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Electronic Supplemental Information for:

Observation of Low Temperature n-p Transition in Individual Titania Nanotubes

Hatem Brahmi,a Ram Neupane,b Lixin Xie,b Shivkant Singh,ac Milad Yarali,a Giwan Katwal,b

Shuo Chen,b Maggie Paulose,b Oomman K. Varghese*b and Anastassios Mavrokefalos*a

aDepartment of Mechanical Engineering, University of Houston, Houston, TX 77204, USA. E-

mail: amavrokefalos@uh.edu

bDepartment of Physics, University of Houston, Houston, TX 77204, USA. E-mail:

okvarghese@uh.edu

cMaterial Science and Engineering Program, University of Houston, Houston, TX 77204, USA

Transmission Electron Microscopy

High resolution transmission electron microscopy (HRTEM) study was done using JEOL

2000-SFX microscope. The results from the actual nanotubes used for thermal and electrical

measurements are shown in Figure 1. Additionally, we tried to find whether there was a distinct

change in the grain dimensions after lattice reduction. For this study, we used nanotubes

prepared under identical conditions as NT1 (reduced) and NT4 (unreduced). Nevertheless, moire

patterns formed by overlapping grains made the task difficult [see Figure S1 (a,b)]. Furthermore,

the grain boundaries were not easily distinguishable [see Figure S1 (c,d)]. The grain size was

determined to be in the range of a few tens of nanometers in both the samples. Although a

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Electronic Supplementary Material (ESI) for Nanoscale. This journal is © The Royal Society of Chemistry 2018

mailto:amavrokefalos@uh.edu mailto:okvarghese@uh.edu

detailed investigation was done by taking lattice images at different regions of nanotubes from

both sample types, no evidences showing the influence of forming gas annealing on grain size

was found. This was not unexpected as both the samples were annealed first at 530 °C and the

sample for lattice reduction (NT1) was subjected to annealing at a lower temperature (500 °C).

Fig. S1. HRTEM images from a nanotube in the class NT1 (a, c) and NT4 (b, d). The dashed

curve in (c) shows a region that is possibly a grain boundary.

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Thermal conductivity modeling

Callaway model developed by Morelli et al.1 was used to interpolate the phonon scattering

rates to the measured thermal conductivity. The model suggests the contribution of both

longitudinal and acoustic modes independently. Phonon group velocity and Debye temperature

are deduced from the bulk phonon dispersion curves. The thermal conductivity is written as:

(1)𝜅 = 𝜅𝐿 + 2𝜅𝑇

where L and T are the longitudinal transverse phonon branch contributions respectively. And

for each branch:

(2)𝜅𝐿(𝑇) = 𝜅𝐿1(𝑇1) + 𝜅𝐿2(𝑇2)

Where the partial thermal conductivities are:

(3) 𝜅𝑖1 =

1 3

𝐶𝑖𝑇 3

𝜃𝑖 𝑇

∫ 0

𝜏 𝑖𝐶(𝑥)𝑥 4𝑒𝑥

(𝑒𝑥 ‒ 1)2 𝑑𝑥

(4)

𝜅𝑖2 = 1 3

𝐶𝑖𝑇 3

[ 𝜃𝑖 𝑇

∫ 0

𝜏 𝑖𝐶(𝑥)𝑥 4𝑒𝑥

𝜏 𝑖𝑁(𝑥)(𝑒 𝑥 ‒ 1)2

𝑑𝑥]2 𝜃𝑖 𝑇

∫ 0

𝜏 𝑖𝐶(𝑥)𝑥 4𝑒𝑥

𝜏 𝑖𝑁(𝑥)𝜏 𝑖 𝑅(𝑥)(𝑒

𝑥 ‒ 1)2 𝑑𝑥

For the calculation, we consider phonon-phonon Umklapp scattering (τU), boundary scattering

(τB), impurity scattering (τI), and normal scattering (τN). Therefore, per Matthiessen's rule the

combined phonon relaxation time (τC) is,

(5)

1 𝜏𝐶

= 1 𝜏𝑈

+ 1

𝜏𝑁 +

1 𝜏𝐵

+ 1 𝜏𝐼

The values of different fitting parameters used in the model are summarized in Table S1.

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Table S1 Fitting parameters used in the thermal conductivity theoretical calculations for each

sample: suspended length (L), Casimir length (Λ0), longitudinal Debye temperature (θL),

transverse Debye temperature (θL), longitudinal phonon group velocity (vL), transverse phonon

group velocity (vL), longitudinal grunneiser parameter (γL), transverse grunneiser parameter (γT),

phonons specularity parameter (p), mass difference scattering parameter (A).

Sample L (μm) Λ0 (nm)

θL(K) θT(K) vL(m/s) vT(m/s) γL γT p A (s3)

NT1 4.29 2 855 390 8127 3715 2.4 2.2 0 6.31x10-43

NT 2 4.21 2 855 390 8127 3715 2.4 2.2 0 6.31x10-43

NT 3 7.85 2 855 390 8127 3715 2.4 2.2 0 2.97x10-43

NT 4 11.94 2 855 390 8127 3715 2.4 2.2 0 2.5x10-43

Thermoelectric modeling

The following equation presents the total Seebeck coefficient and shows both electrons and

holes contributions, which correspond to two-band model Seebeck.

