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phys. stat. sol. (b) 245, No. 12, 2688–2691 (2008) / DOI 10.1002/pssb.200879900 p s sbasic solid state physics

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Spin-density-wave gap in (TMTSF)2PF6 probed by reflection-type terahertz time-domain spectroscopy

Shinichi Watanabe**, 1, Ryusuke Kondo2, Seiichi Kagoshima2, and Ryo Shimano*, 1

1 Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan 2 Department of Basic Science, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8902, Japan

Received 11 June 2008, revised 11 August 2008, accepted 13 August 2008

Published online 20 October 2008

PACS 71.30.+h, 72.80.Le, 75.30.Fv, 78.30.Jw

** Corresponding author: e-mail shimano@phys.s.u-tokyo.ac.jp, Phone: +81 3 5841 4181, Fax: +81 3 5841 4230 ** e-mail swatanabe@phys.s.u-tokyo.ac.jp, Phone: +81 3 5841 4182, Fax: +81 3 5841 4182

© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction The quasi-one-dimensional (1D) con-ducting systems provide a variety of attractive physical properties resulting from their low-dimensional nature of the electronic states. In particular, the nesting instabilities of the Fermi surface bring the formation of density wave states at low temperature accompanied by the metal to in-sulator phase transitions. The spin density wave (SDW) state is one of them, where an electron–electron Coulomb interaction leads to a break in the translational symmetry of the electronic systems. The single particle gap energy of the SDW state is usually in the order of ~10 meV (~80 cm–1 in wave numbers), and the study of the optical conductivity spectra σ(ω) in such an energy or frequency range is particularly important in order to study the gap formation mechanism in detail. Fourier transform spectroscopy is the standard tech-nique to determine σ(ω) in the wide spectral ranges from about 10 cm–1 to 10000 cm–1 [1], where one can observe not only the gap features [2, 3] but also the plasma edges in the metallic states [4], intramolecular vibration modes [4], and interband transitions [5]. This technique is quite pow-erful to determine the overall spectral features of the inves-tigating materials, while some technological difficulties exist. In reflection-type experiments, which are usually performed for opaque materials, what one obtains from the

Fourier transform measurements is the reflectivity ( )R ω only. Accordingly, one additionally needs to determine the phase spectrum ( )φ ω in order to calculate the conductivity spectra σ(ω). Normally, the determination of ( )φ ω is per-formed by applying the Kramers–Krönig relations, which relates ( )R ω and ( )φ ω in the integral forms [6]. As the in-tegral extends from 0ω = to ∞, one needs to measure ( )R ω for the whole spectral ranges in order to deter-

mine σ(ω). In reality, one needs to extrapolate carefully the ( )R ω spectrum in the low-frequency limit, which makes

the Fourier transform spectroscopy very sophisticated and difficult, in particular for the low-frequency range below 1 THz (33 cm–1). The recently developed terahertz time-domain spec-troscopy (THz-TDS) technique overcomes this difficulty by detecting both the reflectivity and the phase simultane-ously [7]. Although the typical frequency range is re-stricted between 10 cm–1 to 200 cm–1 (while sufficient for investigating the spectral gap associated with the SDW ground states), one can directly determine complex con-ductivity σ(ω) from the time-domain waveform without a Kramers–Krönig transformation. Furthermore, THz-TDS can be extended to optical pump and terahertz probe ex-periments with ps temporal resolution. Accordingly, the technique allows one to measure the ultrafast transient

We report on a development of the reflection-type terahertz

time-domain spectroscopy system and apply it to determine

the conductivity spectra of the spin density wave (SDW)

ground states in (TMTSF)2PF

6. We take special care over op-

timizing the spatial resolution of the measurement system in

order to investigate the tiny sample and also for correcting the

phase error of the measurement. We observe a distinct reduc-

tion of the optical conductivity below ~65 cm–1, which is a

clear indication of SDW gap formation with metal– insulator

transition.

