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arXiv:0708.4142v1 [cond-mat.str-el] 30 Aug 2007

Terahertz conductivity of localized photoinduced

carriers in Mott insulator YTiO3at low excitation

density, contrasted with metallic nature in band

semiconductor Si

J. Kitagawa1, Y. Kadoya1, M. Tsubota2, F. Iga1and T.

Takabatake1

1Department of Quantum Matter, ADSM, Hiroshima University, 1-3-1 Kagamiyama,

Higashi-Hiroshima 739-8530, Japan

2Synchrotron Radiation Research Unit, JAEA, Hyogo 679-5148, Japan

E-mail: jkita@hiroshima-u.ac.jp

Abstract.

We performed optical-pump terahertz-probe measurements of a Mott

insulator YTiO3 and a band semiconductor Si using a laser diode (1.47 eV) and a

femtosecond pulse laser (1.55 eV). Both samples possess long energy-relaxation times

(1.5 ms for YTiO3 and 15 µs for Si); therefore, it is possible to extract terahertz

complex conductivities of photoinduced carriers under equilibrium. We observed

highly contrasting behavior - Drude conductivity in Si and localized conductivity

possibly obeying the Jonscher law in YTiO3.The carrier number at the highest

carrier-concentration layer in YTiO3is estimated to be 0.015 per Ti site. Anisotropic

conductivity of YTiO3is determined. Our study indicates that localized carriers might

play an important role in the incipient formation of photoinduced metallic phases in

Mott insulators. In addition, this study shows that the transfer-matrix method is

effective for extracting an optical constant of a sample with a spatially inhomogeneous

carrier distribution.

PACS numbers: 78.47.+p, 72.80.Ga, 71.30.+h

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Terahertz conductivity of localized photoinduced carriers in Mott insulator YTiO3

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Submitted to: J. Phys.: Condens. Matter

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Terahertz conductivity of localized photoinduced carriers in Mott insulator YTiO3

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1. Introduction

Recent discoveries of photoinduced metallic phases in several Mott insulators[1, 2, 3,

4, 5, 6] made us consider the strongly correlated electron physics from a new point

of view. However, the optical properties of photoinduced carriers in Mott insulators

are not well understood even at low excitation densities. Understanding these optical

properties is a prerequisite to understanding the incipient creation of metallic phases.

Drude response by itinerant carriers is observed in the case of band semiconductors with

low excited-carrier density (1014∼1016cm−3)[7, 8, 9, 10]. The comparison between the

optical properties of photoinduced carriers at low excitation densities in Mott insulators

and those in band semiconductors would be important in gaining deeper insight into

strongly correlated electron physics.

The detailed nature of various carrier conductions, exhibiting Drude or hopping

conduction, can be well characterized in the terahertz (THz) regime[11, 12, 13, 14].

THz time-domain spectroscopy (THz-TDS) is a powerful tool for analysing terahertz

conductivity ˜ σ(ω) (= σ1(ω) + iσ2(ω)). The remarkable advantage of THz-TDS is its

simultaneous determination of both the real and imaginary parts of ˜ σ(ω), without using

the Kramers-Kronig transformation[15]. The coherent nature of the THz pulse is also

utilized to investigate photoinduced ˜ σ(ω) by, for instance, optical-pump THz-probe

(OPTP) studies. Recent progress in THz technologies[16, 17] and the methodology

of analysis[18, 19, 20] in OPTP experiments enable evaluation of transient ˜ σ(ω) in

many substances, such as semiconductors, high-Tcsuperconductors, liquids and organic

materials[8, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31].

Because the OPTP method essentially detects non-equilibrium processes, such as

surface recombination and carrier diffusion, the time dependence of a complicated spatial

carrier distribution must be considered[8, 32]. This difficulty is avoided by using thin-

film samples[21, 22, 23, 24, 28, 31] where the optical pump pulse penetrates, and by

analysing the photoinduced phase as a layer with a homogeneous ˜ σ(ω)[8, 22, 24, 28, 33].

Furthermore the extraction of ˜ σ(ω), varying quickly compared with the pulse width

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Terahertz conductivity of localized photoinduced carriers in Mott insulator YTiO3

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of THz probe pulse, only seems possible within some restricted conditions[18, 19, 20].

