arXiv:1302.1087v1 [cond-mat.mes-hall] 26 Jan 2013
C. W. Luo1,∗ C. C. Lee1, H.-J. Chen1, C. M. Tu1, S. A. Ku1, W. Y. Tzeng1, T. T. Yeh1, M. C. Chiang1, H. J. Wang1, W. C. Chu1, J.-Y. Lin2,† K. H. Wu1, J. Y. Juang1, T. Kobayashi1,3, C.-M.
Cheng4, C.-H. Chen4, K.-D. Tsuei4, H. Berger5, R. Sankar6, F. C. Chou6, and H. D. Yang7 1
Department of Electrophysics, National Chiao Tung University, Hsinchu 300, Taiwan, R.O.C.
2
Institute of Physics, National Chiao Tung University, Hsinchu 300, Taiwan, R.O.C.
3
Advanced Ultrafast Laser Research Center, and Department of Engineering Science, Faculty of Informatics and Engineering, University of Electro-Communications,
1-5-1 Chofugaoka, Chofu, Tokyo 182-8585, Japan
4
National Synchrotron Radiation Research Center, Hsinchu 30076, Taiwan, R.O.C.
5
Institute of Physics of Complex Matter, EPFL, 1015 Lausanne, Switzerland
6
Center for Condensed Matter Sciences, National Taiwan University, Taipei 106, Taiwan, R.O.C. and
7
Department of Physics, National Sun Yat-Sen University, Kaohsiung 804, Taiwan, R.O.C. (Dated: February 6, 2013)
This study shows that a terahertz (THz) wave can be generated from the (001) surface of cleaved Bi2Se3 and Cu-doped Bi2Se3 single crystals using 800 nm femtosecond pulses. The generated THz
power is strongly dependent on the carrier concentration of the crystals. An examination of the dependence reveals the two-channel free carrier absorption to which Dirac fermions are indispensable. Dirac fermions in Bi2Se3 are significantly better absorbers of THz radiation than bulk carriers at
room temperature. Moreover, the characteristics of THz emission confirm the existence of a recently proposed surface phonon branch that is normalized by Dirac fermions.
PACS numbers: 71.27.+a, 78.47.D-, 78.68.+m
Three-dimensional topological insulators (TIs) are characterized by a narrow band gap in the bulk and a Dirac cone-like conducting surface state [1–3]. The sur-face state is a new state of quantum matter caused by the strong spin-orbit interaction and protected by time-reversal symmetry. The special properties of TIs have applications in spintronics and quantum computations. Certain TIs with a small band gap are especially use-ful for terahertz (THz) optoelectronics. One of the key issues about TIs has been the identification of the gap-less surface electronic states (Dirac fermions) and the characterizations of their fundamental properties. Angle-resolved photoemission spectroscopy (ARPES) [1, 4–6] and scanning tunneling microscopy (STM) [7–10] have successfully confirmed the existence of Dirac fermions in Bi1-xSex, Bi2Se3, and Bi2Te3. Regarding the trans-port measurements, a metallic channel associated with the protected surface state has been detected by either controlling the gate voltage in TI devices with a suffi-ciently low bulk carrier density or by using very thin TI films [11–14]. However, these experiments used partic-ularly specialized instruments, and their procedures are excessively complex for quick and routine characteriza-tions of Dirac fermions in TIs. Hsieh et al. [15] showed an alternative approach using second harmonic generation (SHG) in arsenic- doped Bi2Se3single crystals associated with Dirac fermions, which showed a new venue for exam-ining Dirac fermions by contact-free optical techniques. However, SHG is highly sensitive to the surface quality of samples and doping. THz waves may in principle be an ideal tool for distinguishing Dirac fermions from bulk carriers because they are not sensitive to the surface