phys. stat. sol. (b) 245, No. 12 (2008) 2689

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response of σ(ω), which is especially suitable to study the photoexcited dynamics in solids. In spite of these apparent advantages, the application of the THz-TDS techniques to the study of the SDW state has not been reported yet, one of the reasons for which, we think, is the difficulty in constructing the reflection-type THz-TDS measurement systems that is necessary for in-vestigating the optically opaque materials. Compared to the widely used transmission-type systems, the reflection geometry has its own difficulty associated with the great sensitivity of the sample position and the incident angle. In particular, the precise determination of the sample position is difficult but essential to properly determine the phase spectra ( )φ ω , where various experimental [8–10] and nu-merical [11, 12] methods to correct the phase errors have been proposed. Another difficulty to study the SDW ground state is the smallness of the sample size. An or-ganic conductor, (TMTSF)2PF6 (TMTSF denotes tetrame-thyltetraselenafulvalene) is the best-known example to ex-hibit the SDW ground state below Tc < 12 K [13–15], whose sample length (or width) is typically less than 1 mm. As the wavelength of the terahertz beam is almost the same as the sample size, diffraction effects are crucial and one needs to construct a proper measurement system in or-der to avoid a shape-dependent influence in the reflectivity spectra. In this paper, we report on the development of the re-flection-type THz-TDS system and apply it to determine the conductivity spectra of the SDW ground state in (TMTSF)2PF6. We take special care with the spatial resolu-tion of the measurement system for investigating the tiny sample and also for the correction of the phase error, in or-der to quantitatively determine the conductivity spectra of (TMTSF)2PF6 in the SDW phase. 2 Experimental setup The experimental setup for the reflection-type THz-TDS measurement system is shown in Fig. 1. A broadband (1–6 THz, 30–200 cm–1) terahertz pulse is obtained by the optical rectification pro- cess and detected by the electro-optic method both uti- lizing GaP crystals [16] illuminated by a mode-locked Ti:sapphire laser of ~90 fs in pulse width. The terahertz beam is focused onto the sample inside a liquid helium cryostat using an off-axis parabolic mirror with an effec-tive focal length of 101.6 mm, which determines the spatial resolution of the measurements with the effective numeri-cal aperture (NA) of about 0.25. The setup is in the nor-mal-incidence reflection geometry where the directions of both incident and reflected terahertz beams are perpendicu-lar to the sample surface. The reflected beam is led to the beam-detection system by a polyethylene terephthalate (PET) film, commercially called Mylar or Lumirror film, which works as a terahertz beam splitter. Special care has to be taken to choose the appropriate thickness of the PET film to avoid the destructive interference between the re-flected waves from the two surfaces of the film. The thick-ness of our beam splitter is 12 μm, where the destructive

Figure 1 Experimental setup of the reflection-type THz-TDS

measurements.

