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[Yusuke Kozuka](https://orcid.org/0000-0001-7674-600X)

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[Quantum transport phenomena at oxide interfaces: Their potential for quantum science and technology](https://mdr.nims.go.jp/datasets/c4860790-7319-4776-be59-362648a7d318)

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New Physics: Sae Mulli,Vol. 75, No. 12, December 2025, pp. 944∼955http://dx.doi.org/10.3938/NPSM.75.944Quantum transport phenomena at oxide interfaces:Their potential for quantum science and technologyYusuke Kozuka∗Research Center for Materials Nanoarchitectonics (MANA),National Institute for Materials Science (NIMS), Tsukuba 305-0047, JapanFaculty of Science and Engineering, Waseda University, Tokyo 169-8555, JapanWPI Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan(Received 16 October 2025 : revised 4 November 2025 : accepted 4 November 2025)High-mobility charge carriers at semiconductor interfaces have provided fertile ground for thestudy of low-dimensional physics, such as the integer and fractional quantum Hall effects. In thesestudies, unique properties belonging to each material have not been explicitly highlighted, asthe fundamental phenomena are believed to be universal and independent of the details of thematerials. Here, we review high-mobility electrons at oxide interfaces, from fabrication to charac-terization. By incorporating the unique aspects of oxide materials as compared to conventionalsemiconductors, we can explore more diverse physics, including characteristic phenomena such assuperconductivity and strong correlation. Finally, we discuss the prospects for potential quantumscience and applications based on oxide interfaces.Keywords: Oxides, Crystal growth, Quantum Hall effect, Superconductivity, Mesoscopic sys-temsI. INTRODUCTIONConventionally, oxides were not regarded as promis-ing platforms for condensed matter physics or majorcomponents of electronic devices because most of themare insulators. However, some researchers were drawn tothe unique properties of conducting transition-metal ox-ides, which stem from the correlated nature of d elec-trons [1–3]. The situation changed dramatically with thediscovery of high transition temperature (high-Tc) su-perconducting cuprates in 1986 [4, 5]. These materialspromised unprecedented applications, such as supercon-ducting magnets and superconducting quantum interfer-ence devices (SQUIDs) that operate above liquid nitro-gen temperatures [6,7]. Furthermore, the high-Tc super-conductivity and other related physical properties couldnot be easily explained within a single-particle picture,∗Correspondence to: KOZUKA.Yusuke@nims.go.jpunderscoring the need to explicitly consider electron cor-relations. Following this discovery, various oxide materi-als were investigated for both electronic applications andfundamental scientific research. This has led to the iden-tification of many notable properties, including colossalmagnetoresistance in manganites [8,9] and a large ther-moelectric effect in cobaltites [10].Despite many intriguing properties, one notably ab-sent in oxides was high carrier mobility (µ), particu-larly in two dimensions, before the early 2000s. The elec-tron correlation in transition-metal oxides might inher-ently limit high electron coherence due to the localiz-ing nature of d electrons. However, the thin-film fab-rication technologies for oxide materials were also un-derdeveloped, impeding the exploration of potentiallyhigh-mobility oxides. This situation stands in stark con-trast to conventional semiconductors like GaAs, whichexhibit extremely high electron mobility, even exceed-ing 10,000,000 cm2/Vs below 4 K [11–13]. In the field ofThis is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License(http://creativecommons.org/licenses/by-nc/4.0) which permits unrestricted non-commercial use, distribution, and reproduction in anymedium, provided the original work is properly cited.http://dx.doi.org/10.3938/NPSM.75.944http://crossmark.crossref.org/dialog/?doi=10.3938/NPSM.75.944&domain=pdf&date_stamp=2025-12-31mailto:KOZUKA.Yusuke@nims.go.jp http://creativecommons.org/licenses/by-nc/4.0Quantum transport phenomena at oxide interfaces: · · · – Yusuke Kozuka 945semiconductors, high-quality thin film and heterostruc-ture fabrication technologies