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Electronic conduction properties of indium tin oxide: single-particle and many-body transport
View the table of contents for this issue, or go to the journal homepage for more 2014 J. Phys.: Condens. Matter 26 343201
(http://iopscience.iop.org/0953-8984/26/34/343201)
J. Phys.: Condens. Matter 26 (2014) 343201 (21pp) doi:10.1088/0953-8984/26/34/343201
Topical Review
Electronic conduction properties of indium
tin oxide: single-particle and many-body
transport
Juhn-Jong Lin
1and Zhi-Qing Li
21NCTU-RIKEN Joint Research Laboratory, Institute of Physics and Department of Electrophysics, National Chiao Tung University, Hsinchu 30010, Taiwan
2Tianjin Key Laboratory of Low Dimensional Materials Physics and Preparing Technology, Department of Physics, Tianjin University, Tianjin 300072, People’s Republic of China
E-mail:[email protected]
Received 1 April 2014, revised 29 June 2014 Accepted for publication 1 July 2014 Published 8 August 2014
Abstract
Indium tin oxide (Sn-doped In2O3−δor ITO) is a very interesting and technologically
important transparent conducting oxide. This class of material has been extensively investigated for decades, with research efforts mostly focusing on the application aspects. The fundamental issues of the electronic conduction properties of ITO from room temperature down to liquid-helium temperatures have rarely been addressed thus far. Studies of the electrical-transport properties over a wide range of temperature are essential to unravelling the underlying electronic dynamics and microscopic electronic parameters. In this topical review, we show that one can learn rich physics in ITO material, including the semi-classical
Boltzmann transport, the quantum-interference electron transport, as well as the many-body Coulomb electron–electron interaction effects in the presence of disorder and inhomogeneity (granularity). To fully reveal the numerous avenues and unique opportunities that the ITO material has provided for fundamental condensed matter physics research, we demonstrate a variety of charge transport properties in different forms of ITO structures, including
homogeneous polycrystalline thin and thick films, homogeneous single-crystalline nanowires and inhomogeneous ultrathin films. In this manner, we not only address new physics
phenomena that can arise in ITO but also illustrate the versatility of the stable ITO material forms for potential technological applications. We emphasize that, microscopically, the novel and rich electronic conduction properties of ITO originate from the inherited robust
free-electron-like energy bandstructure and low-carrier concentration (as compared with that in typical metals) characteristics of this class of material. Furthermore, a low carrier concentration leads to slow electron–phonon relaxation, which in turn causes the
experimentally observed (i) a small residual resistance ratio, (ii) a linear electron diffusion thermoelectric power in a wide temperature range 1–300 K and (iii) a weak electron dephasing rate. We focus our discussion on the metallic-like ITO material.
Keywords: indium tin oxide, transparent conducting oxides, electronic conduction properties, Boltzmann transport, localization, mesoscopic physics, granular systems
1. Introduction
Transparent conducting oxides (TCOs) constitute an appearing and unique class of materials that simultaneously possess high electrical conductivity, σ and high optical transparency at the visible frequencies [1–3]. These combined electrical and optical properties render the TCOs to be widely used, for example, as transparent electrodes in numerous optoelectronic devices, such as flat panel displays, photovoltaic electrochromics, solar cells, energy-efficient windows and resistive touch panes [3–6]. Currently, the major industrial TCO films are made of indium tin oxide (Sn-doped In2O3−δ
or so-called ITO), F-doped tin oxide and group III elements doped zinc oxide. Among them, the ITO films are probably the most widely used TCOs, owing to the ITO’s readiness for fabrication and patterning as well as their high quality and reliability implemented in commercial products.
On the fundamental research side, our current understand-ing of the origins for the combined properties of high elec-trical conductivity and high optical transparency is based on both theoretical and experimental studies [7–33]. The elec-tronic energy bandstructure of ITO has been theoretically cal-culated by several authors [26–29]. It is now known that the bottom of the conduction band of the parent In2O3 is mainly
derived from the hybridization of the In 5s electronic states with the O 2s states. The energy-momentum dispersion near the bottom of the conduction band reveals a parabolic char-acter, manifesting the nature of s-like electronic states (see a schematic in figure1). The Fermi level lies in the middle of the conduction and valence bands, rendering In2O3 a
wide-band-gap insulator. Upon doping, the Sn 5s electrons con-tribute significantly to the electronic states around the bottom of the conduction band, causing the Fermi level to shift upward into the conduction band. Meanwhile, the shape of the con-duction band at the Fermi level faithfully retains the intrinsic parabolic character. This unique material property makes ITO a highly degenerate n-type semiconductor or, alternatively, a low-carrier-concentration metal. As a consequence of the s-like parabolic energy bandstructure, the electronic conduction properties of this class of material demonstrate marked
free-carrier-like characteristics. The charge transport properties of
ITO can thus be quantitatively described by those simple mod-els formulated basing upon a free electron Fermi gas. Indeed, the levels of close quantitative agreement between theoretical calculations and experimental measurements obtained for ITO are not achievable even for alkali (Li, Na, K) and noble (Cu, Ag, Au) metals, as we shall present in this topical review.
In practice, the conduction electron concentration, n, in optimally doped ITO (corresponding to approximately 8 at.% of Sn doping) can reach a level as high as n≈ 1020–1021cm−3
[34,35]. This level of n is two to three orders of magnitude
lower than that (≈1022–1023cm−3 [36]) in typical metals.
The room temperature resistivity can be as low as ρ(300 K) ≈150 µ cm (see table 1). This magnitude is comparable with that of the technologically important titanium–aluminium alloys [37,38]. In terms of the optical properties, the typical plasma frequency is ωp 0.7–1 eV [8], while the typical
energy band gap is Eg 3.7–4.0 eV. Hence, optimally doped
ITO possesses a high optical transparency which exceeds 90% transmittance at the visible light frequencies [39,40]. A value of ωp 1 eV corresponds to a radiation frequency of
fp = ωp/2π 2.4 × 1014Hz, which is approximately one
fifth of the visible light frequency and roughly one fiftieth of the plasma frequency of a typical metal. For optoelectronic applications, on one hand, one would like to dope ITO with a Sn level as high as technologically feasible in order to obtain a high electrical conductivity σ . On the other hand, since
ωp ∝
√
n, one has to keep n sufficiently low such that the visible light can propagate through the ITO structure.
Owing to their technological importance, it is natural that there already exist in the literature a number of review articles on the ITO as well as TCO materials [2,4,41–51]. The early studies up to 1982, covering the deposition methods, crystal structures, scattering mechanisms of conduction electrons and the optical properties of In2O3, SnO2 and ITO, were
reviewed by Jarze¸bski [2]. Hamberg and Granqvist discussed the optical properties of ITO films fabricated by the reactive electron-gun evaporation onto heated glass substrates [42]. The development up to 2000 on the various aspects of utilizing TCOs was summarized in reports considering, for example, characterizations [48], applications and processing [43], criteria for choosing transparent conductors [47], new n- and p-type TCOs [44,45] and the chemical and thin-film strategies for new TCOs [46]. The recent progresses in new TCO materials and TCOs based devices were discussed in [49] and [50]. King and Veal recently surveyed the current theoretical understanding of the effects of defects, impurities and surface states on the electrical conduction in TCOs [51].