interaction occurs at 7.3 THz, assuming the refractive in-

dex of the film as ~1.85 [17]. Thus, the spectral profile of

the beam splitter is almost flat between 1–6 THz, which is

ideal for our measurements. Note that the efficiency de-

pends on the polarization of the incident terahertz beam,

where the P-polarization is less effective in our configura-

tion [18]. A wire grid polarizer is used to measure the po-

larization dependence of the reflection spectra. All the

terahertz beam path is kept in a dry nitrogen environment

in order to avoid absorption by water vapor. The single-crystal (TMTSF)2PF6 sample is grown by electrochemical oxidation, with the size of 4 mm on the a-axis and 0.7 mm on the b-axis, respectively, where the a-axis is the electronic one-dimensional axis [19]. A b′-axis is defined as perpendicular to the a-axis that is slightly tilted from the crystallographic b-axis because the crystal is triclinic. The sample is embedded on a copper plate glued by the epoxy resin Stycast 1266 with the a–b plane on its surface. As the sample size is comparable to the wavelength of the terahertz probe, the spatial resolu-tion of the measurement system is crucial. The Rayleigh resolution criterion for incoherent illumination [20] would be a very rough estimate, which is about 2.5 times the wavelength λ of the terahertz probe in our configuration (0.61 λ/ΝΑ). As the resolution is ~0.7 mm for 1 THz, the reflection measurement above 1 THz in frequency could be feasible for the above size of the sample. However, this estimation is very rough since the beam to be illuminated on the sample is totally coherent and the resolution strongly depends on the phase distribution on and around the sample that would harm the measurements through the constructive or the destructive interference [21]. We care-fully prepare the sample holder that makes the sample ele-vated with respect to the surrounding copper plate prevent-ing such interference, and only the reflected beam from the sample surface is measured. The appropriate monitoring and feedback system of the sample position are useful to correct the sample displace-ment error. We continuously monitor the position of

2690 S. Watanabe et al.: Spin-density-wave gap in (TMTSF)2PF6 probed by reflection-type THz-TDs

© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-b.com

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the sample mount by a CCD laser displacement sensor (Keyence, Model: LK-G85) with 10 ms sampling rate, and the amount of optical delay between the pump and the probe line is corrected numerically according to the meas-ured position. The resolution of the displacement sensor is 0.2 μm, which gives a phase resolution of ~±0.02 radian for 1 THz. The absolute value of the phase shift is finally calibrated by comparing the reflected spectra of two po-larizations parallel (E || a) and perpendicular (E || b′) to the a-axis, of which the method and validity are further dis-cussed later. 3 Experimental results First, we measure the re-flected terahertz waveforms above and below the SDW transition temperature (Tc = 12 K) for E || a polarization. In the frequency range of 30 cm–1 to 100 cm–1, we do not ob-serve any changes on the reflectivity spectrum between T = 20 K and 4 K. On the other hand, for E || b′ polariza-tion, we clearly observe changes of the amplitude and phase in the time-domain waveform of the reflected signal when the temperature is lowered from 20 K to 4 K, as shown by Fig. 2. Since the phase shift arising from SDW formation is small, we correct the displacement error for E || b′ by comparing the reflected waveforms of E || a in which polarization we assumed that there are no phase changes. The validity of this assumption is further dis-cussed in Section 4. In the SDW phase (T = 4 K), we ob-serve a slight reduction of the amplitude as well as a slight advancement of the phase. Figure 3 shows the reflectivity and the phase shift spectra of the SDW ground state with respect to its metallic ground state (T = 20 K). The reflec-tivity decreases below 70 cm–1 (2.1 THz), and the phase is advanced for all the frequency. Next, we estimate the conductivity spectrum σ(ω) of the SDW ground state utilizing both the reflectivity and the phase information in Fig. 3. As we only measure the re-flectivity and the phase change with respect to the values at T = 20 K, the information of the conductivity spectrum at 20 K is necessary for the evaluation. We try to estimate the SDW conductivity spectra at 4 K utilizing two reported Drude parameters of the metallic phase just above Tc [3, 4], and both give almost the same results. Figure 4 shows the estimated conductivity spectrum at 4 K with respect to the spectrum at 20 K. We observe a distinct reduction of the

Figure 2 Reflected terahertz waveforms above (T = 20 K; solid

line) and below (T = 4 K; dashed line) the SDW transition tem-

perature (Tc = 12 K) for E || b′ polarization.

conductivity below ~65 cm–1, which is a clear indication of the SDW gap formation. It is instructive to consider how the reflectivity and the phase change are related to the SDW formation, which provides useful information for the further analysis of the THz-TDS reflection measurements. The amplitude reflec-tivity ( )r ω and the phase spectra ( )φ ω are related to the complex refractive index spectra as

( )( )( ) ( )

( )( ) ( )

