrene, and d=2.36 µm [16] is the absorption depth of the pump light (3.1 ev). ( ) and ( ) are Fourier components of the terahertz electric fields with and without a pump pulse, respectively. S3
S2. Analyses of time evolutions of OD The time evolutions of OD at 294 K and 50 K [Figs. 3(c) and (d)] are replotted in Fig. S2. They can be almost reproduced by the formula, OD= exp + exp +, (S3) as shown by green lines in Fig. S2. In the fittings, the response function of the measurement system corresponding to the time resolution, exp( / ) ( =0.3 ps), is convolved. Obtained parameters are =0.047 ps, =10.6 ps, and = 0.972 : 0.021 : 0.006 at 294 K, and =0.16 ps, = 2.7 ps, and = 0.920 : 0.070 : 0.010 at 50 K. The first term reflects the ultrafast relaxation of hot carriers initially photogenerated. The second term shows the decay of hole carriers responsible for the Drude response shown in Fig. 3(a). The decay time of the second term = 2.7 ps at 50 K is shorter than =10.6 ps at 294 K. This might be attributed to the increase of the encounter rate of a mobile hole and a trapped electron with increase of µh. In this fitting, the third term (the constant term) is necessary to reproduce well the experimental data. The weights of the third terms cannot be neglected as compared to the second terms. The OD signals reflect the conducting hole carriers, so that the introduction of the constant term might not be reasonable from the physical viewpoint. A more realistic model to reproduce the time evolution of OD is the bimolecular recombination model, in which the rate equation of the carrier density, ( ), is expressed as ( ) = ( ), (S4) where α is a bimolecular recombination rate of a hole and a trapped electron. From Eq. S4
(S4), we can obtain a solution for ( ) as ( )= (0) 1+α (0). (S5) The resultant fitting function is given as follows. OD= exp + (0) 1+α (0) (S6) We performed the fitting using this formula and considering the response function of the measurement system exp( / ) ( =0.3 ps) by the convolution. The experimental time evolutions of OD were reproduced well by Eq. S6 as shown by the calculated curves (black broken lines) in Fig. S2. The fitting parameters are / =0.0082, =0.043 ps, (0)=2.2 10 4, α=490 at 294 K, and / =0.0083, =0.14 ps, (0)=9.9 10 4, α=830 at 50 K. We also plotted OD and the same calculated curves (broken lines) as a function of time in Figs. S2(c) and (d). In fact, OD is almost proportional to time t for td>2 ps, following Eq. (S6). The slopes of the broken lines for td>2 ps reflect the magnitudes of α. The larger slope of the broken line at 50 K compared to that at 294 K is attributable to the larger mobility µh of hole carriers. S5
Regenerative amplifier 25 fs, 800 nm, 2.4 mj, 1 khz BS BBO OC 250 Hz DS CP f = 150 BBO Air plasma PM CP BS PM pump Si OC 500 Hz Sample probe PM DS PM ZnTe (10 µm) QWP Purge Box GLP BP Fig. S1 Experimental setup of the optical-pump THz-probe spectroscopy (system I). BS: beam splitter, OC: optical chopper, DS: delay stage, CP: chirped mirror, PM: parabolic mirror, QWP: quarter wave plate, GLP: Glan laser prism. S6
0.0008 0.0006 (a) 294 K 0.002 (b)50 K OD 0.0004 0.0002 OD 0.001 0 0 30000 0 5 10 Delay Time (ps) (c) 294 K 30000 0 5 10 Delay Time (ps) (d) 50 K OD 1 20000 10000 OD 1 20000 10000 0 0 5 10 Delay Time (ps) 0 0 5 10 Delay Time (ps) Fig. S2 (a, b) Time evolutions of the absorption changes ( OD) measured by system II with 1 mm thickness ZnTe (110) at (a) 294 K and (b) 50 K. The excitation photon density x ph is 2.2 10 18 cm 3. The fitting curves by Eq. S3 and Eq. S6 are shown by green lines and black broken lines, respectively. (c, d) Time evolutions of OD 1 at (c) 294 K and (d) 50 K. The fitting curves by Eq. S6 shown in (a) and (b) are also indicated by black broken lines. S7
References [S1] H. -K. Nienhuys and V. Sundström, Phys. Rev. B 71, 235110 (2005). S8