In this topical review, we stress the free-electron-like energy bandstructure and the low-n features (as compared with typical metals) of the ITO material. These inherited intrinsic electronic characteristics make ITO a model system which is ideal for not only revealing the semi-classical Boltzmann transport behaviours (section2) but also studying new physics such as the quantum-interference weak-localization (WL) effect and the universal conduction fluctuations (UCFs) in miniature structures (section 3). The responsible electron dephasing (electron–electron scattering, electron–phonon scattering and spin–orbit scattering) processes are discussed. Furthermore, we show that this class of material provides a very useful platform for experimentally testing the recent theories of granular metals [52–57]. In the last case, ultrathin ITO films can be intentionally made to be slightly inhomogeneous or granular, while the coupling between neighbouring grains remains sufficiently strong so that the system retains global metallic-like conduction (section4). To illustrate the unique and numerous avenues provided by ITO for the studies of the aforementioned semi-classical versus quantum electron transport, as well as homogeneous versus inhomogeneous charge transport, we cover polycrystalline (ultra)thin and thick ITO films and single-crystalline ITO nanowires in this topical review. We demonstrate that high-quality ITO structures can indeed be readily fabricated into various forms which, apart from being powerful for addressing fundamental electronic conduction properties, may be useful for potential technological applications. Furthermore, owing
Figure 1.Schematic energy bandstructure showing the top of the valence band and the bottom of the conduction band for (a) In2O3and (b) Sn-doped In2O3(In2−xSnxO3−δ). This figure was reproduced with permission from [7]. Copyright 1984 by the American Physical Society.
to the similarities in electronic bandstructure between ITO and other TCO materials [3], we expect that the electronic processes and mechanisms discussed in this topical review should be useful for understanding and interpreting the results obtained on general TCOs.
We do not cover insulating or amorphous ITO materials in this topical review, where the electronic conduction processes can be due to thermally excited hopping [58–62]. In addition to the conventional Mott [63] and Efros-Shklovskii [64] hopping conduction mechanisms in homogeneous strongly disordered systems, electronic conduction due to the thermal charging effect [65] and, more recently, the variable-range-hopping pro-cess [66] in inhomogeneous (granular) systems have been dis-cussed in literature. On the other hand, the possible occurrence of superconductivity in ITO has been explored in [67–70].
2. Free-electron-like Boltzmann transport:
homogeneous indium tin oxide films and nanowires
The electrical-transport properties of ITO films have extensively been discussed in the literature. However, previous studies have mainly concentrated on the influences of deposition methods and conditions on the ρ(300 K) values. While those studies have provided useful information for improving the fabrication of high-quality ITO films, they did not deal with the underlying electronic conduction processes in ITO. In section2.1, we first briefly summarize the theoretical calculations of the electronic energy bandstructure of ITO and explain why this class of material behaves like a highly degenerate semiconductor or a low-n metal. In section2.2, we discuss the overall temperature behaviour of resistivity ρ(T ) in ITO and show that ρ(T ) can be well described by the standard Boltzmann transport equation in a wide temperature range. In section2.3, we demonstrate that the thermoelectric power (Seebeck coefficient, or thermopower), S(T ), in ITO follows an approximately linear temperature dependence in the wide temperature range from 1 K up to well above room temperature.
This linear thermoelectric power originates from the diffusion of electrons in the presence of a temperature gradient and provides a powerful, direct manifestation of the robust free-carrier-like characteristic of ITO. The reason why the phonon-drag contribution to thermoelectric power in ITO is absent is heuristically discussed.
2.1. Free-carrier-like bandstructure and relevant electronic parameters
2.1.1. Electronic energy bandstructure. Since the electronic energy bandstructure plays a key role in governing the charge transport properties of a given material, we first discuss the electronic bandstructure of ITO. Based on their x-ray photoemission spectroscopy studies, Fan and Goodenough [71] first suggested a schematic energy band model for the undoped and Sn-doped In2O3 in 1977. A
heuristic energy-band model for ITO was proposed by Hamberg et al [7] in 1984. In their heuristic model (shown in figure1), the bottom (top) of the conduction (valence) band of In2O3was taken to be parabolic. They further proposed that the
shapes of the conduction band and the valence band remained unchanged upon Sn doping. This simple bandstructure model is qualitatively in line with that obtained by later theoretical calculations [26–29].
The first ab initio bandstructure calculations for the ITO material were carried out by Odaka et al [26] and Mryasov and Freeman [27] in 2001. Later on, Medvedeva [28] calculated the bandstructure of In2O3 and Medvedeva and
Hettiarachchi [29] calculated the bandstructure of 6.25 at.% Sn-doped In2O3. Figures 2(a) and (b), respectively, show
the electronic bandstructures of stoichiometric In2O3 and
6.25 at.% Sn-doped In2O3 obtained in [29]. For In2O3,
the conduction band exhibits a free-electron-like, parabolic characteristic around the point, where the bottom of the conduction band originates from the hybridization of In 5s and O 2s electronic states. Medvedeva and Hettiarachchi found that the effective electron mass, m∗, near the point is nearly
Figure 2.Electronic energy bandstructure of (a) undoped In2O3and (b) 6.25 at.% Sn-doped In2O3. In (b), a 6.25 at.% doping level corresponds to a single Sn atom substituting for one of the 16 In atoms in the In2O3supercell. Note the inherent free-electron-like energy-momentum dispersion at the Fermi level. This figure was reproduced with permission from [29]. Copyright 2010 by the American Physical Society.
isotropic. Similar theoretical results were obtained shortly after by Fuchs and Bechstedt [72] and Karazhanov et al [12]. Upon Sn doping, the Sn 5s states further hybridize with the In 5s and O 2s states to form the bottom of the conduction band. Furthermore, the Fermi level in ITO shifts upward into the conduction band, leading to the bandstructure depicted in figure 2(b). Theoretical calculations indicate that the Sn 5s states contribute nearly one fourth of the total electronic density of states at the Fermi level, N (EF), while the In 5s
and O 2s states contribute the rest. At this particular doping level, the s-like symmetry of the original bandstructure around the Fermi level in the parent In2O3 is essentially unaltered.
Thus, the conduction electrons at the Fermi level in ITO possess strong free-carrier-like features. Meanwhile, Fuchs and Bechstedt [72] found that the average effective electron mass increases slightly with increasing carrier concentration
n. At a level of n 1020cm−3, they obtained a value m∗ 0.3 me, where meis the free-electron mass. Their result
agreed with that derived from optical measurements of the Drude term to free carriers [7].
In brief, the combined electronic bandstructure character-istics of a wide energy gap, a small m∗and in particular a low
nas well as a free-carrier-like dispersion at EF, are the crucial
ingredients to make ITO, on one hand, possess high electri-cal conductivity while, on the other hand, reveal high optielectri-cal transparency.
2.1.2. Relevant electronic parameters. Experimentally, a reliable method to check the metal-like energy bandstructure of a material is to examine the temperature T dependence of n. For a metal or a highly degenerate semiconductor, n does not vary with T . Figure 3 shows the variation of n with temperature for a few as-deposited (before annealing) and annealed ITO films studied by Kikuchi et al [73]. It is clear
Figure 3.Carrier concentration as a function of temperature for as-prepared (before annealing) and annealed ITO films. O2-HIP denotes annealing in an O2atmosphere and O3denotes annealing in an O3atmosphere. This figure was reproduced with permission from [73]. Copyright 2000 by Elsevier.
that n remains constant in a wide T range from liquid-helium temperatures up to 300 K. In the as-deposited sample, the n value approaches∼1 × 1021cm−3. Temperature independent nin the ITO material has been reported by a number of groups [61,74–76].