for certain materials un-derwent dramatic advancements in the 1980s and con-tinue to progress even now. These advancements, coupledwith sophisticated microfabrication techniques, have po-sitioned GaAs as the dominant material for quantum de-vices such as quantum point contacts [14] and quantumdots [15].In this review, we discuss advancements in high-mobility electron systems at oxide heterostructures. Asmentioned above, many oxide materials face intrinsicchallenges in achieving high carrier mobility due tostrong electron correlation. However, there are notableexceptions, such as SrTiO3 and ZnO. These materialshave the potential to be utilized in various electronicand quantum devices, including field-effect transistors,quantum wires, and quantum dots, incorporating uniqueproperties that are absent in conventional semiconduc-tors. We also explore the future prospects of oxide-basedquantum devices.II. HIGH-MOBILITY ELECTRONS INTWO DIMENSIONSIn this section, before reviewing oxide semiconductors,we explain the general background of the physics of high-mobility electrons in conventional semiconductors. Highelectron mobility, and consequently a long carrier scat-tering time (τ), is essential for observing quantum inter-ference effects in solid-state materials. One typical quan-tum transport phenomenon is Shubnikov-de Haas (SdH)oscillations. In this phenomenon, circulating electronsform quantized energy levels under a strong magneticfield due to the interference of the wavefunction, result-ing in resistance oscillations. The conditions to observeSdH oscillations are [16]ℏωc ≫ ℏ/τ , kBT ,where ℏ is the Planck constant (h) divided by 2π, ωc(= eB/m∗, e: elementary charge of an electron, B: mag-netic field, m∗: effective mass) is the cyclotron frequency,kB is the Boltzmann constant, and T is the tempera-ture. These conditions dictate that Landau level split-ting energy (ℏωc) should be much larger than the energybroadening of the density of states due to carrier scat-tering (ℏ/τ) and thermal broadening (kBT ). Assuming arelatively high magnetic field of 10 T, these conditionsrequire parameters of µ ∼ 10,000 cm2/Vs and T ∼ 1 K.While SdH oscillations have been utilized to obtain in-formation about the Fermi surfaces in three-dimensionalmaterials [17], more intriguing physics can be observedin two dimensions (2D) under a magnetic field due towell-defined integer filling numbers in the discrete Lan-dau levels [18,19], known as the quantum Hall effect [20].When the Fermi level is located in a mobility edge (local-ized states within a Landau level), all the electrons be-come localized due to circulating orbits [18,21], while anextended one-dimensional state remains along the edgeof the sample [22]. This edge state carries charge currentwithout dissipation because backscattering is completelyprohibited due to the absence of the density of statesof the counter-propagating channel. Simultaneously, op-posing two edge states exhibit a constant difference inchemical potentials, resulting in a quantized Hall effect.The value of this effect is defined by the number of edgestates (ν: filling factor) as Ryx = h/νe2 −∼∼ 25.8 kΩ/ν.Since the accuracy of the quantized Hall resistance is ex-tremely precise, the quantum Hall effect is a promisingcandidate for resistance standards [23–25].III. SrTiO31. Basic propertiesSrTiO3 is known as a wide-gap semiconductor with aband gap of 3.2 eV [26]. Its most remarkable propertyis its high dielectric constant, which is approximately∼300ε0 at room temperature and exceeds 20,000ε0 be-low 10 K (ε0 is the vacuum permittivity) [27]. This largedielectric constant results from its incipient ferroelectricnature, which is suppressed by quantum fluctuations.When doped with Nb, La, or oxygen vacancies, SrTiO3becomes an n-type semiconductor. The electron mobilityis typically around 10 cm2/Vs at room temperature, butit dramatically increases to over 10,000 cm2/Vs at lowtemperatures due to the large dielectric constant effec-tively screening impurity potentials [28–31]. This value946 New Physics: Sae Mulli, Vol. 75, No. 12, December 2025Fig. 1. (Color online) Superconducting temperature as a function of carrier density for various semiconductors. Fortwisted bilayer graphene, the thickness is roughly assumed to be 1 nm. Data are taken from