(6) 𝑆 =

𝑆𝑒𝑛µ𝑒 + 𝑆ℎ𝑝µℎ 𝑛µ𝑒 + 𝑝µℎ

where n and p are the electrons and hole concentration respectively, e and are h electron and

hole mobility. are the Seebeck coefficient of electrons and holes respectively and are 𝑆𝑒(ℎ)

described by the following equations,

(7)

𝑆𝑒(ℎ) =‒ 𝑘𝐵 𝑒 ((𝑟𝑒(ℎ) + 52)𝐹𝑟𝑒(ℎ) + 32( 𝜂𝑒(ℎ))(𝑟𝑒(ℎ) + 32)𝐹𝑟𝑒(ℎ) + 12( 𝜂𝑒(ℎ)) ‒ 𝜂𝑒(ℎ))

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where is the reduced Fermi energy for electrons and is Fermi energy for the same 𝜂𝑒 =

𝐸𝐹 𝑘𝐵𝑇 𝐸𝐹

carrier and is the reduced Fermi energy for holes and is Fermi energy for 𝜂ℎ =

𝐸𝐹ℎ 𝑘𝐵𝑇

=‒ (𝐸𝐹 + 𝐸𝑔)

𝑘𝐵𝑇 𝐸𝐹ℎ

the same carrier. is the bandgap, assumed to be 3.2 eV for the anatase TiO2 and it is relatively 𝐸𝑔

temperature independent based on the empirical eqn (8)2

(8) 𝐸𝑔(𝑇) = 𝐸𝑔(0) ‒

𝛼

𝑒(Θ/𝑇) ‒ 1

is the Fermi Dirac integral of order t and it is calculated using the following eqns:𝐹𝑡

(9) 𝐹𝑡( 𝜂) =

∞

∫ 0

𝑦𝑡𝑑𝑦

𝑒(𝑦 ‒ 𝜂) + 1

In the eqn (7), is the Boltzmann’s constant, e is the electron charge and T is the absolute 𝑘𝐵

temperature.

Besides Ef, the Seebeck coefficient is assumed to be dependent on the electron/hole energy

according to here both and are two constant.3 As reported in previous work 𝜏𝑒/𝑝 = 𝜏0𝐸 𝑟𝑒/𝑝 𝑟𝑒/𝑝 𝜏0

for InSb nanowires,4 from the extracted data of the carrier mobility, that is discussed in the main

text, was found to be limited by either the boundary scattering or ionized impurity scattering in 𝜏𝑒

the NTs, hence = -0.5.𝑟𝑒/𝑝

The electron and hole concentrations were calculated using:

(10) 𝑛 =

4𝜋

ℎ3 (2𝑚 ∗𝑒 𝑘𝐵𝑇)

3/2𝐹1 2

( 𝜂𝑒)

(11) 𝑝 =

4𝜋

ℎ3 (2𝑚 ∗ℎ 𝑘𝐵𝑇)

3/2𝐹1 2

( 𝜂ℎ)

The previous equations describe the two-band model used to fit the measured Seebeck data and

extract the Ef as a function of temperature. Thin film anatase TiO2 data 5 is used for the carrier

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effective masses, and it is expected to be close to what is in NTs. The single band model is

described by eqn (7) for the n-type semiconductor and for a p-type semiconductor. Fig. S2 joins

both models with the measured Seebeck. For the two-band model, we used the bulk values for

electron and hole mobilities.

The determination of the Ef may be accomplished using two solutions for the two-band model

and one solution for the single band model. Those solutions correspond to two different regimes,

one transition regime, and another highly doped regime. Since both solutions match the

measured Seebeck values, the actual value is defined by the associated electrical conductivity

value.

Fig. S2 Seebeck coefficient as a function of the Fermi level (Ef) for Two-band Model (red

curve), Single-band Model (blue curve) and the Experimental Seebeck value (green dashed line)

at T=300 K. Also shown is the conduction and valence band limit for the anatase TiO2.

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The electrical conductivity was calculated according to:

(12)𝜎 = 𝑛𝑒µ𝑒 + 𝑝𝑒µℎ

Based on the two-band model, the Ef is given by the intersection of the measured Seebeck with

the theoretical curve of S from eqn (6) (see Fig. S2). The first solution that associated to the near

transition regime corresponds to very different electrical conductivity value compared to the

actual measured value. The highly doped solution was close or superposed in both models. Fig.

S3 Illustrates the position of various acceptor and donor states that facilitate the fermi level shift

in the NTs 1&2 as a function of temperature.

Fig. S3 Shows the temperature dependent Seebeck coefficient and corresponding energy levels at different Seebeck regime i.e. n-and p-type.

To further compare the fermi levels, the carrier concentration was obtained by the slope of Mott-

Schottky plot Fig. S4. Measurements were performed using Na2SO4 (pH 8.6) in a three electrode

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configuration with TiO2 nanotube array film as the working electrode, platinum as the counter

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