For the convenience of the discussion of charge transport properties in ITO in this topical review, we would like to estimate the values of relevant electronic parameters. Consider a high-quality ITO sample having a value of
ρ(300 K) 150 µ cm, a carrier concentration n 1 × 1021cm−3 and an effective mass m∗ 0.35 m
e. Applying
the free-electron model, we obtain the Fermi wavenumber
kF = (3π2n)1/3 3.1 × 109m−1, the Fermi velocity vF = ¯hkF/m∗ 1.0 × 106m s−1 and the Fermi energy EF = ¯h2k2F/(2m∗) 1.0 eV. The electron mean free
Table 1. Representative values of room temperature resistivity ρ and carrier concentration n for ITO films fabricated by different deposition methods.
Fabrication method ρ (µcm) n(1020cm−3) References Vacuum evaporation ∼150–15 000 ∼5 [35,61,77–79]
Magnetron sputtering ∼100–400 ∼10 [80–83]
Chemical vapour deposition ∼150–500 ∼10 [84–86]
Sol-gel ∼600–4000 ∼1 [87–90]
Spray pyrolysis ∼900–5000 ∼1 [91–93]
time is τ = m∗/(ne2ρ) 8.3 × 10−15s, corresponding to the electron mean free path l = vFτ 8.3 nm. The electron
diffusion constant D = vFl/3 28 cm2s−1. Thus, the
dimensionless product kFl 26. Note that kFlis an important
physical quantity which characterizes the degree of disorder in a conductor. A kFlvalue of order a few tens indicates that
high-quality ITO is a weakly disordered metal and should thus be rich in quantum-interference transport phenomena.
In practice, the ρ and n values in ITO films can vary widely with the deposition methods and conditions, Sn doping levels and the post thermal treatment conditions. In table1, we list some representative values for ITO films prepared by different techniques. This table indicates that those ITO films fabricated by the DC magnetron sputtering method possess relatively high (low) n (ρ) values. Since the films thus prepared are compact and they adhere well to the substrate surface, this low-cost technique is thus the most widely used ITO deposition method in the industrial production nowadays. Recently, researchers have also carried out molecular-beam-epitaxial growth studies of ITO structures [94,95], but the crystal quality obtained was not as high as that previously achieved in the epitaxial films grown by a pulsed-laser deposition technique [96]. We mention in passing that, apart from the bulk properties [97,98], the effect on electronic processes of the surface states due to oxygen vacancies in undoped In2O3−δ [32] as well as
doped TCOs [30,31,51] has recently drawn theoretical and experimental attention.
2.2. Temperature behaviour of electrical resistivity
The temperature dependence of resistivity ρ(T ) from 300 K down to liquid-helium temperatures provides key information for the understanding of the electrical conduction processes in a conductor. Li and Lin [99] have measured ρ(T ) between 0.4 and 300 K in a number of 125 and 240 nm thick polycrystalline ITO films prepared by the standard RF sputtering deposition method. Their films had relatively low values of ρ(300 K) 200 µ cm. Their results are shown in figure4. Li and Lin found that the ρ(T ) data between∼25 and 300 K can be well described by the Bloch–Gr¨uneisen formula
ρ= ρe+ ρe–ph(T ) = ρe+ βT T θD 4 θD/T 0 x5dx (ex− 1)(1 − ex), (1)
where ρe is a residual resistivity, β is an electron–phonon
(e–ph) coupling constant and θDis the Debye temperature. The
solid curves in the main panel of figure4are the theoretical predications of equation (1). This figure demonstrates that
Figure 4.Normalized resistivity, ρ(T )/ρ(300 K), as a function of temperature for a 125 nm (squares) and a 240 nm (triangles) thick ITO films. The symbols are the experimental data and the solid curves are the theoretical predictions of equation (1). For clarity, the data for the 240 nm thick film has been shifted up by 0.01. Inset: normalized sheet resistance,
R(T )/R= [R(T )− R(20.5 K)]/R(20.5 K), as a function of the logarithm of temperature for these two films below 25 K. The straight solid lines are least-squares fits to the theoretical predictions of 2D WL and EEI effects. This figure was reproduced with permission from [99]. Copyright 2004 by the American Institute of Physics.
ITO is a metal, with ρ decreasing with decreasing temperature (or, a positive temperature coefficient of resistivity, i.e. (1/ρ) (dρ/dT ) > 0). In particular, the temperature dependence of ρ(T ) can be well described by the standard Boltzmann transport equation.
The first term on the right hand side of equation (1) originates from the elastic scattering of electrons with defects. The second term originates from the inelastic scattering of electrons with lattice vibrations (phonons). Using the Drude formula σ = ne2τ/m∗, one rewrites ρ = (m∗/ne2)(1/τe+
1/τe–ph) = ρe+ ρe–ph(T ), where e is the electronic charge, τe is the electron elastic mean free time and τe–ph is the
e–ph relaxation time. From figure 4, one finds a small resistivity ratio ρ(300 K)/ρ(25 K)1.1, corresponding to the ratio of scattering rates 1/τe–ph 0.1(1/τe). This observation
explicitly suggests that the e–ph relaxation in the ITO material is weak and hence the contribution of the e–ph scattering to
ρ(300 K) is only approximately one tenth of that of the electron elastic scattering with imperfections. A slow e–ph relaxation rate is a general intrinsic property of low-n conductors, see
below for further discussion3. The presence a moderate
level of disorder in ITO films result in significant quantum-interference weak-localization (WL) and electron–electron interaction (EEI) effects at low temperatures. These two effects cause small corrections to the residual resistivity, which increase with reducing temperature. Close inspection of the inset of figure 4 indicates a well-defined, logarithmic temperature dependent resistivity rise below∼ 25 K. The two-dimensional (2D) WL and EEI effects will be discussed in section3.
In addition to comparatively thick films, present-day RF sputtering deposition technology has advanced such that relatively thin films can be made metallic. In a recent study, Lin et al [103] found that the temperature dependence of ρ(T ) below 300 K for 15 nm thick polycrystalline ITO films can also be described by the Bloch–Gr¨uneisen formula. However, the ρ(T ) curve reaches a minimum around 150 K. At lower temperatures, ρ(T ) increases with decreasing temperature, signifying much more pronounced 2D WL and EEI effects than in thicker films (figure4).
The temperature dependence of resistivity in
single-crystalline ITO nanowire has been investigated by Chiu et al [69]. They measured individual ITO nanowires from 300 K down to 1.5 K employing an electron-beam lithographic four-probe configuration. Figure5shows a plot of the normalized resistivity, ρ(T )/ρ(300 K), as a function of temperature for four ITO nanowires. The solid curves are the theoretical predications of equation (1), indicating that the experimental
ρ(T ) data can be well described by the Bloch–Gr¨uneisen formula. However, it is surprising that, in the wide temperature range 1–300 K, the resistivity drops by no more than∼20%, even though these nanowires are single-crystalline. This observation strongly suggests that these nanowires must contain high levels of point defects which are not detectable by the high-resolution transmission electron microscopy studies [69]. It is worth noting that these nanowires are three-dimensional (3D) with respect to the Boltzmann transport, because the electron elastic mean free paths e = vFτe ≈
5–11 nm are smaller than the nanowire diameters d ≈ 110– 220 nm. On the other hand, the nanowires are one-dimensional (1D) with respect to the WL effect and the UCF phenomena, because the electron dephasing length Lϕ =
Dτϕ > d at
low temperatures, where τϕis the electron dephasing time (see
section3).
From least-squares fits of the measured ρ(T ) to equation (1), several groups have obtained a comparatively high Debye temperature of θD ∼ 1000 K in ITO thick films
[61,99], thin films [103] and nanowires [69]. This magnitude of θDis much higher than those (∼200–400 K [36]) in typical
metals4.
3 For comparison, we note that in typical disordered metals, a measured
small residual resistivity ratio ρ(300 K)/ρ(4 K) is usually due to a large elastic electron scattering rate 1/τe, because the e–ph relaxation is considerably fast
in typical metals, see for example [100–102].