Refs. [33–36] (n-SrTiO3),[36,38] (n-(Ba,Ca,Sr)TiO3), [39] (GeTe), [40] (SnTe), [41] (Ba(Pb,Bi)O3), [42–44] (Diamond), [45] (B:Si), [46] (B:SiC),[47] (Ga:Ge), [48] (twisted bilayer graphene).is extremely high compared to standard doped semicon-ductors, where impurity scattering becomes more dom-inant as the temperature decreases, following µ ∝ T 3/2[32]. Moreover, doped SrTiO3 exhibits superconductiv-ity with a maximum superconducting temperature (Tc)of approximately 0.5 K at a carrier density of about 1020cm−3 [33–36]. Although Tc itself is relatively low, thisvalue is notably high when considering the density ofstates, as the Bardeen-Cooper-Schrieffer theory dictatesthat kBTc ∝ exp(−1/NV ), where N is the density ofstates at the Fermi energy and V is the coupling strengthof a Cooper pair [37]. In fact, SrTiO3 is among the su-perconductors with the lowest carrier densities known todate. For comparison, Fig. 1 shows several superconduct-ing semiconductors with relatively low carrier densities[33–36,38–48].2. Electron-doped SrTiO3 thin filmAlthough bulk n-type SrTiO3 exhibits high electronmobility, fabricating doped SrTiO3 thin films with highelectron mobility was challenging because the fabricationtechniques for SrTiO3 thin films were not well estab-lished until the early 2000s. At that time, the maximumelectron mobility was limited to around 300 cm2/Vs atlow temperature [49, 50]. Instead, there were efforts tofabricate field-effect transistors (FETs) on bulk undopedSrTiO3 to induce 2D electrons at the surface, similar tometal-oxide-semiconductor FETs in Si [51–53]. However,most efforts result in relatively low electron mobility lessthan 100 cm2/Vs, due to damage to the SrTiO3 surfaceduring the deposition of insulators. To mitigate surfacedamage, FET structures using organic insulators or ionicliquids were developed, achieving mobilities of approxi-mately 1,000 cm2/Vs. However, SdH oscillations werenot reported in these structures [54,55].The challenge in growing bulk-quality SrTiO3 thinfilms lies in controlling the Sr/Ti stoichiometry [56,57]. High electron mobility is achieved below a car-rier density of 1019 cm−3 [28–31], which correspondsto ∼0.1 atomic % substitution of Nb in the Ti site.Even slight off-stoichiometry can cause carrier trap-ping at defects. One way to overcome this difficulty isto grow SrTiO3 thin films at extremely high substratetemperatures, above ∼1050 ◦C, where slight Sr/Ti off-stoichiometry can be solved by segregating a part of theoff-stoichiometric components. Oxygen vacancies createdduring high-temperature growth can be filled by anneal-ing in a moderate oxygen atmosphere of ∼1 Pa, consid-ering the defect formation properties of SrTiO3 [58–61].In fact, Nb-doped SrTiO3 thin films grown at 1200 ◦Cby pulsed laser deposition exhibit a high electron mobil-ity of ∼7,000 cm2/Vs with 0.02 atomic % doping of Nb,which is close to the bulk value [62].Another method involves using molecular beam epi-taxy (MBE) with metal-organic gas sources for Ti, whichenables the stoichiometric growth of La-doped SrTiO3thin films. In standard metal-source MBE, precise sto-ichiometric control is challenging unless one of the ele-ments reevaporates to form a stoichiometric film. Both Srand Ti metals have high melting temperatures, makingQuantum transport phenomena at oxide interfaces: · · · – Yusuke Kozuka 947Fig. 2. (Color online) (a) Schematic diagram of the SrTiO3/n-SrTiO3/SrTiO3 heterostructure. (b) Longitudinal resis-tance and (c) Hall resistance as a function of magnetic field of the heterostructure with a carrier density of 1.2 × 1012cm−2, measured at 50 mK. Adapted from Ref. [70].reevaporation negligible at typical growth temperatures.However, using a metal-organic source allows for the self-organizing synthesis of stoichiometric SrTiO3 thin films[63]. The maximum electron mobility of ∼0.001 atomic %La-doped SrTiO3 shows an electron mobility of ∼30,000cm2/Vs [64].3. 