qual-ity of samples with long wavelengths. Furthermore, THz wave have a photon energy (approximately 4 meV) that is significantly lower than the bulk gap (approximately 300 meV) of TIs; thus, THz radiation would allow specific characterizations within the Dirac cone. Aguilar et al. [16] recently showed the THz responses of Dirac fermions in Bi2Se3 thin films. However, this type of THz exper-iments can only be applied to thin TIs of several tens of quintuple layers. This study shows a THz generation from pure Bi2Se3and Cu-doped Bi2Se3single crystals by pumping with femtosecond laser pulses. Dirac fermions were identified to have an indispensable role on the inten-sity of THz emission. Free carrier absorption is a crucial mechanism to the optoelectronic devices of TIs and was revealed from the dependence of generated THz power on the carrier concentration [17]. Moreover, the detailed characteristics of THz generation verified a newly pro-posed phonon branch normalized by Dirac fermions [18]. Single crystals of pure Bi2Se3 and Cu-doped Bi2Se3 were grown using either the Bridgeman, Melt growth, or CVT methods [19]. Single crystals of CuxBi2Se3were ob-tained using a slow-cooling method from 850 to 650◦C at a rate of 2◦C/h and quenching in cold water. Scotch tape was used to cleave the (001) surface of the Bi2Se3 crystals to ensure a flat and bright surface for optical measurements. The carrier concentrations of the samples listed on Table I were obtained using the Hall measure-ments. The mobility was measured using the four-probe method. A reflection-type THz generation scheme was used to generate a THz wave on TIs, as shown in Fig. 1(a) and the inset of Fig. 1(b). An 800 nm Ti:sapphire laser (FemtoLasers, Inc.) beam with a repetition rate of
TABLE I. Carrier concentration and the THz peak amplitude for samples grown by different methods (a
Bridgman,b
CVT,
c
Melt growth). All samples are n-type. Carrier
Code Compounds concentration THz peak amplitude (-1018 cm-3 ) (arb. units.) a #1 Bi2Se3 75.5±13.6 0.72±0.62 b #2 Bi2Se3 34.6±4.37 4.49±0.67 a #3 Bi2Se3 31.0±1.61 3.05±0.51 c #4 Bi2Se3 15.6±10.3 6.43±0.43 a #5 Cu0.02Bi2Se3 3.66±0.16 31.5±0.44 a #6 Cu0.08Bi2Se3 4.23±1.17 32.3±0.51 c #7 Cu0.1Bi2Se3 1.96±0.86 22.8±0.17 c #8 Cu0.125Bi2Se3 1.17±0.56 30.3±0.22
5.2 MHz and a pulse duration of 50 fs was incident at θ = 45◦(to the surface normal) and focused on the surface of the samples with a diameter of 43 µm. The pumping fluence was tuned by varying the laser output power (the typical value for this study was 0.37 mJ/cm2). Follow-ing femtosecond pulse pumpFollow-ing, the generated THz wave was collected using a pair of off-axis parabolic mirrors and focused on a 1-mm-thick ZnTe crystal to allow its detection with electro-optical (EO) sampling [20]. The entire generation and detection systems were sealed in a nitrogen-filled plastic box to reduce the humidity to < 6.0%. All optical measurements were performed at room temperature.
Fig. 1(b) shows the typical THz waveform generated from pure Bi2Se3 and Cu-doped Bi2Se3 single crystals with a reflection-type setup (THz radiation cannot be de-tected after TIs). The amplitude of the THz wave gen-erated from pure Bi2Se3 single crystals is significantly smaller than that from a Cu0.02Bi2Se3 single crystal. Certain Bi2Se3crystals such as sample #1 produce nearly zero amplitude (below the S/N ratio in the detection sys-tem). The THz generation intensity is strongly depen-dent on carrier concentration and doping. In general, the THz waveform is composed of a large single pulse and a damped oscillation, which is due to the interference be-tween THz electric fields from different positions of the crystal. These electric fields are caused by the mismatch between the THz phase velocity and the group velocity of the optical pumping pulse [21]. Time-domain THz wave-forms (Fig. 1(b)) can be converted to frequency domain spectra (Fig. 2) by using fast Fourier transform (FFT). The central frequency for Cu0.02Bi2Se3and bandwidth is approximately 1.2 THz and 1.6 THz, respectively. The intensity of pure Bi2Se3 THz spectra is relatively small, corresponding to small THz signals in the time domain.