4 In applying equation (1) to describe the ρ(T ) data in figures4and 5,
we have focused on the temperature regime below room temperature. At room temperature and above, the interaction of electrons with polar optical phonons is strong. By taking into consideration electron–polar optical phonon interaction, Preissler et al [104] obtained a value of θD 700 K from studies
of Hall mobility in In2O3. These studies suggest a high Debye temperature in
the In2O3based material.
Figure 5.Normalized resistivity, ρ(T )/ρ(300 K), as a function of temperature for four single-crystalline ITO nanowires with diameters ranging from 110 to 220 nm. The symbols are the experimental data and the solid curves are the theoretical predictions of equation (1). At low temperatures, disorder effects cause resistivity rises with reducing temperature. The ITO-g nanowire reveals a possible superconducting transition. The inset shows the measured resistivity as described by the sum of the Bloch–Gr¨uneisen law (solid curve) and a disorder-induced correction to the residual resistivity ρ0(dashed curve) for the ITO-c-R nanowire. This figure was reproduced with permission from [69].
In addition to films and nanowires, nanoscale ITO particles can be made metallic. Ederth et al [105] studied the temperature behaviour of porous thin films comprising of ITO nanoparticles. Their films were produced by spin coating a dispersion of ITO nanoparticles (mean grain size ≈16 nm) onto glass substrates, followed by post thermal treatment. They found that the temperature coefficient of resistivity was negative (i.e. (1/ρ)(dρ/dT ) < 0) between 77 and 300 K. However, their ρ(T ) data obeyed the ‘thermally fluctuation-induced-tunnelling conduction’ (FITC) process [106–108]. Figure6shows the normalized resistivity, ρ(T )/ρ(273 K), as a function of temperature for four ITO nanoparticle films studied by Ederth et al. The symbols are the experimental data and the solid curves are the FITC theory predictions. Theoretically, the FITC model considered the global electrical conduction of an inhomogeneous system consisting of metal grains separated by very thin insulating barriers. The thin insulating barriers were modelled as mesoscopic tunnel junctions. Hence, an observation of the FITC processes occurring in porous ITO films implies that the constituent ITO nanoparticles are metallic. Indeed, in section4, we will discuss that the metallic feature of ITO nanoparticles has provided a powerful platform to experimentally test the recent theories of granular metals [52–57,109].
We notice in passing that the overall temperature behaviour of resistivity in other TCOs, such as Al-doped
Figure 6.Normalized resistivity, ρ(T )/ρ(273 K), as a function of temperature for four ITO nanoparticle films. The nanoparticles have a mean grain size≈16 nm and the films have an approximate thickness∼1.1 µm. The symbols are the experimental data and the solid curves are the theoretical predictions of the thermally fluctuation-induced-tunnelling conduction process. This figure was reproduced with permission from [105]. Copyright 2003 by the American Physical Society.
ZnO [110–112], Ga-doped ZnO [113,114], Nb-doped TiO2
[115,116] and F-doped SnO2[117,118], can also be described
by the standard Boltzmann transport equation (1).
2.3. Linear temperature dependence of thermoelectric power The thermoelectric power is an important physical quantity which describes the electronic conduction behaviours in the presence of a temperature gradient and under the open circuit situation. Studies of the temperature dependence of thermopower, S(T ), can provide useful information about the electronic density of states at the Fermi level N (EF),
the magnitude of EF, the responsible carrier types (electrons
and/or holes), as well as the electron and phonon-phonon relaxation processes in the material. In a metal, the thermopower arises from two contributions and can be expressed as S(T ) = Sd(T )+ Sg(T ), where Sd(T ) is the
electron-diffusion contribution and Sg(T )is the phonon-drag
contribution [119,120].
2.3.1. Electron-diffusion thermopower. The electron-diffusion contribution stems from the electron-diffusion of thermal electrons in the presence of a temperature gradient. A general form is given by the Mott formula [119]
Sd(T )= − π2k2 BT 3|e|EF dln σ (E) dln E E=EF , (2)
where kB is the Boltzmann constant and σ (E) is the
conductivity of electrons that have energy E. The Mott formula is derived under the assumption that the phonon
distribution is itself in overall equilibrium at temperature T . Note that in the case of hole conduction the minus sign in equation (4) should be replaced by a plus sign.
Consider a free electron Fermi gas. By substituting the Einstein relation σ (E) = N(E)e2D(E) into equation (2),
where D(E)= v2(E)τ (E)/3 is the electron diffusion constant
in a 3D conductor with respect to the Boltzmann transport and
v(E)is the electron velocity, one obtains
Sd(T )= − π2kB2T 3|e|EF 3 2+ dln τ (E) dln E E=EF . (3)
Equation (3) predicts a linear temperature dependence of Sd.
The slope of this linear T dependence varies inversely with EF
and its precise value is governed by the energy dependence of mean-free time τ (E)∝ Eq, where q is an exponent of order
unity.
The temperature behaviour of Sd in the low temperature
limit (which is pertinent to ITO) can be approximated as follows. At T θD and in the presence of notable defect
scattering such that the electron mean free path l(E) =
v(E)τ (E)is nearly a constant, i.e. τ (E)∝ 1/v(E) ∝ 1/√E, equation (3) reduces to
Sd = −
π2k2BT
3|e|EF
. (4)
Since the typical EFvalue in ITO is one order of magnitude
smaller than that in a typical metal, the Sd value in the
former is thus approximately one order of magnitude larger than that in the latter. Alternatively, equation (4) can be rewritten in the following form: Sd = −2Ce/(3n|e|), where Ce = π2nkB2T /(2EF)is the electronic specific heat per unit
volume. This expression will be used in equation (5). The temperature behaviour of thermopower in ITO films has been studied by several groups [99,103,121,122]. Figure7shows the measured S(T ) data between 5 and 300 K for one as-grown and three annealed ITO films. This figure clearly indicates that S is negative and varies essentially
linearly with T in the wide temperature range 5–300 K. The
negative sign confirms that electrons are the major charge carriers in ITO.
Recall the fact that the Debye temperature θD ∼ 1000 K
in ITO [69,99,104]. Therefore, one may safely ascribe the measured S below 300 K (figure 7) mainly to the diffusion thermopower Sd(T ). The straight solid lines in figure 7are
least-squares fits to equation (4). From the extracted slopes, one can compute the EF value in each sample. The value
of electron concentration n can thus be deduced through the free-electron-model expression EF= (¯h2/2m∗)(3π2n)2/3. In
ITO structures, the extracted values of EF generally lie in
the range≈0.5–1 eV [70,99,121], corresponding to values of
n≈ 1020–1021cm−3. Therefore, ITO can be treated as a highly degenerate semiconductor or a low-n metal, as mentioned.
It is worth noting that the n values in ITO films obtained from S(T ) measurements agree well with those obtained from the Hall coefficient, RH = 1/(nHe), measurements.
Figure 7.Thermoelectric power as a function of temperature for one as-grown and three O2annealed ITO films. The films were 21 nm thick and the thermal annealing was performed at three different temperatures, as indicated. The straight solid lines are least-squares fits to equation (4). This figure was reproduced with permission from [121]. Copyright 2010 by the American Institute of Physics.
Figure 8.Variation in carrier concentration and Fermi energy with annealing temperature for a good number of 21 nm thick ITO films. Open (closed) squares: n for air (oxygen) annealed films; open (closed) circles: nHfor air (oxygen) annealed films. This figure was
reproduced with permission from [121]. Copyright 2010 by the American Institute of Physics.