Therefore, to analyse photoinduced ˜ σ(ω) for a wide variety of Mott insulators, free

from the restrictions in sample preparation and analysis, we initially examined a nearly

equilibrium state of photoexcited bulk material with a longer energy-relaxation time,

τ. The photoexcitation of a material with longer τ creates a quasi-equilibrium state

averaging over various non-equilibrium processes, making it easy to obtain the optical

constants of the highest carrier-concentration region in the material.

Our analysis also required extracting ˜ σ(ω) of materials with inhomogeneous carrier

distributions in a more rigorous manner. One good candidate for accomplishing this

is the transfer-matrix method, which expresses inhomogeneous carrier distribution by

a multi-layer system. This method has been briefly commented on in the literature[8].

The promise that the transfer-matrix method can incorporate inhomogeneity is seen

in the analyses of reflectivity in optical pump-probe studies[34, 35]. Although it is

a versatile method, its effectiveness has not been thoroughly discussed, especially in

THz-TDS studies.

In this study, we found that τ of a Mott insulator YTiO3 with Mott gap of

approximately 1 eV[36] is 1.5 ms at 1.47 eV photoexcitation. We characterized the

photoinduced ˜ σ(ω) by comparing it with that of a band semiconductor Si with a bandgap

of 1.1 eV[37]. We also present a more detailed discussion of the transfer-matrix method.

2. Experimental method

Single-crystalline samples of YTiO3, with the orthorhombic perovskite GdFeO3-type

structure, were grown by the floating zone method[38]. The Si sample was commercial

high-resistivity Si.

Photoconductivity was measured to assess τ. The light emitted from a multimode

continuous wave (CW) laser diode (LD) with photon energy of 1.47 eV was modulated

and used for illuminating the sample under an electric field of about 0.3 kV/cm. The

photocurrent Iphflowing through a 100 Ω resistance, connected in series with the sample,

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Terahertz conductivity of localized photoinduced carriers in Mott insulator YTiO3

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was lock-in detected.

OPTP experiment was performed by a transmission THz-TDS system described in

detail elsewhere[39, 40]. The thicknesses of platelet samples are 420 µm for YTiO3and

512 µm for Si. The optical pulses were generated by a mode-locked Ti-sapphire laser with

a repetition rate of 76 MHz and central wavelength of 800 nm. Both the THz emitter

and detector were low-temperature grown GaAs photoconductive antennas. Si lenses

were attached to the antennas to enhance the emission power and collection efficiency

of THz pulses. The THz spectral range in this experiment was between 0.5 and 8 meV.

The THz-wave-emission sides of the samples were photoexcited by the multi-mode CW

LD with an incident angle of 45◦. The pump-beam power was 0.8 W for Si and 1.2 W

for YTiO3. For YTiO3, the polarization of THz electric field ETHz is parallel to the

b-axis. The beam diameter of the LD light was about 8 mm and larger than that of the

THz probe pulse, which is energy dependent (e.g. 2 mm at 2 meV and 1 mm at 4 meV).

The fluence rate was 1.6 W/cm2for Si and 2.4 W/cm2for YTiO3, respectively. The

temperature rise resulting from the thermalisation by photoexcitation[41] is estimated

not to exceed 1 K.

OPTP measurements were also performed with the 1.55 eV optical-pump pulses

split from the Ti-sapphire laser to investigate the nature of conduction of carriers induced

by light with photon energy larger than that of the CW LD. This experiment studied

the anisotropy of photoinduced ˜ σ(ω) in YTiO3. The optical-pump pulse power was 230

mW. The optical-pump pulse beam diameter was about 2 mm (fluence: 96 nJ/cm2) -

slightly smaller than that of the THz probe pulse below 2 meV. The ETHzwas applied

along either the b- or c-axis, maintaining the polarization of the optical-pump pulse

electric field E1.55eV parallel to the b- or c-axis.

All measurements were performed at room temperature.