THz signals were measured at various azimuth angles ϕ along the surface normal to understand the THz genera-tion mechanism in TIs (inset of Fig. 1(b)). The data from inset of Fig. 3 clearly show that the THz peak amplitude
0 1 2 3 4 5 (b) #1 Bi2Se3 #2 Bi2Se3 #3 Bi2Se3 #5 Cu0.02Bi2Se3 InAs x 0.01 Delay Time (ps) T H z fi e ld ( a . u .) fs laser pulse THz pulse TI Dirac-cone surface state (a)
FIG. 1. (color online) (a) Schematic illustration of THz gen-eration and FCA inside topological insulators. The dots in-dicate the free carriers. (b) THz waveform generated from various Bi2Se3 and Cu0.02Bi2Se3single crystals, and an InAs
wafer. Inset: schematic illustration of THz generation and the detection scheme.
is virtually independent of ϕ and at odds with the opti-cal rectification simulation curve with six-fold symmetry [22]. Therefore, optical rectification is not the dominant mechanism for THz generation in TIs, and the nonlinear effect does not mainly contribute to THz generation in TIs.
The band gap of 0.3 eV in Bi2Se3 is significantly smaller than the pumping photon energy of 1.55 eV. Free carriers are generated when the femtosecond laser illu-minates Bi2Se3 or Cu-doped Bi2Se3 crystals. These ex-cited carriers are located inside the bulk within 100 nm [23]. When the electric field is built inside the crystals, the excited carriers in the bulk are driven and form the currents. Two types of built-in electric fields are nor-mally present in semiconductors. The surface depletion field results from the bending of the conduction band on the semiconductor surface as in GaAs [24]. The photo-Dember field is caused by the inhomogeneous distribution of holes and electrons [25], which is usually present in narrow bandgap semiconductors, such as InAs and InSb
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 F o u ri e r p o w e r (a . u .) Frequency (THz) InAs x 10-4 #5 Cu 0.02Bi2Se3 #2 Bi 2Se3 #3 Bi 2Se3 0.0 0.1 0.2 0.3 0.4 0.5 1.0 1.5 2.0 2.5 3.0 #5 Cu0.02Bi2Se3 Fluence(mJ/cm2 ) TH z pe a k a m plit ude ( a . u.)
FIG. 2. (color online) Fourier power spectra in the frequency domain converted from the time-domain THz signals in Fig. 1(b) using fast Fourier transform for Bi2Se3and Cu0.02Bi2Se3
single crystals, and an InAs wafer. Inset: the THz peak am-plitude of a Cu0.02Bi2Se3 single crystal as a function of the
pumping fluence.
with a bandgaps of 0.35 eV and 0.17 eV, respectively. The bandgap is approximately 0.3 eV for Bi2Se3 single crystals, which is close to that of InAs (reference sample in Fig. 1). Additionally, the intrinsic charge inhomogene-ity in the vicininhomogene-ity of the surface and, as a result, band-bending effects were reported [26]. Consequently, both photo-Dember and surface depletion effects are possible mechanism for THz generation in Bi2Se3 and Cu-doped Bi2Se3 single crystals. The currents formed by the ex-cited carriers are suppressed within several picoseconds, due to carrier scattering with impurities (e.g., Se vacan-cies) or with the layer boundary. The transient current further generates THz radiation by ETHz(t) ∝ ∂J(t)/∂t. Higher pumping fluences generate more free carriers, leading to larger changes of the transient current; thus , a stronger THz radiation should be generated. The THz peak amplitude increases linearly with the pumping fluences (inset of Fig. 2) [27].
The excited carriers in the bulk can diffuse either along the [001] direction or on the (001) plane to form two types of currents. In principle, the diffusion along the [001] direction is suppressed by the layer boundary or im-purity scattering to generate p-polarized THz radiation, and the diffusion on the (001) plane is suppressed mainly by impurity scattering to generate s-polarized THz ra-diation. The addition of a single wire-grid polarizer be-tween the sample and an EO detection system allows the p- and s-polarized THz radiation from TIs (e.g., from the Cu0.08Bi2Se3single crystal) to be distinguished (Fig. 3). Intriguingly, the difference between p-polarized and s-polarized THz radiation is not only in the central fre-quency, but also in the shape of the spectra. Subtraction of the s-polarized THz spectrum from the p-polarized
0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 50 100 150 200 250300 350 0 5 10 15 PTHz-STHz PTHz STHz #6 Cu0.08Bi2Se3 N o rm a li z e d F o u ri e r p o w e r Frequency (THz) Cu0.08Bi2Se3 Similation T H z p e a k a m p li tu d e (a . u .) I(degree) 0.0 0.2 0.4 0.6 qX (Å -1)
FIG. 3. (color online) Polarization-dependent Fourier power spectra of a Cu0.08Bi2Se3 single crystal. PTHz (STHz): The
electric field of the p-(s-)polarized THz radiation is parallel (perpendicular) to the plane of incidence (see the inset of Fig. 1(b)). The solid squares are the surface topological phonon dispersion data taken from Ref. [18]. Inset: THz peak am-plitude for a Cu0.02Bi2Se3 single crystal as a function of the
azimuth angles φ along the [001] direction, as shown in the inset of Fig. 1(b).