Hall concentration nH (circles) for a number of as-grown
and annealed ITO films [121]. It is seen that the n values agree with the nH values to within 30% or better (except
for the films annealed at 200◦C, see discussion in [121]).
This observation provides a strong experimental support for the validity of the theoretical predictions of a free-carrier-like energy bandstructure in ITO. In fact, such kind of prevailing
linearity in S(T ) from liquid-helium temperatures all the way
up to at least 300 K (figure7) is seldom seen in any textbook simple metals, where the phonon-drag contribution Sg(T )
often causes profound, non-monotonic temperature behaviour of S(T ) (see, for example, [120] and figures 7.10 and 7.12 in reference [123]). Thus, ITO does serve as a model system for studying electronic conduction phenomena and extracting reliable electronic parameters.
2.3.2. Phonon-drag thermopower. We would like to comment on the negligible phonon-drag contribution to the measured S(T ) in the ITO material. The phonon-drag term stems from the interaction between heat conducting phonons with conduction electrons. In ITO (figure7), the prevailing linearity over a wide range of temperature is a direct and strong indication of the absence of the phonon-drag contribution. The reason for the practically complete suppression of the phonon-drag term can be explained as follows. Considering the phonon scattering processes and ignoring their frequency dependence, the phonon-drag thermopower Sg(T )at T < θD
can be approximated by [119,123] Sg − Cg 3n|e| τph τph+ τph−e − Cg 3n|e| τph τph−e 1 2 τph τe–ph Sd, (5)
where Cgis the lattice specific heat per unit volume, τphis the
phonon relaxation time due to all kinds of phonon scattering processes (such as phonon–phonon (ph–ph) scattering, phonon scattering with imperfections, etc) except the phonon–electron (ph–e) scattering and τph–e is the ph–e scattering time. In
writing equation (5), we have assumed that τph τph–e.
Note that we have also applied the energy-balance equation
Ce/τe–ph = Cg/τph–e( [124,125]) to replace τph–eby τe–ph.
Consider a representative temperature of 100 K ∼ 0.1θD in ITO. We take the phonon mean free path to be
few nanometres long [126–128], which corresponds to a relaxation time τph(100 K)∼ 10−12s, with a sound velocity vp 4400 m s−1 in ITO [126]. According to our previous
studies of the weak-localization effect in ITO films [129], we estimate τe−ph(100 K)∼ 10−11s. Thus, equation (5) indicates
that the phonon-drag term would contribute only a few percent to the measured thermopower at a temperature of 100 K. The underlying physics for the smallness of the phonon-drag term
Sgcan further be reasoned as follows. (i) The value of τph in
ITO is generally very short due to the presence of a moderately high level of disorder in this class of material. (ii) Since the e–ph coupling strength in a conductor is proportional to the carrier concentration n [101,129], the relaxation time τe–ph
in ITO is thus notably long compared with that in typical metals. (See further discussion in subsection3.1.2). These two intrinsic material characteristics combine to cause a small
τph/τe–ph ratio and hence Sg Sd in the ITO material. By
negligible contribution from Sghas recently been observed in
F-doped SnO2films [118].
3. Quantum-interference transport at low
temperature: homogeneous indium tin oxide films and nanowires
In section2, we have examined the temperature dependence of electrical resistivity and thermoelectric power over a wide temperature range to demonstrate that the electronic conduction properties of metallic ITO obey the standard Boltzmann transport equation. In particular, being inherited with a free-carrier-like energy bandstructure, the essential electronic parameters can be reliably extracted from combined
ρ(T ), S(T ) and Hall coefficient RH measurements. In this
section, we show that metallic ITO also opens avenues for the studies of quantum electron transport properties. We shall focus on the quantum-interference weak-localization (WL) effect and the universal conductance fluctuation (UCF) phenomenon, which manifest in ITO films and nanowires at low temperatures. The many-body electron–electron interaction (EEI) effect in homogeneous disordered systems will not be explicitly discussed in this topical review, but will be briefly mentioned where appropriate.
3.1. Weak-localization effect and electron dephasing time The WL effect and electron dephasing in disordered conductors have been studied for three decades [130–135]. During this time, the mesoscopic and nanoscale physics underlying these processes has witnessed significant theoretical and experimental advances. Over years, the WL effect has also been explored in a few TCO materials, including ITO [67, 129, 136–140] and ZnO based materials [141–143]. In this subsection, we address the experimental 3D, 2D and 1D WL effects in ITO thick films, thin films and nanowires, respectively. In particular, we show that ITO has a relatively long electron dephasing (phase-breaking) length,
Lϕ(T )=
Dτϕ and a relatively weak e–ph relaxation rate
1/τe–ph, where D is the electron diffusion constant and τϕ
is the electron dephasing time. As a consequence, the WL effect in ITO can persist up to a high measurement temperature of ∼100 K. For comparison, in typical normal metals, the WL effect can often be observed only up to ∼20–30 K, due to a comparatively strong e–ph relaxation rate as the temperature increases to above liquid-helium temperatures [135]. Furthermore, as a consequence of the small 1/τe–ph,
one may use ITO thick films to explicitly examine the 3D small-energy-transfer electron–electron (e–e) scattering rate, 1/τN
ee,3D, for the first time in the literature [140]. A long Lϕ also causes the 1D WL effect and the UCF phenomenon
to significantly manifest in ITO nanowires with diameters
d < Lϕ. Since the electronic parameters, such as EFand D, are
well known in ITO, the value of τϕcan be reliably extracted and
closely compared with the theoretical calculations. Such levels of close comparison between experimental and theoretical values are nontrivial for many typical metals.
Figure 9.Change in the normalized conductivity with the magnetic field for a 7.5 nm thick ITO film at three temperatures, as indicated. The magnetic field was applied perpendicular to the film plane. The solid curves are the predictions of the 2D WL theory. Note that the quantum-interference WL effect persists up to liquid-nitrogen temperatures in the ITO material. This figure was reproduced with permission from [136]. Copyright 1983 by the Physical Society of Japan.
3.1.1. Weak-localization magnetoresistance in various dimensions. As discussed in section2, ρ(T ) of ITO samples decrease by small amounts (10% in polycrystalline films and20% in single-crystalline nanowires) as the temperature decreases from 300 K down to helium (or liquid-nitrogen) temperatures, suggesting the presence of moderately high levels of disorder in all kinds of ITO materials. Thus, the WL effect must prevail in ITO. In 1983, Ohyama et al [136] measured ITO thin films and found negative magnetoresistance (MR) and logarithmic temperature dependence of resistance in a wide temperature range 1.5–100 K. They explained the negative MR in terms of the 2D WL effect and the logarithmic temperature dependence of resistance in terms of a sum of the 2D WL and EEI effects. Figure9shows a plot of the positive magnetoconductance (i.e. negative MR) induced by the WL effect in a 7.5 nm thick ITO film measured by Ohyama and coworkers. It is seen that the experimental data (symbols) can be well described by the 2D WL theory predictions (solid curves).
Recently, with the advances of nanoscience and technology, the 1D WL effect has been investigated in single-crystalline ITO nanowires [137–139]. In particular, since Lϕ
is relatively long in the ITO material at low temperatures (see below), the quasi-1D dimensional criterion Lϕ > dis readily
Figure 10.Normalized magnetoresistance, R(B)/R(0)= [R(B) − R(0)]/R(0), as a function of magnetic field of (a) a 60 nm diameter ITO nanowire at (from bottom up): 0.25, 5.0, 12, 20, 30 and 40 K and (b) a 72 nm diameter ITO nanowire at (from top down): 0.26, 1.0, 2.0 and 4.0 K. The magnetic field was applied perpendicular to the nanowire axis. The symbols are the experimental data and the solid curves are the 1D WL theory predictions. Note that the MRs are negative in (a) (i.e. the weak-localization effect), while positive in (b) (i.e. the weak-antilocalization effect) in small magnetic fields. This figure was reproduced with permission from [138]. Copyright 2010 by the American Physical Society.