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Terahertz conductivity of localized photoinduced carriers in Mott insulator YTiO3

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3. Results and Discussion

Figure 1 shows the modulation-frequency dependence of Iph. Increasing the modulation

frequency causes Iph to decrease, according to

B

(1+(ωτ)2), where B is the proportional

coefficient. The obtained B and τ are listed in the figure. Longer τ (∼ 15 µs for Si and

∼ 1.5 ms for YTiO3) indicates that the samples are in quasi-equilibrium photoinduced

state during the THz-TDS measurements. τ of YTiO3 is almost excitation-intensity

independent; this implies that a thermal effect does not dominate the relaxation process.

Although the direction of Iphand of electric field of the CW LD pump are not identified

accurately in YTiO3, a huge anisotropic τ that depends on the direction of Iphand the

polarization of the excitation light is not anticipated. The OPTP results shown later

support this (see Fig. 7).

The temporal evolution of ETHz transmitted through Si and YTiO3 are shown

in Fig. 2(a) and 2(b), respectively, with and without LD excitations (1.47 eV).

Photoexcitation attenuates both THz waves implying THz wave absorption is by the

photoinduced carriers. THz energy dependence of transmission T(ω) and phase shift

∆φ(ω) are obtained by Fourier transformation of THz waves, as shown in Fig. 3(a)

and 3(b).They are calculated with the equations T(ω) =

E(ω)

Eref(ω)and ∆φ(ω) =

φ(ω) − φref(ω), where E(ω)(Eref(ω)) and φ(ω)(φref(ω)) are the Fourier transformed

amplitude and phase, with and without excitation, respectively. THz wave absorption

by photoinduced carriers is responsible for T(ω) decreasing below 1 for both samples,

but the two exhibit different energy dependencies. As the THz energy increases, T(ω) of

Si approaches 1, while that of YTiO3gradually decreases. The opposite sign of ∆φ(ω)

of the two samples strongly indicates different fundamental conduction mechanisms of

the photoinduced carriers. Negative (positive) ∆φ roughly means the refractive index

is reduced (increased), compared with an unexcited state, which influences the negative

(positive) real part of the dielectric constant of photoinduced carriers. As explained

below, these results indicate a metallic nature below a plasma frequency in Si, and a

localized nature, such as hopping carriers, in YTiO3.

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Terahertz conductivity of localized photoinduced carriers in Mott insulator YTiO3

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Before showing the photoinduced ˜ σ(ω) of Si and YTiO3, we mention the detailed

procedure of the transfer-matrix method. A spatially inhomogeneous distribution of

photoinduced carriers is initially regarded as exponentially decaying. Subsequently a

non-exponentially decaying distribution is introduced. The photoinduced phase with

an exponentially decaying carrier distribution is divided into many thin slabs (see Fig.

4). Each slab is supposed to have a uniform complex refractive index ˜ njwhose value is

set to reproduce the exponential decay of the photoinduced carrier concentration. The

transfer-matrix of each slab is described by

Mj=

cosδisinδ/˜ nj

i˜ njsinδ cosδ

,(1)

where j is the number index of the slab, δ =2π

λ˜ njd, λ is the incident THz wavelength

in vacuum, and d is the slab thickness[42, 43]. The ˜ njof each slab is calculated from

the complex dielectric constant ˜ ǫ using,

˜ n2

j= ˜ ǫno+ ˜ ǫsurexp

?

−zj

dp

?

,(2)

where ˜ ǫnois ˜ ǫ without excitation, ˜ ǫsur, the parameter to be optimized in this analysis,

is ˜ ǫ resulting from carriers at the photoexcited surface of the sample, zj(= j ×d) is the

depth from the photoexcited surface into the sample along the THz wave propagation,

and dpis the optical penetration depth. For Si, frequency independent ˜ ǫno[44] of 11.7 is

used. For YTiO3, ˜ ǫnois experimentally determined by the THz-TDS measurement and

is weakly energy dependent (e.g. 16.5+0.4i at 2 meV and 17+0.4i at 4 meV). dpof Si at

1.47 eV is determined to be 8.4 µm using the absorption coefficient from an optical data

handbook[45]. That of YTiO3at 1.47 eV was calculated to be 0.22 µm from the reported

reflectivity spectra[46] (0.05-40 eV) combined with the Kramers-Kronig transformation.