2D electron system in SrTiO3Conventionally, 2D electron systems in semiconduc-tors are formed at the interface with a larger band gapmaterial, such as in GaAs/(Al,Ga)As structures, wheresubstituting Al with Ga widens the band gap and con-fines electrons within the GaAs layer [65]. However, bandengineering is not well established in the case of SrTiO3.Instead, homoepitaxial SrTiO3/n-SrTiO3/SrTiO3 het-erostructures, resembling δ-doped structures [66], areused to effectively confine electrons within the dopedlayer, where the n-SrTiO3 layer is doped with either Nbor La [67,68]. When the n-SrTiO3 layer is thin enough,below a typical de Broglie wavelength, the electron layeris regarded as 2D, and the angle dependence of SdH os-cillations indicates a 2D Fermi surface [69]. However,the carrier density estimated from SdH oscillations isabout one order of magnitude smaller than that esti-mated from the Hall effect. This discrepancy arises be-cause the sheet carrier density cannot be reduced be-low ∼1013 cm−2 to maintain metallic conduction and,at this carrier density, multiple subbands are still par-tially filled. This issue was resolved by fabricating higher-quality heterostructures using metal-organic gas sourceMBE at high substrate temperatures, achieving a carrierdensity below 1012 cm−2 [70]. The quantum Hall effectwas finally observed by modulating the carrier densityusing the SrTiO3 substrate as a gate insulator as shownin Fig. 2 [70]. It is important to note that precise tuningof carrier density is essential, particularly considering thehysteretic gate dependence characteristic of the SrTiO3substrate as a gate insulator [71–73].As with bulk-doped SrTiO3, 2D electrons in SrTiO3heterostructures also exhibit superconductivity. By vary-ing the thickness of the conducting layer, a 3D-2Dcrossover of superconductivity, as well as of electronicstructures, were observed as shown in Fig. 3 [74]. Specifi-cally, the dimensionality of superconductivity can be de-duced from the temperature dependence of the uppercritical field (Bc2) with magnetic fields applied in twodifferent directions. When the magnetic field is perpen-dicular to the conducting plane, Bc2 is expressed asB⊥c2(T ) =ϕ02πξGL(0)2(1− TTc),while for magnetic field parallel to the plane,B∥c2(T ) =ϕ0√122πξGL(0)d(1− TTc)1/2,where ϕ0 is the magnetic flux quantum, ξGL(0) is theGinzburg-Landau coherence length at low temperature,and d is the thickness of the superconducting layer. Thesecond expression is valid for the two-dimensional su-perconductivity, where d ≪ ξGL. These expressions ofBc2 assume that Cooper pairs are broken by the vortexformation. In the d → 0 limit, it becomes difficult for948 New Physics: Sae Mulli, Vol. 75, No. 12, December 2025Fig. 3. (Color online) Electron mobility (µ) and su-perconducting transition temperature (Tc) as a func-tion of doped layer thickness (dNSTO) of the SrTiO3/n-SrTiO3/SrTiO3 heterostructure. 3D-2D crossovers ofelectronic structure and superconductivity are alsoshown. I: insulator; 2DM: two-dimensional metal; 3DM:three-dimensional metal; 2DSC: two-dimensional su-perconductor; 3DSC: three-dimensional superconductor.Reprinted Fig. 5 with permission from [M. Kim et al.,Fermi surface and superconductivity in low-density high-mobility δ-doped SrTiO3, Phys. Rev. Lett. 107, 106801(2011)] Copyright (2011) by the American Physical So-ciety.vortices to form under a parallel magnetic field. Instead,Zeeman energy breaks the pair potential of the Cooperpairs (Pauli paramagnetic limit), which is expressed as[75,76]BPc2 =∆0√2µB=1.76kBTc√2µB,where BPc2 is the Pauli limiting field, ∆0 is the supercon-ducting gap energy in the weak-coupling limit, and µBis the Bohr magneton. The superconductivity of SrTiO3exceeds the Pauli paramagnetic limit by a factor of morethan 4. This is believed to be due to spin-orbit interac-tion [77] as reported in the case of metal bilayer films[78]. The concept of breaking the Pauli paramagneticlimit through spin-orbit interaction has since been ex-tended to other material systems, including transition-metal dichalcogenides [79,80].4. Prospects for quantum devices of SrTiO3As mentioned above, doped SrTiO3 is quite uniquein that it exhibits superconductivity at very low carrierFig. 