ITO nanowires. Indeed, figure 10(a) shows a plot of the negative MR due to the 1D WL effect in a 60 nm diameter ITO nanowire studied by Hsu et al [138]. This nanowire had a low resistivity value of ρ(10 K) 185 µ cm. The magnetic field was applied perpendicular to the nanowire axis. The data (symbols) is well described by the 1D WL theory predictions (solid curves). The extracted dephasing lengths are
Lϕ(0.25 K) 520 nm and Lϕ(40 K) 150 nm. Similarly, the
negative MR in the 3D WL effect can be observed in ITO thick films and is well described by the 3D WL theory predictions. (The explicit theoretical predictions for the 1D, 2D and 3D MR in the WL effect can be found in [143] and references therein.)
3.1.2. Electron dephasing time. Measurements of MR in the WL effect allows one to extract the value of τϕ. Detailed
studies of the electron dephasing processes in ITO thin films have recently been carried out by Wu et al [129]. They have measured the negative MR due to the 2D WL effect and extracted the τϕ values in two series of 15 and 21 nm thick
ITO films in a wide temperature range 0.3–90 K. Figure 11
shows a plot of representative variation of extracted 1/τϕ
with temperature. In general, the responsible dephasing processes are determined by the sample dimensionality, level of disorder and measurement temperature [133,135,144]. In 3D weakly disordered metals, e–ph scattering is often the dominant dephasing mechanism [100,135,145], while in reduced dimensions (2D and 1D), the e–e scattering is the major dephasing process [135,144,146,147]. As T→ 0 K, a constant or very weakly temperature dependent dephasing process may exist in a given sample, the physical origin for which is yet to be fully identified [135,148–152]. In ITO, as already mentioned, the e–ph relaxation rate is very weak.
The total electron dephasing rate 1/τϕ(T ) (the solid
curves) in figure11for the 2D ITO thin films studied by Wu
Figure 11.Electron dephasing rate 1/τϕas a function of
temperature for four 15 nm thick ITO films. The solid curves are least-squares fits to equation (6). For clarity, the data for the n3, A3 and A1 films have been shifted up by multiplying by factors of 2, 4 and 8, respectively. This figure was reproduced with permission from [129]. Copyright 2012 by the American Physical Society. et al [129] is described by 1 τϕ(T ) = 1 τ0 ϕ + ANee,2DT + Aee,2DT2ln EF kBT , (6)
where the first, second and third terms on the right-hand side of the equation stand for the ‘saturation’ term, the small-energy-transfer (Nyquist) e–e scattering term and the large-energy-transfer e–e scattering term, respectively. The small-energy-transfer term is dominant at low temperatures of T < ¯h/(kBτe), while the large-energy-transfer term is
Figure 12.(a) Electron dephasing rate 1/τϕas a function of temperature for four ITO thick films. Note that 1/τϕvaries with a T3/2
temperature dependence. (b) Variation of 1/τϕwith k−5/2F l−3/2for a good number of ITO thick films at 5 and 15 K. In the measurement temperature range 4–35 K, the dephasing rate 1/τϕ 1/τee,3DN = A
N
ee,3DT3/2. In (a) and (b), the solid lines are linear fits. This figure was reproduced with permission from [140].
dominant at high temperatures of T > ¯h/(kBτe). By
comparing their measured 1/τϕ(T ) with equation (11), Wu
et al found that their extracted values of the e–e scattering
strengths AN
ee,2D ≈ 3 × 109K−1 s−1 and Aee,2D ≈ 9 ×
106K−2s−1are consistent with the theoretical values to within
a factor of ∼ 3 and ∼ 5, respectively5. Considering that
the ITO material is a disordered In2−xSnxO3−δ with random
Sn dopants and possible oxygen vacancies, such levels of agreement between experimental and theoretical values are satisfactory. The good theoretical estimates must derive from the free-carrier-like energy bandstructure characteristics of ITO, which renders evaluations of the electronic parameters reliable. In terms of dephasing length, figure 11 gives rise to relatively long length scales of Lϕ(0.3 K) ≈ 500 nm and
Lϕ(60 K)≈ 45 nm.
The e–e scattering rate in other-dimensional ITO samples has also been studied. In the case of 1D nanowires, due to the sample dimensionality effect, the Nyquist e–e scattering rate obeys a 1/τee,1DN ∝ T2/3temperature law [135,144]. This
scattering process is largely responsible for the 1D WL MR shown in figures 10(a) and (b), as analyzed and discussed in [138]. In the case of 3D thick films, the temperature dependence of the Nyquist rate changes to the 1/τN
ee,3D∝ T3/2
temperature law [153,154]. Owing to the intrinsic weak e–ph coupling in this material, ITO provides a valuable platform for detailed study of the 3D small-energy-transfer e–e scattering process over wide ranges of temperature and disorder, as discussed below.
In a 3D weakly disordered metal, the e–e scattering rate has been calculated by Schmid in 1974 and his result is given
5 The theoretical expressions for the small-energy-transfer and
large-energy-transfer e–e scattering strengths, respectively, are AN ee,2D =
(e2/2π ¯h2)R
kBln(π ¯h/e2R)and Aee,2D= πkB2/(2¯hEF), where Ris the
sheet resistance. In the comparison of experiment with theory, the Rvalue was directly measured and the EFvalue was extracted from thermoelectric
power measurement. by [153] 1 τee =π 8 (kBT )2 ¯hEF + √ 3 2¯h√EF kBT kFl 3/2 . (7)
A similar result has also been obtained by Altshuler and Aronov [154]. The first term on the right-hand side of equation (7) is the e–e scattering rate in a perfect, periodic potential, while the second term is the enhanced contribution due to the presence of imperfections (defects, impurities, interfaces, etc) in the sample. Microscopically, the second term stands for the Nyquist e–e scattering process and is dominant at low temperatures of T < ¯h/(kBτe), while the first term
represents the large-energy-transfer process and dominates at high temperatures of T > ¯h/(kBτe)([131,154]). We shall
denote the second term by 1/τee,3DN = ANee,3DT3/2. In 3D weakly disordered typical metals, the e–ph scattering is strong and dominates over the e–e scattering [135]. Thus, equation (7) has been difficult to test in a quantitative manner for decades, even though the mesoscopic physics has witnessed marvellous advances.
Very recently, Zhang et al [140] have measured the low magnetic field MRs in a series of 3D ITO films with thicknesses exceeding 1 micrometre. Their polycrystalline samples were prepared by the standard RF sputtering deposition method in an Ar and O2mixture. During deposition, the oxygen content,
together with the substrate temperature, was varied to ‘tune’ the electron concentration as well as the amount of disorder. By comparing the MR data with the 3D WL theory, Zhang et al extracted the dephasing rate 1/τϕ as plotted in figure 12(a).
Clearly, one observes a strict 1/τϕ ∝ T3/2 temperature
dependence in a wide T range 4–35 K. Quantitatively, the scattering rate of the first term in equation (7) is about one order of magnitude smaller than that of the second term even at T = 35 K in ITO. Thus, the contribution of the first term can be safely ignored. The straight solid lines in figures12(a) are described by 1/τϕ = 1/τϕ0 + A
N
a constant and AN
ee,3D (2.1–2.8)×108K−3/2s−1for various
samples. These experimental AN
ee,3Dvalues are within a factor
of ∼3 of the theoretical values given by the second term of equation (7).