THz spectrum results in the gray area between 0.53 THz and 1.72 THz (Fig. 3), indicating additional absorp-tion for the s-polarized THz wave. Zhu et al. [18] re-cently measured the surface phonon dispersion in Bi2Se3 crystals using the coherent helium beam surface scatter-ing technique. They discovered a low-energy isotropic convex-dispersive surface phonon branch normalized by Dirac fermions and with a frequency range from 1.8 to 0.74 THz. This frequency range is consistent with that of the gray area in Fig. 3. Therefore, the missing power of an electromagnetic wave with an electrical field parallel to the surface manifests the additional effect due to the Dirac-fermion-normalized surface phonons.
Table I shows that the carrier concentration decreases by more than one order of magnitude when Cu is doped into Bi2Se3 crystals (n-type, caused by Se va-cancies) because Cu replaces Bi in the lattice [28–30]. The carrier concentrations from 15.6±10.3×1018cm-3to 75.5±13.6×1018cm-3 were observed for the pure Bi2Se3 crystals, potentially because of the different growing con-ditions and methods. The THz signals from pure Bi2Se3 crystals are generally relatively small. Conversely, the Cu0.02Bi2Se3 crystal with a lower carrier concentration produces stronger THz emission than pure Bi2Se3 crys-tals. The spectral weight of the FFT spectra in Fig. 2 was plotted as a function of the carrier concentration to further quantify these results. Fig. 4 clearly shows that the THz output power (i.e., the spectral weight of the FFT spectrum) increases with the decreasing carrier concentration.
The THz wave generated inside the bulk likely suffers free carrier absorption (FCA) during propagation. This
effect can generally be reduced by suppressing the carrier concentration to increase the output intensity of the THz, as demonstrated in III-V semiconductors such as InAs [31]. The dependence of the THz intensity on the carrier concentration can be phenomenologically described using the Beer-Lambert Law:
ITHz= I0,THze−σln (1)
where ITHz and I0,THzare the transmitted THz intensi-ties outside and inside the samples, respectively, as shown in Fig. 1(a), l is the path length, n is the carrier concen-tration, and σ is the absorption cross-section. The exper-imental data in Fig. 4 can be fitted as the black dashed line by Eq. (1), and is qualitatively similar to the case of InAs [31]. However, a closer inspection reveals that the fit by Eq. (1) leads to a larger deviation from the data in the low concentration regime. Consequently, the experimental data in Fig. 4 cannot be explained solely by FCA due to the bulk carriers.
It is known that Dirac fermions exist in Bi2Se3 crys-tals [1]. The Dirac band contribution to FCA on the THz emission must be considered. Fig. 1(a) shows that the THz wave can be absorbed by the electrons on the Dirac cone. The effective two-channel FCA including the con-tribution from both the bulk carriers and Dirac fermions is written as a modified Beer-Lambert equation:
ITHz= I0,THze−(σblbnb+σsls ns
ls) (2)
where σbis the absorption cross-section of bulk, lbis the path length of the bulk, nb is the carrier concentration in bulk, σs is the absorption cross-section of the surface state, ls is the path length of the surface state, and ns is the carrier concentration in the surface state. The empirical relation between nb and ns can be described by ns×10-13=0.51+2.10[1-exp(-nb×lb×10-13/20.48)] as shown in the inset of Fig. 4 [32]. The observed spec-tral weight of THz in Fig. 4 is significantly better fit by Eq. (2) (the red solid line) with ls = 2 nm (thickness of the surface state) [33] and lb = 23.5 nm (THz emit-ted from the bulk within 23.5 nm; i.e., the penetration depth of 800 nm pumping light) [34]. The acquired fitting parameters are σb=2.59×10-14cm2and σs=1.46×10-13 cm2, and the ratio of σ
s/σb= 5.64. Therefore, this study concludes that FCA caused by Dirac fermions is more ef-ficient than that caused by bulk carriers (at least at room temperature).