4. (Color online) Advancements of electron mobility(µ) as a function of carrier density n for (Mg,Zn)O/ZnOheterostructures. Adapted from Ref. [106]. Referencenumbers [11], [30], [31], and [15] in the figure correspondto Refs. [97], [103], [108], and [105] in this paper, respec-tively.density. This property allows for superconductor-metal-insulator transitions to be controlled solely by electro-static gating. By incorporating a split gate structure, su-perconducting weak links can be electrostatically formedas a foundation for quantum devices [81–83]. A similardevice using other materials has only recently becomepossible with twisted bilayer graphene [84].IV. ZnO1. Crystal growth of ZnO thin filmsZnO is considered one of the typical II-VI semiconduc-tors with a wide band gap of 3.37 eV. In terms of elec-tronic applications, bulk ZnO has traditionally been usedas a varistor [85]. However, until around the late 1990s,ZnO was not considered a promising material for elec-tronic and quantum devices due to the immature growthtechniques and the crystal quality, which was not com-parable to that of other semiconductors. Below, the ad-vancements in ZnO thin film growth are surveyed, assummarized in Fig. 4, in terms of electron mobility.Research on ZnO thin films began to aim at appli-cations such as transparent transistors [86, 87] or blueQuantum transport phenomena at oxide interfaces: · · · – Yusuke Kozuka 949and ultraviolet light-emitting diodes [88,89], leveragingits wide band gap. ZnO thin films were initially fab-ricated using pulsed laser deposition, and fundamentalproperties, including band gap modulation through Mgor Cd substitution, were investigated [90, 91]. Mg andCd are isovalent with Zn, and their substitution leadsto structural changes and, consequently, band gap mod-ifications, similar to Al or In substitution in GaAs. Atthe first stage, sapphire substrates were mostly used forgrowing ZnO thin films due to their commercial avail-ability. However, the mobility of ZnO thin films waslimited to around 100 cm2/Vs due to the large latticemismatch [92], which caused high-density dislocations.Later, more lattice-matched ScAlMgO4 (SCAM) sub-strates were employed, leading to a maximum mobilityof 5,000 cm2/Vs, particularly when a high-temperatureannealed Mg-substituted ZnO layer was used as a bufferlayer [93]. The significant advancements in growth tech-niques, along with p-type doping using nitrogen, enabledthe notable demonstration of a blue light-emitting diode[94,95].Using the same growth technique and substrate,high-mobility 2D electrons were found to form at theZnO/(Mg,Zn)O interface [96, 97]. The accumulation of2D electrons results from the relaxation of potential di-vergence, which is caused by the polarization disconti-nuity across the interface. ZnO has a Wurtzite crystalstructure with inversion asymmetry, inherently possess-ing spontaneous polarization. The magnitude of this po-larization varies with Mg substitution due to changesin the lattice constant and internal atomic coordinates.At the ZnO/(Mg,Zn)O interface, uncompensated chargesare present, leading to the accumulation of 2D electrons[97,98]. Because of the high electron mobility, the quan-tum Hall effect was observed at low temperatures [97].To further improve crystal quality and develop amethod for mass production, two major advancementswere made: the development of single-crystal ZnO sub-strates for homoepitaxy, and the use of molecular beamepitaxy (MBE). Single-crystal ZnO substrates weregrown using the hydrothermal method [99]. The surfaceof the substrate was etched by diluted hydrochloric acidto prepare an epitaxy-ready surface [100]. In MBE, high-purity Zn and Mg metal sources were used along withoxygen plasma as the oxygen source. Compared to thesingle-crystal ZnO target used in pulsed laser deposition,the use of high-purity Zn and Mg metals significantly re-duced the incorporation of impurities [101–103]. As aresult, the electron mobility of the 2D electrons at the(Mg,Zn)O/ZnO interface increased to 100,000 cm2/Vs[104]. Further improvements included optimizing the Mgcontent in the (Mg,Zn)O layer [105] and employing apure ozone generator instead of oxygen plasma for theoxygen source [106]. Notably, using pure ozone expandedthe growth temperature window to lower temperatures,from ∼900 ◦C with oxygen plasma to ∼750 ◦C, reducingthe incorporation of impurities from surrounding com-ponents of the substrate. These efforts led to a recordmobility of over 1,000,000 cm2/Vs [107].2. Quantum Hall effect in ZnOAs mentioned above, the integer quantum Hall effectwas first observed in ZnO heterostructures with an elec-tron mobility of ∼5,000 cm2/Vs, fabricated using pulsedlaser deposition [97]. By employing MBE, the electronmobility increased to 100,000 cm2/Vs, leading to the ob-servation of the fractional quantum Hall effect [108,109].In this phenomenon, electron correlation plays an essen-tial role in forming fractionally charged quasiparticlesknown