Furthermore, applying the free-electron model, Zhang
et al [140] rewrote the second term on the right hand of equation (7) into the form 1/τN
ee,3D = A
N
ee,3DT3/2 = (1.22√m∗/¯h2)(kBT )3/2k−5/2
F l−3/2. This expression allows
one to check the combined disorder (k−3/2F l−3/2) and carrier concentration (kF−1) dependence of 1/τN
ee,3D at a given
temperature. Figure 12(b) shows a plot of the variation of the extracted 1/τϕ with k−5/2F l−3/2at two T values of 5 and
15 K. Obviously, a variation 1/τϕ ∝ kF−5/2l−3/2is observed.
Quantitatively, the experimental slopes ( 1.2 × 1019 and
3.7× 1019m−1s−1at 5 and 15 K, respectively) in figure12(b) are within a factor of∼ 5 of the theoretical values. Thus, the experimental dephasing rate 1/τϕ 1/τee,3DN = ANee,3DT3/2
in ITO thick films quantitatively confirms the temperature, disorder and carrier concentration dependences of the Schmid-Altshuler-Aronov theory of 3D small-energy-transfer e–e scattering in disordered metals [153,154].
Electron–phonon relaxation rate. We would like to comment on the reason why the e–e scattering dominates the electron dephasing rate in 3D ITO thick films (figure12) in a wide T range up to several tens of degrees of kelvin. The reason is owing to the fact that the ITO material possesses relatively low n values which result in a greatly suppressed 1/τe–ph
1/τee,3DN . Theoretically, it is established that the electron scattering by transverse vibrations of defects and impurities dominates the e–ph relaxation. In the quasi-ballistic limit (qTl >1, where qTis the wavenumber of a thermal phonon)6,
the electron-transverse phonon scattering rate is given by [101,145,155] 1 τe−t,ph = 3π2k2 Bβt (pFut)(pFl) T2, (8)
where βt = (2EF/3)2N (EF)/(2ρmu2t) is the
electron-transverse phonon coupling constant, pF is the Fermi
momentum, ut is the transverse sound velocity and ρm is
the mass density. Since the electronic parameters EF, pF,
N (EF)and l in ITO samples are known, the theoretical value
of equation (8) can be computed and is of the magnitude 1/τe−t,ph ∼ 4×106T2K−2s−1. Note that this relaxation rate
is about one order of magnitude smaller than 1/τN
ee,3Deven at
a relatively high temperature of 40 K. A weak e–ph relaxation rate allows the quantum-interference WL effect and UCF phenomena to persist up to a few tens of degrees of kelvin in ITO7.
6 In high-quality ITO structures, q
Tl≈ 0.1 T [129,140] and hence the
quasi-ballistic limit is valid above∼10 K. In disordered normal metals, due to a relatively short electron mean free path l = 3π2¯h/(e2k2
Fρ)∝ 1/kF2for a same
ρvalue, the quasi-ballistic regime is more difficult to realize in experiment. For example, a polycrystalline Ti73Al27 alloy [38] (an amorphous CuZrAl
alloy [157]) with ρ≈ 225 µ cm (≈ 200 µ cm) has a value of qTl≈ 0.006 T
(≈0.01 T).
7 The electron dephasing length L
ϕ = Dτϕ Dτe–ph above a few
degrees of kelvin is much shorter in a typical disordered metal than in ITO, due to both a much shorter τe–ph and a smaller diffusion constant
D∝ 1/(N(EF)ρ)∝ 1/N(EF)for a same ρ value in the former.
We reiterate that equation (8) predicts a relaxation rate 1/τe−t,ph ∝ n. On the other hand, equation (7) predicts a scattering rate 1/τee,3DN ∝ n−5/6. Thus, the ratio of these two scattering rates varies approximately inversely with the square of n, namely, (1/τN
ee,3D)/(1/τe−t,ph)∝ n−2. Since the n
values in ITO samples are relatively low, the 3D small-energy-transfer e–e scattering rate can thus be enhanced over the e–ph relaxation rate. This observation can be extended to other TCO materials and is worth of further investigations.
We also would like to note that, in recent studies of superconducting hot electron bolometers, a weak e–ph relaxation rate has been observed in quasi-2D heterostructures containing ultrathin La2−xSrxCuO4 (LSCO) layers [156].
LSCO has a n value about two orders of magnitude lower that in the conventional superconductor NbN and hence τe–ph(LSCO)
is nearly two orders of magnitude longer than τe–ph(NbN). In
short, we remark that slow e–ph relaxation is a general intrinsic property of low-n conductors. Generally speaking, one may keep in mind that the relaxation rate varies approximately as 1/τe–ph∝ n ( [155,156]).
Spin–orbit scattering time. According to the recent measure-ments on a good number of ITO films [129] and nanowires [138] down to as low as 0.25 K, only negative MR was ob-served (see, for example, figure 10(a)). This result sug-gests that the spin–orbit scattering rate, 1/τso, is relatively
weak in ITO. Even at sub-kelvin temperatures where the in-elastic electron scattering events are scarce, one still obtains 1/τso<1/τeeN(0.25 K) in many ITO samples. In other words,
the ITO material possesses an inherent long spin–orbit scatter-ing length Lso =
√
Dτso. In typical ITO films [129], the
ex-tracted length scale is Lso>500 nm, corresponding to a
scat-tering time τso >250 ps. This τso value is one to two orders
of magnitude longer than those in typical metals, such as Ag films [158] and Sn-doped Ti73Al27alloys [38].
In practice, the strength of spin–orbit coupling in a given metal can be tuned by varying the level of disorder. In general, the spin–orbit scattering rate can be approximately expressed by 1/τso ∝ Z4/τe ∝ ρ, where Z is the atomic number of
the relevant (heavy) scatterer. Indeed, an enhancement of the spin–orbit scattering rate has been achieved in an ITO nanowire which was intentionally made to have a high resistivity value of ρ(10 K) = 1030 µ cm [138]. Hsu et al then observed positive MR at temperatures T < 4 K in low magnetic fields, see figure 10(b). A positive MR is a direct manifestation of the weak-antilocalization effect which results from the scattering rates 1/τso > 1/τee,1DN at T < 4 K. At higher
temperatures, a negative MR was recovered, suggesting that 1/τso < 1/τee,1DN at T > 4 K. In this high-ρ ITO nanowire,
Hsu et al obtained a moderate length scale Lso ≈ 95 nm,
corresponding to a scattering time τso ≈ 15 ps. The capability
of tuning the spin–orbit coupling strength might be useful for the future implementation of nanoscale spintronic devices [159]. Recently, Shinozaki et al [160] have observed an increasing ratio (1/τso)/(1/τee,3DN )with increasing ρ in a series
of amorphous indium-zinc-oxide and indium-(tin,gallium)-zinc-oxide thick films.
Figure 13.(a) Universal conductance fluctuations for three∼20 nm wide (Ga, Mn)As wires with different lengths L 100, 200 and 300 nm. The inset shows an electron micrograph of the 100 nm long wire. (b) Conductance G versus magnetic field B for the 200 nm long wire at several temperatures below 1 K. The magnetic field was applied perpendicular to the wire axis. This figure was reproduced with permission from [168]. Copyright 2006 by the American Physical Society.