FCA can be explained in the context of the Drude model. The FCA cross-section
σ = eµ
ncε0(1 + ω2τ2) (3)
where e is the electron charge, µ is the mobility, ω is the angular frequency of the THz wave, τ is the scatter-ing time of the carriers, n is the refractive index of the material, c is the seed of light, and ε0 is the permittiv-ity constant. Equation (3) indicates that the observed
0.0 2.0x1019 4.0x1019 6.0x1019 8.0x1019 #8 #5 #4 #2 #1 #3 #6 S p e c tr a l w e ig h t o f F F T s p e c tr u m ( a .u .) Carrier concentration (cm-3) #7 -2 0 2 4 6 8 10 12 14 0.4 0.6 0.8 1.0 1.2 1.4 1.6 ns ( x 1 0 1 3 c m -2 ) n bxlb(x10 13 cm-2 )
FIG. 4. (color online) Spectral weight of the fast Fourier transform (FFT) spectra vs. the carrier concentration. The black dashed line represents the fit by Eq. (1). The red solid line represents the fit by Eq. (2). Inset: the car-rier concentration ns of the Dirac fermions vs. nb of the
bulk carriers. The dotted line represents the empirical fit of ns×10
-13
=0.51+2.10[1-exp(-nb×lb×10 -13
/20.48)].
cross-sectional ratio is as follows: σs σb = µs µb (1 + ω2τ2 b) (1 + ω2τ2 s) (4) where s denotes the physical properties of the Dirac fermions of the surface state and b denotes the physi-cal properties of the bulk carriers. In this study, µb ≈ 1500 cm2/V s at room temperature for the typical Bi2Se3 single crystals. The mobility can be expressed as µb = eτ /m, where m is the effective mass of the carriers. With the effective mass of the bulk carrier mb=0.14m0[35, 36], where m0is the electron bare mass, τb≈0.12 ps at room temperature. According to Ref. 36 and the references therein, τs≪τbat low temperatures (low T ); so is it as-sumed at room temperature. The above analysis results in ωτb≈1.2 and ωτs≪1. Usually, µs is smaller than µb at low T in Bi2Se3 single crystals of approximately 100 µm in thickness [11, 35, 36]. However, the experimen-tal data for µsat room temperature have been relatively rare. Equation (4) indicates that the ratio σs/σb = 5.64 may lead to comparable values for µs and µb at room temperature. Generally, the effective mass varies weakly with T. Conversely, τb decreases rapidly with increasing T because of electron-phonon scattering. The scattering of Dirac fermions may be more dominated by the im-purity scattering than that of the bulk carriers; thus, τs is less temperature dependent and does not decrease as rapidly with increasing T compared to τb. Therefore, the values of µs and µbmay be comparable at room temper-ature. These results strongly suggest that a THz wave generated inside the Bi2Se3crystals can be used not only
to identify but also to effectively examine fundamental properties of Dirac fermions.
In summary, THz radiation can be generated from Bi2Se3 and Cu-doped Bi2Se3 single crystals. Dirac fermions of the surface state are indispensable to explain-ing the strong dependence of the THz emission power on the carrier concentration. Furthermore, the detailed characteristics of the THz emission confirmed the pres-ence of a Dirac-fermion-normalized phonon branch and valuable information regarding the fundamental proper-ties of Dirac fermions.
This work was support by the National Science Coun-cil of Taiwan, under grant: Nos. NSC101-2112-M-009-016-MY2, NSC101-2112-M-009-017-MY2 and NSC 100-2112-M110-004-MY3, and by the MOEATU program at NCTU of Taiwan, R.O.C. Technical help from C. K. Wen and discussions with H. T. Jeng and T. M. Uen are ap-preciated.
∗ cwluo@mail.nctu.edu.tw † ago@nctu.edu.tw
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