as composite fermions.Although the quantum Hall effect is a universal phe-nomenon regardless of the material, the specifics of theinteger and fractional quantum Hall effects significantlydepend on various factors, such as the strength of elec-tron correlations. The strength of the correlation for itin-erant electrons is expressed based on the ratio of theCoulomb energy to the kinetic energy asrs =m∗e24πεℏ2√nπ,where ε is the dielectric constant and n is the sheet car-rier density. As shown in Fig. 5 [11–13, 106, 109–120],rs parameter of ZnO is significantly larger than that ofother typical high-mobility 2D systems while maintain-ing high scattering time. One notable consequence of thisis the enhancement of electron susceptibility, which is theproduct of the effective g-factor and the effective mass[121,122]. The electron susceptibility was experimentally950 New Physics: Sae Mulli, Vol. 75, No. 12, December 2025Fig. 5. (Color online) Mapping of transport scatter-ing time (τ) and electron interaction parameter (rs) forseveral high-mobility 2D carriers. Data are taken fromRefs. [11–13,110–112] (GaAs), [113,114] (Si/SiGe), [115](CdTe), [116–118] (AlAs), [119] (GaN), [120] (bilayergraphene), and [106] (ZnO).estimated using the coincidence method, where the cy-clotron energy is an integer multiple of the Zeeman en-ergy, leading to a double period of SdH oscillations as afunction of 1/B.Another notable finding is the even-denominator frac-tional quantum Hall effect [123]. Usually, the filling fac-tor has an odd denominator to comply with the Pauliexclusion principle [124, 125]. However, when compos-ite fermions form pairs due to effective attraction, thefractional quantum Hall effect with an even denomi-nator filling can be observed [126]. This phenomenonhas been observed in extremely high-quality GaAs het-erostructures. ZnO has become another example exhibit-ing the even-denominator fractional quantum Hall effect[127,128]. Notably, ZnO shows a greater variety of even-denominator fractional quantum Hall effects by rotatingthe conducting plane with respect to the magnetic field,including ν = 3/2, 5/2 and 7/2. In contrast, GaAs mostlyshows ν = 5/2 and, in very rare cases, ν = 7/2 [129], butthe signal easily diminishes with increasing in-plane mag-netic field [130]. This variety of fractional states in ZnOis attributed to the strong electron correlation and theresulting complex energy diagram, which originates fromlarge and nonlinear electron susceptibility [131].3. Prospects for quantum devicesA high-quality 2D electron system is essential for con-structing more complex quantum devices, such as quan-tum point contacts or quantum dots. ZnO, in particular,possesses promising properties for the application of spinqubits. Firstly, only 2% of natural Zn isotopes (67Zn)include nuclear spins, which minimizes the loss of spininformation [132]. Secondly, ZnO is a direct band gapsemiconductor, and its conduction band consists of a sin-gle electron pocket, thus eliminating interband scatter-ing, unlike in Si. These two factors combine the advanta-geous aspects of both GaAs (single electron pocket) andSi (low-density nuclear spins). The spin coherence timeof ZnO has been theoretically predicted to be relativelylong [133], even compared with Si and diamond, whichare under intensive study for quantum sensing applica-tions. Experimentally, a long longitudinal spin coherencetime (T1) of ∼500 ms has been observed for electronsbound around Al donors [134,135]. Additionally, electro-statically defined quantum dots have been demonstrated,marking an important step toward realizing spin qubitsusing ZnO heterostructures [136]. The unique proper-ties of ZnO, combined with these experimental advance-ments, are expected to lead to superior functionalities inquantum devices.ACKNOWLEDGEMENTSThe author thanks all the collaborators and particu-larly acknowledges C. Bell, J. Falson, Y. Hikita, H. Y.Hwang, M. Kawasaki, M. Kim, D. Maryenko, and A.Tsukazaki. This work is partly supported by JST FOR-EST Program (Grant Number: JPMJFR203D) and byJSPS KAKENHI (23K26482, 25H00613). MANA andAIMR are supported byWorld Premier International Re-search Center Initiative (WPI), MEXT, Japan.REFERENCES[1] C. N. R. Rao and G. V. S. Rao, Electrical con-duction in metal oxides, Phys. Stat. Sol. 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