3.2. Universal conductance fluctuations
Universal conductance fluctuations (UCFs) are a fundamental phenomenon in mesoscopic physics. The UCFs originate from the quantum interference between electron partial waves that propagate along different trajectories in a miniature system in which classical self-averaging is absent or incomplete [161–164]. Thus, the shape of the UCF patterns (called ‘magneto-fingerprints’) is very sensitive to the specific
impurity configuration of a given sample. The UCFs have
previously been experimentally observed in lithographic metal and semiconductor mesoscopic structures at low temperatures [163,165,166], where the electron dephasing length Lϕ is
comparable to the sample size. Recently, UCFs have been observed in new artificial materials, including epitaxial InAs nanowires [167], lithographic ferromagnets [168], carbon nanotubes [169], graphene [170] and topological insulators [171,172]. These new observations in artificially synthesized materials have enriched and deepened quantum electron transport physics.
Wagner et al [168] have measured the UCFs in lithographically defined ferromagnetic (Ga, Mn)As nanowires. Figure 13(a) shows their measured conductance G as a function of magnetic field B for three wires at T = 20 mK.
The wires were ∼20 nm wide and 100, 200, or 300 nm long. Figure 13(b) shows G versus B at several different temperatures between 20 mK and 1 K for the 200 nm long wire. The magnetic field was applied perpendicular to the wire axis. Figure13(b) clearly reveals that the UCFs are observable below ∼0.5 K. Figure13(a) demonstrates that the UCF amplitude decreases with increasing sample length, suggesting a fairly short dephasing length of Lϕ(20 mK) ≈ 100 nm. For the
100 nm long wire, the peak-to-peak UCF amplitude reaches a value of e2/ hat 20 mK, where h is the Planck constant. Impurity reconfiguration. Let us return to the case of ITO. Since Lϕ can reach≈500 nm at low temperatures, the ITO
nanowires are very useful for the investigations of the 1D UCF phenomena. Yang et al [139] have recently carried out the magneto-transport measurements on individual ITO nanowires with a focus on studying the UCFs. Their nanowires were made by implanting Sn ions into In2O3−δ nanowires.
Figures 14(a)–(d) show four plots of the variation of the UCFs, denoted by δGUCF(T , B), with magnetic field B for
a 110 nm diameter ITO nanowire at several temperatures8.
The magnetic field was applied perpendicular to the nanowire axis. Here, after the first run at liquid-helium temperatures, the nanowire was thermally cycled to room temperature, at which it stayed overnight and cooled down again for the magneto-transport measurements at liquid-helium temperatures. The thermal cycling to room temperature was repeated twice and the sample was thus measured for three times at three different cooldowns. The idea was that a thermal cycling to 300 K could possibly induce impurity reconfiguration in the given nanowire. A new impurity configuration must lead to differing trajectories of the propagating electron partial waves, which in turn cause distinct quantum interference. As a result, the shape of the UCF patterns should be completely changed. Figure14(a) shows δGUCF(T , B)as a function of B at several
temperatures measured at the first cooldown. Figure 14(b) shows δGUCF(T , B)as a function of B at several temperatures
measured at the second cooldown and figure14(c) shows those measured at the third cooldown.
A number of important UCF features and the underlying physics can be learned from close inspection of these figures. (i) Inspection of figures 14(a)–(c) indicates that the UCF magnitudes decrease with increasing temperature and disappear at ∼25 K. Thus, these quantum conductance fluctuations are distinctly different from the classical thermal noise whose resistance fluctuation magnitudes increase with increasing temperature.
(ii) During a given cooldown, the shape of the UCF patterns at different temperatures remains the same to a large extent. This observation implies that the impurity configuration is frozen for a considerable period of time if the nanowire is constantly kept at liquid-helium temperatures. A given impurity configuration gives rise to a specific ‘magneto-fingerprint’, strongly suggesting that the UCF phenomena is a robust manifestation of an intrinsic quantum-interference effect.
8 The universal conductance fluctuation δG
UCF(T , B) is defined by
subtracting a smooth magneto-conductance background (including the WL MR contribution) from the measured G(T , B).
Figure 14.Variation of the UCFs, δGUCF(T , B), with magnetic field at several temperatures for a 110 nm diameter and 1.2 µm long ITO nanowire at (a) first cooldown, (b) second cooldown and (c) third cooldown. (d) The δGUCF(T= 0.26 K) curves taken from panel (a) (top curve) and panel (b) (middle curve) and their difference (bottom curve), as a function of magnetic field. The magnetic field was applied perpendicular to the nanowire axis. In panels (a)–(d), the UCF curves are vertically offset for clarity. This figure was reproduced with permission from [139]. Copyright 2012 by the American Physical Society.
(iii) At a given temperature, the UCFs among different cooldowns reveal similar peak-to-peak magnitudes. (iv) Figure14(d) shows a plot of the δGUCF(T = 0.26 K,B)
curves taken from figure14(a) (top curve) and figure14(b) (middle curve) and their difference (bottom curve). This figure is convenient for close inspection and comparison. The top two curves reveal completely different shapes of the UCF patterns, strongly reflecting that a thermal cycling to 300 K has induced an impurity reconfiguration. On the other hand, the UCF magnitudes of these two curves retain similar, with a peak-to-peak value of δGUCF(T =
0.26 K) ≈0.5 e2/ h for both curves. The reason for
retaining a similar UCF magnitude is as follows. The UCF magnitudes in a given nanowire are governed by the Lϕ values, which are determined by the level of
disorder, i.e. the ρ value (or the R value in 2D), see section3.1.2. The ρ (R) value of a sample is determined by the total number of impurities, but insensitive to the specific spatial distribution of the impurities (provided that the impurity concentration is uniform throughout the sample)9.
9 The UCF studies also allow extractions of the L
ϕ(T )values in a miniature
sample. The values thus obtained are in fair accord with those extracted from the WL MR measurements. In addition to Lϕ, the thermal diffusion length LT plays a key role in governing the UCF magnitudes.
Classical self-averaging and thermal averaging at finite temperatures. In the case of a quasi-1D wire with length
L, the UCF theory predicts a root-mean-square conductance fluctuation magnitude of(δGUCF)2 0.73 e2/ h in the
limit of T → 0 K [161,162,164]. At this low T limit, the wire behaves as a single phase-coherent regime. As the temperature gradually increases from absolute zero, Lϕ(T )
becomes progressively shorter and one has to take into account the classical self-averaging effect. That is, the phase-coherent regime is expected to be cut off by Lϕ and the
UCF magnitude(δGUCF)2 is predicted to be suppressed
by a factor (Lϕ/L)3/2 under the condition Lϕ < LT, where
LT =
√
D¯h/kBT ∝ 1/
√
T is the thermal diffusion length defined in the EEI theory. The suppression of the UCF magnitudes originates from the fact that the UCFs of different phase-coherent regimes fluctuate statistically independently. If the temperature further increases such that LT < Lϕ
or, equivalently, the thermal energy exceeds the Thouless energy kBT > ¯h/τϕ, one also has to take into account
the thermal averaging effect. That is, the phase-coherent regime is now expected to be cut off by LT and the UCF
magnitude(δGUCF)2 is predicted to be suppressed by a
factor (LT/L)
Lϕ/L. These theoretical concepts have been
well accepted by the mesoscopic physics communities for three decades, but have rarely been experimentally tested in
![Figure 10. Normalized magnetoresistance, R(B)/R(0) = [R(B) − R(0)]/R(0), as a function of magnetic field of (a) a 60 nm diameter ITO nanowire at (from bottom up): 0.25, 5.0, 12, 20, 30 and 40 K and (b) a 72 nm diameter ITO nanowire at (from top down): 0.2](https://thumb-ap.123doks.com/thumbv2/9libinfo/7609440.130029/11.892.123.768.99.381/normalized-magnetoresistance-function-magnetic-diameter-nanowire-diameter-nanowire.webp)