2 2

2 2

1

1

n

r

n

ω κ ω

ω

ω κ ω

- +=

+ +

, (1)

( )( )

( ) ( )2 2

2tan

1 n

κ ωφ ω

ω κ ω

-

=

- -

, (2)

where n(ω) and κ(ω) are the frequency-dependent real and imaginary parts of the refractive index, respectively. In the metallic state, the terahertz frequency is well below the scattering rate, where we can apply the formulation for the Hagen–Rubens regime as [22]

( ) ( )

1/ 2

dc

0

.2

n

σ

ω κ ω

ε ω

Ê ˆª ª Á ˜Ë ¯ (3)

where σdc is the conductivity at ω = 0. While n(ω) ~ κ(ω) → ∞, thus r(ω) → 1 and φ → –π for ω → 0, these do not take such ultimate values in the terahertz frequency range, especially when σdc is not so large, as in the case of

Figure 3 (a) Reflectivity and (b) phase-shift

spectra of the SDW state (T = 4 K) with re-

spect to the metallic state (T = 20 K). Fre-

quency ν is related to the angular frequency

ω as ν = ω/2π. The error bars are determined

by the signal to noise ratio and the resolution

of the displacement sensor described in the

text. The lines are guides to the eye.

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Figure 4 Conductivity spectrum at T = 4 K normalized by that of

T = 20 K. The error bars are determined by the measured ones in

Fig. 3. The line is a guide to the eye.

the E || b′ conduction in (TMTSF)2PF6 single crystals [4]. Therefore, φ is slightly less than –π at T = 20 K. The most drastic change in the SDW ground state at T = 4 K could be the lack of loss below the gap frequency, where κ(ω) → 0 can be applied. Therefore, the phase φ becomes –π regardless of the n(ω) values, which brings the ad-vancement of the phase in the SDW state as shown in Fig. 3b. The reduction of κ(ω) in the SDW phase results in decreasing r(ω) in Eq. (1), which is also consistent with the measured spectra in Fig. 3a. 4 Discussion The reason why the SDW gap is not evident in the reflectivity for the E || a polarization may be attributed to a large value of ( )n ω both in the metallic phase above Tc and in the SDW phase, which could over-come the contribution from the change of conductiv-ity σ(ω) and hence κ(ω), resulting in a very small ref-lectivity change and phase shift between T = 20 K and 4 K. Since what we measure is the reflectivity and the phase changes, if the refractive index at 20 K is quite high, φ is almost –π already at 20 K, thus the phase change associated with the SDW formation would not be observed. Additionally, the reflectivity change below Tc could be very small even if κ(ω) goes to 0 when the value of n(ω) remains large. In addition, the invariance of E || a re-flectivity on the SDW formation may imply that σ(ω) for a-axis direction is insensitive to SDW formation in the present frequency range of 30 cm–1 to 70 cm–1, as reported by Vescoli et al. [3] and Schwartz et al. [5] based on Fourier transform infrared spectroscopy measurements, in which the tail of the absorption band located around 200 cm–1 is considered to obscure the SDW gap. We need further justification for the interpretation of the E || a response, and THz-TDS technique with much broader spectral ranges [23] will give further information to clarify this is-sue.

5 Conclusion We report on a development of the re-flection-type THz-TDS system with a diffraction-limited spatial resolution of ~2.5 times the wavelength. The reso-lution of the phase shift is ~±0.02 rad at 1 THz, which is sufficient to observe a very small phase shift accompanied by the transition from metallic to SDW insulating phase in (TMTSF)2PF6. A distinct reduction of the conductivity be-low ~65 cm–1 is observed, which is a clear indication of the SDW gap formation.

Acknowledgements S.W. would like to thank Dr. Y.

Fuseya for valuable discussions. This work is in part supported

by a Grant-in-Aid for Scientific Research (B) (No. 18340082) of

the Ministry of Education, Culture, Sports, Science and Tech-

nologie, Japan.

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