Then the total matrix Mtis described as

Mt=

0?

j=k

Mj,(3)

where k is the total number of photoexcited slabs.Finally, the THz complex

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Terahertz conductivity of localized photoinduced carriers in Mott insulator YTiO3

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transmission is given by

T(ω)exp(i∆φ(ω)) =tw/LD

two/LD

=(Q + iP)wo/LD

(Q + iP)w/LD

, (4)

where tw/LDand two/LDmean the THz wave transmission with and without excitation,

respectively, Q = Re((Mt11+Mt12)√˜ ǫno+Mt21+Mt22) and P = Im((Mt11+Mt12)√˜ ǫno+

Mt21+Mt22). The fitting of experimental T(ω) and ∆φ(ω) following the above mentioned

procedure provides ˜ σ(ω) resulting from carriers at the photoexcited surface through

˜ σ(ω) = iωǫ0˜ ǫsur(ω),(5)

where ǫ0 is the vacuum permittivity.Note that the convergence of transmission is

checked carefully by decreasing the thickness or by increasing the number of slabs. A

thickness of the photoexcited phase (= k × d, k=100) 5 to 10 times thicker than dpis

typically employed.

Figure 5 shows photoinduced ˜ σ(ω) of Si and YTiO3. ˜ σ(ω) of Si can be interpreted

by the Drude model as

σ(ω) =

nceµ

1 − iω/Γ,

e

m∗Γ,

(6)

µ=

(7)

where ncis the carrier density, µ is the mobility, Γ is the carrier collision rate and m∗is

the effective mass. The photoexcitation introduces both electrons and holes; therefore,

the tentatively assigned m∗value is 0.26m0for electrons and 0.37m0for holes[44], where

m0is the free-electron mass. Hereafter, µ of each carrier is denoted as µefor electrons

and µhfor holes. We have considered the following two cases, neither of which can be

excluded at the present stage. One is the two-carrier model of electrons and holes. The

other takes only electrons into consideration, assuming that holes with heavy m∗do not

contribute to ˜ σ(ω). The solid lines in Fig. 5(a) represent the calculated ˜ σ(ω) for the

two-carrier model and the broken lines represent the electron-only model. The curves

are in agreement with the experimental ˜ σ(ω). This suggests that the itinerant carriers

are certainly photogenerated in Si. The obtained µeand µhare 2410 (±210) cm2/Vs

and 500 (±90) cm2/Vs for the two-carrier model, and µeis 1820 (±100) cm2/Vs for the

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other model. They are roughly consistent with the literature values[37], but it is to be

noted that µein both models might be larger than the predicted ones. The ambiguity

of m∗may be responsible for this deviation.

The most striking feature in ˜ σ(ω) of YTiO3 is the negative σ2. It suggests an

existence of localized carriers[27, 39, 47], which is very different from Si. The localization

may arise from the on-site strong Coulomb interaction between 3d electrons in YTiO3.

To explain ˜ σ(ω), we used the empirical Jonscher law[48], which expresses ˜ σ(ω) of many

materials with hopping carriers. The Jonscher law is given by[48, 49]

σ1(ω) = σdc+ Aωs,(8)

σ2(ω) = − Aωstansπ

2, (9)

where σdc is the DC conductivity, A the proportional coefficient and s is restricted

between 0 and 1. As shown in Fig. 5(b), the solid curves from the Jonscher law

seem to agree with experimental ˜ σ(ω). In the solid curves, s, σdc and A are 0.95,

235 (±10) Ω−1cm−1and 2.40 (±0.05)×10−11Ω−1cm−1s0.95, respectively. The allowed

s ranges from 0.91 to 0.99, and corresponding σdcand A are 210 (±10) Ω−1cm−1and

1.41 (±0.03)×10−10Ω−1cm−1s0.91, and 260 (±10) Ω−1cm−1and 1.43 (±0.03)×10−12

Ω−1cm−1s0.99, respectively.

Note that ˜ σ(ω) can be also fitted by a two-component model, such as the Drude-

Lorentz. The estimated photoinduced carrier number at the surface layer is about 0.015

per Ti site. Photoexcited YTiO3with the derived carrier density would be equivalent

to chemically hole-doped Y1−xCaxTiO3 with x much less than 0.1 given in Ref. 36.

The σ1(ω) spectrum of Y1−xCaxTiO3in this composition region is very different than a

Drude response. Therefore, it would be difficult to expect a Drude component to exist.

Clarifying this point might require broadband spectroscopic information obtained under

photoexcitation or the temperature dependence of ˜ σ(ω).

Since long relaxation times, τ, are observed in both samples, a diffusion or a

surface-recombination process, making the carrier distribution a non-exponential decay

type, must be considered, and the analysis method modified. The carrier number n(z)

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Terahertz conductivity of localized photoinduced carriers in Mott insulator YTiO3

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along the THz wave propagation in a quasi-equilibrium state is obtained using a one-

dimensional diffusion equation[50] as follows:

∂n(z,t)

∂t

= D∂2n(z,t)

∂z2

−n(z,t)

τ

+ δ(t)exp

?

−z

dp

?

, (10)

where n(z,t) depends on the time t and the position z along the THz wave propagation,

and δ(t) is the δ-function. D is the diffusion coefficient and given by

D =µbikBTs

|e|

, (11)

where 1/µbiis equal to 1/µe+1/µh, kBis the Boltzmann constant and Tsis the sample

temperature equal to 300 K. The solution[50] of eq. (10) is

n(z,t) = exp

?

−z2

4Dt

?

1

2

f

t

D+

?√Dt

dp

−

z

2√Dt

?

+

D

dp+ vs

D

dp− vsf

?

?√Dt

dp

+

z

2√Dt

?

−

vs

D

dp− vsf

vs

?

z

2√Dt

exp

?

−t

τ

, (12)

where vsis the surface recombination velocity and f(z) is related to the error function

by f(z) = exp(z2)(1−erf(z)). The carrier number in the quasi-equilibrium state requires

the integration of n(z,t) with respect to t,

n(z) =

?∞

0

n(z,t)dt.(13)

Therefore, with the assumption of a conduction model and the knowledge of n(z)

determined by appropriate µbi and vs, T(ω)ei∆φ(ω)can be calculated using eq. (4).

In this case, eq. (2) is replaced by

˜ n2

j= ˜ ǫno+ ˜ ǫmaxn(zj)

nmax,(14)

where nmax and ˜ ǫmax are n(z) and ˜ ǫ of the highest carrier-concentration layer,

respectively.

After the determination of µbiof Si (= 346 cm2/Vs) using the literature values[37],

vsis varied between 1×104cm/s and 1×106cm/s. Representative n(z) normalized at

nmaxare shown in the inset of Fig. 6(a). The nmaxis observed around 1∼2 µm. Assuming

that both electrons and holes obeying the Drude conductivity are responsible for ˜ σ(ω),

T(ω)ei∆φ(ω)is confirmed as being consistent with experimental data for both n(z) (see

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Terahertz conductivity of localized photoinduced carriers in Mott insulator YTiO3

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Fig. 6(a)). The estimated ncis 5.2 (±0.3)×1016cm−3and is comparable to that obtained

by the previous model. This indicates that, at the highest carrier-concentration layer

almost the same nccan be obtained, irrespective of the carrier distribution decay type.

For YTiO3, both vsand µbiare unknown parameters. The wide range sweep of vsand µbi

gives various n(z) curves as depicted in the inset of Fig. 6(b) with peak positions around

0.1 µm. For each n(z), the experimental T(ω)ei∆φ(ω)is well reproduced by the Jonscher

law, where s is restricted within the same range obtained in Fig. 5(b) (0.91∼0.99).

Typical examples are shown in Fig. 6(b) with s of 0.95. The other parameters (σdc

in Ω−1cm−1and A in Ω−1cm−1s0.95) for the dotted-solid, solid and broken lines are

125(±10) and 1.40(±0.05)×10−11, 125(±10) and 1.2(±0.1)×10−11, and 90(±10) and

1.0(±0.1)×10−11, respectively. σ1(ω) and σ2(ω) calculated from the parameters are half

to two-thirds of those in Fig. 5(b). Thus, for YTiO3, the ˜ σ(ω) extracted from the

model with exponentially-decaying carrier distribution roughly represents the highest

carrier-concentration layer in the model using eq. (14).

The photoinduced carrier number at the highest carrier-concentration layer in

YTiO3 is calculated as 0.015 per Ti site[51].It can be proposed, therefore, that

a phase with localized carriers would emerge initially at the photogeneration of the

metallic phase in Mott insulators. Photoexcitation creates both electrons and holes,

which differs from chemical doping, and a comparison of ˜ σ(ω) between photoexcited

YTiO3 and a hole-doped Y1−xCaxTiO3 is discussed. The absolute value of σ1(ω) of

photoexcited state might be much larger than that of corresponding Y1−xCaxTiO3if

the extrapolation of σ1(ω) in Y1−xCaxTiO3is carried out toward the THz energy. The

preservation of spectral weight implies that the localization energy of photoinduced

carriers would be much lower than for holes in Y1−xCaxTiO3, even if both holes and

electrons contribute to ˜ σ(ω) in photoexcited YTiO3. As it is not clear that the large

difference in localization energy originates from only holes in such a low-carrier system,

it is plausible that electrons with small localization energies also contribute to ˜ σ(ω).

Therefore ˜ σ(ω) of photoexcited YTiO3would be supported by bound electrons as well

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as holes.

In a halogen-bridged Ni one-dimensional chain compound [Ni(chxn)2Br]Br2(chxn =

cyclohexanediamine), which is compared with YTiO3composed of a three-dimensional

Ti network, the localized σ1is determined at a lower excitation density[3]. Despite being

in a different energy region, carrier localization in photoexcited Mott insulators at low

excitation densities may be the general phenomenon, irrespective of the dimensionality.

The fact that one-dimensional Mott insulators, such as [Ni(chxn)2Br]Br2[3] and

Sr2CuO3[52], exhibit τ in the order of pico-seconds may suggest that dimensionality is

a decisive factor of τ.

Figure 7 shows T(ω) and ∆φ(ω) obtained by OPTP experiments using a

femtosecond-pulse laser (1.55 eV) for YTiO3. Since the period of optical-pump arrival

time (13 ns) is much shorter than τ, the photoinduced carriers are also in quasi-

equilibrium state. In both polarizations of E1.55eV, it is found that the degree of variation

from the unexcited state in THz wave amplitude and phase is larger for ETHz||b within

the measured THz energy range. This implies that the absolute values of σ1 and σ2

for ETHz||b are larger than those for ETHz||c. The anisotropy would reflect the crystal

symmetry of YTiO3 or the 3d-orbital state at Ti site. Comparing with the OPTP

results to those using the CW LD, the ˜ σ(ω) does not seem to depend strongly on the

optical-photon energy.

4. Summary

We have optically characterized photoinduced carriers of Mott insulator YTiO3at low

excitation densities in the THz regime by OPTP measurements, and compared the

experimental results with those for band semiconductor Si. The τ of the photoinduced

carriers in YTiO3is about 1.5 ms. The inhomogeneous carrier distribution along the

THz wave propagation can be treated accurately using the transfer-matrix method. This

method successfully determined ˜ σ(ω) of the highest carrier-concentration layer under the

quasi-equilibrium states. YTiO3shows localized ˜ σ(ω), possibly with the Jonscher law,

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Terahertz conductivity of localized photoinduced carriers in Mott insulator YTiO3

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whereas Si exhibits the Drude response. Anisotropic ˜ σ(ω) in YTiO3is determined. Our

study demonstrates that localized carriers might play an important role in the incipient

formation of metallic phases in photoexcited Mott insulators. Although the exact origin

of the localization in YTiO3remains an open question, THz-TDS under photoexcitation

with another photon energy or for another Mott insulator might provide the answer.

We note here that a preliminary THz-TDS experiment of YTiO3excited by a CW LD

of 1.9 eV also leads to localized ˜ σ(ω).

Acknowledgments

This work was supported by Casio Science Foundation, and Strategic Information and

Communications R&D Promotion Programme of Ministry of Public Management, Home

Affairs, Posts and Telecommunications.

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